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API for Regularizers

This page contains documentation of the public API of the RegularizedLeastSquares. In the Julia REPL one can access this documentation by entering the help mode with ?

RegularizedLeastSquares.L21Regularization — Type
L21Regularization

Regularization term implementing the proximal map for group-soft-thresholding.

Arguments

  • λ - regularization paramter

Keywords

  • slices=1 - number of elements per group
source
RegularizedLeastSquares.LLRRegularization — Type
LLRRegularization

Regularization term implementing the proximal map for locally low rank (LLR) regularization using singular-value-thresholding.

Arguments

  • λ - regularization paramter

Keywords

  • shape::Tuple{Int} - dimensions of the image
  • blockSize::Tuple{Int}=(2,2) - size of patches to perform singular value thresholding on
  • randshift::Bool=true - randomly shifts the patches to ensure translation invariance
  • fullyOverlapping::Bool=false - choose between fully overlapping block or non-overlapping blocks
source
RegularizedLeastSquares.NuclearRegularization — Type
NuclearRegularization

Regularization term implementing the proximal map for singular value soft-thresholding.

Arguments:

  • λ - regularization paramter

Keywords

  • svtShape::NTuple - size of the underlying matrix
source
RegularizedLeastSquares.TVRegularization — Type
TVRegularization

Regularization term implementing the proximal map for TV regularization. Calculated with the Condat algorithm if the TV is calculated only along one real-valued dimension and with the Fast Gradient Projection algorithm otherwise.

Reference for the Condat algorithm: https://lcondat.github.io/publis/Condat-fast_TV-SPL-2013.pdf

Reference for the FGP algorithm: A. Beck and T. Teboulle, "Fast Gradient-Based Algorithms for Constrained Total Variation Image Denoising and Deblurring Problems", IEEE Trans. Image Process. 18(11), 2009

Arguments

  • λ::T - regularization parameter

Keywords

  • shape::NTuple - size of the underlying image
  • dims - Dimension to perform the TV along. If Integer, the Condat algorithm is called, and the FDG algorithm otherwise.
  • iterationsTV=20 - number of FGP iterations
source

Projection Regularization

Nested Regularization

RegularizedLeastSquares.innerreg — Method
innerreg(reg::AbstractNestedRegularization)

return the inner regularization term of reg. Nested regularization terms also implement the iteration interface.

source

Scaled Regularization

Misc. Nested Regularization

RegularizedLeastSquares.MaskedRegularization — Type
MaskedRegularization

Nested regularization term that only applies prox! and norm to elements of x for which the mask is true.

Examples

julia> positive = PositiveRegularization();
+
+julia> masked = MaskedRegularization(reg, [true, false, true, false]);
+
+julia> prox!(masked, fill(-1, 4))
+4-element Vector{Float64}:
+  0.0
+ -1.0
+  0.0
+ -1.0
source
RegularizedLeastSquares.TransformedRegularization — Type
TransformedRegularization(reg, trafo)

Nested regularization term that applies prox! or norm on z = trafo * x and returns (inplace) x = adjoint(trafo) * z.

Example

julia> core = L1Regularization(0.8)
+L1Regularization{Float64}(0.8)
+
+julia> wop = WaveletOp(Float32, shape = (32,32));
+
+julia> reg = TransformedRegularization(core, wop);
+
+julia> prox!(reg, randn(32*32)); # Apply soft-thresholding in Wavelet domain
source
RegularizedLeastSquares.PlugAndPlayRegularization — Type
    PlugAndPlayRegularization

Regularization term implementing a given plug-and-play proximal mapping. The actual regularization term is indirectly defined by the learned proximal mapping and as such there is no norm implemented.

Arguments

  • λ - regularization paramter

Keywords

  • model - model applied to the image
  • shape - dimensions of the image
  • input_transform - transform of image before model
source

Miscellaneous Functions

RegularizedLeastSquares.prox! — Method
prox!(reg::AbstractParameterizedRegularization, x)

perform the proximal mapping defined by reg on x. Uses the regularization parameter defined for reg.

source
RegularizedLeastSquares.prox! — Method
prox!(regType::Type{<:AbstractParameterizedRegularization}, x, λ; kwargs...)

construct a regularization term of type regType with given λ and kwargs and apply its prox! on x

source
LinearAlgebra.norm — Method
norm(reg::AbstractParameterizedRegularization, x)

returns the value of the reg regularization term on x. Uses the regularization parameter defined for reg.

source
LinearAlgebra.norm — Method
norm(regType::Type{<:AbstractParameterizedRegularization}, x, λ; kwargs...)

construct a regularization term of type regType with given λ and kwargs and apply its norm on x

source
diff --git a/previews/PR92/API/solvers/index.html b/previews/PR92/API/solvers/index.html new file mode 100644 index 00000000..671e865f --- /dev/null +++ b/previews/PR92/API/solvers/index.html @@ -0,0 +1,49 @@ + +Solvers · RegularizedLeastSquares.jl

API for Solvers

This page contains documentation of the public API of the RegularizedLeastSquares. In the Julia REPL one can access this documentation by entering the help mode with ?

solve!

RegularizedLeastSquares.solve! — Method
solve!(solver::AbstractLinearSolver, b; x0 = 0, callbacks = (_, _) -> nothing)

Solves an inverse problem for the data vector b using solver.

Required Arguments

  • solver::AbstractLinearSolver - linear solver (e.g., ADMM or FISTA), containing forward/normal operator and regularizer
  • b::AbstractVector - data vector if A was supplied to the solver, back-projection of the data otherwise

Optional Keyword Arguments

  • x0::AbstractVector - initial guess for the solution; default is zero
  • callbacks - (optionally a vector of) function or callable struct that takes the two arguments callback(solver, iteration) and, e.g., stores, prints, or plots the intermediate solutions or convergence parameters. Be sure not to modify solver or iteration in the callback function as this would japaridze convergence. The default does nothing.

Examples

The optimization problem

\[ argmin_x ||Ax - b||_2^2 + λ ||x||_1\]

can be solved with the following lines of code:

julia> using RegularizedLeastSquares
+
+julia> A = [0.831658  0.96717
+            0.383056  0.39043
+            0.820692  0.08118];
+
+julia> x = [0.5932234523399985; 0.2697534345340015];
+
+julia> b = A * x;
+
+julia> S = ADMM(A);
+
+julia> x_approx = solve!(S, b)
+2-element Vector{Float64}:
+ 0.5932234523399984
+ 0.26975343453400163

Here, we use L1Regularization, which is default for ADMM. All regularization options can be found in API for Regularizers.

The following example solves the same problem, but stores the solution x of each interation in tr:

julia> tr = Dict[]
+Dict[]
+
+julia> store_trace!(tr, solver, iteration) = push!(tr, Dict("iteration" => iteration, "x" => solver.x, "beta" => solver.β))
+store_trace! (generic function with 1 method)
+
+julia> x_approx = solve!(S, b; callbacks=(solver, iteration) -> store_trace!(tr, solver, iteration))
+2-element Vector{Float64}:
+ 0.5932234523399984
+ 0.26975343453400163
+
+julia> tr[3]
+Dict{String, Any} with 3 entries:
+  "iteration" => 2
+  "x"         => [0.593223, 0.269753]
+  "beta"      => [1.23152, 0.927611]

The last example show demonstrates how to plot the solution at every 10th iteration and store the solvers convergence metrics:

julia> using Plots
+
+julia> conv = StoreConvergenceCallback()
+
+julia> function plot_trace(solver, iteration)
+         if iteration % 10 == 0
+           display(scatter(solver.x))
+         end
+       end
+plot_trace (generic function with 1 method)
+
+julia> x_approx = solve!(S, b; callbacks = [conv, plot_trace]);

The keyword callbacks allows you to pass a (vector of) callable objects that takes the arguments solver and iteration and prints, stores, or plots intermediate result.

See also StoreSolutionCallback, StoreConvergenceCallback, CompareSolutionCallback for a number of provided callback options.

source
RegularizedLeastSquares.init! — Method
init!(solver::AbstractLinearSolver, state::AbstractSolverState, b::AbstractMatrix; scheduler = SequentialState, kwargs...)

Initialize the solver with each column of b and pass the corresponding states to the scheduler.

source

ADMM

RegularizedLeastSquares.ADMM — Type
ADMM(A; AHA = A'*A, precon = Identity(), reg = L1Regularization(zero(real(eltype(AHA)))), regTrafo = opEye(eltype(AHA), size(AHA,1)), normalizeReg = NoNormalization(), rho = 1e-1, vary_rho = :none, iterations = 10, iterationsCG = 10, absTol = eps(real(eltype(AHA))), relTol = eps(real(eltype(AHA))), tolInner = 1e-5, verbose = false)
+ADMM( ; AHA = ,     precon = Identity(), reg = L1Regularization(zero(real(eltype(AHA)))), regTrafo = opEye(eltype(AHA), size(AHA,1)), normalizeReg = NoNormalization(), rho = 1e-1, vary_rho = :none, iterations = 10, iterationsCG = 10, absTol = eps(real(eltype(AHA))), relTol = eps(real(eltype(AHA))), tolInner = 1e-5, verbose = false)

Creates an ADMM object for the forward operator A or normal operator AHA.

Required Arguments

  • A - forward operator

OR

  • AHA - normal operator (as a keyword argument)

Optional Keyword Arguments

  • AHA - normal operator is optional if A is supplied
  • precon - preconditionner for the internal CG algorithm
  • reg::AbstractParameterizedRegularization - regularization term; can also be a vector of regularization terms
  • regTrafo - transformation to a space in which reg is applied; if reg is a vector, regTrafo has to be a vector of the same length. Use opEye(eltype(AHA), size(AHA,1)) if no transformation is desired.
  • normalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()
  • rho::Real - penalty of the augmented Lagrangian
  • vary_rho::Symbol - vary rho to balance primal and dual feasibility; options :none, :balance, :PnP
  • iterations::Int - maximum number of (outer) ADMM iterations
  • iterationsCG::Int - maximum number of (inner) CG iterations
  • absTol::Real - absolute tolerance for stopping criterion
  • relTol::Real - relative tolerance for stopping criterion
  • tolInner::Real - relative tolerance for CG stopping criterion
  • verbose::Bool - print residual in each iteration

ADMM differs from ISTA-type algorithms in the sense that the proximal operation is applied separately from the transformation to the space in which the penalty is applied. This is reflected by the interface which has reg and regTrafo as separate arguments. E.g., for a TV penalty, you should NOT set reg=TVRegularization, but instead use reg=L1Regularization(λ), regTrafo=RegularizedLeastSquares.GradientOp(Float64; shape=(Nx,Ny,Nz)).

See also createLinearSolver, solve!.

source

CGNR

RegularizedLeastSquares.CGNR — Type
CGNR(A; AHA = A' * A, reg = L2Regularization(zero(real(eltype(AHA)))), normalizeReg = NoNormalization(), iterations = 10, relTol = eps(real(eltype(AHA))))
+CGNR( ; AHA = ,       reg = L2Regularization(zero(real(eltype(AHA)))), normalizeReg = NoNormalization(), iterations = 10, relTol = eps(real(eltype(AHA))))

creates an CGNR object for the forward operator A or normal operator AHA.

Required Arguments

  • A - forward operator

OR

  • AHA - normal operator (as a keyword argument)

Optional Keyword Arguments

  • AHA - normal operator is optional if A is supplied
  • reg::AbstractParameterizedRegularization - regularization term; can also be a vector of regularization terms
  • normalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()
  • iterations::Int - maximum number of iterations
  • relTol::Real - tolerance for stopping criterion

See also createLinearSolver, solve!.

source

Kaczmarz

RegularizedLeastSquares.Kaczmarz — Type
Kaczmarz(A; reg = L2Regularization(0), normalizeReg = NoNormalization(), randomized=false, subMatrixFraction=0.15, shuffleRows=false, seed=1234, iterations=10)

Creates a Kaczmarz object for the forward operator A.

Required Arguments

  • A - forward operator

Optional Keyword Arguments

  • reg::AbstractParameterizedRegularization - regularization term
  • normalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()
  • randomized::Bool - randomize Kacmarz algorithm
  • subMatrixFraction::Real - fraction of rows used in randomized Kaczmarz algorithm
  • shuffleRows::Bool - randomize Kacmarz algorithm
  • seed::Int - seed for randomized algorithm
  • iterations::Int - number of iterations

See also createLinearSolver, solve!.

source

FISTA

RegularizedLeastSquares.FISTA — Type
FISTA(A; AHA=A'*A, reg=L1Regularization(zero(real(eltype(AHA)))), normalizeReg=NoNormalization(), iterations=50, verbose = false, rho = 0.95 / power_iterations(AHA), theta=1, relTol=eps(real(eltype(AHA))), restart = :none)
+FISTA( ; AHA=,     reg=L1Regularization(zero(real(eltype(AHA)))), normalizeReg=NoNormalization(), iterations=50, verbose = false, rho = 0.95 / power_iterations(AHA), theta=1, relTol=eps(real(eltype(AHA))), restart = :none)

creates a FISTA object for the forward operator A or normal operator AHA.

Required Arguments

  • A - forward operator

OR

  • AHA - normal operator (as a keyword argument)

Optional Keyword Arguments

  • AHA - normal operator is optional if A is supplied
  • precon - preconditionner for the internal CG algorithm
  • reg::AbstractParameterizedRegularization - regularization term; can also be a vector of regularization terms
  • normalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()
  • rho::Real - step size for gradient step; the default is 0.95 / max_eigenvalue as determined with power iterations.
  • theta::Real - parameter for predictor-corrector step
  • relTol::Real - tolerance for stopping criterion
  • iterations::Int - maximum number of iterations
  • restart::Symbol - :none, :gradient options for restarting
  • verbose::Bool - print residual in each iteration

See also createLinearSolver, solve!.

source

OptISTA

RegularizedLeastSquares.OptISTA — Type
OptISTA(A; AHA=A'*A, reg=L1Regularization(zero(real(eltype(AHA)))), normalizeReg=NoNormalization(), iterations=50, verbose = false, rho=0.95 / power_iterations(AHA), theta=1, relTol=eps(real(eltype(AHA))))
+OptISTA( ; AHA=,     reg=L1Regularization(zero(real(eltype(AHA)))), normalizeReg=NoNormalization(), iterations=50, verbose = false, rho=0.95 / power_iterations(AHA), theta=1, relTol=eps(real(eltype(AHA))))

creates a OptISTA object for the forward operator A or normal operator AHA. OptISTA has a 2x better worst-case bound than FISTA, but actual performance varies by application. It stores 2 extra intermediate variables the size of the image compared to FISTA.

Reference:

  • Uijeong Jang, Shuvomoy Das Gupta, Ernest K. Ryu, "Computer-Assisted Design of Accelerated Composite Optimization Methods: OptISTA," arXiv:2305.15704, 2023, [https://arxiv.org/abs/2305.15704]

Required Arguments

  • A - forward operator

OR

  • AHA - normal operator (as a keyword argument)

Optional Keyword Arguments

  • AHA - normal operator is optional if A is supplied
  • reg::AbstractParameterizedRegularization - regularization term
  • normalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()
  • rho::Real - step size for gradient step; the default is 0.95 / max_eigenvalue as determined with power iterations.
  • theta::Real - parameter for predictor-corrector step
  • relTol::Real - tolerance for stopping criterion
  • iterations::Int - maximum number of iterations
  • verbose::Bool - print residual in each iteration

See also createLinearSolver, solve!.

source

POGM

RegularizedLeastSquares.POGM — Type
POGM(A; AHA = A'*A, reg = L1Regularization(zero(real(eltype(AHA)))), normalizeReg = NoNormalization(), iterations = 50, verbose = false, rho = 0.95 / power_iterations(AHA), theta = 1, sigma_fac = 1, relTol = eps(real(eltype(AHA))), restart = :none)
+POGM( ; AHA = ,     reg = L1Regularization(zero(real(eltype(AHA)))), normalizeReg = NoNormalization(), iterations = 50, verbose = false, rho = 0.95 / power_iterations(AHA), theta = 1, sigma_fac = 1, relTol = eps(real(eltype(AHA))), restart = :none)

Creates a POGM object for the forward operator A or normal operator AHA. POGM has a 2x better worst-case bound than FISTA, but actual performance varies by application. It stores 3 extra intermediate variables the size of the image compared to FISTA. Only gradient restart scheme is implemented for now.

References:

  • A.B. Taylor, J.M. Hendrickx, F. Glineur, "Exact worst-case performance of first-order algorithms for composite convex optimization," Arxiv:1512.07516, 2015, SIAM J. Opt. 2017 [http://doi.org/10.1137/16m108104x]

  • Kim, D., & Fessler, J. A. (2018). Adaptive Restart of the Optimized Gradient Method for Convex Optimization. Journal of Optimization Theory and Applications, 178(1), 240–263. [https://doi.org/10.1007/s10957-018-1287-4]

    Required Arguments

    • A - forward operator

    OR

    • AHA - normal operator (as a keyword argument)

    Optional Keyword Arguments

    • AHA - normal operator is optional if A is supplied
    • reg::AbstractParameterizedRegularization - regularization term
    • normalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()
    • rho::Real - step size for gradient step; the default is 0.95 / max_eigenvalue as determined with power iterations.
    • theta::Real - parameter for predictor-corrector step
    • sigma_fac::Real - parameter for decreasing γ-momentum ∈ [0,1]
    • relTol::Real - tolerance for stopping criterion
    • iterations::Int - maximum number of iterations
    • restart::Symbol - :none, :gradient options for restarting
    • verbose::Bool - print residual in each iteration

See also createLinearSolver, solve!.

source

SplitBregman

RegularizedLeastSquares.SplitBregman — Type
SplitBregman(A; AHA = A'*A, precon = Identity(), reg = L1Regularization(zero(real(eltype(AHA)))), regTrafo = opEye(eltype(AHA), size(AHA,1)), normalizeReg = NoNormalization(), rho = 1e-1, iterations = 10, iterationsInner = 10, iterationsCG = 10, absTol = eps(real(eltype(AHA))), relTol = eps(real(eltype(AHA))), tolInner = 1e-5, verbose = false)
+SplitBregman( ; AHA = ,     precon = Identity(), reg = L1Regularization(zero(real(eltype(AHA)))), regTrafo = opEye(eltype(AHA), size(AHA,1)), normalizeReg = NoNormalization(), rho = 1e-1, iterations = 10, iterationsInner = 10, iterationsCG = 10, absTol = eps(real(eltype(AHA))), relTol = eps(real(eltype(AHA))), tolInner = 1e-5, verbose = false)

Creates a SplitBregman object for the forward operator A or normal operator AHA.

Required Arguments

  • A - forward operator

OR

  • AHA - normal operator (as a keyword argument)

Optional Keyword Arguments

  • AHA - normal operator is optional if A is supplied
  • precon - preconditionner for the internal CG algorithm
  • reg::AbstractParameterizedRegularization - regularization term; can also be a vector of regularization terms
  • regTrafo - transformation to a space in which reg is applied; if reg is a vector, regTrafo has to be a vector of the same length. Use opEye(eltype(AHA), size(AHA,1)) if no transformation is desired.
  • normalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()
  • rho::Real - weights for condition on regularized variables; can also be a vector for multiple regularization terms
  • iterations::Int - maximum number of outer iterations. Set to 1 for unconstraint split Bregman (equivalent to ADMM)
  • iterationsInner::Int - maximum number of inner iterations
  • iterationsCG::Int - maximum number of (inner) CG iterations
  • absTol::Real - absolute tolerance for stopping criterion
  • relTol::Real - relative tolerance for stopping criterion
  • tolInner::Real - relative tolerance for CG stopping criterion
  • verbose::Bool - print residual in each iteration

This algorithm solves the constraint problem (Eq. (4.7) in Tom Goldstein and Stanley Osher), i.e. ||R(x)||₁ such that ||Ax -b||₂² < σ². In order to solve the unconstraint problem (Eq. (4.8) in Tom Goldstein and Stanley Osher), i.e. ||Ax -b||₂² + λ ||R(x)||₁, you can either set iterations=1 or use ADMM instead, which is equivalent (iterations=1 in SplitBregman in implied in ADMM and the SplitBregman variable iterationsInner is simply called iterations in ADMM)

Like ADMM, SplitBregman differs from ISTA-type algorithms in the sense that the proximal operation is applied separately from the transformation to the space in which the penalty is applied. This is reflected by the interface which has reg and regTrafo as separate arguments. E.g., for a TV penalty, you should NOT set reg=TVRegularization, but instead use reg=L1Regularization(λ), regTrafo=RegularizedLeastSquares.GradientOp(Float64; shape=(Nx,Ny,Nz)).

See also createLinearSolver, solve!.

source

Miscellaneous

RegularizedLeastSquares.createLinearSolver — Function
createLinearSolver(solver::AbstractLinearSolver, A; kargs...)

This method creates a solver. The supported solvers are methods typically used for solving regularized linear systems. All solvers return an approximate solution to Ax = b.

TODO: give a hint what solvers are available

source
RegularizedLeastSquares.isapplicable — Function
isapplicable(solverType::Type{<:AbstractLinearSolver}, A, x, reg)

return true if a solver of type solverType is applicable to system matrix A, data x and regularization terms reg.

source
diff --git a/previews/PR92/assets/documenter.js b/previews/PR92/assets/documenter.js new file mode 100644 index 00000000..82252a11 --- /dev/null +++ b/previews/PR92/assets/documenter.js @@ -0,0 +1,1064 @@ +// Generated by Documenter.jl +requirejs.config({ + paths: { + 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia.min', + 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', + 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.13.2/jquery-ui.min', + 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/contrib/auto-render.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.0/jquery.min', + 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', + 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min', + 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min', + 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia-repl.min', + }, + shim: { + "highlight-julia": { + "deps": [ + "highlight" + ] + }, + "katex-auto-render": { + "deps": [ + "katex" + ] + }, + "headroom-jquery": { + "deps": [ + "jquery", + "headroom" + ] + }, + "highlight-julia-repl": { + "deps": [ + "highlight" + ] + } +} +}); +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'katex', 'katex-auto-render'], function($, katex, renderMathInElement) { +$(document).ready(function() { + renderMathInElement( + document.body, + { + "delimiters": [ + { + "left": "$", + "right": "$", + "display": false + }, + { + "left": "$$", + "right": "$$", + "display": true + }, + { + "left": "\\[", + "right": "\\]", + "display": true + } + ] +} + + ); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($) { +$(document).ready(function() { + hljs.highlightAll(); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +let timer = 0; +var isExpanded = true; + +$(document).on( + "click", + ".docstring .docstring-article-toggle-button", + function () { + let articleToggleTitle = "Expand docstring"; + const parent = $(this).parent(); + + debounce(() => { + if (parent.siblings("section").is(":visible")) { + parent + .find("a.docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + } else { + parent + .find("a.docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + articleToggleTitle = "Collapse docstring"; + } + + parent + .children(".docstring-article-toggle-button") + .prop("title", articleToggleTitle); + parent.siblings("section").slideToggle(); + }); + } +); + +$(document).on("click", ".docs-article-toggle-button", function (event) { + let articleToggleTitle = "Expand docstring"; + let navArticleToggleTitle = "Expand all docstrings"; + let animationSpeed = event.noToggleAnimation ? 0 : 400; + + debounce(() => { + if (isExpanded) { + $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down"); + $("a.docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + + isExpanded = false; + + $(".docstring section").slideUp(animationSpeed); + } else { + $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up"); + $("a.docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + isExpanded = true; + articleToggleTitle = "Collapse docstring"; + navArticleToggleTitle = "Collapse all docstrings"; + + $(".docstring section").slideDown(animationSpeed); + } + + $(this).prop("title", navArticleToggleTitle); + $(".docstring-article-toggle-button").prop("title", articleToggleTitle); + }); +}); + +function debounce(callback, timeout = 300) { + if (Date.now() - timer > timeout) { + callback(); + } + + clearTimeout(timer); + + timer = Date.now(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require([], function() { +function addCopyButtonCallbacks() { + for (const el of document.getElementsByTagName("pre")) { + const button = document.createElement("button"); + button.classList.add("copy-button", "fa-solid", "fa-copy"); + button.setAttribute("aria-label", "Copy this code block"); + button.setAttribute("title", "Copy"); + + el.appendChild(button); + + const success = function () { + button.classList.add("success", "fa-check"); + button.classList.remove("fa-copy"); + }; + + const failure = function () { + button.classList.add("error", "fa-xmark"); + button.classList.remove("fa-copy"); + }; + + button.addEventListener("click", function () { + copyToClipboard(el.innerText).then(success, failure); + + setTimeout(function () { + button.classList.add("fa-copy"); + button.classList.remove("success", "fa-check", "fa-xmark"); + }, 5000); + }); + } +} + +function copyToClipboard(text) { + // clipboard API is only available in secure contexts + if (window.navigator && window.navigator.clipboard) { + return window.navigator.clipboard.writeText(text); + } else { + return new Promise(function (resolve, reject) { + try { + const el = document.createElement("textarea"); + el.textContent = text; + el.style.position = "fixed"; + el.style.opacity = 0; + document.body.appendChild(el); + el.select(); + document.execCommand("copy"); + + resolve(); + } catch (err) { + reject(err); + } finally { + document.body.removeChild(el); + } + }); + } +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", addCopyButtonCallbacks); +} else { + addCopyButtonCallbacks(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) { + +// Manages the top navigation bar (hides it when the user starts scrolling down on the +// mobile). +window.Headroom = Headroom; // work around buggy module loading? +$(document).ready(function () { + $("#documenter .docs-navbar").headroom({ + tolerance: { up: 10, down: 10 }, + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let meta = $("div[data-docstringscollapsed]").data(); + + if (meta?.docstringscollapsed) { + $("#documenter-article-toggle-button").trigger({ + type: "click", + noToggleAnimation: true, + }); + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +/* +To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc + +PSEUDOCODE: + +Searching happens automatically as the user types or adjusts the selected filters. +To preserve responsiveness, as much as possible of the slow parts of the search are done +in a web worker. Searching and result generation are done in the worker, and filtering and +DOM updates are done in the main thread. The filters are in the main thread as they should +be very quick to apply. This lets filters be changed without re-searching with minisearch +(which is possible even if filtering is on the worker thread) and also lets filters be +changed _while_ the worker is searching and without message passing (neither of which are +possible if filtering is on the worker thread) + +SEARCH WORKER: + +Import minisearch + +Build index + +On message from main thread + run search + find the first 200 unique results from each category, and compute their divs for display + note that this is necessary and sufficient information for the main thread to find the + first 200 unique results from any given filter set + post results to main thread + +MAIN: + +Launch worker + +Declare nonconstant globals (worker_is_running, last_search_text, unfiltered_results) + +On text update + if worker is not running, launch_search() + +launch_search + set worker_is_running to true, set last_search_text to the search text + post the search query to worker + +on message from worker + if last_search_text is not the same as the text in the search field, + the latest search result is not reflective of the latest search query, so update again + launch_search() + otherwise + set worker_is_running to false + + regardless, display the new search results to the user + save the unfiltered_results as a global + update_search() + +on filter click + adjust the filter selection + update_search() + +update_search + apply search filters by looping through the unfiltered_results and finding the first 200 + unique results that match the filters + + Update the DOM +*/ + +/////// SEARCH WORKER /////// + +function worker_function(documenterSearchIndex, documenterBaseURL, filters) { + importScripts( + "https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min.js" + ); + + let data = documenterSearchIndex.map((x, key) => { + x["id"] = key; // minisearch requires a unique for each object + return x; + }); + + // list below is the lunr 2.1.3 list minus the intersect with names(Base) + // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) + // ideally we'd just filter the original list but it's not available as a variable + const stopWords = new Set([ + "a", + "able", + "about", + "across", + "after", + "almost", + "also", + "am", + "among", + "an", + "and", + "are", + "as", + "at", + "be", + "because", + "been", + "but", + "by", + "can", + "cannot", + "could", + "dear", + "did", + "does", + "either", + "ever", + "every", + "from", + "got", + "had", + "has", + "have", + "he", + "her", + "hers", + "him", + "his", + "how", + "however", + "i", + "if", + "into", + "it", + "its", + "just", + "least", + "like", + "likely", + "may", + "me", + "might", + "most", + "must", + "my", + "neither", + "no", + "nor", + "not", + "of", + "off", + "often", + "on", + "or", + "other", + "our", + "own", + "rather", + "said", + "say", + "says", + "she", + "should", + "since", + "so", + "some", + "than", + "that", + "the", + "their", + "them", + "then", + "there", + "these", + "they", + "this", + "tis", + "to", + "too", + "twas", + "us", + "wants", + "was", + "we", + "were", + "what", + "when", + "who", + "whom", + "why", + "will", + "would", + "yet", + "you", + "your", + ]); + + let index = new MiniSearch({ + fields: ["title", "text"], // fields to index for full-text search + storeFields: ["location", "title", "text", "category", "page"], // fields to return with results + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + + word = word.toLowerCase(); + } + + return word ?? null; + }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not + // find anything if searching for "add!", only for the entire qualification + tokenize: (string) => string.split(/[\s\-\.]+/), + // options which will be applied during the search + searchOptions: { + prefix: true, + boost: { title: 100 }, + fuzzy: 2, + }, + }); + + index.addAll(data); + + /** + * Used to map characters to HTML entities. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const htmlEscapes = { + "&": "&", + "<": "<", + ">": ">", + '"': """, + "'": "'", + }; + + /** + * Used to match HTML entities and HTML characters. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const reUnescapedHtml = /[&<>"']/g; + const reHasUnescapedHtml = RegExp(reUnescapedHtml.source); + + /** + * Escape function from lodash + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + function escape(string) { + return string && reHasUnescapedHtml.test(string) + ? string.replace(reUnescapedHtml, (chr) => htmlEscapes[chr]) + : string || ""; + } + + /** + * RegX escape function from MDN + * Refer: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Regular_Expressions#escaping + */ + function escapeRegExp(string) { + return string.replace(/[.*+?^${}()|[\]\\]/g, "\\$&"); // $& means the whole matched string + } + + /** + * Make the result component given a minisearch result data object and the value + * of the search input as queryString. To view the result object structure, refer: + * https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ + function make_search_result(result, querystring) { + let search_divider = `
`; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } + searchstring = escapeRegExp(querystring); + let textindex = new RegExp(`${searchstring}`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + text = text.length ? escape(text) : ""; + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`${escape(searchstring)}`, "i"), // For first occurrence + '$&' + ) + + "..." + : ""; // highlights the match + + let in_code = false; + if (!["page", "section"].includes(result.category.toLowerCase())) { + in_code = true; + } + + // We encode the full url to escape some special characters which can lead to broken links + let result_div = ` + +
+
${escape(result.title)}
+
${result.category}
+
+

+ ${display_result} +

+
+ ${display_link} +
+ + ${search_divider} + `; + + return result_div; + } + + self.onmessage = function (e) { + let query = e.data; + let results = index.search(query, { + filter: (result) => { + // Only return relevant results + return result.score >= 1; + }, + combineWith: "AND", + }); + + // Pre-filter to deduplicate and limit to 200 per category to the extent + // possible without knowing what the filters are. + let filtered_results = []; + let counts = {}; + for (let filter of filters) { + counts[filter] = 0; + } + let present = {}; + + for (let result of results) { + cat = result.category; + cnt = counts[cat]; + if (cnt < 200) { + id = cat + "---" + result.location; + if (present[id]) { + continue; + } + present[id] = true; + filtered_results.push({ + location: result.location, + category: cat, + div: make_search_result(result, query), + }); + } + } + + postMessage(filtered_results); + }; +} + +// `worker = Threads.@spawn worker_function(documenterSearchIndex)`, but in JavaScript! +const filters = [ + ...new Set(documenterSearchIndex["docs"].map((x) => x.category)), +]; +const worker_str = + "(" + + worker_function.toString() + + ")(" + + JSON.stringify(documenterSearchIndex["docs"]) + + "," + + JSON.stringify(documenterBaseURL) + + "," + + JSON.stringify(filters) + + ")"; +const worker_blob = new Blob([worker_str], { type: "text/javascript" }); +const worker = new Worker(URL.createObjectURL(worker_blob)); + +/////// SEARCH MAIN /////// + +// Whether the worker is currently handling a search. This is a boolean +// as the worker only ever handles 1 or 0 searches at a time. +var worker_is_running = false; + +// The last search text that was sent to the worker. This is used to determine +// if the worker should be launched again when it reports back results. +var last_search_text = ""; + +// The results of the last search. This, in combination with the state of the filters +// in the DOM, is used compute the results to display on calls to update_search. +var unfiltered_results = []; + +// Which filter is currently selected +var selected_filter = ""; + +$(document).on("input", ".documenter-search-input", function (event) { + if (!worker_is_running) { + launch_search(); + } +}); + +function launch_search() { + worker_is_running = true; + last_search_text = $(".documenter-search-input").val(); + worker.postMessage(last_search_text); +} + +worker.onmessage = function (e) { + if (last_search_text !== $(".documenter-search-input").val()) { + launch_search(); + } else { + worker_is_running = false; + } + + unfiltered_results = e.data; + update_search(); +}; + +$(document).on("click", ".search-filter", function () { + if ($(this).hasClass("search-filter-selected")) { + selected_filter = ""; + } else { + selected_filter = $(this).text().toLowerCase(); + } + + // This updates search results and toggles classes for UI: + update_search(); +}); + +/** + * Make/Update the search component + */ +function update_search() { + let querystring = $(".documenter-search-input").val(); + + if (querystring.trim()) { + if (selected_filter == "") { + results = unfiltered_results; + } else { + results = unfiltered_results.filter((result) => { + return selected_filter == result.category.toLowerCase(); + }); + } + + let search_result_container = ``; + let modal_filters = make_modal_body_filters(); + let search_divider = `
`; + + if (results.length) { + let links = []; + let count = 0; + let search_results = ""; + + for (var i = 0, n = results.length; i < n && count < 200; ++i) { + let result = results[i]; + if (result.location && !links.includes(result.location)) { + search_results += result.div; + count++; + links.push(result.location); + } + } + + if (count == 1) { + count_str = "1 result"; + } else if (count == 200) { + count_str = "200+ results"; + } else { + count_str = count + " results"; + } + let result_count = `
${count_str}
`; + + search_result_container = ` +
+ ${modal_filters} + ${search_divider} + ${result_count} +
+ ${search_results} +
+
+ `; + } else { + search_result_container = ` +
+ ${modal_filters} + ${search_divider} +
0 result(s)
+
+
No result found!
+ `; + } + + if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").removeClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(search_result_container); + } else { + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(` +
Type something to get started!
+ `); + } +} + +/** + * Make the modal filter html + * + * @returns string + */ +function make_modal_body_filters() { + let str = filters + .map((val) => { + if (selected_filter == val.toLowerCase()) { + return `${val}`; + } else { + return `${val}`; + } + }) + .join(""); + + return ` +
+ Filters: + ${str} +
`; +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Modal settings dialog +$(document).ready(function () { + var settings = $("#documenter-settings"); + $("#documenter-settings-button").click(function () { + settings.toggleClass("is-active"); + }); + // Close the dialog if X is clicked + $("#documenter-settings button.delete").click(function () { + settings.removeClass("is-active"); + }); + // Close dialog if ESC is pressed + $(document).keyup(function (e) { + if (e.keyCode == 27) settings.removeClass("is-active"); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let search_modal_header = ` +
+
+

+ + + + +

+
+
+ +
+
+ `; + + let initial_search_body = ` +
Type something to get started!
+ `; + + let search_modal_footer = ` + + `; + + $(document.body).append( + ` + + ` + ); + + document.querySelector(".docs-search-query").addEventListener("click", () => { + openModal(); + }); + + document + .querySelector(".close-search-modal") + .addEventListener("click", () => { + closeModal(); + }); + + $(document).on("click", ".search-result-link", function () { + closeModal(); + }); + + document.addEventListener("keydown", (event) => { + if ((event.ctrlKey || event.metaKey) && event.key === "/") { + openModal(); + } else if (event.key === "Escape") { + closeModal(); + } + + return false; + }); + + // Functions to open and close a modal + function openModal() { + let searchModal = document.querySelector("#search-modal"); + + searchModal.classList.add("is-active"); + document.querySelector(".documenter-search-input").focus(); + } + + function closeModal() { + let searchModal = document.querySelector("#search-modal"); + let initial_search_body = ` +
Type something to get started!
+ `; + + searchModal.classList.remove("is-active"); + document.querySelector(".documenter-search-input").blur(); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".documenter-search-input").val(""); + $(".search-modal-card-body").html(initial_search_body); + } + + document + .querySelector("#search-modal .modal-background") + .addEventListener("click", () => { + closeModal(); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Manages the showing and hiding of the sidebar. +$(document).ready(function () { + var sidebar = $("#documenter > .docs-sidebar"); + var sidebar_button = $("#documenter-sidebar-button"); + sidebar_button.click(function (ev) { + ev.preventDefault(); + sidebar.toggleClass("visible"); + if (sidebar.hasClass("visible")) { + // Makes sure that the current menu item is visible in the sidebar. + $("#documenter .docs-menu a.is-active").focus(); + } + }); + $("#documenter > .docs-main").bind("click", function (ev) { + if ($(ev.target).is(sidebar_button)) { + return; + } + if (sidebar.hasClass("visible")) { + sidebar.removeClass("visible"); + } + }); +}); + +// Resizes the package name / sitename in the sidebar if it is too wide. +// Inspired by: https://github.com/davatron5000/FitText.js +$(document).ready(function () { + e = $("#documenter .docs-autofit"); + function resize() { + var L = parseInt(e.css("max-width"), 10); + var L0 = e.width(); + if (L0 > L) { + var h0 = parseInt(e.css("font-size"), 10); + e.css("font-size", (L * h0) / L0); + // TODO: make sure it survives resizes? + } + } + // call once and then register events + resize(); + $(window).resize(resize); + $(window).on("orientationchange", resize); +}); + +// Scroll the navigation bar to the currently selected menu item +$(document).ready(function () { + var sidebar = $("#documenter .docs-menu").get(0); + var active = $("#documenter .docs-menu .is-active").get(0); + if (typeof active !== "undefined") { + sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Theme picker setup +$(document).ready(function () { + // onchange callback + $("#documenter-themepicker").change(function themepick_callback(ev) { + var themename = $("#documenter-themepicker option:selected").attr("value"); + if (themename === "auto") { + // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light'); + window.localStorage.removeItem("documenter-theme"); + } else { + // set_theme(themename); + window.localStorage.setItem("documenter-theme", themename); + } + // We re-use the global function from themeswap.js to actually do the swapping. + set_theme_from_local_storage(); + }); + + // Make sure that the themepicker displays the correct theme when the theme is retrieved + // from localStorage + if (typeof window.localStorage !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if (theme !== null) { + $("#documenter-themepicker option").each(function (i, e) { + e.selected = e.value === theme; + }); + } + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// update the version selector with info from the siteinfo.js and ../versions.js files +$(document).ready(function () { + // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the + // siteinfo.js file, we just return immediately and not display the version selector. + if ( + typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" && + DOCUMENTER_VERSION_SELECTOR_DISABLED + ) { + return; + } + + var version_selector = $("#documenter .docs-version-selector"); + var version_selector_select = $("#documenter .docs-version-selector select"); + + version_selector_select.change(function (x) { + target_href = version_selector_select + .children("option:selected") + .get(0).value; + window.location.href = target_href; + }); + + // add the current version to the selector based on siteinfo.js, but only if the selector is empty + if ( + typeof DOCUMENTER_CURRENT_VERSION !== "undefined" && + $("#version-selector > option").length == 0 + ) { + var option = $( + "" + ); + version_selector_select.append(option); + } + + if (typeof DOC_VERSIONS !== "undefined") { + var existing_versions = version_selector_select.children("option"); + var existing_versions_texts = 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html.theme--catppuccin-frappe .content kbd.button.is-outlined{background-color:transparent;border-color:#414559;box-shadow:none;color:#414559}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#414559}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #414559 #414559 !important}html.theme--catppuccin-frappe 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.button.is-primary.is-inverted.is-outlined.is-focused,html.theme--catppuccin-frappe .docstring>section>a.button.is-inverted.is-outlined.is-focused.docs-sourcelink{background-color:#fff;color:#8caaee}html.theme--catppuccin-frappe .button.is-primary.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .docstring>section>a.button.is-inverted.is-outlined.is-loading.docs-sourcelink:hover::after,html.theme--catppuccin-frappe .button.is-primary.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .docstring>section>a.button.is-inverted.is-outlined.is-loading.is-hovered.docs-sourcelink::after,html.theme--catppuccin-frappe .button.is-primary.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .docstring>section>a.button.is-inverted.is-outlined.is-loading.docs-sourcelink:focus::after,html.theme--catppuccin-frappe .button.is-primary.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-frappe 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.button.is-primary.is-light.is-hovered,html.theme--catppuccin-frappe .docstring>section>a.button.is-light.is-hovered.docs-sourcelink{background-color:#e2eafb;border-color:transparent;color:#153a8e}html.theme--catppuccin-frappe .button.is-primary.is-light:active,html.theme--catppuccin-frappe .docstring>section>a.button.is-light.docs-sourcelink:active,html.theme--catppuccin-frappe .button.is-primary.is-light.is-active,html.theme--catppuccin-frappe .docstring>section>a.button.is-light.is-active.docs-sourcelink{background-color:#d7e1f9;border-color:transparent;color:#153a8e}html.theme--catppuccin-frappe .button.is-link{background-color:#8caaee;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-link:hover,html.theme--catppuccin-frappe .button.is-link.is-hovered{background-color:#81a2ec;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-link:focus,html.theme--catppuccin-frappe 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.button.is-link.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#8caaee}html.theme--catppuccin-frappe .button.is-link.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-link.is-outlined{background-color:transparent;border-color:#8caaee;color:#8caaee}html.theme--catppuccin-frappe .button.is-link.is-outlined:hover,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-link.is-outlined:focus,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-focused{background-color:#8caaee;border-color:#8caaee;color:#fff}html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading::after{border-color:transparent transparent #8caaee #8caaee !important}html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-link.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-link.is-outlined{background-color:transparent;border-color:#8caaee;box-shadow:none;color:#8caaee}html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe 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.button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #81c8be #81c8be !important}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-light{background-color:#f1f9f8;color:#2d675f}html.theme--catppuccin-frappe .button.is-info.is-light:hover,html.theme--catppuccin-frappe .button.is-info.is-light.is-hovered{background-color:#e8f5f3;border-color:transparent;color:#2d675f}html.theme--catppuccin-frappe .button.is-info.is-light:active,html.theme--catppuccin-frappe .button.is-info.is-light.is-active{background-color:#dff1ef;border-color:transparent;color:#2d675f}html.theme--catppuccin-frappe 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.button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6d189 #a6d189 !important}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-light{background-color:#f4f9f0;color:#446a29}html.theme--catppuccin-frappe .button.is-success.is-light:hover,html.theme--catppuccin-frappe .button.is-success.is-light.is-hovered{background-color:#edf6e7;border-color:transparent;color:#446a29}html.theme--catppuccin-frappe .button.is-success.is-light:active,html.theme--catppuccin-frappe .button.is-success.is-light.is-active{background-color:#e6f2de;border-color:transparent;color:#446a29}html.theme--catppuccin-frappe .button.is-warning{background-color:#e5c890;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:hover,html.theme--catppuccin-frappe .button.is-warning.is-hovered{background-color:#e3c386;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:focus,html.theme--catppuccin-frappe .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:focus:not(:active),html.theme--catppuccin-frappe .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(229,200,144,0.25)}html.theme--catppuccin-frappe .button.is-warning:active,html.theme--catppuccin-frappe .button.is-warning.is-active{background-color:#e0be7b;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning{background-color:#e5c890;border-color:#e5c890;box-shadow:none}html.theme--catppuccin-frappe .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted:hover,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-warning.is-outlined{background-color:transparent;border-color:#e5c890;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-outlined:hover,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-warning.is-outlined:focus,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-focused{background-color:#e5c890;border-color:#e5c890;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #e5c890 #e5c890 !important}html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-outlined{background-color:transparent;border-color:#e5c890;box-shadow:none;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #e5c890 #e5c890 !important}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-light{background-color:#fbf7ee;color:#78591c}html.theme--catppuccin-frappe .button.is-warning.is-light:hover,html.theme--catppuccin-frappe .button.is-warning.is-light.is-hovered{background-color:#f9f2e4;border-color:transparent;color:#78591c}html.theme--catppuccin-frappe .button.is-warning.is-light:active,html.theme--catppuccin-frappe .button.is-warning.is-light.is-active{background-color:#f6edda;border-color:transparent;color:#78591c}html.theme--catppuccin-frappe .button.is-danger{background-color:#e78284;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:hover,html.theme--catppuccin-frappe .button.is-danger.is-hovered{background-color:#e57779;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:focus,html.theme--catppuccin-frappe .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:focus:not(:active),html.theme--catppuccin-frappe .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(231,130,132,0.25)}html.theme--catppuccin-frappe .button.is-danger:active,html.theme--catppuccin-frappe .button.is-danger.is-active{background-color:#e36d6f;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger{background-color:#e78284;border-color:#e78284;box-shadow:none}html.theme--catppuccin-frappe .button.is-danger.is-inverted{background-color:#fff;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted:hover,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-frappe .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined{background-color:transparent;border-color:#e78284;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-outlined:hover,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-danger.is-outlined:focus,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-focused{background-color:#e78284;border-color:#e78284;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #e78284 #e78284 !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-outlined{background-color:transparent;border-color:#e78284;box-shadow:none;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #e78284 #e78284 !important}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-light{background-color:#fceeee;color:#9a1e20}html.theme--catppuccin-frappe .button.is-danger.is-light:hover,html.theme--catppuccin-frappe .button.is-danger.is-light.is-hovered{background-color:#fae3e4;border-color:transparent;color:#9a1e20}html.theme--catppuccin-frappe .button.is-danger.is-light:active,html.theme--catppuccin-frappe .button.is-danger.is-light.is-active{background-color:#f8d8d9;border-color:transparent;color:#9a1e20}html.theme--catppuccin-frappe .button.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-frappe .button.is-small:not(.is-rounded),html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-frappe .button.is-normal{font-size:1rem}html.theme--catppuccin-frappe .button.is-medium{font-size:1.25rem}html.theme--catppuccin-frappe .button.is-large{font-size:1.5rem}html.theme--catppuccin-frappe .button[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button{background-color:#737994;border-color:#626880;box-shadow:none;opacity:.5}html.theme--catppuccin-frappe .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-frappe .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-frappe .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-frappe .button.is-static{background-color:#292c3c;border-color:#626880;color:#838ba7;box-shadow:none;pointer-events:none}html.theme--catppuccin-frappe .button.is-rounded,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-frappe .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-frappe .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-frappe .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-frappe .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-frappe .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-frappe .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-frappe .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-frappe .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-frappe .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-frappe .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-frappe .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-frappe .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-frappe .buttons.has-addons .button:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-frappe .buttons.has-addons .button:focus,html.theme--catppuccin-frappe .buttons.has-addons .button.is-focused,html.theme--catppuccin-frappe .buttons.has-addons .button:active,html.theme--catppuccin-frappe .buttons.has-addons .button.is-active,html.theme--catppuccin-frappe .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-frappe .buttons.has-addons .button:focus:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-frappe .buttons.has-addons .button:active:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-frappe .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-frappe .buttons.is-centered{justify-content:center}html.theme--catppuccin-frappe .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-frappe .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-frappe .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-frappe .button.is-responsive.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-frappe .button.is-responsive,html.theme--catppuccin-frappe .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-frappe .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-frappe .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-frappe .button.is-responsive.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-frappe .button.is-responsive,html.theme--catppuccin-frappe .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-frappe .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-frappe .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-frappe .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-frappe .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-frappe .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-frappe .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--catppuccin-frappe .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--catppuccin-frappe .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-frappe .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-frappe .content li+li{margin-top:0.25em}html.theme--catppuccin-frappe .content p:not(:last-child),html.theme--catppuccin-frappe .content dl:not(:last-child),html.theme--catppuccin-frappe .content ol:not(:last-child),html.theme--catppuccin-frappe .content ul:not(:last-child),html.theme--catppuccin-frappe .content blockquote:not(:last-child),html.theme--catppuccin-frappe .content pre:not(:last-child),html.theme--catppuccin-frappe .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-frappe .content h1,html.theme--catppuccin-frappe .content h2,html.theme--catppuccin-frappe .content h3,html.theme--catppuccin-frappe .content h4,html.theme--catppuccin-frappe .content h5,html.theme--catppuccin-frappe .content h6{color:#c6d0f5;font-weight:600;line-height:1.125}html.theme--catppuccin-frappe .content h1{font-size:2em;margin-bottom:0.5em}html.theme--catppuccin-frappe .content h1:not(:first-child){margin-top:1em}html.theme--catppuccin-frappe .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--catppuccin-frappe .content h2:not(:first-child){margin-top:1.1428em}html.theme--catppuccin-frappe .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--catppuccin-frappe .content h3:not(:first-child){margin-top:1.3333em}html.theme--catppuccin-frappe .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--catppuccin-frappe .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--catppuccin-frappe .content h6{font-size:1em;margin-bottom:1em}html.theme--catppuccin-frappe .content blockquote{background-color:#292c3c;border-left:5px solid #626880;padding:1.25em 1.5em}html.theme--catppuccin-frappe .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-frappe .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-frappe .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-frappe .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-frappe .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-frappe .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-frappe .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-frappe .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--catppuccin-frappe .content ul ul ul{list-style-type:square}html.theme--catppuccin-frappe .content dd{margin-left:2em}html.theme--catppuccin-frappe .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--catppuccin-frappe .content figure:not(:first-child){margin-top:2em}html.theme--catppuccin-frappe .content figure:not(:last-child){margin-bottom:2em}html.theme--catppuccin-frappe .content figure img{display:inline-block}html.theme--catppuccin-frappe .content figure figcaption{font-style:italic}html.theme--catppuccin-frappe .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--catppuccin-frappe .content sup,html.theme--catppuccin-frappe .content sub{font-size:75%}html.theme--catppuccin-frappe .content table{width:100%}html.theme--catppuccin-frappe .content table td,html.theme--catppuccin-frappe .content table th{border:1px solid #626880;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--catppuccin-frappe .content table th{color:#b0bef1}html.theme--catppuccin-frappe .content table th:not([align]){text-align:inherit}html.theme--catppuccin-frappe .content table thead td,html.theme--catppuccin-frappe .content table thead th{border-width:0 0 2px;color:#b0bef1}html.theme--catppuccin-frappe .content table tfoot td,html.theme--catppuccin-frappe .content table tfoot th{border-width:2px 0 0;color:#b0bef1}html.theme--catppuccin-frappe .content table tbody tr:last-child td,html.theme--catppuccin-frappe .content table tbody tr:last-child th{border-bottom-width:0}html.theme--catppuccin-frappe .content .tabs li+li{margin-top:0}html.theme--catppuccin-frappe .content.is-small,html.theme--catppuccin-frappe #documenter 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select:not([multiple]){padding-right:2.5em}html.theme--catppuccin-frappe .select select[multiple]{height:auto;padding:0}html.theme--catppuccin-frappe .select select[multiple] option{padding:0.5em 1em}html.theme--catppuccin-frappe .select:not(.is-multiple):not(.is-loading):hover::after{border-color:#99d1db}html.theme--catppuccin-frappe .select.is-white:not(:hover)::after{border-color:#fff}html.theme--catppuccin-frappe .select.is-white select{border-color:#fff}html.theme--catppuccin-frappe .select.is-white select:hover,html.theme--catppuccin-frappe .select.is-white select.is-hovered{border-color:#f2f2f2}html.theme--catppuccin-frappe .select.is-white select:focus,html.theme--catppuccin-frappe .select.is-white select.is-focused,html.theme--catppuccin-frappe .select.is-white select:active,html.theme--catppuccin-frappe .select.is-white select.is-active{box-shadow:0 0 0 0.125em rgba(255,255,255,0.25)}html.theme--catppuccin-frappe 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select.is-focused,html.theme--catppuccin-frappe .select.is-dark select:active,html.theme--catppuccin-frappe .content kbd.select select:active,html.theme--catppuccin-frappe .select.is-dark select.is-active,html.theme--catppuccin-frappe .content kbd.select select.is-active{box-shadow:0 0 0 0.125em rgba(65,69,89,0.25)}html.theme--catppuccin-frappe .select.is-primary:not(:hover)::after,html.theme--catppuccin-frappe .docstring>section>a.select.docs-sourcelink:not(:hover)::after{border-color:#8caaee}html.theme--catppuccin-frappe .select.is-primary select,html.theme--catppuccin-frappe .docstring>section>a.select.docs-sourcelink select{border-color:#8caaee}html.theme--catppuccin-frappe .select.is-primary select:hover,html.theme--catppuccin-frappe .docstring>section>a.select.docs-sourcelink select:hover,html.theme--catppuccin-frappe .select.is-primary select.is-hovered,html.theme--catppuccin-frappe .docstring>section>a.select.docs-sourcelink select.is-hovered{border-color:#769aeb}html.theme--catppuccin-frappe .select.is-primary select:focus,html.theme--catppuccin-frappe .docstring>section>a.select.docs-sourcelink select:focus,html.theme--catppuccin-frappe .select.is-primary select.is-focused,html.theme--catppuccin-frappe .docstring>section>a.select.docs-sourcelink select.is-focused,html.theme--catppuccin-frappe .select.is-primary select:active,html.theme--catppuccin-frappe .docstring>section>a.select.docs-sourcelink select:active,html.theme--catppuccin-frappe .select.is-primary select.is-active,html.theme--catppuccin-frappe .docstring>section>a.select.docs-sourcelink select.is-active{box-shadow:0 0 0 0.125em rgba(140,170,238,0.25)}html.theme--catppuccin-frappe .select.is-link:not(:hover)::after{border-color:#8caaee}html.theme--catppuccin-frappe .select.is-link select{border-color:#8caaee}html.theme--catppuccin-frappe .select.is-link select:hover,html.theme--catppuccin-frappe .select.is-link select.is-hovered{border-color:#769aeb}html.theme--catppuccin-frappe .select.is-link select:focus,html.theme--catppuccin-frappe .select.is-link select.is-focused,html.theme--catppuccin-frappe .select.is-link select:active,html.theme--catppuccin-frappe .select.is-link select.is-active{box-shadow:0 0 0 0.125em rgba(140,170,238,0.25)}html.theme--catppuccin-frappe .select.is-info:not(:hover)::after{border-color:#81c8be}html.theme--catppuccin-frappe .select.is-info select{border-color:#81c8be}html.theme--catppuccin-frappe .select.is-info select:hover,html.theme--catppuccin-frappe .select.is-info select.is-hovered{border-color:#6fc0b5}html.theme--catppuccin-frappe .select.is-info select:focus,html.theme--catppuccin-frappe .select.is-info select.is-focused,html.theme--catppuccin-frappe .select.is-info select:active,html.theme--catppuccin-frappe .select.is-info select.is-active{box-shadow:0 0 0 0.125em rgba(129,200,190,0.25)}html.theme--catppuccin-frappe 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a.navbar-item:focus,html.theme--catppuccin-frappe a.navbar-item:focus-within,html.theme--catppuccin-frappe a.navbar-item:hover,html.theme--catppuccin-frappe a.navbar-item.is-active,html.theme--catppuccin-frappe .navbar-link:focus,html.theme--catppuccin-frappe .navbar-link:focus-within,html.theme--catppuccin-frappe .navbar-link:hover,html.theme--catppuccin-frappe .navbar-link.is-active{background-color:rgba(0,0,0,0);color:#8caaee}html.theme--catppuccin-frappe .navbar-item{flex-grow:0;flex-shrink:0}html.theme--catppuccin-frappe .navbar-item img{max-height:1.75rem}html.theme--catppuccin-frappe .navbar-item.has-dropdown{padding:0}html.theme--catppuccin-frappe .navbar-item.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-frappe .navbar-item.is-tab{border-bottom:1px solid transparent;min-height:4rem;padding-bottom:calc(0.5rem - 1px)}html.theme--catppuccin-frappe .navbar-item.is-tab:focus,html.theme--catppuccin-frappe .navbar-item.is-tab:hover{background-color:rgba(0,0,0,0);border-bottom-color:#8caaee}html.theme--catppuccin-frappe .navbar-item.is-tab.is-active{background-color:rgba(0,0,0,0);border-bottom-color:#8caaee;border-bottom-style:solid;border-bottom-width:3px;color:#8caaee;padding-bottom:calc(0.5rem - 3px)}html.theme--catppuccin-frappe .navbar-content{flex-grow:1;flex-shrink:1}html.theme--catppuccin-frappe .navbar-link:not(.is-arrowless){padding-right:2.5em}html.theme--catppuccin-frappe .navbar-link:not(.is-arrowless)::after{border-color:#fff;margin-top:-0.375em;right:1.125em}html.theme--catppuccin-frappe .navbar-dropdown{font-size:0.875rem;padding-bottom:0.5rem;padding-top:0.5rem}html.theme--catppuccin-frappe .navbar-dropdown .navbar-item{padding-left:1.5rem;padding-right:1.5rem}html.theme--catppuccin-frappe .navbar-divider{background-color:rgba(0,0,0,0.2);border:none;display:none;height:2px;margin:0.5rem 0}@media screen and (max-width: 1055px){html.theme--catppuccin-frappe 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a.navbar-item:hover{background-color:rgba(0,0,0,0);color:#838ba7}html.theme--catppuccin-frappe .navbar-dropdown a.navbar-item.is-active{background-color:rgba(0,0,0,0);color:#8caaee}.navbar.is-spaced html.theme--catppuccin-frappe .navbar-dropdown,html.theme--catppuccin-frappe .navbar-dropdown.is-boxed{border-radius:8px;border-top:none;box-shadow:0 8px 8px rgba(10,10,10,0.1), 0 0 0 1px rgba(10,10,10,0.1);display:block;opacity:0;pointer-events:none;top:calc(100% + (-4px));transform:translateY(-5px);transition-duration:86ms;transition-property:opacity, transform}html.theme--catppuccin-frappe .navbar-dropdown.is-right{left:auto;right:0}html.theme--catppuccin-frappe .navbar-divider{display:block}html.theme--catppuccin-frappe .navbar>.container .navbar-brand,html.theme--catppuccin-frappe .container>.navbar .navbar-brand{margin-left:-.75rem}html.theme--catppuccin-frappe .navbar>.container .navbar-menu,html.theme--catppuccin-frappe .container>.navbar 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body.has-spaced-navbar-fixed-bottom{padding-bottom:6rem}html.theme--catppuccin-frappe a.navbar-item.is-active,html.theme--catppuccin-frappe .navbar-link.is-active{color:#8caaee}html.theme--catppuccin-frappe a.navbar-item.is-active:not(:focus):not(:hover),html.theme--catppuccin-frappe .navbar-link.is-active:not(:focus):not(:hover){background-color:rgba(0,0,0,0)}html.theme--catppuccin-frappe .navbar-item.has-dropdown:focus .navbar-link,html.theme--catppuccin-frappe .navbar-item.has-dropdown:hover .navbar-link,html.theme--catppuccin-frappe .navbar-item.has-dropdown.is-active .navbar-link{background-color:rgba(0,0,0,0)}}html.theme--catppuccin-frappe .hero.is-fullheight-with-navbar{min-height:calc(100vh - 4rem)}html.theme--catppuccin-frappe .pagination{font-size:1rem;margin:-.25rem}html.theme--catppuccin-frappe .pagination.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.pagination{font-size:.75rem}html.theme--catppuccin-frappe .pagination.is-medium{font-size:1.25rem}html.theme--catppuccin-frappe .pagination.is-large{font-size:1.5rem}html.theme--catppuccin-frappe .pagination.is-rounded .pagination-previous,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.pagination .pagination-previous,html.theme--catppuccin-frappe .pagination.is-rounded .pagination-next,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.pagination .pagination-next{padding-left:1em;padding-right:1em;border-radius:9999px}html.theme--catppuccin-frappe .pagination.is-rounded .pagination-link,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.pagination .pagination-link{border-radius:9999px}html.theme--catppuccin-frappe .pagination,html.theme--catppuccin-frappe .pagination-list{align-items:center;display:flex;justify-content:center;text-align:center}html.theme--catppuccin-frappe .pagination-previous,html.theme--catppuccin-frappe .pagination-next,html.theme--catppuccin-frappe .pagination-link,html.theme--catppuccin-frappe .pagination-ellipsis{font-size:1em;justify-content:center;margin:.25rem;padding-left:.5em;padding-right:.5em;text-align:center}html.theme--catppuccin-frappe .pagination-previous,html.theme--catppuccin-frappe .pagination-next,html.theme--catppuccin-frappe .pagination-link{border-color:#626880;color:#8caaee;min-width:2.5em}html.theme--catppuccin-frappe .pagination-previous:hover,html.theme--catppuccin-frappe .pagination-next:hover,html.theme--catppuccin-frappe .pagination-link:hover{border-color:#737994;color:#99d1db}html.theme--catppuccin-frappe .pagination-previous:focus,html.theme--catppuccin-frappe .pagination-next:focus,html.theme--catppuccin-frappe .pagination-link:focus{border-color:#737994}html.theme--catppuccin-frappe .pagination-previous:active,html.theme--catppuccin-frappe .pagination-next:active,html.theme--catppuccin-frappe .pagination-link:active{box-shadow:inset 0 1px 2px 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768px){html.theme--catppuccin-frappe .pagination{flex-wrap:wrap}html.theme--catppuccin-frappe .pagination-previous,html.theme--catppuccin-frappe .pagination-next{flex-grow:1;flex-shrink:1}html.theme--catppuccin-frappe .pagination-list li{flex-grow:1;flex-shrink:1}}@media screen and (min-width: 769px),print{html.theme--catppuccin-frappe .pagination-list{flex-grow:1;flex-shrink:1;justify-content:flex-start;order:1}html.theme--catppuccin-frappe .pagination-previous,html.theme--catppuccin-frappe .pagination-next,html.theme--catppuccin-frappe .pagination-link,html.theme--catppuccin-frappe .pagination-ellipsis{margin-bottom:0;margin-top:0}html.theme--catppuccin-frappe .pagination-previous{order:2}html.theme--catppuccin-frappe .pagination-next{order:3}html.theme--catppuccin-frappe .pagination{justify-content:space-between;margin-bottom:0;margin-top:0}html.theme--catppuccin-frappe .pagination.is-centered .pagination-previous{order:1}html.theme--catppuccin-frappe .pagination.is-centered 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rgba(10,10,10,0.25)}html.theme--catppuccin-latte .button.is-black:active,html.theme--catppuccin-latte .button.is-black.is-active{background-color:#000;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-black[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-black{background-color:#0a0a0a;border-color:#0a0a0a;box-shadow:none}html.theme--catppuccin-latte .button.is-black.is-inverted{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black.is-inverted:hover,html.theme--catppuccin-latte .button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-black.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black.is-outlined:hover,html.theme--catppuccin-latte .button.is-black.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-black.is-outlined:focus,html.theme--catppuccin-latte .button.is-black.is-outlined.is-focused{background-color:#0a0a0a;border-color:#0a0a0a;color:#fff}html.theme--catppuccin-latte .button.is-black.is-outlined.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-latte .button.is-black.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-black.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-black.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-black.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-black.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-light{background-color:#f5f5f5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light:hover,html.theme--catppuccin-latte .button.is-light.is-hovered{background-color:#eee;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light:focus,html.theme--catppuccin-latte .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light:focus:not(:active),html.theme--catppuccin-latte .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(245,245,245,0.25)}html.theme--catppuccin-latte .button.is-light:active,html.theme--catppuccin-latte .button.is-light.is-active{background-color:#e8e8e8;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-light{background-color:#f5f5f5;border-color:#f5f5f5;box-shadow:none}html.theme--catppuccin-latte .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-latte .button.is-light.is-inverted:hover,html.theme--catppuccin-latte .button.is-light.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-latte .button.is-light.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-latte .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;color:#f5f5f5}html.theme--catppuccin-latte .button.is-light.is-outlined:hover,html.theme--catppuccin-latte .button.is-light.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-light.is-outlined:focus,html.theme--catppuccin-latte .button.is-light.is-outlined.is-focused{background-color:#f5f5f5;border-color:#f5f5f5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-latte .button.is-light.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-light.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-light.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-light.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-latte .button.is-light.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark,html.theme--catppuccin-latte .content kbd.button{background-color:#ccd0da;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark:hover,html.theme--catppuccin-latte .content kbd.button:hover,html.theme--catppuccin-latte .button.is-dark.is-hovered,html.theme--catppuccin-latte .content kbd.button.is-hovered{background-color:#c5c9d5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark:focus,html.theme--catppuccin-latte .content kbd.button:focus,html.theme--catppuccin-latte .button.is-dark.is-focused,html.theme--catppuccin-latte .content kbd.button.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark:focus:not(:active),html.theme--catppuccin-latte .content kbd.button:focus:not(:active),html.theme--catppuccin-latte .button.is-dark.is-focused:not(:active),html.theme--catppuccin-latte .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(204,208,218,0.25)}html.theme--catppuccin-latte .button.is-dark:active,html.theme--catppuccin-latte .content kbd.button:active,html.theme--catppuccin-latte .button.is-dark.is-active,html.theme--catppuccin-latte .content kbd.button.is-active{background-color:#bdc2cf;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark[disabled],html.theme--catppuccin-latte .content kbd.button[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-dark,fieldset[disabled] html.theme--catppuccin-latte .content kbd.button{background-color:#ccd0da;border-color:#ccd0da;box-shadow:none}html.theme--catppuccin-latte .button.is-dark.is-inverted,html.theme--catppuccin-latte .content kbd.button.is-inverted{background-color:rgba(0,0,0,0.7);color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-inverted:hover,html.theme--catppuccin-latte .content kbd.button.is-inverted:hover,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-hovered,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark.is-inverted[disabled],html.theme--catppuccin-latte .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-dark.is-inverted,fieldset[disabled] html.theme--catppuccin-latte .content kbd.button.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-loading::after,html.theme--catppuccin-latte .content kbd.button.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-latte .button.is-dark.is-outlined,html.theme--catppuccin-latte .content kbd.button.is-outlined{background-color:transparent;border-color:#ccd0da;color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-outlined:hover,html.theme--catppuccin-latte .content kbd.button.is-outlined:hover,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-hovered,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-dark.is-outlined:focus,html.theme--catppuccin-latte .content kbd.button.is-outlined:focus,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-focused,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-focused{background-color:#ccd0da;border-color:#ccd0da;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark.is-outlined.is-loading::after,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #ccd0da #ccd0da !important}html.theme--catppuccin-latte .button.is-dark.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-latte .button.is-dark.is-outlined[disabled],html.theme--catppuccin-latte .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-dark.is-outlined,fieldset[disabled] html.theme--catppuccin-latte .content kbd.button.is-outlined{background-color:transparent;border-color:#ccd0da;box-shadow:none;color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ccd0da #ccd0da !important}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined[disabled],html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-primary,html.theme--catppuccin-latte .docstring>section>a.button.docs-sourcelink{background-color:#1e66f5;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-primary:hover,html.theme--catppuccin-latte .docstring>section>a.button.docs-sourcelink:hover,html.theme--catppuccin-latte .button.is-primary.is-hovered,html.theme--catppuccin-latte .docstring>section>a.button.is-hovered.docs-sourcelink{background-color:#125ef4;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-primary:focus,html.theme--catppuccin-latte .docstring>section>a.button.docs-sourcelink:focus,html.theme--catppuccin-latte .button.is-primary.is-focused,html.theme--catppuccin-latte .docstring>section>a.button.is-focused.docs-sourcelink{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-primary:focus:not(:active),html.theme--catppuccin-latte .docstring>section>a.button.docs-sourcelink:focus:not(:active),html.theme--catppuccin-latte .button.is-primary.is-focused:not(:active),html.theme--catppuccin-latte .docstring>section>a.button.is-focused.docs-sourcelink:not(:active){box-shadow:0 0 0 0.125em rgba(30,102,245,0.25)}html.theme--catppuccin-latte .button.is-primary:active,html.theme--catppuccin-latte .docstring>section>a.button.docs-sourcelink:active,html.theme--catppuccin-latte .button.is-primary.is-active,html.theme--catppuccin-latte .docstring>section>a.button.is-active.docs-sourcelink{background-color:#0b57ef;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-primary[disabled],html.theme--catppuccin-latte .docstring>section>a.button.docs-sourcelink[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-primary,fieldset[disabled] html.theme--catppuccin-latte .docstring>section>a.button.docs-sourcelink{background-color:#1e66f5;border-color:#1e66f5;box-shadow:none}html.theme--catppuccin-latte .button.is-primary.is-inverted,html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.docs-sourcelink{background-color:#fff;color:#1e66f5}html.theme--catppuccin-latte .button.is-primary.is-inverted:hover,html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.docs-sourcelink:hover,html.theme--catppuccin-latte .button.is-primary.is-inverted.is-hovered,html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.is-hovered.docs-sourcelink{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-primary.is-inverted[disabled],html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.docs-sourcelink[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-primary.is-inverted,fieldset[disabled] html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.docs-sourcelink{background-color:#fff;border-color:transparent;box-shadow:none;color:#1e66f5}html.theme--catppuccin-latte .button.is-primary.is-loading::after,html.theme--catppuccin-latte .docstring>section>a.button.is-loading.docs-sourcelink::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-primary.is-outlined,html.theme--catppuccin-latte .docstring>section>a.button.is-outlined.docs-sourcelink{background-color:transparent;border-color:#1e66f5;color:#1e66f5}html.theme--catppuccin-latte .button.is-primary.is-outlined:hover,html.theme--catppuccin-latte .docstring>section>a.button.is-outlined.docs-sourcelink:hover,html.theme--catppuccin-latte .button.is-primary.is-outlined.is-hovered,html.theme--catppuccin-latte .docstring>section>a.button.is-outlined.is-hovered.docs-sourcelink,html.theme--catppuccin-latte .button.is-primary.is-outlined:focus,html.theme--catppuccin-latte .docstring>section>a.button.is-outlined.docs-sourcelink:focus,html.theme--catppuccin-latte .button.is-primary.is-outlined.is-focused,html.theme--catppuccin-latte 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.button.is-primary.is-light.is-active,html.theme--catppuccin-latte .docstring>section>a.button.is-light.is-active.docs-sourcelink{background-color:#d3e1fd;border-color:transparent;color:#0a52e1}html.theme--catppuccin-latte .button.is-link{background-color:#1e66f5;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-link:hover,html.theme--catppuccin-latte .button.is-link.is-hovered{background-color:#125ef4;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-link:focus,html.theme--catppuccin-latte .button.is-link.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-link:focus:not(:active),html.theme--catppuccin-latte .button.is-link.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(30,102,245,0.25)}html.theme--catppuccin-latte .button.is-link:active,html.theme--catppuccin-latte .button.is-link.is-active{background-color:#0b57ef;border-color:transparent;color:#fff}html.theme--catppuccin-latte 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.button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-outlined{background-color:transparent;border-color:#40a02b;box-shadow:none;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-focused{background-color:#fff;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #40a02b #40a02b !important}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-success.is-light{background-color:#f1fbef;color:#40a12b}html.theme--catppuccin-latte .button.is-success.is-light:hover,html.theme--catppuccin-latte .button.is-success.is-light.is-hovered{background-color:#e8f8e5;border-color:transparent;color:#40a12b}html.theme--catppuccin-latte .button.is-success.is-light:active,html.theme--catppuccin-latte .button.is-success.is-light.is-active{background-color:#e0f5db;border-color:transparent;color:#40a12b}html.theme--catppuccin-latte .button.is-warning{background-color:#df8e1d;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:hover,html.theme--catppuccin-latte .button.is-warning.is-hovered{background-color:#d4871c;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:focus,html.theme--catppuccin-latte .button.is-warning.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:focus:not(:active),html.theme--catppuccin-latte .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(223,142,29,0.25)}html.theme--catppuccin-latte .button.is-warning:active,html.theme--catppuccin-latte .button.is-warning.is-active{background-color:#c8801a;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning{background-color:#df8e1d;border-color:#df8e1d;box-shadow:none}html.theme--catppuccin-latte .button.is-warning.is-inverted{background-color:#fff;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted:hover,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-warning.is-outlined{background-color:transparent;border-color:#df8e1d;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-outlined:hover,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-warning.is-outlined:focus,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-focused{background-color:#df8e1d;border-color:#df8e1d;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #df8e1d #df8e1d !important}html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-outlined{background-color:transparent;border-color:#df8e1d;box-shadow:none;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-focused{background-color:#fff;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #df8e1d #df8e1d !important}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-light{background-color:#fdf6ed;color:#9e6515}html.theme--catppuccin-latte .button.is-warning.is-light:hover,html.theme--catppuccin-latte .button.is-warning.is-light.is-hovered{background-color:#fbf1e2;border-color:transparent;color:#9e6515}html.theme--catppuccin-latte .button.is-warning.is-light:active,html.theme--catppuccin-latte .button.is-warning.is-light.is-active{background-color:#faebd6;border-color:transparent;color:#9e6515}html.theme--catppuccin-latte .button.is-danger{background-color:#d20f39;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:hover,html.theme--catppuccin-latte .button.is-danger.is-hovered{background-color:#c60e36;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:focus,html.theme--catppuccin-latte .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:focus:not(:active),html.theme--catppuccin-latte .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(210,15,57,0.25)}html.theme--catppuccin-latte .button.is-danger:active,html.theme--catppuccin-latte .button.is-danger.is-active{background-color:#ba0d33;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger{background-color:#d20f39;border-color:#d20f39;box-shadow:none}html.theme--catppuccin-latte .button.is-danger.is-inverted{background-color:#fff;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted:hover,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-danger.is-outlined{background-color:transparent;border-color:#d20f39;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-outlined:hover,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-danger.is-outlined:focus,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-focused{background-color:#d20f39;border-color:#d20f39;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #d20f39 #d20f39 !important}html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-outlined{background-color:transparent;border-color:#d20f39;box-shadow:none;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #d20f39 #d20f39 !important}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-light{background-color:#feecf0;color:#e9113f}html.theme--catppuccin-latte .button.is-danger.is-light:hover,html.theme--catppuccin-latte .button.is-danger.is-light.is-hovered{background-color:#fde0e6;border-color:transparent;color:#e9113f}html.theme--catppuccin-latte .button.is-danger.is-light:active,html.theme--catppuccin-latte .button.is-danger.is-light.is-active{background-color:#fcd4dd;border-color:transparent;color:#e9113f}html.theme--catppuccin-latte .button.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-latte .button.is-small:not(.is-rounded),html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-latte .button.is-normal{font-size:1rem}html.theme--catppuccin-latte .button.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .button.is-large{font-size:1.5rem}html.theme--catppuccin-latte .button[disabled],fieldset[disabled] html.theme--catppuccin-latte .button{background-color:#9ca0b0;border-color:#acb0be;box-shadow:none;opacity:.5}html.theme--catppuccin-latte .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-latte .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-latte .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-latte .button.is-static{background-color:#e6e9ef;border-color:#acb0be;color:#8c8fa1;box-shadow:none;pointer-events:none}html.theme--catppuccin-latte .button.is-rounded,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-latte .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-latte .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-latte .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-latte .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-latte .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-latte .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-latte .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-latte .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-latte .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-latte .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-latte .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-latte .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-latte .buttons.has-addons .button:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-latte .buttons.has-addons .button:focus,html.theme--catppuccin-latte .buttons.has-addons .button.is-focused,html.theme--catppuccin-latte .buttons.has-addons .button:active,html.theme--catppuccin-latte .buttons.has-addons .button.is-active,html.theme--catppuccin-latte .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-latte .buttons.has-addons .button:focus:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-latte .buttons.has-addons .button:active:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-latte .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .buttons.is-centered{justify-content:center}html.theme--catppuccin-latte .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-latte .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-latte .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-latte .button.is-responsive.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-latte .button.is-responsive,html.theme--catppuccin-latte .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-latte .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-latte .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-latte .button.is-responsive.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-latte .button.is-responsive,html.theme--catppuccin-latte .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-latte .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-latte .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-latte .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-latte .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-latte .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-latte .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--catppuccin-latte .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--catppuccin-latte .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-latte .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-latte .content li+li{margin-top:0.25em}html.theme--catppuccin-latte .content p:not(:last-child),html.theme--catppuccin-latte .content dl:not(:last-child),html.theme--catppuccin-latte .content ol:not(:last-child),html.theme--catppuccin-latte .content ul:not(:last-child),html.theme--catppuccin-latte .content blockquote:not(:last-child),html.theme--catppuccin-latte .content pre:not(:last-child),html.theme--catppuccin-latte .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-latte .content h1,html.theme--catppuccin-latte .content h2,html.theme--catppuccin-latte .content h3,html.theme--catppuccin-latte .content h4,html.theme--catppuccin-latte .content h5,html.theme--catppuccin-latte .content h6{color:#4c4f69;font-weight:600;line-height:1.125}html.theme--catppuccin-latte .content h1{font-size:2em;margin-bottom:0.5em}html.theme--catppuccin-latte .content h1:not(:first-child){margin-top:1em}html.theme--catppuccin-latte .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--catppuccin-latte .content h2:not(:first-child){margin-top:1.1428em}html.theme--catppuccin-latte .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--catppuccin-latte .content h3:not(:first-child){margin-top:1.3333em}html.theme--catppuccin-latte .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--catppuccin-latte .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--catppuccin-latte .content h6{font-size:1em;margin-bottom:1em}html.theme--catppuccin-latte .content blockquote{background-color:#e6e9ef;border-left:5px solid #acb0be;padding:1.25em 1.5em}html.theme--catppuccin-latte .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-latte .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-latte .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-latte .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-latte .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-latte .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-latte .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-latte .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--catppuccin-latte .content ul ul ul{list-style-type:square}html.theme--catppuccin-latte .content dd{margin-left:2em}html.theme--catppuccin-latte .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--catppuccin-latte .content figure:not(:first-child){margin-top:2em}html.theme--catppuccin-latte .content figure:not(:last-child){margin-bottom:2em}html.theme--catppuccin-latte .content figure img{display:inline-block}html.theme--catppuccin-latte .content figure figcaption{font-style:italic}html.theme--catppuccin-latte .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--catppuccin-latte .content sup,html.theme--catppuccin-latte .content sub{font-size:75%}html.theme--catppuccin-latte .content table{width:100%}html.theme--catppuccin-latte .content table td,html.theme--catppuccin-latte .content table th{border:1px solid #acb0be;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--catppuccin-latte .content table th{color:#41445a}html.theme--catppuccin-latte .content table 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#documenter .docs-sidebar form.docs-search>input.file{font-size:.75rem}html.theme--catppuccin-latte .file.is-normal{font-size:1rem}html.theme--catppuccin-latte .file.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .file.is-medium .file-icon .fa{font-size:21px}html.theme--catppuccin-latte .file.is-large{font-size:1.5rem}html.theme--catppuccin-latte .file.is-large .file-icon .fa{font-size:28px}html.theme--catppuccin-latte .file.has-name .file-cta{border-bottom-right-radius:0;border-top-right-radius:0}html.theme--catppuccin-latte .file.has-name .file-name{border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-latte .file.has-name.is-empty .file-cta{border-radius:.4em}html.theme--catppuccin-latte .file.has-name.is-empty .file-name{display:none}html.theme--catppuccin-latte .file.is-boxed .file-label{flex-direction:column}html.theme--catppuccin-latte .file.is-boxed .file-cta{flex-direction:column;height:auto;padding:1em 3em}html.theme--catppuccin-latte .file.is-boxed .file-name{border-width:0 1px 1px}html.theme--catppuccin-latte .file.is-boxed .file-icon{height:1.5em;width:1.5em}html.theme--catppuccin-latte .file.is-boxed .file-icon .fa{font-size:21px}html.theme--catppuccin-latte .file.is-boxed.is-small .file-icon .fa,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-boxed .file-icon .fa{font-size:14px}html.theme--catppuccin-latte .file.is-boxed.is-medium .file-icon .fa{font-size:28px}html.theme--catppuccin-latte .file.is-boxed.is-large .file-icon .fa{font-size:35px}html.theme--catppuccin-latte .file.is-boxed.has-name .file-cta{border-radius:.4em .4em 0 0}html.theme--catppuccin-latte .file.is-boxed.has-name .file-name{border-radius:0 0 .4em .4em;border-width:0 1px 1px}html.theme--catppuccin-latte .file.is-centered{justify-content:center}html.theme--catppuccin-latte .file.is-fullwidth .file-label{width:100%}html.theme--catppuccin-latte .file.is-fullwidth .file-name{flex-grow:1;max-width:none}html.theme--catppuccin-latte .file.is-right{justify-content:flex-end}html.theme--catppuccin-latte .file.is-right .file-cta{border-radius:0 .4em .4em 0}html.theme--catppuccin-latte .file.is-right .file-name{border-radius:.4em 0 0 .4em;border-width:1px 0 1px 1px;order:-1}html.theme--catppuccin-latte .file-label{align-items:stretch;display:flex;cursor:pointer;justify-content:flex-start;overflow:hidden;position:relative}html.theme--catppuccin-latte .file-label:hover .file-cta{background-color:#c5c9d5;color:#41445a}html.theme--catppuccin-latte .file-label:hover .file-name{border-color:#a5a9b8}html.theme--catppuccin-latte .file-label:active .file-cta{background-color:#bdc2cf;color:#41445a}html.theme--catppuccin-latte .file-label:active .file-name{border-color:#9ea2b3}html.theme--catppuccin-latte .file-input{height:100%;left:0;opacity:0;outline:none;position:absolute;top:0;width:100%}html.theme--catppuccin-latte .file-cta,html.theme--catppuccin-latte .file-name{border-color:#acb0be;border-radius:.4em;font-size:1em;padding-left:1em;padding-right:1em;white-space:nowrap}html.theme--catppuccin-latte .file-cta{background-color:#ccd0da;color:#4c4f69}html.theme--catppuccin-latte .file-name{border-color:#acb0be;border-style:solid;border-width:1px 1px 1px 0;display:block;max-width:16em;overflow:hidden;text-align:inherit;text-overflow:ellipsis}html.theme--catppuccin-latte .file-icon{align-items:center;display:flex;height:1em;justify-content:center;margin-right:.5em;width:1em}html.theme--catppuccin-latte .file-icon .fa{font-size:14px}html.theme--catppuccin-latte .label{color:#41445a;display:block;font-size:1rem;font-weight:700}html.theme--catppuccin-latte .label:not(:last-child){margin-bottom:0.5em}html.theme--catppuccin-latte .label.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.label{font-size:.75rem}html.theme--catppuccin-latte .label.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .label.is-large{font-size:1.5rem}html.theme--catppuccin-latte .help{display:block;font-size:.75rem;margin-top:0.25rem}html.theme--catppuccin-latte .help.is-white{color:#fff}html.theme--catppuccin-latte .help.is-black{color:#0a0a0a}html.theme--catppuccin-latte .help.is-light{color:#f5f5f5}html.theme--catppuccin-latte .help.is-dark,html.theme--catppuccin-latte .content kbd.help{color:#ccd0da}html.theme--catppuccin-latte .help.is-primary,html.theme--catppuccin-latte .docstring>section>a.help.docs-sourcelink{color:#1e66f5}html.theme--catppuccin-latte .help.is-link{color:#1e66f5}html.theme--catppuccin-latte .help.is-info{color:#179299}html.theme--catppuccin-latte .help.is-success{color:#40a02b}html.theme--catppuccin-latte .help.is-warning{color:#df8e1d}html.theme--catppuccin-latte .help.is-danger{color:#d20f39}html.theme--catppuccin-latte .field:not(:last-child){margin-bottom:0.75rem}html.theme--catppuccin-latte .field.has-addons{display:flex;justify-content:flex-start}html.theme--catppuccin-latte .field.has-addons .control:not(:last-child){margin-right:-1px}html.theme--catppuccin-latte .field.has-addons .control:not(:first-child):not(:last-child) .button,html.theme--catppuccin-latte .field.has-addons .control:not(:first-child):not(:last-child) .input,html.theme--catppuccin-latte .field.has-addons .control:not(:first-child):not(:last-child) #documenter .docs-sidebar form.docs-search>input,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control:not(:first-child):not(:last-child) form.docs-search>input,html.theme--catppuccin-latte .field.has-addons .control:not(:first-child):not(:last-child) .select select{border-radius:0}html.theme--catppuccin-latte .field.has-addons .control:first-child:not(:only-child) .button,html.theme--catppuccin-latte .field.has-addons .control:first-child:not(:only-child) .input,html.theme--catppuccin-latte .field.has-addons .control:first-child:not(:only-child) #documenter .docs-sidebar form.docs-search>input,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control:first-child:not(:only-child) form.docs-search>input,html.theme--catppuccin-latte .field.has-addons .control:first-child:not(:only-child) .select select{border-bottom-right-radius:0;border-top-right-radius:0}html.theme--catppuccin-latte .field.has-addons .control:last-child:not(:only-child) .button,html.theme--catppuccin-latte .field.has-addons .control:last-child:not(:only-child) .input,html.theme--catppuccin-latte .field.has-addons .control:last-child:not(:only-child) #documenter .docs-sidebar form.docs-search>input,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control:last-child:not(:only-child) form.docs-search>input,html.theme--catppuccin-latte .field.has-addons .control:last-child:not(:only-child) .select select{border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-latte .field.has-addons .control .button:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .button.is-hovered:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .input:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input:not([disabled]):hover,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .input.is-hovered:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input.is-hovered:not([disabled]),html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input.is-hovered:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .select select:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .select select.is-hovered:not([disabled]){z-index:2}html.theme--catppuccin-latte .field.has-addons .control .button:not([disabled]):focus,html.theme--catppuccin-latte .field.has-addons .control .button.is-focused:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .button:not([disabled]):active,html.theme--catppuccin-latte .field.has-addons .control .button.is-active:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .input:not([disabled]):focus,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input:not([disabled]):focus,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input:not([disabled]):focus,html.theme--catppuccin-latte .field.has-addons .control .input.is-focused:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input.is-focused:not([disabled]),html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input.is-focused:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .input:not([disabled]):active,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input:not([disabled]):active,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input:not([disabled]):active,html.theme--catppuccin-latte .field.has-addons .control .input.is-active:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input.is-active:not([disabled]),html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input.is-active:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .select select:not([disabled]):focus,html.theme--catppuccin-latte .field.has-addons .control .select select.is-focused:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .select select:not([disabled]):active,html.theme--catppuccin-latte .field.has-addons .control .select select.is-active:not([disabled]){z-index:3}html.theme--catppuccin-latte .field.has-addons .control .button:not([disabled]):focus:hover,html.theme--catppuccin-latte .field.has-addons .control .button.is-focused:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .button:not([disabled]):active:hover,html.theme--catppuccin-latte .field.has-addons .control .button.is-active:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .input:not([disabled]):focus:hover,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input:not([disabled]):focus:hover,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input:not([disabled]):focus:hover,html.theme--catppuccin-latte .field.has-addons .control .input.is-focused:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input.is-focused:not([disabled]):hover,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input.is-focused:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .input:not([disabled]):active:hover,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input:not([disabled]):active:hover,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input:not([disabled]):active:hover,html.theme--catppuccin-latte .field.has-addons .control .input.is-active:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input.is-active:not([disabled]):hover,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input.is-active:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .select select:not([disabled]):focus:hover,html.theme--catppuccin-latte .field.has-addons .control .select select.is-focused:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .select select:not([disabled]):active:hover,html.theme--catppuccin-latte .field.has-addons .control .select select.is-active:not([disabled]):hover{z-index:4}html.theme--catppuccin-latte .field.has-addons .control.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .field.has-addons.has-addons-centered{justify-content:center}html.theme--catppuccin-latte .field.has-addons.has-addons-right{justify-content:flex-end}html.theme--catppuccin-latte .field.has-addons.has-addons-fullwidth .control{flex-grow:1;flex-shrink:0}html.theme--catppuccin-latte .field.is-grouped{display:flex;justify-content:flex-start}html.theme--catppuccin-latte .field.is-grouped>.control{flex-shrink:0}html.theme--catppuccin-latte .field.is-grouped>.control:not(:last-child){margin-bottom:0;margin-right:.75rem}html.theme--catppuccin-latte .field.is-grouped>.control.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .field.is-grouped.is-grouped-centered{justify-content:center}html.theme--catppuccin-latte .field.is-grouped.is-grouped-right{justify-content:flex-end}html.theme--catppuccin-latte .field.is-grouped.is-grouped-multiline{flex-wrap:wrap}html.theme--catppuccin-latte .field.is-grouped.is-grouped-multiline>.control:last-child,html.theme--catppuccin-latte .field.is-grouped.is-grouped-multiline>.control:not(:last-child){margin-bottom:0.75rem}html.theme--catppuccin-latte .field.is-grouped.is-grouped-multiline:last-child{margin-bottom:-0.75rem}html.theme--catppuccin-latte .field.is-grouped.is-grouped-multiline:not(:last-child){margin-bottom:0}@media screen and (min-width: 769px),print{html.theme--catppuccin-latte .field.is-horizontal{display:flex}}html.theme--catppuccin-latte .field-label .label{font-size:inherit}@media screen and (max-width: 768px){html.theme--catppuccin-latte .field-label{margin-bottom:0.5rem}}@media screen and (min-width: 769px),print{html.theme--catppuccin-latte .field-label{flex-basis:0;flex-grow:1;flex-shrink:0;margin-right:1.5rem;text-align:right}html.theme--catppuccin-latte .field-label.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.field-label{font-size:.75rem;padding-top:0.375em}html.theme--catppuccin-latte .field-label.is-normal{padding-top:0.375em}html.theme--catppuccin-latte .field-label.is-medium{font-size:1.25rem;padding-top:0.375em}html.theme--catppuccin-latte .field-label.is-large{font-size:1.5rem;padding-top:0.375em}}html.theme--catppuccin-latte .field-body .field .field{margin-bottom:0}@media screen and (min-width: 769px),print{html.theme--catppuccin-latte .field-body{display:flex;flex-basis:0;flex-grow:5;flex-shrink:1}html.theme--catppuccin-latte .field-body .field{margin-bottom:0}html.theme--catppuccin-latte .field-body>.field{flex-shrink:1}html.theme--catppuccin-latte .field-body>.field:not(.is-narrow){flex-grow:1}html.theme--catppuccin-latte .field-body>.field:not(:last-child){margin-right:.75rem}}html.theme--catppuccin-latte .control{box-sizing:border-box;clear:both;font-size:1rem;position:relative;text-align:inherit}html.theme--catppuccin-latte .control.has-icons-left .input:focus~.icon,html.theme--catppuccin-latte .control.has-icons-left #documenter .docs-sidebar form.docs-search>input:focus~.icon,html.theme--catppuccin-latte #documenter .docs-sidebar .control.has-icons-left form.docs-search>input:focus~.icon,html.theme--catppuccin-latte .control.has-icons-left .select:focus~.icon,html.theme--catppuccin-latte .control.has-icons-right .input:focus~.icon,html.theme--catppuccin-latte .control.has-icons-right #documenter .docs-sidebar form.docs-search>input:focus~.icon,html.theme--catppuccin-latte #documenter .docs-sidebar .control.has-icons-right form.docs-search>input:focus~.icon,html.theme--catppuccin-latte .control.has-icons-right .select:focus~.icon{color:#ccd0da}html.theme--catppuccin-latte .control.has-icons-left .input.is-small~.icon,html.theme--catppuccin-latte .control.has-icons-left #documenter .docs-sidebar form.docs-search>input~.icon,html.theme--catppuccin-latte #documenter .docs-sidebar .control.has-icons-left form.docs-search>input~.icon,html.theme--catppuccin-latte .control.has-icons-left .select.is-small~.icon,html.theme--catppuccin-latte .control.has-icons-right .input.is-small~.icon,html.theme--catppuccin-latte .control.has-icons-right #documenter .docs-sidebar form.docs-search>input~.icon,html.theme--catppuccin-latte #documenter .docs-sidebar .control.has-icons-right form.docs-search>input~.icon,html.theme--catppuccin-latte .control.has-icons-right .select.is-small~.icon{font-size:.75rem}html.theme--catppuccin-latte .control.has-icons-left .input.is-medium~.icon,html.theme--catppuccin-latte .control.has-icons-left #documenter .docs-sidebar form.docs-search>input.is-medium~.icon,html.theme--catppuccin-latte #documenter .docs-sidebar .control.has-icons-left form.docs-search>input.is-medium~.icon,html.theme--catppuccin-latte .control.has-icons-left .select.is-medium~.icon,html.theme--catppuccin-latte .control.has-icons-right .input.is-medium~.icon,html.theme--catppuccin-latte .control.has-icons-right #documenter .docs-sidebar form.docs-search>input.is-medium~.icon,html.theme--catppuccin-latte #documenter .docs-sidebar .control.has-icons-right form.docs-search>input.is-medium~.icon,html.theme--catppuccin-latte .control.has-icons-right .select.is-medium~.icon{font-size:1.25rem}html.theme--catppuccin-latte .control.has-icons-left 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kbd.button.is-outlined{background-color:transparent;border-color:#363a4f;box-shadow:none;color:#363a4f}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-macchiato .content 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.button.is-info.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-info.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-focused{background-color:#8bd5ca;border-color:#8bd5ca;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #8bd5ca #8bd5ca !important}html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-info.is-outlined{background-color:transparent;border-color:#8bd5ca;box-shadow:none;color:#8bd5ca}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#8bd5ca}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #8bd5ca #8bd5ca !important}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-info.is-light{background-color:#f0faf8;color:#276d62}html.theme--catppuccin-macchiato .button.is-info.is-light:hover,html.theme--catppuccin-macchiato .button.is-info.is-light.is-hovered{background-color:#e7f6f4;border-color:transparent;color:#276d62}html.theme--catppuccin-macchiato .button.is-info.is-light:active,html.theme--catppuccin-macchiato .button.is-info.is-light.is-active{background-color:#ddf3f0;border-color:transparent;color:#276d62}html.theme--catppuccin-macchiato .button.is-success{background-color:#a6da95;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success:hover,html.theme--catppuccin-macchiato .button.is-success.is-hovered{background-color:#9ed78c;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success:focus,html.theme--catppuccin-macchiato .button.is-success.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success:focus:not(:active),html.theme--catppuccin-macchiato .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(166,218,149,0.25)}html.theme--catppuccin-macchiato .button.is-success:active,html.theme--catppuccin-macchiato .button.is-success.is-active{background-color:#96d382;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success{background-color:#a6da95;border-color:#a6da95;box-shadow:none}html.theme--catppuccin-macchiato .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined{background-color:transparent;border-color:#a6da95;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-success.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-focused{background-color:#a6da95;border-color:#a6da95;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6da95 #a6da95 !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-outlined{background-color:transparent;border-color:#a6da95;box-shadow:none;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6da95 #a6da95 !important}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-light{background-color:#f2faf0;color:#386e26}html.theme--catppuccin-macchiato .button.is-success.is-light:hover,html.theme--catppuccin-macchiato .button.is-success.is-light.is-hovered{background-color:#eaf6e6;border-color:transparent;color:#386e26}html.theme--catppuccin-macchiato .button.is-success.is-light:active,html.theme--catppuccin-macchiato .button.is-success.is-light.is-active{background-color:#e2f3dd;border-color:transparent;color:#386e26}html.theme--catppuccin-macchiato .button.is-warning{background-color:#eed49f;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:hover,html.theme--catppuccin-macchiato .button.is-warning.is-hovered{background-color:#eccf94;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:focus,html.theme--catppuccin-macchiato .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:focus:not(:active),html.theme--catppuccin-macchiato .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(238,212,159,0.25)}html.theme--catppuccin-macchiato .button.is-warning:active,html.theme--catppuccin-macchiato .button.is-warning.is-active{background-color:#eaca89;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning{background-color:#eed49f;border-color:#eed49f;box-shadow:none}html.theme--catppuccin-macchiato .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined{background-color:transparent;border-color:#eed49f;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-warning.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-focused{background-color:#eed49f;border-color:#eed49f;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #eed49f #eed49f !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-outlined{background-color:transparent;border-color:#eed49f;box-shadow:none;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #eed49f #eed49f !important}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-light{background-color:#fcf7ee;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-warning.is-light:hover,html.theme--catppuccin-macchiato .button.is-warning.is-light.is-hovered{background-color:#faf2e3;border-color:transparent;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-warning.is-light:active,html.theme--catppuccin-macchiato .button.is-warning.is-light.is-active{background-color:#f8eed8;border-color:transparent;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-danger{background-color:#ed8796;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:hover,html.theme--catppuccin-macchiato .button.is-danger.is-hovered{background-color:#eb7c8c;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:focus,html.theme--catppuccin-macchiato .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:focus:not(:active),html.theme--catppuccin-macchiato .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(237,135,150,0.25)}html.theme--catppuccin-macchiato .button.is-danger:active,html.theme--catppuccin-macchiato .button.is-danger.is-active{background-color:#ea7183;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger{background-color:#ed8796;border-color:#ed8796;box-shadow:none}html.theme--catppuccin-macchiato .button.is-danger.is-inverted{background-color:#fff;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-macchiato .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined{background-color:transparent;border-color:#ed8796;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-danger.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-focused{background-color:#ed8796;border-color:#ed8796;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #ed8796 #ed8796 !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-outlined{background-color:transparent;border-color:#ed8796;box-shadow:none;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ed8796 #ed8796 !important}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-light{background-color:#fcedef;color:#971729}html.theme--catppuccin-macchiato .button.is-danger.is-light:hover,html.theme--catppuccin-macchiato .button.is-danger.is-light.is-hovered{background-color:#fbe2e6;border-color:transparent;color:#971729}html.theme--catppuccin-macchiato .button.is-danger.is-light:active,html.theme--catppuccin-macchiato .button.is-danger.is-light.is-active{background-color:#f9d7dc;border-color:transparent;color:#971729}html.theme--catppuccin-macchiato .button.is-small,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-macchiato .button.is-small:not(.is-rounded),html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-macchiato .button.is-normal{font-size:1rem}html.theme--catppuccin-macchiato .button.is-medium{font-size:1.25rem}html.theme--catppuccin-macchiato .button.is-large{font-size:1.5rem}html.theme--catppuccin-macchiato .button[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button{background-color:#6e738d;border-color:#5b6078;box-shadow:none;opacity:.5}html.theme--catppuccin-macchiato .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-macchiato .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-macchiato .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-macchiato .button.is-static{background-color:#1e2030;border-color:#5b6078;color:#8087a2;box-shadow:none;pointer-events:none}html.theme--catppuccin-macchiato .button.is-rounded,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-macchiato .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-macchiato .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-macchiato .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-macchiato .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-macchiato .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-macchiato .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-macchiato .buttons.are-small 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.container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-macchiato .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-macchiato .content li+li{margin-top:0.25em}html.theme--catppuccin-macchiato .content p:not(:last-child),html.theme--catppuccin-macchiato .content dl:not(:last-child),html.theme--catppuccin-macchiato .content ol:not(:last-child),html.theme--catppuccin-macchiato .content ul:not(:last-child),html.theme--catppuccin-macchiato .content blockquote:not(:last-child),html.theme--catppuccin-macchiato .content pre:not(:last-child),html.theme--catppuccin-macchiato .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-macchiato .content h1,html.theme--catppuccin-macchiato .content h2,html.theme--catppuccin-macchiato .content h3,html.theme--catppuccin-macchiato .content h4,html.theme--catppuccin-macchiato .content h5,html.theme--catppuccin-macchiato .content 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ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-macchiato .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-macchiato .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-macchiato .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-macchiato .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-macchiato .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-macchiato .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-macchiato .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--catppuccin-macchiato .content ul ul ul{list-style-type:square}html.theme--catppuccin-macchiato .content dd{margin-left:2em}html.theme--catppuccin-macchiato .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--catppuccin-macchiato 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form.docs-search>input.is-active{box-shadow:0 0 0 0.125em rgba(255,255,255,0.25)}html.theme--catppuccin-macchiato .is-black.textarea,html.theme--catppuccin-macchiato .is-black.input,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.is-black{border-color:#0a0a0a}html.theme--catppuccin-macchiato .is-black.textarea:focus,html.theme--catppuccin-macchiato .is-black.input:focus,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.is-black:focus,html.theme--catppuccin-macchiato .is-black.is-focused.textarea,html.theme--catppuccin-macchiato .is-black.is-focused.input,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.is-focused,html.theme--catppuccin-macchiato .is-black.textarea:active,html.theme--catppuccin-macchiato .is-black.input:active,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.is-black:active,html.theme--catppuccin-macchiato 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html.theme--catppuccin-mocha .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-link.is-light{background-color:#ebf3fe;color:#063c93}html.theme--catppuccin-mocha .button.is-link.is-light:hover,html.theme--catppuccin-mocha .button.is-link.is-light.is-hovered{background-color:#dfebfe;border-color:transparent;color:#063c93}html.theme--catppuccin-mocha .button.is-link.is-light:active,html.theme--catppuccin-mocha .button.is-link.is-light.is-active{background-color:#d3e3fd;border-color:transparent;color:#063c93}html.theme--catppuccin-mocha .button.is-info{background-color:#94e2d5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info:hover,html.theme--catppuccin-mocha .button.is-info.is-hovered{background-color:#8adfd1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info:focus,html.theme--catppuccin-mocha .button.is-info.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info:focus:not(:active),html.theme--catppuccin-mocha .button.is-info.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(148,226,213,0.25)}html.theme--catppuccin-mocha .button.is-info:active,html.theme--catppuccin-mocha .button.is-info.is-active{background-color:#80ddcd;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info{background-color:#94e2d5;border-color:#94e2d5;box-shadow:none}html.theme--catppuccin-mocha .button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-inverted:hover,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-info.is-outlined{background-color:transparent;border-color:#94e2d5;color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-outlined:hover,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-info.is-outlined:focus,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-focused{background-color:#94e2d5;border-color:#94e2d5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #94e2d5 #94e2d5 !important}html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info.is-outlined{background-color:transparent;border-color:#94e2d5;box-shadow:none;color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #94e2d5 #94e2d5 !important}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-light{background-color:#effbf9;color:#207466}html.theme--catppuccin-mocha .button.is-info.is-light:hover,html.theme--catppuccin-mocha .button.is-info.is-light.is-hovered{background-color:#e5f8f5;border-color:transparent;color:#207466}html.theme--catppuccin-mocha .button.is-info.is-light:active,html.theme--catppuccin-mocha .button.is-info.is-light.is-active{background-color:#dbf5f1;border-color:transparent;color:#207466}html.theme--catppuccin-mocha .button.is-success{background-color:#a6e3a1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:hover,html.theme--catppuccin-mocha .button.is-success.is-hovered{background-color:#9de097;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:focus,html.theme--catppuccin-mocha .button.is-success.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:focus:not(:active),html.theme--catppuccin-mocha .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(166,227,161,0.25)}html.theme--catppuccin-mocha .button.is-success:active,html.theme--catppuccin-mocha .button.is-success.is-active{background-color:#93dd8d;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success{background-color:#a6e3a1;border-color:#a6e3a1;box-shadow:none}html.theme--catppuccin-mocha .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted:hover,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-success.is-outlined{background-color:transparent;border-color:#a6e3a1;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-outlined:hover,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-success.is-outlined:focus,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-focused{background-color:#a6e3a1;border-color:#a6e3a1;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6e3a1 #a6e3a1 !important}html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-outlined{background-color:transparent;border-color:#a6e3a1;box-shadow:none;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6e3a1 #a6e3a1 !important}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-light{background-color:#f0faef;color:#287222}html.theme--catppuccin-mocha .button.is-success.is-light:hover,html.theme--catppuccin-mocha .button.is-success.is-light.is-hovered{background-color:#e7f7e5;border-color:transparent;color:#287222}html.theme--catppuccin-mocha .button.is-success.is-light:active,html.theme--catppuccin-mocha .button.is-success.is-light.is-active{background-color:#def4dc;border-color:transparent;color:#287222}html.theme--catppuccin-mocha .button.is-warning{background-color:#f9e2af;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:hover,html.theme--catppuccin-mocha .button.is-warning.is-hovered{background-color:#f8dea3;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:focus,html.theme--catppuccin-mocha .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:focus:not(:active),html.theme--catppuccin-mocha .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(249,226,175,0.25)}html.theme--catppuccin-mocha .button.is-warning:active,html.theme--catppuccin-mocha .button.is-warning.is-active{background-color:#f7d997;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning{background-color:#f9e2af;border-color:#f9e2af;box-shadow:none}html.theme--catppuccin-mocha .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted:hover,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined{background-color:transparent;border-color:#f9e2af;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-outlined:hover,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-warning.is-outlined:focus,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-focused{background-color:#f9e2af;border-color:#f9e2af;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #f9e2af #f9e2af !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-outlined{background-color:transparent;border-color:#f9e2af;box-shadow:none;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f9e2af #f9e2af !important}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-light{background-color:#fef8ec;color:#8a620a}html.theme--catppuccin-mocha .button.is-warning.is-light:hover,html.theme--catppuccin-mocha .button.is-warning.is-light.is-hovered{background-color:#fdf4e0;border-color:transparent;color:#8a620a}html.theme--catppuccin-mocha .button.is-warning.is-light:active,html.theme--catppuccin-mocha .button.is-warning.is-light.is-active{background-color:#fcf0d4;border-color:transparent;color:#8a620a}html.theme--catppuccin-mocha .button.is-danger{background-color:#f38ba8;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:hover,html.theme--catppuccin-mocha .button.is-danger.is-hovered{background-color:#f27f9f;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:focus,html.theme--catppuccin-mocha .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:focus:not(:active),html.theme--catppuccin-mocha .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(243,139,168,0.25)}html.theme--catppuccin-mocha .button.is-danger:active,html.theme--catppuccin-mocha .button.is-danger.is-active{background-color:#f17497;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger{background-color:#f38ba8;border-color:#f38ba8;box-shadow:none}html.theme--catppuccin-mocha .button.is-danger.is-inverted{background-color:#fff;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted:hover,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-mocha .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined{background-color:transparent;border-color:#f38ba8;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-outlined:hover,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-danger.is-outlined:focus,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-focused{background-color:#f38ba8;border-color:#f38ba8;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #f38ba8 #f38ba8 !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-outlined{background-color:transparent;border-color:#f38ba8;box-shadow:none;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f38ba8 #f38ba8 !important}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-light{background-color:#fdedf1;color:#991036}html.theme--catppuccin-mocha .button.is-danger.is-light:hover,html.theme--catppuccin-mocha .button.is-danger.is-light.is-hovered{background-color:#fce1e8;border-color:transparent;color:#991036}html.theme--catppuccin-mocha .button.is-danger.is-light:active,html.theme--catppuccin-mocha .button.is-danger.is-light.is-active{background-color:#fbd5e0;border-color:transparent;color:#991036}html.theme--catppuccin-mocha .button.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-mocha .button.is-small:not(.is-rounded),html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-mocha .button.is-normal{font-size:1rem}html.theme--catppuccin-mocha .button.is-medium{font-size:1.25rem}html.theme--catppuccin-mocha .button.is-large{font-size:1.5rem}html.theme--catppuccin-mocha .button[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button{background-color:#6c7086;border-color:#585b70;box-shadow:none;opacity:.5}html.theme--catppuccin-mocha .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-mocha .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-mocha .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-mocha .button.is-static{background-color:#181825;border-color:#585b70;color:#7f849c;box-shadow:none;pointer-events:none}html.theme--catppuccin-mocha .button.is-rounded,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-mocha .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-mocha .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-mocha .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-mocha .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-mocha .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-mocha .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-mocha .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-mocha .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-mocha .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-mocha .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-mocha .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-mocha .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-mocha .buttons.has-addons .button:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-mocha .buttons.has-addons .button:focus,html.theme--catppuccin-mocha .buttons.has-addons .button.is-focused,html.theme--catppuccin-mocha .buttons.has-addons .button:active,html.theme--catppuccin-mocha .buttons.has-addons .button.is-active,html.theme--catppuccin-mocha .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-mocha .buttons.has-addons .button:focus:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-mocha .buttons.has-addons .button:active:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-mocha .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-mocha .buttons.is-centered{justify-content:center}html.theme--catppuccin-mocha .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-mocha .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-mocha .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-mocha .button.is-responsive.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-mocha .button.is-responsive,html.theme--catppuccin-mocha .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-mocha .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-mocha .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-mocha .button.is-responsive.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-mocha .button.is-responsive,html.theme--catppuccin-mocha .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-mocha .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-mocha .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-mocha .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-mocha .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-mocha .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-mocha .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--catppuccin-mocha .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--catppuccin-mocha .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-mocha .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-mocha .content li+li{margin-top:0.25em}html.theme--catppuccin-mocha .content p:not(:last-child),html.theme--catppuccin-mocha .content dl:not(:last-child),html.theme--catppuccin-mocha .content ol:not(:last-child),html.theme--catppuccin-mocha .content ul:not(:last-child),html.theme--catppuccin-mocha .content blockquote:not(:last-child),html.theme--catppuccin-mocha .content pre:not(:last-child),html.theme--catppuccin-mocha .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-mocha .content h1,html.theme--catppuccin-mocha .content h2,html.theme--catppuccin-mocha .content h3,html.theme--catppuccin-mocha .content h4,html.theme--catppuccin-mocha .content h5,html.theme--catppuccin-mocha .content h6{color:#cdd6f4;font-weight:600;line-height:1.125}html.theme--catppuccin-mocha .content h1{font-size:2em;margin-bottom:0.5em}html.theme--catppuccin-mocha .content h1:not(:first-child){margin-top:1em}html.theme--catppuccin-mocha .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--catppuccin-mocha .content h2:not(:first-child){margin-top:1.1428em}html.theme--catppuccin-mocha .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--catppuccin-mocha .content h3:not(:first-child){margin-top:1.3333em}html.theme--catppuccin-mocha .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--catppuccin-mocha .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--catppuccin-mocha .content h6{font-size:1em;margin-bottom:1em}html.theme--catppuccin-mocha .content blockquote{background-color:#181825;border-left:5px solid #585b70;padding:1.25em 1.5em}html.theme--catppuccin-mocha .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-mocha .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-mocha .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-mocha .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-mocha .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-mocha .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-mocha .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-mocha .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--catppuccin-mocha .content ul ul ul{list-style-type:square}html.theme--catppuccin-mocha .content dd{margin-left:2em}html.theme--catppuccin-mocha .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--catppuccin-mocha .content figure:not(:first-child){margin-top:2em}html.theme--catppuccin-mocha .content figure:not(:last-child){margin-bottom:2em}html.theme--catppuccin-mocha .content figure img{display:inline-block}html.theme--catppuccin-mocha .content figure figcaption{font-style:italic}html.theme--catppuccin-mocha .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--catppuccin-mocha .content sup,html.theme--catppuccin-mocha .content sub{font-size:75%}html.theme--catppuccin-mocha .content table{width:100%}html.theme--catppuccin-mocha .content table td,html.theme--catppuccin-mocha .content table th{border:1px solid #585b70;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--catppuccin-mocha .content table th{color:#b8c5ef}html.theme--catppuccin-mocha .content table th:not([align]){text-align:inherit}html.theme--catppuccin-mocha .content table thead td,html.theme--catppuccin-mocha .content table thead th{border-width:0 0 2px;color:#b8c5ef}html.theme--catppuccin-mocha .content table tfoot td,html.theme--catppuccin-mocha .content table tfoot th{border-width:2px 0 0;color:#b8c5ef}html.theme--catppuccin-mocha .content table tbody tr:last-child td,html.theme--catppuccin-mocha .content table tbody tr:last-child th{border-bottom-width:0}html.theme--catppuccin-mocha .content .tabs li+li{margin-top:0}html.theme--catppuccin-mocha .content.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--catppuccin-mocha .content.is-normal{font-size:1rem}html.theme--catppuccin-mocha .content.is-medium{font-size:1.25rem}html.theme--catppuccin-mocha .content.is-large{font-size:1.5rem}html.theme--catppuccin-mocha .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--catppuccin-mocha .icon.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.icon{height:1rem;width:1rem}html.theme--catppuccin-mocha .icon.is-medium{height:2rem;width:2rem}html.theme--catppuccin-mocha .icon.is-large{height:3rem;width:3rem}html.theme--catppuccin-mocha .icon-text{align-items:flex-start;color:inherit;display:inline-flex;flex-wrap:wrap;line-height:1.5rem;vertical-align:top}html.theme--catppuccin-mocha .icon-text .icon{flex-grow:0;flex-shrink:0}html.theme--catppuccin-mocha .icon-text .icon:not(:last-child){margin-right:.25em}html.theme--catppuccin-mocha .icon-text .icon:not(:first-child){margin-left:.25em}html.theme--catppuccin-mocha div.icon-text{display:flex}html.theme--catppuccin-mocha .image,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img{display:block;position:relative}html.theme--catppuccin-mocha .image img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--catppuccin-mocha .image img.is-rounded,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--catppuccin-mocha .image.is-fullwidth,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--catppuccin-mocha .image.is-square img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--catppuccin-mocha .image.is-square .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--catppuccin-mocha .image.is-1by1 img,html.theme--catppuccin-mocha #documenter .docs-sidebar 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.has-ratio,html.theme--catppuccin-mocha .image.is-5by3 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-5by3 img,html.theme--catppuccin-mocha .image.is-5by3 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-5by3 .has-ratio,html.theme--catppuccin-mocha .image.is-16by9 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-16by9 img,html.theme--catppuccin-mocha .image.is-16by9 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-16by9 .has-ratio,html.theme--catppuccin-mocha .image.is-2by1 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-2by1 img,html.theme--catppuccin-mocha .image.is-2by1 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-2by1 .has-ratio,html.theme--catppuccin-mocha .image.is-3by1 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by1 img,html.theme--catppuccin-mocha 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img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by5 img,html.theme--catppuccin-mocha .image.is-3by5 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by5 .has-ratio,html.theme--catppuccin-mocha .image.is-9by16 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-9by16 img,html.theme--catppuccin-mocha .image.is-9by16 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-9by16 .has-ratio,html.theme--catppuccin-mocha .image.is-1by2 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-1by2 img,html.theme--catppuccin-mocha .image.is-1by2 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-1by2 .has-ratio,html.theme--catppuccin-mocha .image.is-1by3 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-1by3 img,html.theme--catppuccin-mocha .image.is-1by3 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-1by3 .has-ratio{height:100%;width:100%}html.theme--catppuccin-mocha .image.is-square,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-square,html.theme--catppuccin-mocha .image.is-1by1,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-1by1{padding-top:100%}html.theme--catppuccin-mocha .image.is-5by4,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-5by4{padding-top:80%}html.theme--catppuccin-mocha .image.is-4by3,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-4by3{padding-top:75%}html.theme--catppuccin-mocha .image.is-3by2,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by2{padding-top:66.6666%}html.theme--catppuccin-mocha .image.is-5by3,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-5by3{padding-top:60%}html.theme--catppuccin-mocha .image.is-16by9,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-16by9{padding-top:56.25%}html.theme--catppuccin-mocha .image.is-2by1,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-2by1{padding-top:50%}html.theme--catppuccin-mocha .image.is-3by1,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by1{padding-top:33.3333%}html.theme--catppuccin-mocha .image.is-4by5,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-4by5{padding-top:125%}html.theme--catppuccin-mocha .image.is-3by4,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by4{padding-top:133.3333%}html.theme--catppuccin-mocha .image.is-2by3,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-2by3{padding-top:150%}html.theme--catppuccin-mocha .image.is-3by5,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by5{padding-top:166.6666%}html.theme--catppuccin-mocha .image.is-9by16,html.theme--catppuccin-mocha #documenter 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.button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-light{background-color:#ecf0f1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:hover,html.theme--documenter-dark .button.is-light.is-hovered{background-color:#e5eaec;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:focus,html.theme--documenter-dark .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:focus:not(:active),html.theme--documenter-dark .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(236,240,241,0.25)}html.theme--documenter-dark .button.is-light:active,html.theme--documenter-dark .button.is-light.is-active{background-color:#dde4e6;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light{background-color:#ecf0f1;border-color:#ecf0f1;box-shadow:none}html.theme--documenter-dark .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted:hover,html.theme--documenter-dark .button.is-light.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-light.is-outlined{background-color:transparent;border-color:#ecf0f1;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-outlined:hover,html.theme--documenter-dark .button.is-light.is-outlined.is-hovered,html.theme--documenter-dark .button.is-light.is-outlined:focus,html.theme--documenter-dark .button.is-light.is-outlined.is-focused{background-color:#ecf0f1;border-color:#ecf0f1;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-light.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-light.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-outlined{background-color:transparent;border-color:#ecf0f1;box-shadow:none;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-dark,html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:hover,html.theme--documenter-dark .content kbd.button:hover,html.theme--documenter-dark .button.is-dark.is-hovered,html.theme--documenter-dark .content kbd.button.is-hovered{background-color:#232829;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus,html.theme--documenter-dark .content kbd.button:focus,html.theme--documenter-dark .button.is-dark.is-focused,html.theme--documenter-dark .content kbd.button.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus:not(:active),html.theme--documenter-dark .content kbd.button:focus:not(:active),html.theme--documenter-dark .button.is-dark.is-focused:not(:active),html.theme--documenter-dark .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(40,47,47,0.25)}html.theme--documenter-dark .button.is-dark:active,html.theme--documenter-dark .content kbd.button:active,html.theme--documenter-dark .button.is-dark.is-active,html.theme--documenter-dark .content kbd.button.is-active{background-color:#1d2122;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark[disabled],html.theme--documenter-dark .content kbd.button[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark,fieldset[disabled] html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:#282f2f;box-shadow:none}html.theme--documenter-dark .button.is-dark.is-inverted,html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted:hover,html.theme--documenter-dark .content kbd.button.is-inverted:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-dark.is-inverted[disabled],html.theme--documenter-dark .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-loading::after,html.theme--documenter-dark .content kbd.button.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined,html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-outlined.is-focused{background-color:#282f2f;border-color:#282f2f;color:#fff}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-primary,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink{background-color:#375a7f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-primary:hover,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink:hover,html.theme--documenter-dark .button.is-primary.is-hovered,html.theme--documenter-dark .docstring>section>a.button.is-hovered.docs-sourcelink{background-color:#335476;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-primary:focus,html.theme--documenter-dark 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.docstring>section>a.button.is-outlined.is-hovered.docs-sourcelink,html.theme--documenter-dark .button.is-primary.is-outlined:focus,html.theme--documenter-dark .docstring>section>a.button.is-outlined.docs-sourcelink:focus,html.theme--documenter-dark .button.is-primary.is-outlined.is-focused,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-focused.docs-sourcelink{background-color:#375a7f;border-color:#375a7f;color:#fff}html.theme--documenter-dark .button.is-primary.is-outlined.is-loading::after,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-loading.docs-sourcelink::after{border-color:transparent transparent #375a7f #375a7f !important}html.theme--documenter-dark .button.is-primary.is-outlined.is-loading:hover::after,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-loading.docs-sourcelink:hover::after,html.theme--documenter-dark .button.is-primary.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-loading.is-hovered.docs-sourcelink::after,html.theme--documenter-dark .button.is-primary.is-outlined.is-loading:focus::after,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-loading.docs-sourcelink:focus::after,html.theme--documenter-dark .button.is-primary.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-loading.is-focused.docs-sourcelink::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-primary.is-outlined[disabled],html.theme--documenter-dark .docstring>section>a.button.is-outlined.docs-sourcelink[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-primary.is-outlined,fieldset[disabled] html.theme--documenter-dark .docstring>section>a.button.is-outlined.docs-sourcelink{background-color:transparent;border-color:#375a7f;box-shadow:none;color:#375a7f}html.theme--documenter-dark 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.button.is-link.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-link.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-link.is-outlined{background-color:transparent;border-color:#1abc9c;box-shadow:none;color:#1abc9c}html.theme--documenter-dark .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-link.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-link.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-focused{background-color:#fff;color:#1abc9c}html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #1abc9c #1abc9c !important}html.theme--documenter-dark .button.is-link.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-link.is-light{background-color:#edfdf9;color:#15987e}html.theme--documenter-dark .button.is-link.is-light:hover,html.theme--documenter-dark .button.is-link.is-light.is-hovered{background-color:#e2fbf6;border-color:transparent;color:#15987e}html.theme--documenter-dark .button.is-link.is-light:active,html.theme--documenter-dark 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.button.is-info.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-focused{background-color:#fff;color:#3c5dcd}html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #3c5dcd #3c5dcd !important}html.theme--documenter-dark .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark 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.button.is-success.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#259a12}html.theme--documenter-dark .button.is-success.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-success.is-outlined{background-color:transparent;border-color:#259a12;color:#259a12}html.theme--documenter-dark .button.is-success.is-outlined:hover,html.theme--documenter-dark .button.is-success.is-outlined.is-hovered,html.theme--documenter-dark .button.is-success.is-outlined:focus,html.theme--documenter-dark .button.is-success.is-outlined.is-focused{background-color:#259a12;border-color:#259a12;color:#fff}html.theme--documenter-dark .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #259a12 #259a12 !important}html.theme--documenter-dark .button.is-success.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-success.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-success.is-outlined{background-color:transparent;border-color:#259a12;box-shadow:none;color:#259a12}html.theme--documenter-dark .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-success.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-focused{background-color:#fff;color:#259a12}html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #259a12 #259a12 !important}html.theme--documenter-dark .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-success.is-light{background-color:#effded;color:#2ec016}html.theme--documenter-dark .button.is-success.is-light:hover,html.theme--documenter-dark 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rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-outlined{background-color:transparent;border-color:#f4c72f;box-shadow:none;color:#f4c72f}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f4c72f}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f4c72f #f4c72f !important}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-warning.is-light{background-color:#fefaec;color:#8c6e07}html.theme--documenter-dark .button.is-warning.is-light:hover,html.theme--documenter-dark .button.is-warning.is-light.is-hovered{background-color:#fdf7e0;border-color:transparent;color:#8c6e07}html.theme--documenter-dark .button.is-warning.is-light:active,html.theme--documenter-dark .button.is-warning.is-light.is-active{background-color:#fdf3d3;border-color:transparent;color:#8c6e07}html.theme--documenter-dark .button.is-danger{background-color:#cb3c33;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger:hover,html.theme--documenter-dark .button.is-danger.is-hovered{background-color:#c13930;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger:focus,html.theme--documenter-dark .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger:focus:not(:active),html.theme--documenter-dark .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(203,60,51,0.25)}html.theme--documenter-dark .button.is-danger:active,html.theme--documenter-dark .button.is-danger.is-active{background-color:#b7362e;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger{background-color:#cb3c33;border-color:#cb3c33;box-shadow:none}html.theme--documenter-dark .button.is-danger.is-inverted{background-color:#fff;color:#cb3c33}html.theme--documenter-dark .button.is-danger.is-inverted:hover,html.theme--documenter-dark .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#cb3c33}html.theme--documenter-dark .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-danger.is-outlined{background-color:transparent;border-color:#cb3c33;color:#cb3c33}html.theme--documenter-dark .button.is-danger.is-outlined:hover,html.theme--documenter-dark .button.is-danger.is-outlined.is-hovered,html.theme--documenter-dark 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html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-danger.is-light{background-color:#fbefef;color:#c03930}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#f8e6e5;border-color:transparent;color:#c03930}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#f6dcda;border-color:transparent;color:#c03930}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark .button.is-normal{font-size:1rem}html.theme--documenter-dark .button.is-medium{font-size:1.25rem}html.theme--documenter-dark .button.is-large{font-size:1.5rem}html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#8c9b9d;border-color:#5e6d6f;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#282f2f;border-color:#5e6d6f;color:#dbdee0;box-shadow:none;pointer-events:none}html.theme--documenter-dark .button.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 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.buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--documenter-dark .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--documenter-dark .buttons.has-addons .button:last-child{margin-right:0}html.theme--documenter-dark .buttons.has-addons .button:hover,html.theme--documenter-dark .buttons.has-addons .button.is-hovered{z-index:2}html.theme--documenter-dark .buttons.has-addons .button:focus,html.theme--documenter-dark .buttons.has-addons .button.is-focused,html.theme--documenter-dark .buttons.has-addons .button:active,html.theme--documenter-dark .buttons.has-addons .button.is-active,html.theme--documenter-dark .buttons.has-addons .button.is-selected{z-index:3}html.theme--documenter-dark .buttons.has-addons .button:focus:hover,html.theme--documenter-dark .buttons.has-addons .button.is-focused:hover,html.theme--documenter-dark .buttons.has-addons .button:active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--documenter-dark .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--documenter-dark .buttons.is-centered{justify-content:center}html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--documenter-dark .buttons.is-right{justify-content:flex-end}html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:1rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1.25rem}}html.theme--documenter-dark .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--documenter-dark .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--documenter-dark .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--documenter-dark .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--documenter-dark .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--documenter-dark .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--documenter-dark .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--documenter-dark .content li+li{margin-top:0.25em}html.theme--documenter-dark .content p:not(:last-child),html.theme--documenter-dark .content dl:not(:last-child),html.theme--documenter-dark .content ol:not(:last-child),html.theme--documenter-dark .content ul:not(:last-child),html.theme--documenter-dark .content blockquote:not(:last-child),html.theme--documenter-dark .content pre:not(:last-child),html.theme--documenter-dark .content table:not(:last-child){margin-bottom:1em}html.theme--documenter-dark .content h1,html.theme--documenter-dark .content h2,html.theme--documenter-dark .content h3,html.theme--documenter-dark .content h4,html.theme--documenter-dark .content h5,html.theme--documenter-dark .content h6{color:#f2f2f2;font-weight:600;line-height:1.125}html.theme--documenter-dark .content h1{font-size:2em;margin-bottom:0.5em}html.theme--documenter-dark .content h1:not(:first-child){margin-top:1em}html.theme--documenter-dark .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--documenter-dark .content h2:not(:first-child){margin-top:1.1428em}html.theme--documenter-dark .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--documenter-dark .content h3:not(:first-child){margin-top:1.3333em}html.theme--documenter-dark .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--documenter-dark .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content blockquote{background-color:#282f2f;border-left:5px solid #5e6d6f;padding:1.25em 1.5em}html.theme--documenter-dark .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ol:not([type]){list-style-type:decimal}html.theme--documenter-dark .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--documenter-dark .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--documenter-dark .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--documenter-dark .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--documenter-dark .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--documenter-dark .content ul ul ul{list-style-type:square}html.theme--documenter-dark .content dd{margin-left:2em}html.theme--documenter-dark .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--documenter-dark .content figure:not(:first-child){margin-top:2em}html.theme--documenter-dark .content figure:not(:last-child){margin-bottom:2em}html.theme--documenter-dark .content figure img{display:inline-block}html.theme--documenter-dark .content figure figcaption{font-style:italic}html.theme--documenter-dark .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--documenter-dark .content sup,html.theme--documenter-dark .content sub{font-size:75%}html.theme--documenter-dark .content table{width:100%}html.theme--documenter-dark .content table td,html.theme--documenter-dark .content table th{border:1px solid #5e6d6f;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--documenter-dark .content table th{color:#f2f2f2}html.theme--documenter-dark .content table th:not([align]){text-align:inherit}html.theme--documenter-dark .content table thead td,html.theme--documenter-dark .content table thead th{border-width:0 0 2px;color:#f2f2f2}html.theme--documenter-dark .content table tfoot td,html.theme--documenter-dark .content table tfoot th{border-width:2px 0 0;color:#f2f2f2}html.theme--documenter-dark .content table tbody tr:last-child td,html.theme--documenter-dark .content table tbody tr:last-child th{border-bottom-width:0}html.theme--documenter-dark .content .tabs li+li{margin-top:0}html.theme--documenter-dark .content.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--documenter-dark .content.is-normal{font-size:1rem}html.theme--documenter-dark .content.is-medium{font-size:1.25rem}html.theme--documenter-dark .content.is-large{font-size:1.5rem}html.theme--documenter-dark 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img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--documenter-dark .image img.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--documenter-dark .image.is-fullwidth,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--documenter-dark .image.is-square img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--documenter-dark .image.is-square .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--documenter-dark .image.is-1by1 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--documenter-dark .image.is-1by1 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--documenter-dark .image.is-5by4 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.docs-logo>img.is-3by1{padding-top:33.3333%}html.theme--documenter-dark .image.is-4by5,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-4by5{padding-top:125%}html.theme--documenter-dark .image.is-3by4,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by4{padding-top:133.3333%}html.theme--documenter-dark .image.is-2by3,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-2by3{padding-top:150%}html.theme--documenter-dark .image.is-3by5,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by5{padding-top:166.6666%}html.theme--documenter-dark .image.is-9by16,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-9by16{padding-top:177.7777%}html.theme--documenter-dark .image.is-1by2,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by2{padding-top:200%}html.theme--documenter-dark .image.is-1by3,html.theme--documenter-dark #documenter .docs-sidebar 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.hljs-string{color:#38a}.hljs-emphasis{font-style:italic}.hljs-strong{font-weight:bold}.gap-4{gap:1rem} diff --git a/previews/PR92/assets/themeswap.js b/previews/PR92/assets/themeswap.js new file mode 100644 index 00000000..9f5eebe6 --- /dev/null +++ b/previews/PR92/assets/themeswap.js @@ -0,0 +1,84 @@ +// Small function to quickly swap out themes. Gets put into the tag.. +function set_theme_from_local_storage() { + // Initialize the theme to null, which means default + var theme = null; + // If the browser supports the localstorage and is not disabled then try to get the + // documenter theme + if (window.localStorage != null) { + // Get the user-picked theme from localStorage. May be `null`, which means the default + // theme. + theme = window.localStorage.getItem("documenter-theme"); + } + // Check if the users preference is for dark color scheme + var darkPreference = + window.matchMedia("(prefers-color-scheme: dark)").matches === true; + // Initialize a few variables for the loop: + // + // - active: will contain the index of the theme that should be active. Note that there + // is no guarantee that localStorage contains sane values. If `active` stays `null` + // we either could not find the theme or it is the default (primary) theme anyway. + // Either way, we then need to stick to the primary theme. + // + // - disabled: style sheets that should be disabled (i.e. all the theme style sheets + // that are not the currently active theme) + var active = null; + var disabled = []; + var primaryLightTheme = null; + var primaryDarkTheme = null; + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // To distinguish the default (primary) theme, it needs to have the data-theme-primary + // attribute set. + if (ss.ownerNode.getAttribute("data-theme-primary") !== null) { + primaryLightTheme = themename; + } + // Check if the theme is primary dark theme so that we could store its name in darkTheme + if (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null) { + primaryDarkTheme = themename; + } + // If we find a matching theme (and it's not the default), we'll set active to non-null + if (themename === theme) active = i; + // Store the style sheets of inactive themes so that we could disable them + if (themename !== theme) disabled.push(ss); + } + var activeTheme = null; + if (active !== null) { + // If we did find an active theme, we'll (1) add the theme--$(theme) class to + document.getElementsByTagName("html")[0].className = "theme--" + theme; + activeTheme = theme; + } else { + // If we did _not_ find an active theme, then we need to fall back to the primary theme + // which can either be dark or light, depending on the user's OS preference. + var activeTheme = darkPreference ? primaryDarkTheme : primaryLightTheme; + // In case it somehow happens that the relevant primary theme was not found in the + // preceding loop, we abort without doing anything. + if (activeTheme === null) { + console.error("Unable to determine primary theme."); + return; + } + // When switching to the primary light theme, then we must not have a class name + // for the tag. That's only for non-primary or the primary dark theme. + if (darkPreference) { + document.getElementsByTagName("html")[0].className = + "theme--" + activeTheme; + } else { + document.getElementsByTagName("html")[0].className = ""; + } + } + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // we'll disable all the stylesheets, except for the active one + ss.disabled = !(themename == activeTheme); + } +} +set_theme_from_local_storage(); diff --git a/previews/PR92/assets/warner.js b/previews/PR92/assets/warner.js new file mode 100644 index 00000000..3f6f5d00 --- /dev/null +++ b/previews/PR92/assets/warner.js @@ -0,0 +1,52 @@ +function maybeAddWarning() { + // DOCUMENTER_NEWEST is defined in versions.js, DOCUMENTER_CURRENT_VERSION and DOCUMENTER_STABLE + // in siteinfo.js. + // If either of these are undefined something went horribly wrong, so we abort. + if ( + window.DOCUMENTER_NEWEST === undefined || + window.DOCUMENTER_CURRENT_VERSION === undefined || + window.DOCUMENTER_STABLE === undefined + ) { + return; + } + + // Current version is not a version number, so we can't tell if it's the newest version. 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zmi5rqY}R(Z1SDGo&kuh#-*ESoSC;B~4jV;4Rz}J0+DS-(P0+`bfMf@%mh0;3J~Ta| z08GWJF`X7IEfH`V>Zvs_>7vLP z(bmD=)!m(;~U8vrx*@#9mcPF0Ld5a5eKrenF&2C=3hLsOM{be#3ZaTT&8NMMAr?w8$dWv1`| zoQXicE&|QK>p)a%DJ#PfuLfx?sM6FKkHSK&Np{Kj;x z5*t>0auhNdIrf?;rz~J~5*~pVcx{3OZwto%?rcZ$%7Mrj?x}T-Y*bBycGPXnr2|{p zy4Yi_>!k@xIBE&u9*l{@o5BLpifBmN1rB6;`# literal 0 HcmV?d00001 diff --git a/previews/PR92/generated/examples/computed_tomography/index.html b/previews/PR92/generated/examples/computed_tomography/index.html new file mode 100644 index 00000000..e1354e0a --- /dev/null +++ b/previews/PR92/generated/examples/computed_tomography/index.html @@ -0,0 +1,65 @@ + +Computed Tomography · RegularizedLeastSquares.jl

Computed Tomography Example

In this example we will go through a simple example from the field of Computed Tomography. In addtion to RegularizedLeastSquares.jl, we will need the packages LinearOperatorCollection.jl, ImagePhantoms.jl, ImageGeoms.jl and RadonKA.jl, as well as CairoMakie.jl for visualization. We can install them the same way we did RegularizedLeastSquares.jl.

RadonKA is a package for the computation of the Radon transform and its adjoint. It is implemented with KernelAbstractions.jl and supports GPU acceleration. See the GPU acceleration how-to for more information.

Preparing the Inverse Problem

To get started, let us generate a simple phantom using the ImagePhantoms and ImageGeom packages:

using ImagePhantoms, ImageGeoms
+N = 256
+image = shepp_logan(N, SheppLoganToft())
+size(image)
(256, 256)

This produces a 64x64 image of a Shepp-Logan phantom.

using RadonKA, LinearOperatorCollection
+angles = collect(range(0, π, 256))
+sinogram = Array(RadonKA.radon(image, angles))
255×256 Matrix{Float32}:
+ 0.0      0.0      0.0      0.0      …  0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0      0.0      0.0      0.0         0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0      0.0      0.0      0.0         0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0      0.0      0.0      0.0         0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0      0.0      0.0      0.0         0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0      0.0      0.0      0.0      …  0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0      0.0      0.0      0.0         0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0      0.0      0.0      0.0         0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0      0.0      0.0      0.0         0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 2.88889  2.88911  2.88977  2.89086     0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ ⋮                                   ⋱       ⋮                        ⋮
+ 0.0      0.0      0.0      0.0         0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0      0.0      0.0      0.0         0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0      0.0      0.0      0.0         0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0      0.0      0.0      0.0         0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0      0.0      0.0      0.0      …  0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0      0.0      0.0      0.0         0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0      0.0      0.0      0.0         0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0      0.0      0.0      0.0         0.0  0.0  0.0  0.0  0.0  0.0  0.0
+ 0.0      0.0      0.0      0.0         0.0  0.0  0.0  0.0  0.0  0.0  0.0

Afterwards we build a Radon operator implementing both the forward and adjoint Radon transform

A = RadonOp(eltype(image); angles, shape = size(image));

To visualize our image we can use CairoMakie:

using CairoMakie
+function plot_image(figPos, img; title = "", width = 150, height = 150)
+  ax = CairoMakie.Axis(figPos[1, 1]; yreversed=true, title, width, height)
+  hidedecorations!(ax)
+  hm = heatmap!(ax, img)
+  Colorbar(figPos[2, 1], hm, vertical = false, flipaxis = false)
+end
+fig = Figure()
+plot_image(fig[1,1], image, title = "Image")
+plot_image(fig[1,2], sinogram, title = "Sinogram")
+plot_image(fig[1,3], backproject(sinogram, angles), title = "Backprojection")
+resize_to_layout!(fig)
+fig
Example block output

In the figure we can see our original image, the sinogram, and the backprojection of the sinogram. The goal of the inverse problem is to recover the original image from the sinogram.

Solving the Inverse Problem

To recover the image from the measurement vector, we solve the $l^2_2$-regularized least squares problem

\[\begin{equation} + \underset{\mathbf{x}}{argmin} \frac{1}{2}\vert\vert \mathbf{A}\mathbf{x}-\mathbf{b} \vert\vert_2^2 + \vert\vert\mathbf{x}\vert\vert^2_2 . +\end{equation}\]

For this purpose we build a $l^2_2$ with regularization parameter $λ=0.001$

using RegularizedLeastSquares
+reg = L2Regularization(0.001);

To solve this inverse problem, the Conjugate Gradient Normal Residual (CGNR) algorithm can be used. This solver is based on the normal operator of the Radon operator and uses both the forward and adjoint Radon transform internally. We now build the corresponding solver

solver = createLinearSolver(CGNR, A; reg=reg, iterations=20);

and apply it to our measurement vector

img_approx = solve!(solver, vec(sinogram))
65536-element Vector{Float32}:
+ 0.0
+ 0.0
+ 0.0
+ 0.0
+ 0.0
+ 0.0
+ 0.0
+ 0.0
+ 0.0
+ 0.0
+ ⋮
+ 0.0
+ 0.0
+ 0.0
+ 0.0
+ 0.0
+ 0.0
+ 0.0
+ 0.0
+ 0.0

To visualize the reconstructed image, we need to reshape the result vector to the correct shape. Afterwards we can use CairoMakie again:

img_approx = reshape(img_approx,size(image));
+plot_image(fig[1,4], img_approx, title = "Reconstructed Image")
+resize_to_layout!(fig)
+fig
Example block output

This page was generated using Literate.jl.

diff --git a/previews/PR92/generated/examples/getting_started/index.html b/previews/PR92/generated/examples/getting_started/index.html new file mode 100644 index 00000000..813e69bc --- /dev/null +++ b/previews/PR92/generated/examples/getting_started/index.html @@ -0,0 +1,12 @@ + +Getting Started · RegularizedLeastSquares.jl

Getting Started

In this example we will go through a simple random inverse problem to get familiar with RegularizedLeastSquares.jl.

Installation

To install RegularizedLeastSquares.jl, you can use the Julia package manager. Open a Julia REPL and run the following command:

using Pkg
+Pkg.add("RegularizedLeastSquares")

This will download and install the RegularizedLeastSquares.jl package and its dependencies. To install a different version, please consult the Pkg documentation.

Once the installation is complete, you can import the package with the using keyword:

using RegularizedLeastSquares

RegularizedLeastSquares aims to solve inverse problems of the form:

\[\begin{equation} + \underset{\mathbf{x}}{argmin} \frac{1}{2}\vert\vert \mathbf{A}\mathbf{x}-\mathbf{b} \vert\vert_2^2 + \mathbf{R(x)} . +\end{equation}\]

where $\mathbf{A}$ is a linear operator, $\mathbf{y}$ is the measurement vector, and $\mathbf{R(x)}$ is an (optional) regularization term. The goal is to retrieve an approximation of the unknown vector $\mathbf{x}$. In this first exampel we will just work with simple random arrays. For more advanced examples, please refer to the examples.

A = rand(32, 16)
+x = rand(16)
+b = A*x;

To approximate x from b, we can use the Conjugate Gradient Normal Residual (CGNR) algorithm. We first build the corresponding solver:

solver = createLinearSolver(CGNR, A; iterations=32);

and apply it to our measurement vector

x_approx = solve!(solver, b)
+isapprox(x, x_approx, rtol = 0.001)
true

Usually the inverse problems are ill-posed and require regularization. RegularizedLeastSquares.jl provides a variety of regularization terms. The CGNR algorithm can solve optimzation problems of the form:

\[\begin{equation} + \underset{\mathbf{x}}{argmin} \frac{1}{2}\vert\vert \mathbf{A}\mathbf{x}-\mathbf{b} \vert\vert_2^2 + \vert\vert\mathbf{x}\vert\vert^2_2 . +\end{equation}\]

The corresponding solver can be built with the L2 regularization term:

solver = createLinearSolver(CGNR, A; reg = L2Regularization(0.0001), iterations=32);
+x_approx = solve!(solver, b)
+isapprox(x, x_approx, rtol = 0.001)
true

This page was generated using Literate.jl.

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znG#71avA8=J09F4703^e0j15>o4U7t0RS~*&Y8MZGjU&SJBE4IOCn(0}( z)QOqF3XYrB9si%hG+4lt07BBxRpau +Regularization · RegularizedLeastSquares.jl

Regularization

When formulating inverse problems, a regulariser is formulated as an additional cost function which has to be minimised. Many algorithms deal with a regularizers $g$, by computing the proximal map:

\[\begin{equation} + prox_g (\mathbf{x}) = \underset{\mathbf{u}}{argmin} \frac{1}{2}\vert\vert \mathbf{u}-\mathbf{x} \vert {\vert}^2 + g(\mathbf{x}). +\end{equation}\]

For many regularizers, the proximal map can be computed efficiently in a closed form.

In order to implement these proximal mappings, RegularizedLeastSquares.jl defines the following type hierarchy:

abstract type AbstractRegularization
+prox!(reg::AbstractRegularization, x)
+norm(reg::AbstractRegularization, x)

Here prox!(reg, x) is an in-place function which computes the proximal map on the input-vector x. The function norm computes the value of the corresponding term in the inverse problem. RegularizedLeastSquares.jl provides AbstractParameterizedRegularization and AbstractProjectionRegularization as core regularization types.

Parameterized Regularization Terms

This group of regularization terms features a regularization parameter λ that is used during the prox! and normcomputations. Examples of this regulariztion group are L1, L2 or LLR (locally low rank) regularization terms.

These terms are constructed by supplying a λ and optionally term specific keyword arguments:

using RegularizedLeastSquares
+l2 = L2Regularization(0.3)
L2Regularization{Float64}(0.3)

Parameterized regularization terms implement:

prox!(reg::AbstractParameterizedRegularization, x, λ)
+norm(reg::AbstractParameterizedRegularization, x, λ)

where λ by default is filled with the value used during construction.

Invoking λ on a parameterized term retrieves its regularization parameter. This can be used in a solver to scale and overwrite the parameter as follows:

prox!(l2, [1.0])
1-element Vector{Float64}:
+ 0.625
param = λ(l2)
+prox!(l2, [1.0], param*0.2)
1-element Vector{Float64}:
+ 0.8928571428571428

Projection Regularization Terms

This group of regularization terms implement projections, such as a positivity constraint or a projection with a given convex projection function. These are essentially proximal maps where $g(\mathbf{x})$ is the indicator function of a convex set.

positive = PositiveRegularization()
+prox!(positive, [2.0, -0.2])
2-element Vector{Float64}:
+ 2.0
+ 0.0

Nested Regularization Terms

Nested regularization terms are terms that act as decorators to the core regularization terms. These terms can be nested around other terms and add functionality to a regularization term, such as scaling λ based on the provided operator or applying a transform, such as the Wavelet, to x. As an example, we can nest a L1 regularization term around a Wavelet operator.

First we generate an image and apply a wavelet operator to it:

using Wavelets, LinearOperatorCollection, ImagePhantoms, ImageGeoms
+N = 256
+image = shepp_logan(N, SheppLoganToft())
+wop = WaveletOp(Float32, shape = size(image))
+
+wavelet_image = reshape(wop*vec(image), size(image))
+wavelet_image = log.(abs.(wavelet_image) .+ 0.01)
256×256 Matrix{Float64}:
+  3.4488     1.59865    1.90295    0.372817  …  -4.60517  -4.60517  -4.60517
+  2.5399     0.468954  -0.785108   1.34906      -4.60517  -4.60517  -4.60517
+ -0.733675   1.94682    1.57167    1.56522      -4.60517  -4.60517  -4.60517
+  2.30907    2.05713    1.25204    1.01251      -4.60517  -4.60517  -4.60517
+  1.34696    1.87188    1.8159    -2.05018      -4.60517  -4.60517  -4.60517
+  0.713718   1.74888    0.398654  -2.13583   …  -4.60517  -4.60517  -4.60517
+ -0.692972   0.637682   0.532763   1.12284      -4.60517  -4.60517  -4.60517
+  1.11718    1.58921    0.404766   0.677583     -4.60517  -4.60517  -4.60517
+ -4.24067   -2.45816   -0.730035  -1.37884      -4.60517  -4.60517  -4.60517
+ -0.445031   1.51269    0.754944   1.39635      -4.60517  -4.60517  -4.60517
+  ⋮                                          ⋱                       ⋮
+ -4.60517   -4.60517   -4.60517   -4.60517      -4.60517  -4.60517  -4.60517
+ -4.60517   -4.60517   -4.60517   -4.60517      -4.60517  -4.60517  -4.60517
+ -4.60517   -4.60517   -4.60517   -4.60517      -4.60517  -4.60517  -4.60517
+ -4.60517   -4.60517   -4.60517   -4.60517   …  -4.60517  -4.60517  -4.60517
+ -4.60517   -4.60517   -4.60517   -4.60517      -4.60517  -4.60517  -4.60517
+ -4.60517   -4.60517   -4.60517   -4.60517      -4.60517  -4.60517  -4.60517
+ -4.60517   -4.60517   -4.60517   -4.60517      -4.60517  -4.60517  -4.60517
+ -4.60517   -4.60517   -4.60517   -4.60517      -4.60517  -4.60517  -4.60517
+ -4.60517   -4.60517   -4.60517   -4.60517   …  -4.60517  -4.60517  -4.60517

We will use CairoMakie for visualization:

using CairoMakie
+function plot_image(figPos, img; title = "", width = 150, height = 150)
+  ax = CairoMakie.Axis(figPos; yreversed=true, title, width, height)
+  hidedecorations!(ax)
+  heatmap!(ax, img)
+end
+fig = Figure()
+plot_image(fig[1,1], image, title = "Image")
+plot_image(fig[1,2], wavelet_image, title = "Wavelet Image")
+resize_to_layout!(fig)
+fig
Example block output

To apply soft-thresholding in the wavelet domain, we can nest a L1 regularization term around the Wavelet operator:

core = L1Regularization(0.1)
+reg = TransformedRegularization(core, wop);

We can then apply the proximal map to the image or the image in the wavelet domain:

img_prox_image = prox!(core, copy(vec(image)));
+img_prox_wavelet = prox!(reg, copy(vec(image)));

We can visualize the result:

img_prox_image = reshape(img_prox_image, size(image))
+img_prox_wavelet = reshape(img_prox_wavelet, size(image))
+plot_image(fig[1,3], img_prox_image, title = "Reg. Image Domain")
+plot_image(fig[1,4], img_prox_wavelet, title = "Reg. Wavelet Domain")
+resize_to_layout!(fig)
+fig
Example block output

Generally, regularization terms can be nested arbitrarly deep, adding new functionality with each layer. Each nested regularization term can return its inner regularization term. Furthermore, all regularization terms implement the iteration interface to iterate over the nesting. The innermost regularization term of a nested term must be a core regularization term and it can be returned by the sink function:

RegularizedLeastSquares.innerreg(reg) == core
true
sink(reg) == core
true
foreach(r -> println(nameof(typeof(r))), reg)
TransformedRegularization
+L1Regularization

This page was generated using Literate.jl.

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    Callbacks

    For certain optimization it is important to monitor the internal state of the solver. RegularizedLeastSquares.jl provides a callback mechanism to allow developres to access this state after each iteration. The package provides a variety of default callbacks, which for example store the solution after each iteration. More information can be found in the API reference for the solve! function.

    In this example we will revist the compressed sensing compressed sensing example and implement a custom callback using the do-syntax of the solve! function. This allows us to implement our callback inline and access the solver state after each iteration.

    We first recreate the operator A and the measurement vector b:

    using ImagePhantoms, ImageGeoms
+N = 256
+image = shepp_logan(N, SheppLoganToft())
+size(image)
+using Random, LinearOperatorCollection
+randomIndices = shuffle(eachindex(image))
+sampledIndices = sort(randomIndices[1:div(end, 3)])
+A = SamplingOp(eltype(image), pattern = sampledIndices , shape = size(image));
+b = A*vec(image);

    Next we prepare our visualization helper function:

    using CairoMakie
+function plot_image(figPos, img; title = "", width = 150, height = 150, clim = extrema(img))
+  ax = CairoMakie.Axis(figPos; yreversed=true, title, width, height)
+  hidedecorations!(ax)
+  heatmap!(ax, img, colorrange = clim)
+end
    plot_image (generic function with 1 method)

    Now we construct the solver with the TV regularization term:

    using RegularizedLeastSquares
+reg = TVRegularization(0.01; shape=size(image));
+solver = createLinearSolver(FISTA, A; reg=reg, iterations=20);

    We will now implement a callback that plots the solution every four iteration:

    fig = Figure()
+idx = 1
+solve!(solver, b) do solver, iteration
+  if iteration % 4 == 0
+    img_approx = copy(solversolution(solver))
+    img_approx = reshape(img_approx, size(image))
+    plot_image(fig[div(idx -1, 3) + 1, mod1(idx, 3)], img_approx, clim = extrema(image), title = "$iteration")
+    global idx += 1
+  end
+end
    65536-element Vector{Float32}:
+ 8.485987f-19
+ 1.4974684f-18
+ 3.218811f-18
+ 6.962047f-18
+ 1.4528301f-17
+ 2.9826816f-17
+ 6.144824f-17
+ 1.2836768f-16
+ 2.7374405f-16
+ 5.7480397f-16
+ ⋮
+ 3.2739965f-16
+ 1.5413677f-16
+ 6.9313745f-17
+ 3.038671f-17
+ 1.3158814f-17
+ 5.651829f-18
+ 2.4297326f-18
+ 1.0793823f-18
+ 5.860541f-19

    In the callback we have to copy the solution, as the solver will update it in place. As is explained in the solver section, each features fields that are intended to be immutable during a solve! call and a state that is modified in each iteration. Depending on the solvers parameters and the measurement input, the state can differ in its fields and their type. Ideally, one tries to avoid accessing the state directly and uses the provided functions to access the state.

    The resulting figure shows the reconstructed image after 0, 4, 8, 12, 16 and 20 iterations:

    resize_to_layout!(fig)
+fig
    Example block output

    This page was generated using Literate.jl.

    diff --git a/previews/PR92/generated/howto/efficient_kaczmarz/index.html b/previews/PR92/generated/howto/efficient_kaczmarz/index.html new file mode 100644 index 00000000..4f5a5d1b --- /dev/null +++ b/previews/PR92/generated/howto/efficient_kaczmarz/index.html @@ -0,0 +1,48 @@ + +Efficient Kaczmarz · RegularizedLeastSquares.jl

    Efficient Kaczmarz

    Unlike many of the other solvers provided by RegularizedLeastSquares.jl, the Kaczmarz method does not utilize a matrix-vector product with the operator $\mathbf{A}$ nor the normal operator $\mathbf{A*A}$. Instead, it uses the rows of $\mathbf{A}$ to update the solution iteratively. Efficient Kaczmarz implementation therefore require very efficient dot products with the rows of $\mathbf{A}$. In RegularizedLeastSquares.jl, this is achieved with the dot_with_matrix_row function.

    using RegularizedLeastSquares
+A = randn(256, 256)
+x = randn(256)
+b = A*x;

    The dot_with_matrix_row function calculates the dot product between a row of A and the current approximate solution of x:

    row = 1
+isapprox(RegularizedLeastSquares.dot_with_matrix_row(A, x, row), sum(A[row, :] .* x))
    true

    Since in Julia, dense arrays are stored in column-major order, such a row-based operation is quite inefficient. A workaround is to transpose the matrix then pass it to a Kaczmarz solver.

    At = collect(transpose(A))
+A_eff = transpose(At)
    256×256 transpose(::Matrix{Float64}) with eltype Float64:
+ -0.643807    -0.124905    1.69779   …   0.786185   0.523156  -1.56588
+ -0.574667    -0.670973   -1.16691      -0.514316   0.443007  -0.430666
+ -1.59326     -1.64937     1.47003      -1.09443   -2.4363     0.0384111
+  0.549962    -0.48861     1.56325       1.2575     0.660812  -0.294459
+  1.2651      -0.146657   -1.13553      -0.985312  -0.284547  -0.600972
+  0.204547     0.741245   -1.75309   …   1.66562    0.57055    3.27423
+  0.067074    -1.25213    -0.225855      0.59542    0.22696    2.1312
+ -0.336004    -0.0590082   0.952705     -0.506446   1.36659    1.12518
+ -0.00599911  -0.726714    1.11414       0.147164  -1.63973   -0.612732
+  0.057356    -0.111582   -0.473511     -0.584454   0.540707   1.10184
+  ⋮                                  ⋱                         ⋮
+ -0.446105    -0.314711   -1.57024      -0.52868   -0.696636  -0.230257
+ -0.650605    -0.0643963   0.663417     -1.74822   -0.688242   0.0146301
+ -0.977737    -0.958501    1.81901       0.452491   1.25678   -0.341416
+ -1.31304     -0.523236    0.086328  …   0.255264  -0.274285  -1.07043
+  0.431185     0.701168   -0.507113      0.692781  -0.147133   0.394625
+ -0.346674     0.848182   -0.494203     -0.753903   0.287102  -2.02961
+  1.33714     -0.483075    0.86367      -0.628352   0.498603   0.560838
+  0.680813     0.145508   -0.211347      1.15326    1.05285    0.657485
+ -0.502064     0.50922     0.504781  …  -0.738186  -0.352926  -1.56952

    Note that the transpose function can return a lazy transpose object, so we first collect the transpose into a dense matrix. Then we transpose it again to get the efficient representation of the matrix.

    We can compare the performance using the BenchmarkTools.jl package. First for the original matrix:

    using BenchmarkTools
+solver = createLinearSolver(Kaczmarz, A; reg = L2Regularization(0.0001), iterations=100)
+@benchmark solve!(solver, b) samples = 100
    BenchmarkTools.Trial: 100 samples with 1 evaluation.
+ Range (min … max):  25.646 ms …  27.122 ms  ┊ GC (min … max): 0.00% … 0.00%
+ Time  (median):     27.002 ms               ┊ GC (median):    0.00%
+ Time  (mean ± σ):   26.980 ms ± 146.705 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%
+
+                                                       ▂ ▂█▃    
+  ▄▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▄▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▆█▇███▇▆ ▄
+  25.6 ms       Histogram: log(frequency) by time      27.1 ms <
+
+ Memory estimate: 17.31 KiB, allocs estimate: 504.

    And then for the efficient matrix:

    solver_eff = createLinearSolver(Kaczmarz, A_eff; reg = L2Regularization(0.0001), iterations=100)
+@benchmark solve!(solver_eff, b) samples = 100
    BenchmarkTools.Trial: 100 samples with 1 evaluation.
+ Range (min … max):  1.907 ms …  2.028 ms  ┊ GC (min … max): 0.00% … 0.00%
+ Time  (median):     1.933 ms              ┊ GC (median):    0.00%
+ Time  (mean ± σ):   1.937 ms ± 18.773 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%
+
+           █ ▂ ▄▂ ▆▆   █          ▂                           
+  ▄▄▁▄▄▁▆▁▆█▆███████▄▆███▄▆▄██▄▁▄▆█▁▄█▄▁▁▁▄█▁▄▄▁▄▄▁▁▁▁▁▄▁▁▆▄ ▄
+  1.91 ms        Histogram: frequency by time        1.98 ms <
+
+ Memory estimate: 17.34 KiB, allocs estimate: 506.

    We can also combine the efficient matrix with a weighting matrix, as is shown in the Weighting example.

    Custom operators need to implement the dot_with_matrix_row function to be used with the Kaczmarz solver. Ideally, such an implementation is allocation free.

    This page was generated using Literate.jl.

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    GPU Acceleration

    RegularizedLeastSquares.jl supports generic GPU acceleration. This means that the user can use any GPU array type that supports the GPUArrays interface. This includes CUDA.jl, AMDGPU.jl, and Metal.jl. In this example we will use the package JLArrays.jl which provides a reference implementation for GPUArrays, that can runs on CPUs.

    using JLArrays
+gpu = JLArray;

    To use the following examples on an actual GPU, load the appropraite package replace JLArray with the respective GPU array type, for example:

    using CUDA
+gpu = CuArray

    At first we will look at an example of dense GPU arrays.

    using RegularizedLeastSquares
+A = gpu(rand(Float32, 32, 16))
+x = gpu(rand(Float32, 16))
+b = A*x;

    Solvers adapt their states based on the type of the given measurement vector. This means that the solver will automatically switch to GPU acceleration if a GPU array is passed as the measurement vector.

    solver = createLinearSolver(CGNR, A; reg = L2Regularization(0.0001), iterations=32);
+x_approx = solve!(solver, b)
    16-element JLArrays.JLArray{Float32, 1}:
+ 0.76989436
+ 0.09178298
+ 0.2045388
+ 0.31862575
+ 0.19223855
+ 0.034395475
+ 0.6367978
+ 0.019584134
+ 0.6599743
+ 0.50930434
+ 0.11515529
+ 0.41526586
+ 0.45584652
+ 0.95969784
+ 0.03715256
+ 0.2970575

    This adaption does not include the operator. So if we want to compare with CPU result, we need to construct a new solver with a CPU operator.

    solver = createLinearSolver(CGNR, Array(A); reg = L2Regularization(0.0001), iterations=32);
+x_cpu = solve!(solver, Array(b))
+isapprox(Array(x_approx), x_cpu)
    true

    Matrix-Free Operators

    A special case is the usage of matrix-free operators. Since these operators do not have a concrete matrix representation, their GPU support depends on their implementation. Since not all multiplications within a solver approximation are in-place, the operator also needs to support the * operation and construct an appropriate result vector. For matrix-free operators based on LinearOperators.jl, this can be achieved by implementing the LinearOperators.storage_type method.

    In the following, we will take another look at the CS example and execute it on the GPU. Note that for the JLArray example we chose a small phantom, since the JLArray implementation is not optimized for performance:

    using ImagePhantoms, ImageGeoms
+N = 32
+image = shepp_logan(N, SheppLoganToft())
+
+using Random, LinearOperatorCollection
+randomIndices = shuffle(eachindex(image))
+sampledIndices = sort(randomIndices[1:div(end, 3)]);

    To construct the operator, we need to convert the indices to a GPU array. We also need to specify the correct storage type. In both LinearOperators.jl and LinearOperatorCollection.jl this is done with the S keyword argument.

    gpu_indices = gpu(sampledIndices)
+A = SamplingOp(eltype(image), pattern = gpu_indices, shape = size(image), S = typeof(b));

    Let's inspect the storage type of the operator:

    using LinearOperatorCollection.LinearOperators
+LinearOperators.storage_type(A)
    JLArrays.JLArray{Float32, 1}

    Afterwards we can use the operator as usual:

    b = A*vec(gpu(image));

    And use it in the solver:

    using RegularizedLeastSquares
+reg = TVRegularization(0.01; shape=size(image))
+solver = createLinearSolver(FISTA, A; reg=reg, iterations=20)
+img_approx = solve!(solver,b);

    To visualize the reconstructed image, we need to reshape the result vector to the correct shape and convert it to a CPU array:

    img_approx = reshape(Array(img_approx),size(image))
    32×32 Matrix{Float32}:
+ 0.00182596  0.00182079  0.00180018  …  0.00170522  0.00151066  0.00139194
+ 0.00194785  0.00196028  0.00196445     0.0019336   0.00173191  0.00160886
+ 0.00220521  0.00223796  0.0022853      0.00240092  0.00216919  0.0020296
+ 0.00255223  0.0026198   0.00276246     0.00314142  0.00281662  0.00263576
+ 0.00296817  0.00313723  0.00350533     0.00424536  0.00372654  0.00345912
+ 0.00359499  0.00398958  0.00474067  …  0.00577368  0.00499637  0.00456198
+ 0.00467823  0.00534572  0.00635574     0.0075705   0.00662761  0.00597426
+ 0.0061956   0.00691098  0.00765569     0.00917998  0.00850771  0.00777992
+ 0.0079152   0.00837985  0.00829485     0.0104648   0.0103646   0.0100378
+ 0.00976733  0.00973923  0.0097611      0.0111948   0.0119605   0.0125794
+ ⋮                                   ⋱              ⋮           
+ 0.00818123  0.00878127  0.00938514     0.0100501   0.00900169  0.00831158
+ 0.0067967   0.00756416  0.00850452     0.00845737  0.00729177  0.00660801
+ 0.00547793  0.00612875  0.00708301  …  0.00646632  0.00557981  0.00514117
+ 0.00424285  0.00466339  0.00540846     0.0047201   0.00419738  0.00401492
+ 0.00317613  0.00341177  0.00385285     0.00341493  0.00318044  0.00314506
+ 0.00236193  0.00249397  0.00273293     0.00253895  0.00245715  0.00246623
+ 0.00178943  0.00186363  0.00200414     0.00197298  0.00194269  0.00194867
+ 0.00139031  0.0014287   0.00151914  …  0.00162275  0.00160093  0.00159668
+ 0.00116179  0.00118196  0.00124816     0.00144526  0.00142355  0.00141422

    We will again use CairoMakie for visualization:

    using CairoMakie
+function plot_image(figPos, img; title = "", width = 150, height = 150)
+  ax = CairoMakie.Axis(figPos; yreversed=true, title, width, height)
+  hidedecorations!(ax)
+  heatmap!(ax, img)
+end
+fig = Figure()
+plot_image(fig[1,1], image, title = "Image")
+samplingMask = fill(false, size(image))
+samplingMask[sampledIndices] .= true
+plot_image(fig[1,2], image .* samplingMask, title = "Sampled Image")
+plot_image(fig[1,3], img_approx, title = "Reconstructed Image")
+resize_to_layout!(fig)
+fig
    Example block output

    This page was generated using Literate.jl.

    diff --git a/previews/PR92/generated/howto/multi_threading/index.html b/previews/PR92/generated/howto/multi_threading/index.html new file mode 100644 index 00000000..fa52a5a0 --- /dev/null +++ b/previews/PR92/generated/howto/multi_threading/index.html @@ -0,0 +1,28 @@ + +Multi-Threading · RegularizedLeastSquares.jl

    Multi-Threading

    There are different ways multi-threading can be used with RegularizedLeastSquares.jl. To use multi-threading in Julia, one needs to start their session with multi-threads, see the Julia documentation for more information.

    Solver Based Multi-Threading

    This type of multi-threading is transparent to the solver and is applicable if the total solution is composed of individual solutions that can be solved in parallel. In particular, this approach also allows for using solvers with different parameters, such as their operator or regularization parameters.

    using RegularizedLeastSquares
+As = [rand(32, 16) for _ in 1:4]
+xs = [rand(16) for _ in 1:4]
+bs = [A*x for (A, x) in zip(As, xs)]
+
+xs_approx = similar(xs)
+Threads.@threads for i in 1:4
+  solver = createLinearSolver(CGNR, As[i]; iterations=32)
+  xs_approx[i] = solve!(solver, bs[i])
+end

    Operator Based Multi-Threading

    This type of multi-threading involves linear operators or proximal maps that can be implemnted in parallel. Examples of this include the proximal map of the TV regularization term, which is based on the multi-threaded GradientOp from LinearOperatorCollection. GPU acceleration also falls under this approach, see GPU Acceleration for more information.

    Measurement Based Multi-Threading

    This level of multi-threading applies the same solver (and its parameters) to multiple measurement vectors or rather a measurement matrix B. This is useful in the case of multiple measurements that can be solved in parallel and can reuse the same solver. This approach is not applicable if the operator is stateful.

    To use this approach we first build a measurement matrix B and a corresponding solver:

    A = first(As)
+B = mapreduce(x -> A*x, hcat, xs)
+solver = createLinearSolver(CGNR, A; iterations=32)
    CGNR{Matrix{Float64}, Matrix{Float64}, L2Regularization{Float64}, Vector{Any}}([0.5522964464282882 0.057967659145914285 … 0.44454909909974416 0.6572841488411938; 0.8724836761034838 0.8814319753082565 … 0.9270247390018734 0.4184112511224466; … ; 0.14903530400804987 0.9808112713209824 … 0.14469174934369433 0.4074920700464463; 0.00788791769537489 0.22565144415014737 … 0.426311979994105 0.07486720041580652], [9.630725574230791 7.799436811825599 … 8.443403684936504 6.314897612703452; 7.799436811825599 11.237405536890643 … 9.351294069243538 6.783206080272212; … ; 8.443403684936504 9.351294069243538 … 12.007186664392426 7.459403612518464; 6.314897612703452 6.783206080272212 … 7.459403612518464 7.775950989526439], L2Regularization{Float64}(0.0), Any[], NoNormalization(), 32, RegularizedLeastSquares.CGNRState{Float64, Float64, Vector{Float64}}([5.0e-324, NaN, 1.5e-323, 2.0e-323, NaN, 3.0e-323, 3.5e-323, NaN, NaN, 5.0e-323, 5.4e-323, 6.0e-323, 6.4e-323, 7.0e-323, 7.4e-323, NaN], [5.0e-324, 1.0e-323, 1.5e-323, 2.0e-323, 2.5e-323, 3.0e-323, 3.5e-323, 4.0e-323, 4.4e-323, 5.0e-323, 5.4e-323, 6.0e-323, 6.4e-323, 7.0e-323, 7.4e-323, 8.0e-323], [1.5e-323, 2.5e-323, 6.90978611664797e-310, 6.90978611930406e-310, 3.0e-323, 4.0e-323, 6.90978611665113e-310, 6.9097861166543e-310, 4.4e-323, 4.4e-323, 6.90978611665745e-310, 6.9097861166606e-310, 5.0e-323, 8.0e-323, 6.9097861166638e-310, 6.90978611666694e-310], [5.0e-324, 0.0, 0.0, 5.0e-324, 0.0, 5.0e-324, 5.0e-324, 0.0, 0.0, 0.0, 5.0e-324, 5.0e-324, 5.0e-324, 5.0e-324, 5.0e-324, 0.0], 0.0, 0.0, 0.0, 0, 2.220446049250313e-16, 0.0))

    We can then simply pass the measurement matrix to the solver. The result will be the same as if we passed each colument of B seperately:

    x_approx = solve!(solver, B)
+size(x_approx)
    (16, 4)

    The previous solve! call was still executed sequentially. To execute it in parallel, we have to specify a multi-threaded scheduler as a keyword-argument of the solve! call. RegularizedLeastSquares.jl provides a MultiThreadingState scheduler that can be used for this purpose. This scheduler is based on the Threads.@threads macro:

    x_multi = solve!(solver, B; scheduler = MultiThreadingState)
+x_approx == x_multi
    true

    Custom Scheduling

    It is possible to implement custom scheduling. The following code shows how to implement this for the Threads.@spawn macro. Usually one this to implement multi-threading with a package such as FLoop.jl or ThreadPools.jl for thread pinning:

    Since most solver have conv. criteria, they can finish at different iteration numbers, which we track this information with flags.

     mutable struct SpawnState{S, ST <: AbstractSolverState{S}} <: RegularizedLeastSquares.AbstractMatrixSolverState{S}
+   states::Vector{ST}
+   active::Vector{Bool}
+   SpawnState(states::Vector{ST}) where {S, ST <: AbstractSolverState{S}} = new{S, ST}(states, fill(true, length(states)))
+ end

    To hook into the existing init! code we only have to supply a method that gets a copyable "vector" state. This will invoke our SpawnState constructor with copies of the given state.

     prepareMultiStates(solver::AbstractLinearSolver, state::SpawnState, b::AbstractMatrix) = prepareMultiStates(solver, first(state.states), b);

    We specialise the iterate function which is called with the idx of still active states

    function Base.iterate(solver::AbstractLinearSolver, state::SpawnState, activeIdx)
+  @sync Threads.@spawn for i in activeIdx
+    res = iterate(solver, state.states[i])
+    if isnothing(res)
+      state.active[i] = false
+    end
+  end
+  return state.active, state
+end

    Now we can simply use the SpawnState scheduler in the solve! call:

    x_custom = solve!(solver, B; scheduler = SpawnState)
+x_approx == x_multi
    true

    This page was generated using Literate.jl.

    diff --git a/previews/PR92/generated/howto/normal_operator/index.html b/previews/PR92/generated/howto/normal_operator/index.html new file mode 100644 index 00000000..3ad4ed5a --- /dev/null +++ b/previews/PR92/generated/howto/normal_operator/index.html @@ -0,0 +1,68 @@ + +Normal Operator · RegularizedLeastSquares.jl

    Normal Operator

    Many solvers in RegularizedLeastSquares.jl are based on the normal operator $\mathbf{A}^*\mathbf{A}$ of the linear operator $\mathbf{A}$.

    Solvers such as ADMM, FISTA and POGM generally solve optimization problems of the form:

    \[\begin{equation} + \underset{\mathbf{x}}{argmin} \mathbf{f(x)}+ \mathbf{R(x)} +\end{equation}\]

    and require the gradient of the function $\mathbf{f(x)}$. In this package we specialise the function $\mathbf{f(x)}$ to the least squares norm:

    \[\begin{equation} + \mathbf{f(x)} = \frac{1}{2}\vert\vert \mathbf{A}\mathbf{x}-\mathbf{b} \vert\vert^2_2 +\end{equation}\]

    The gradient of this function is:

    \[\begin{equation} + \nabla \mathbf{f(x)} = \mathbf{A}^*(\mathbf{A}\mathbf{x}-\mathbf{b}) = \mathbf{A}^*\mathbf{Ax} - \mathbf{A}^*\mathbf{b} +\end{equation}\]

    Similarily, the conjugate gradient normal residual (CGNR) algorithm applies conjugate gradient algorithm to:

    \[\begin{equation} + \mathbf{A}^*\mathbf{A}\mathbf{x} = \mathbf{A}^*\mathbf{b} +\end{equation}\]

    The normal operator can be passed directly to these solvers, otherwise it is computed internally.

    using RegularizedLeastSquares
+A = randn(32, 16)
+x = randn(16)
+b = A*x
+
+solver = createLinearSolver(CGNR, A; AHA = adjoint(A) * A, reg = L2Regularization(0.0001), iterations=32);
+x_approx = solve!(solver, b)
    16-element Vector{Float64}:
+  0.9476327004064123
+  0.2909593014660477
+  1.5617433787181285
+ -0.4832102295169169
+  0.40728981484468824
+ -0.5808991768866963
+  0.04958063559988508
+ -1.3036488059908822
+  0.8457195775886119
+  1.1327975395110086
+  0.7586031313687629
+  0.3953932290654046
+ -0.5604711251844071
+ -0.6528532400965956
+  1.1891089165505841
+ -0.3028868503003121

    The normal operator can also be computed using the normalOperator function from LinearOperatorCollection.jl. This is useful if the normal operator is not directly available or shouldn't be stored in memory. This function is opinionated and attempts to optimize the resulting operator for iterative applications. Specifying a custom method for a custom operator allows one to control this optimization.

    An example of such an optimization is a matrix-free weighting of $\mathbf{A}$ as shown in the Weighting example:

    using LinearOperatorCollection
+weights = rand(32)
+WA = ProdOp(WeightingOp(weights), A)
+AHA = LinearOperatorCollection.normalOperator(WA)
    Linear operator
+  nrow: 16
+  ncol: 16
+  eltype: Float64
+  symmetric: false
+  hermitian: false
+  nprod:   0
+  ntprod:  0
+  nctprod: 0
+
+

    Without an optimization a matrix-free product would apply the following operator each iteration:

    \[\begin{equation} + (\mathbf{WA})^*\mathbf{WA} = \mathbf{A}^*\mathbf{W}^*\mathbf{W}\mathbf{A} +\end{equation}\]

    This is not efficient and instead the normal operator can be optimized by initially computing the weights:

    \[\begin{equation} + \tilde{\mathbf{W}} = \mathbf{W}^*\mathbf{W} +\end{equation}\]

    and then applying the following each iteration:

    \[\begin{equation} + \mathbf{A}^*\tilde{\mathbf{W}}\mathbf{A} +\end{equation}\]

    The optimized normal operator can then be passed to the solver:

    solver = createLinearSolver(CGNR, WA; AHA = AHA, reg = L2Regularization(0.0001), iterations=32);
+x_approx2 = solve!(solver, weights .* b)
    16-element Vector{Float64}:
+  0.947572518870901
+  0.2910097085095847
+  1.5616937478013226
+ -0.4832224770348008
+  0.407297903088875
+ -0.5809345232797501
+  0.049522979484697555
+ -1.3036941661409458
+  0.84573787532379
+  1.132837106775444
+  0.7585804435038358
+  0.39537107523528126
+ -0.5604656214986606
+ -0.6528156854499366
+  1.1890385349995984
+ -0.30284417399641844

    Of course it is also possible to optimize a normal operator with other means and pass it to the solver via the AHA keyword argument.

    It is also possible to only supply the normal operator to these solvers, however on then needs to supply $\mathbf{A^*b}$ intead of $\mathbf{b}$.

    This page was generated using Literate.jl.

    diff --git a/previews/PR92/generated/howto/plug-and-play/index.html b/previews/PR92/generated/howto/plug-and-play/index.html new file mode 100644 index 00000000..8fcde03b --- /dev/null +++ b/previews/PR92/generated/howto/plug-and-play/index.html @@ -0,0 +1,23 @@ + +Plug-and-Play Regularization · RegularizedLeastSquares.jl

    Plug-and-Play Regularization

    A group of regularization terms that can not be directly written down as function are learned plug-and-play (PnP) priors. These are terms based on deep neural networks, which are trainted to implement the proximal map corresponding to the regularization term. Such a PnP prior can be used in the same way as any other regularization term.

    The following example shows how to use a PnP prior in the context of the Kaczmarz solver.

    using RegularizedLeastSquares
+A = randn(32, 16)
+x = randn(16)
+b = A*x;

    For the documentation we will just use the identity function as a placeholder for the PnP prior.

    model = identity
    identity (generic function with 1 method)

    In practice, you would replace this with a neural network:

    using Flux
+model = Flux.loadmodel!(model, ...)

    The model can then be used together with the PnPRegularization term:

    reg = PnPRegularization(1.0; model = model, shape = [16]);

    Since models often expect a specific input range, we can use the MinMaxTransform to normalize the input:

    reg = PnPRegularization(1.0; model = model, shape = [16], input_transform = RegularizedLeastSquares.MinMaxTransform);

    Custom input transforms can be implemented by passing something callable as the input_transform keyword argument. For more details see the PnPRegularization documentation.

    The regularization term can then be used in the solver:

    solver = createLinearSolver(Kaczmarz, A; reg = reg, iterations = 32)
+x_approx = solve!(solver, b)
    16-element Vector{Float64}:
+ -0.500506740286645
+  0.09619227453944945
+  2.0880887259218133
+ -0.2791009333337865
+ -1.1048322863071698
+  0.12252884375050987
+ -0.3449229727058336
+  1.0257063561304396
+  0.934009634300548
+ -0.025072508193380028
+  0.71648044046844
+ -0.8937409934458287
+ -0.1749943482952453
+  2.2365318256411104
+  0.2942241606480469
+ -0.13622280294445321

    This page was generated using Literate.jl.

    diff --git a/previews/PR92/generated/howto/weighting/index.html b/previews/PR92/generated/howto/weighting/index.html new file mode 100644 index 00000000..c8b791c5 --- /dev/null +++ b/previews/PR92/generated/howto/weighting/index.html @@ -0,0 +1,15 @@ + +Weighting · RegularizedLeastSquares.jl

    Weighting

    Often time one wants to solve a weighted least squares problem of the form:

    \[\begin{equation} + \underset{\mathbf{x}}{argmin} \frac{1}{2}\vert\vert \mathbf{A}\mathbf{x}-\mathbf{b} \vert\vert^2_{\mathbf{W}} + \mathbf{R(x)} . +\end{equation}\]

    where $\mathbf{W}$ is a symmetric, positive weighting matrix and $\vert\vert\mathbf{y}\vert\vert^2_\mathbf{W}$ denotes the weighted Euclidean norm. An example of such a weighting matrix is a noise whitening matrix. Another example could be a scaling of the matrix rows by the reciprocal of their row energy.

    In the following, we will solve a weighted least squares problem of the form:

    \[\begin{equation} + \underset{\mathbf{x}}{argmin} \frac{1}{2}\vert\vert \mathbf{A}\mathbf{x}-\mathbf{b} \vert\vert_\mathbf{W}^2 + \vert\vert\mathbf{x}\vert\vert^2_2 . +\end{equation}\]

    using RegularizedLeastSquares, LinearOperatorCollection, LinearAlgebra
+A = rand(32, 16)
+x = rand(16)
+b = A*x;

    As a weighting matrix, we will use the reciprocal of the row energy of the matrix A.

    weights = map(row -> 1/rownorm²(A, row), 1:size(A, 1));

    First, let us solve the problem with matrices we manually weighted.

    WA = diagm(weights) * A
+solver = createLinearSolver(Kaczmarz, WA; reg = L2Regularization(0.0001), iterations=10)
+x_approx = solve!(solver, weights .* b);

    The operator A is not always a dense matrix and the product between the operator and the weighting matrix is not always efficient or possible. The package LinearOperatorCollection.jl provides a matrix-free implementation of a diagonal weighting matrix, as well as a matrix free product between two matrices. This weighted operator has efficient implementations of the normal operator and also for the row-action operations of the Kaczmarz solver.

    W = WeightingOp(weights)
+P = ProdOp(W, A)
+solver = createLinearSolver(Kaczmarz, P; reg = L2Regularization(0.0001), iterations=10)
+x_approx2 = solve!(solver, W * b)
+isapprox(x_approx, x_approx2)
    true

    This page was generated using Literate.jl.

    diff --git a/previews/PR92/index.html b/previews/PR92/index.html new file mode 100644 index 00000000..47fe2e99 --- /dev/null +++ b/previews/PR92/index.html @@ -0,0 +1,2 @@ + +Home · RegularizedLeastSquares.jl

    RegularizedLeastSquares.jl

    Solvers for Linear Inverse Problems using Regularization Techniques

    Introduction

    RegularizedLeastSquares.jl is a Julia package for solving large linear systems using various types of algorithms. Ill-conditioned problems arise in many areas of practical interest. Regularisation techniques and nonlinear problem formulations are often used to solve these problems. This package provides implementations for a variety of solvers used in areas such as MPI and MRI. In particular, this package serves as the optimization backend of the Julia packages MPIReco.jl and MRIReco.jl.

    The implemented methods range from the $l^2_2$-regularized CGNR method to more general optimizers such as the Alternating Direction of Multipliers Method (ADMM) or the Split-Bregman method.

    For convenience, implementations of popular regularizers, such as $l_1$-regularization and TV regularization, are provided. On the other hand, hand-crafted regularizers can be used quite easily.

    Depending on the problem, it becomes unfeasible to store the full system matrix at hand. For this purpose, RegularizedLeastSquares.jl allows for the use of matrix-free operators. Such operators can be realized using the interface provided by the package LinearOperators.jl. Other interfaces can be used as well, as long as the product *(A,x) and the adjoint adjoint(A) are provided. A number of common matrix-free operators are provided by the package LinearOperatorColection.jl.

    Features

    • Variety of optimization algorithms optimized for least squares problems
    • Support for matrix-free operators
    • GPU support

    Usage

    See also

    Packages:

    Organizations:

    diff --git a/previews/PR92/literate/examples/compressed_sensing.jl b/previews/PR92/literate/examples/compressed_sensing.jl new file mode 100644 index 00000000..4e81d0fe --- /dev/null +++ b/previews/PR92/literate/examples/compressed_sensing.jl @@ -0,0 +1,69 @@ +# # Compressed Sensing Example +# In this example we will go through a simple example from the field of Compressed Sensing. +# In addtion to RegularizedLeastSquares.jl, we will need the packages [LinearOperatorCollection.jl](https://github.com/JuliaImageRecon/LinearOperatorCollection.jl), [ImagePhantoms.jl](https://github.com/JuliaImageRecon/ImagePhantoms.jl), [ImageGeoms.jl](https://github.com/JuliaImageRecon/ImageGeoms.jl) and Random.jl, as well as [CairoMakie.jl](https://docs.makie.org/stable/) for visualization. We can install them the same way we did RegularizedLeastSquares.jl. + +# ## Preparing the Inverse Problem +# To get started, let us generate a simple phantom using the ImagePhantoms and ImageGeom packages: + +using ImagePhantoms, ImageGeoms +N = 256 +image = shepp_logan(N, SheppLoganToft()) +size(image) + +# This produces a 256x256 image of a Shepp-Logan phantom. + +# In this example, we consider a problem in which we randomly sample a third of the pixels in the image. Such a problem and the corresponding measurement can be constructed with the packages LinearOperatorCollection and Random: + +# We first randomly shuffle the indices of the image and then select the first third of the indices to sample. +using Random, LinearOperatorCollection +randomIndices = shuffle(eachindex(image)) +sampledIndices = sort(randomIndices[1:div(end, 3)]) + +# Afterwards we build a sampling operator which samples the image at the selected indices. +A = SamplingOp(eltype(image), pattern = sampledIndices , shape = size(image)); + +# Then we apply the sampling operator to the vectorized image to obtain the sampled measurement vector +b = A*vec(image); + +# To visualize our image we can use CairoMakie: +using CairoMakie +function plot_image(figPos, img; title = "", width = 150, height = 150) + ax = CairoMakie.Axis(figPos; yreversed=true, title, width, height) + hidedecorations!(ax) + heatmap!(ax, img) +end +fig = Figure() +plot_image(fig[1,1], image, title = "Image") +samplingMask = fill(false, size(image)) +samplingMask[sampledIndices] .= true +plot_image(fig[1,2], image .* samplingMask, title = "Sampled Image") +resize_to_layout!(fig) +fig + +# As we can see in the right image, only a third of the pixels are sampled. The goal of the inverse problem is to recover the original image from this measurement vector. + +# ## Solving the Inverse Problem +# To recover the image from the measurement vector, we solve the TV-regularized least squares problem: +# ```math +# \begin{equation} +# \underset{\mathbf{x}}{argmin} \frac{1}{2}\vert\vert \mathbf{A}\mathbf{x}-\mathbf{b} \vert\vert_2^2 + \vert\vert\mathbf{x}\vert\vert_{\lambda\text{TV}} . +# \end{equation} +# ``` + +# For this purpose we build a TV regularizer with regularization parameter $λ=0.01$ + +using RegularizedLeastSquares +reg = TVRegularization(0.01; shape=size(image)); + +# We will use the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) to solve our inverse problem. Thus, we build the corresponding solver + +solver = createLinearSolver(FISTA, A; reg=reg, iterations=20); + +# and apply it to our measurement vector +img_approx = solve!(solver,b) + +# To visualize the reconstructed image, we need to reshape the result vector to the correct shape. Afterwards we can use CairoMakie again: +img_approx = reshape(img_approx,size(image)); +plot_image(fig[1,3], img_approx, title = "Reconstructed Image") +resize_to_layout!(fig) +fig \ No newline at end of file diff --git a/previews/PR92/literate/examples/computed_tomography.jl b/previews/PR92/literate/examples/computed_tomography.jl new file mode 100644 index 00000000..2d1fca28 --- /dev/null +++ b/previews/PR92/literate/examples/computed_tomography.jl @@ -0,0 +1,66 @@ +# # Computed Tomography Example +# In this example we will go through a simple example from the field of Computed Tomography. +# In addtion to RegularizedLeastSquares.jl, we will need the packages [LinearOperatorCollection.jl](https://github.com/JuliaImageRecon/LinearOperatorCollection.jl), [ImagePhantoms.jl](https://github.com/JuliaImageRecon/ImagePhantoms.jl), [ImageGeoms.jl](https://github.com/JuliaImageRecon/ImageGeoms.jl) +# and [RadonKA.jl](https://github.com/roflmaostc/RadonKA.jl/tree/main), as well as [CairoMakie.jl](https://docs.makie.org/stable/) for visualization. We can install them the same way we did RegularizedLeastSquares.jl. + +# RadonKA is a package for the computation of the Radon transform and its adjoint. It is implemented with KernelAbstractions.jl and supports GPU acceleration. See the GPU acceleration [how-to](../howto/gpu_acceleration.md) for more information. + +# ## Preparing the Inverse Problem +# To get started, let us generate a simple phantom using the ImagePhantoms and ImageGeom packages: + +using ImagePhantoms, ImageGeoms +N = 256 +image = shepp_logan(N, SheppLoganToft()) +size(image) + +# This produces a 64x64 image of a Shepp-Logan phantom. + +using RadonKA, LinearOperatorCollection +angles = collect(range(0, π, 256)) +sinogram = Array(RadonKA.radon(image, angles)) + +# Afterwards we build a Radon operator implementing both the forward and adjoint Radon transform +A = RadonOp(eltype(image); angles, shape = size(image)); + +# To visualize our image we can use CairoMakie: +using CairoMakie +function plot_image(figPos, img; title = "", width = 150, height = 150) + ax = CairoMakie.Axis(figPos[1, 1]; yreversed=true, title, width, height) + hidedecorations!(ax) + hm = heatmap!(ax, img) + Colorbar(figPos[2, 1], hm, vertical = false, flipaxis = false) +end +fig = Figure() +plot_image(fig[1,1], image, title = "Image") +plot_image(fig[1,2], sinogram, title = "Sinogram") +plot_image(fig[1,3], backproject(sinogram, angles), title = "Backprojection") +resize_to_layout!(fig) +fig + +# In the figure we can see our original image, the sinogram, and the backprojection of the sinogram. The goal of the inverse problem is to recover the original image from the sinogram. + +# ## Solving the Inverse Problem +# To recover the image from the measurement vector, we solve the $l^2_2$-regularized least squares problem +# ```math +# \begin{equation} +# \underset{\mathbf{x}}{argmin} \frac{1}{2}\vert\vert \mathbf{A}\mathbf{x}-\mathbf{b} \vert\vert_2^2 + \vert\vert\mathbf{x}\vert\vert^2_2 . +# \end{equation} +# ``` + +# For this purpose we build a $l^2_2$ with regularization parameter $λ=0.001$ + +using RegularizedLeastSquares +reg = L2Regularization(0.001); + +# To solve this inverse problem, the Conjugate Gradient Normal Residual (CGNR) algorithm can be used. This solver is based on the normal operator of the Radon operator and uses both the forward and adjoint Radon transform internally. +# We now build the corresponding solver +solver = createLinearSolver(CGNR, A; reg=reg, iterations=20); + +# and apply it to our measurement vector +img_approx = solve!(solver, vec(sinogram)) + +# To visualize the reconstructed image, we need to reshape the result vector to the correct shape. Afterwards we can use CairoMakie again: +img_approx = reshape(img_approx,size(image)); +plot_image(fig[1,4], img_approx, title = "Reconstructed Image") +resize_to_layout!(fig) +fig \ No newline at end of file diff --git a/previews/PR92/literate/examples/getting_started.jl b/previews/PR92/literate/examples/getting_started.jl new file mode 100644 index 00000000..750d3341 --- /dev/null +++ b/previews/PR92/literate/examples/getting_started.jl @@ -0,0 +1,51 @@ +# # Getting Started + +# In this example we will go through a simple random inverse problem to get familiar with RegularizedLeastSquares.jl. + +# ## Installation + +# To install RegularizedLeastSquares.jl, you can use the Julia package manager. Open a Julia REPL and run the following command: + +# ```julia +# using Pkg +# Pkg.add("RegularizedLeastSquares") +# ``` +# This will download and install the RegularizedLeastSquares.jl package and its dependencies. To install a different version, please consult the [Pkg documentation](https://pkgdocs.julialang.org/dev/managing-packages/#Adding-packages). + +# Once the installation is complete, you can import the package with the `using` keyword: + + +using RegularizedLeastSquares + +# RegularizedLeastSquares aims to solve inverse problems of the form: +# ```math +# \begin{equation} +# \underset{\mathbf{x}}{argmin} \frac{1}{2}\vert\vert \mathbf{A}\mathbf{x}-\mathbf{b} \vert\vert_2^2 + \mathbf{R(x)} . +# \end{equation} +# ``` +# where $\mathbf{A}$ is a linear operator, $\mathbf{y}$ is the measurement vector, and $\mathbf{R(x)}$ is an (optional) regularization term. +# The goal is to retrieve an approximation of the unknown vector $\mathbf{x}$. In this first exampel we will just work with simple random arrays. For more advanced examples, please refer to the examples. + +A = rand(32, 16) +x = rand(16) +b = A*x; + +# To approximate x from b, we can use the Conjugate Gradient Normal Residual (CGNR) algorithm. We first build the corresponding solver: +solver = createLinearSolver(CGNR, A; iterations=32); + +# and apply it to our measurement vector +x_approx = solve!(solver, b) +isapprox(x, x_approx, rtol = 0.001) + +# Usually the inverse problems are ill-posed and require regularization. RegularizedLeastSquares.jl provides a variety of regularization terms. +# The CGNR algorithm can solve optimzation problems of the form: +# ```math +# \begin{equation} +# \underset{\mathbf{x}}{argmin} \frac{1}{2}\vert\vert \mathbf{A}\mathbf{x}-\mathbf{b} \vert\vert_2^2 + \vert\vert\mathbf{x}\vert\vert^2_2 . +# \end{equation} +# ``` + +# The corresponding solver can be built with the L2 regularization term: +solver = createLinearSolver(CGNR, A; reg = L2Regularization(0.0001), iterations=32); +x_approx = solve!(solver, b) +isapprox(x, x_approx, rtol = 0.001) diff --git a/previews/PR92/literate/explanations/regularization.jl b/previews/PR92/literate/explanations/regularization.jl new file mode 100644 index 00000000..bd9b6c27 --- /dev/null +++ b/previews/PR92/literate/explanations/regularization.jl @@ -0,0 +1,96 @@ +# # Regularization +# When formulating inverse problems, a regulariser is formulated as an additional cost function which has to be minimised. +# Many algorithms deal with a regularizers $g$, by computing the proximal map: + +# ```math +# \begin{equation} +# prox_g (\mathbf{x}) = \underset{\mathbf{u}}{argmin} \frac{1}{2}\vert\vert \mathbf{u}-\mathbf{x} \vert {\vert}^2 + g(\mathbf{x}). +# \end{equation} +# ``` +# For many regularizers, the proximal map can be computed efficiently in a closed form. + +# In order to implement these proximal mappings, RegularizedLeastSquares.jl defines the following type hierarchy: +# ```julia +# abstract type AbstractRegularization +# prox!(reg::AbstractRegularization, x) +# norm(reg::AbstractRegularization, x) +# ``` +# Here `prox!(reg, x)` is an in-place function which computes the proximal map on the input-vector `x`. The function `norm` computes the value of the corresponding term in the inverse problem. RegularizedLeastSquares.jl provides `AbstractParameterizedRegularization` and `AbstractProjectionRegularization` as core regularization types. + +# ## Parameterized Regularization Terms +# This group of regularization terms features a regularization parameter `λ` that is used during the `prox!` and `norm `computations. Examples of this regulariztion group are `L1`, `L2` or `LLR` (locally low rank) regularization terms. + +# These terms are constructed by supplying a `λ` and optionally term specific keyword arguments: +using RegularizedLeastSquares +l2 = L2Regularization(0.3) +# Parameterized regularization terms implement: +# ```julia +# prox!(reg::AbstractParameterizedRegularization, x, λ) +# norm(reg::AbstractParameterizedRegularization, x, λ) +# ``` +# where `λ` by default is filled with the value used during construction. + +# Invoking `λ` on a parameterized term retrieves its regularization parameter. This can be used in a solver to scale and overwrite the parameter as follows: +prox!(l2, [1.0]) +# +param = λ(l2) +prox!(l2, [1.0], param*0.2) + + +# ## Projection Regularization Terms +# This group of regularization terms implement projections, such as a positivity constraint or a projection with a given convex projection function. +# These are essentially proximal maps where $g(\mathbf{x})$ is the indicator function of a convex set. + +positive = PositiveRegularization() +prox!(positive, [2.0, -0.2]) + +# ## Nested Regularization Terms +# Nested regularization terms are terms that act as decorators to the core regularization terms. +# These terms can be nested around other terms and add functionality to a regularization term, such as scaling `λ` based on the provided operator or applying a transform, such as the Wavelet, to `x`. +# As an example, we can nest a `L1` regularization term around a `Wavelet` operator. + +# First we generate an image and apply a wavelet operator to it: +using Wavelets, LinearOperatorCollection, ImagePhantoms, ImageGeoms +N = 256 +image = shepp_logan(N, SheppLoganToft()) +wop = WaveletOp(Float32, shape = size(image)) + +wavelet_image = reshape(wop*vec(image), size(image)) +wavelet_image = log.(abs.(wavelet_image) .+ 0.01) + +# We will use CairoMakie for visualization: +using CairoMakie +function plot_image(figPos, img; title = "", width = 150, height = 150) + ax = CairoMakie.Axis(figPos; yreversed=true, title, width, height) + hidedecorations!(ax) + heatmap!(ax, img) +end +fig = Figure() +plot_image(fig[1,1], image, title = "Image") +plot_image(fig[1,2], wavelet_image, title = "Wavelet Image") +resize_to_layout!(fig) +fig + +# To apply soft-thresholding in the wavelet domain, we can nest a `L1` regularization term around the `Wavelet` operator: +core = L1Regularization(0.1) +reg = TransformedRegularization(core, wop); + +# We can then apply the proximal map to the image or the image in the wavelet domain: +img_prox_image = prox!(core, copy(vec(image))); +img_prox_wavelet = prox!(reg, copy(vec(image))); + +# We can visualize the result: +img_prox_image = reshape(img_prox_image, size(image)) +img_prox_wavelet = reshape(img_prox_wavelet, size(image)) +plot_image(fig[1,3], img_prox_image, title = "Reg. Image Domain") +plot_image(fig[1,4], img_prox_wavelet, title = "Reg. Wavelet Domain") +resize_to_layout!(fig) +fig + +# Generally, regularization terms can be nested arbitrarly deep, adding new functionality with each layer. Each nested regularization term can return its `inner` regularization term. +# Furthermore, all regularization terms implement the iteration interface to iterate over the nesting. The innermost regularization term of a nested term must be a core regularization term and it can be returned by the `sink` function: +RegularizedLeastSquares.innerreg(reg) == core +# +sink(reg) == core +# +foreach(r -> println(nameof(typeof(r))), reg) \ No newline at end of file diff --git a/previews/PR92/literate/howto/callbacks.jl b/previews/PR92/literate/howto/callbacks.jl new file mode 100644 index 00000000..ea25aa9f --- /dev/null +++ b/previews/PR92/literate/howto/callbacks.jl @@ -0,0 +1,49 @@ +# # Callbacks +# For certain optimization it is important to monitor the internal state of the solver. RegularizedLeastSquares.jl provides a callback mechanism to allow developres to access this state after each iteration. +# The package provides a variety of default callbacks, which for example store the solution after each iteration. More information can be found in the API reference for the `solve!` function. + +# In this example we will revist the compressed sensing compressed sensing [example](../examples/compressed_sensing.md) and implement a custom callback using the do-syntax of the `solve!` function. +# This allows us to implement our callback inline and access the solver state after each iteration. + +# We first recreate the operator `A` and the measurement vector `b`: +using ImagePhantoms, ImageGeoms +N = 256 +image = shepp_logan(N, SheppLoganToft()) +size(image) +using Random, LinearOperatorCollection +randomIndices = shuffle(eachindex(image)) +sampledIndices = sort(randomIndices[1:div(end, 3)]) +A = SamplingOp(eltype(image), pattern = sampledIndices , shape = size(image)); +b = A*vec(image); + +# Next we prepare our visualization helper function: +using CairoMakie +function plot_image(figPos, img; title = "", width = 150, height = 150, clim = extrema(img)) + ax = CairoMakie.Axis(figPos; yreversed=true, title, width, height) + hidedecorations!(ax) + heatmap!(ax, img, colorrange = clim) +end + +# Now we construct the solver with the TV regularization term: +using RegularizedLeastSquares +reg = TVRegularization(0.01; shape=size(image)); +solver = createLinearSolver(FISTA, A; reg=reg, iterations=20); + +# We will now implement a callback that plots the solution every four iteration: +fig = Figure() +idx = 1 +solve!(solver, b) do solver, iteration + if iteration % 4 == 0 + img_approx = copy(solversolution(solver)) + img_approx = reshape(img_approx, size(image)) + plot_image(fig[div(idx -1, 3) + 1, mod1(idx, 3)], img_approx, clim = extrema(image), title = "$iteration") + global idx += 1 + end +end +# In the callback we have to copy the solution, as the solver will update it in place. +# As is explained in the solver section, each features fields that are intended to be immutable during a `solve!` call and a state that is modified in each iteration. +# Depending on the solvers parameters and the measurement input, the state can differ in its fields and their type. Ideally, one tries to avoid accessing the state directly and uses the provided functions to access the state. + +# The resulting figure shows the reconstructed image after 0, 4, 8, 12, 16 and 20 iterations: +resize_to_layout!(fig) +fig diff --git a/previews/PR92/literate/howto/efficient_kaczmarz.jl b/previews/PR92/literate/howto/efficient_kaczmarz.jl new file mode 100644 index 00000000..51b16ac3 --- /dev/null +++ b/previews/PR92/literate/howto/efficient_kaczmarz.jl @@ -0,0 +1,31 @@ +# # Efficient Kaczmarz +# Unlike many of the other solvers provided by RegularizedLeastSquares.jl, the Kaczmarz method does not utilize a matrix-vector product with the operator $\mathbf{A}$ nor the normal operator $\mathbf{A*A}$. Instead, it uses the rows of $\mathbf{A}$ to update the solution iteratively. +# Efficient Kaczmarz implementation therefore require very efficient dot products with the rows of $\mathbf{A}$. In RegularizedLeastSquares.jl, this is achieved with the `dot_with_matrix_row` function. +using RegularizedLeastSquares +A = randn(256, 256) +x = randn(256) +b = A*x; + +# The `dot_with_matrix_row` function calculates the dot product between a row of A and the current approximate solution of x: +row = 1 +isapprox(RegularizedLeastSquares.dot_with_matrix_row(A, x, row), sum(A[row, :] .* x)) + +# Since in Julia, dense arrays are stored in column-major order, such a row-based operation is quite inefficient. A workaround is to transpose the matrix then pass it to a Kaczmarz solver. +At = collect(transpose(A)) +A_eff = transpose(At) + +# Note that the transpose function can return a lazy transpose object, so we first collect the transpose into a dense matrix. +# Then we transpose it again to get the efficient representation of the matrix. + +# We can compare the performance using the BenchmarkTools.jl package. First for the original matrix: +using BenchmarkTools +solver = createLinearSolver(Kaczmarz, A; reg = L2Regularization(0.0001), iterations=100) +@benchmark solve!(solver, b) samples = 100 + +# And then for the efficient matrix: +solver_eff = createLinearSolver(Kaczmarz, A_eff; reg = L2Regularization(0.0001), iterations=100) +@benchmark solve!(solver_eff, b) samples = 100 + +# We can also combine the efficient matrix with a weighting matrix, as is shown in the [Weighting](weighting.md) example. + +# Custom operators need to implement the `dot_with_matrix_row` function to be used with the Kaczmarz solver. Ideally, such an implementation is allocation free. \ No newline at end of file diff --git a/previews/PR92/literate/howto/gpu_acceleration.jl b/previews/PR92/literate/howto/gpu_acceleration.jl new file mode 100644 index 00000000..b6a5fa1f --- /dev/null +++ b/previews/PR92/literate/howto/gpu_acceleration.jl @@ -0,0 +1,77 @@ +# # GPU Acceleration +# RegularizedLeastSquares.jl supports generic GPU acceleration. This means that the user can use any GPU array type that supports the GPUArrays interface. This includes [CUDA.jl](https://github.com/JuliaGPU/CUDA.jl), [AMDGPU.jl](https://github.com/JuliaGPU/AMDGPU.jl), and [Metal.jl](https://github.com/JuliaGPU/Metal.jl). +# In this example we will use the package JLArrays.jl which provides a reference implementation for GPUArrays, that can runs on CPUs. +using JLArrays +gpu = JLArray; + +# To use the following examples on an actual GPU, load the appropraite package replace `JLArray` with the respective GPU array type, for example: +# ```julia +# using CUDA +# gpu = CuArray +# ``` + +# At first we will look at an example of dense GPU arrays. +using RegularizedLeastSquares +A = gpu(rand(Float32, 32, 16)) +x = gpu(rand(Float32, 16)) +b = A*x; + +# Solvers adapt their states based on the type of the given measurement vector. This means that the solver will automatically switch to GPU acceleration if a GPU array is passed as the measurement vector. +solver = createLinearSolver(CGNR, A; reg = L2Regularization(0.0001), iterations=32); +x_approx = solve!(solver, b) + +# This adaption does not include the operator. So if we want to compare with CPU result, we need to construct a new solver with a CPU operator. +solver = createLinearSolver(CGNR, Array(A); reg = L2Regularization(0.0001), iterations=32); +x_cpu = solve!(solver, Array(b)) +isapprox(Array(x_approx), x_cpu) + +# ## Matrix-Free Operators +# A special case is the usage of matrix-free operators. Since these operators do not have a concrete matrix representation, their GPU support depends on their implementation. +# Since not all multiplications within a solver approximation are in-place, the operator also needs to support the `*` operation and construct an appropriate result vector. +# For matrix-free operators based on LinearOperators.jl, this can be achieved by implementing the `LinearOperators.storage_type` method. + +# In the following, we will take another look at the CS example and execute it on the GPU. Note that for the JLArray example we chose a small phantom, since the JLArray implementation is not optimized for performance: +using ImagePhantoms, ImageGeoms +N = 32 +image = shepp_logan(N, SheppLoganToft()) + +using Random, LinearOperatorCollection +randomIndices = shuffle(eachindex(image)) +sampledIndices = sort(randomIndices[1:div(end, 3)]); + +# To construct the operator, we need to convert the indices to a GPU array. +# We also need to specify the correct storage type. In both LinearOperators.jl and LinearOperatorCollection.jl this is done with the `S` keyword argument. +gpu_indices = gpu(sampledIndices) +A = SamplingOp(eltype(image), pattern = gpu_indices, shape = size(image), S = typeof(b)); + +# Let's inspect the storage type of the operator: +using LinearOperatorCollection.LinearOperators +LinearOperators.storage_type(A) + +# Afterwards we can use the operator as usual: +b = A*vec(gpu(image)); + +# And use it in the solver: +using RegularizedLeastSquares +reg = TVRegularization(0.01; shape=size(image)) +solver = createLinearSolver(FISTA, A; reg=reg, iterations=20) +img_approx = solve!(solver,b); + +# To visualize the reconstructed image, we need to reshape the result vector to the correct shape and convert it to a CPU array: +img_approx = reshape(Array(img_approx),size(image)) + +# We will again use CairoMakie for visualization: +using CairoMakie +function plot_image(figPos, img; title = "", width = 150, height = 150) + ax = CairoMakie.Axis(figPos; yreversed=true, title, width, height) + hidedecorations!(ax) + heatmap!(ax, img) +end +fig = Figure() +plot_image(fig[1,1], image, title = "Image") +samplingMask = fill(false, size(image)) +samplingMask[sampledIndices] .= true +plot_image(fig[1,2], image .* samplingMask, title = "Sampled Image") +plot_image(fig[1,3], img_approx, title = "Reconstructed Image") +resize_to_layout!(fig) +fig \ No newline at end of file diff --git a/previews/PR92/literate/howto/multi_threading.jl b/previews/PR92/literate/howto/multi_threading.jl new file mode 100644 index 00000000..84b53897 --- /dev/null +++ b/previews/PR92/literate/howto/multi_threading.jl @@ -0,0 +1,68 @@ +# # Multi-Threading +# There are different ways multi-threading can be used with RegularizedLeastSquares.jl. To use multi-threading in Julia, one needs to start their session with multi-threads, see the [Julia documentation](https://docs.julialang.org/en/v1/manual/multi-threading/) for more information. + +# ## Solver Based Multi-Threading +# This type of multi-threading is transparent to the solver and is applicable if the total solution is composed of individual solutions that can be solved in parallel. +# In particular, this approach also allows for using solvers with different parameters, such as their operator or regularization parameters. + +using RegularizedLeastSquares +As = [rand(32, 16) for _ in 1:4] +xs = [rand(16) for _ in 1:4] +bs = [A*x for (A, x) in zip(As, xs)] + +xs_approx = similar(xs) +Threads.@threads for i in 1:4 + solver = createLinearSolver(CGNR, As[i]; iterations=32) + xs_approx[i] = solve!(solver, bs[i]) +end + +# ## Operator Based Multi-Threading +# This type of multi-threading involves linear operators or proximal maps that can be implemnted in parallel. +# Examples of this include the proximal map of the TV regularization term, which is based on the multi-threaded GradientOp from LinearOperatorCollection. +# GPU acceleration also falls under this approach, see [GPU Acceleration](gpu_acceleration.md) for more information. + +# ## Measurement Based Multi-Threading +# This level of multi-threading applies the same solver (and its parameters) to multiple measurement vectors or rather a measurement matrix B. +# This is useful in the case of multiple measurements that can be solved in parallel and can reuse the same solver. This approach is not applicable if the operator is stateful. + +# To use this approach we first build a measurement matrix B and a corresponding solver: +A = first(As) +B = mapreduce(x -> A*x, hcat, xs) +solver = createLinearSolver(CGNR, A; iterations=32) + +# We can then simply pass the measurement matrix to the solver. The result will be the same as if we passed each colument of B seperately: +x_approx = solve!(solver, B) +size(x_approx) + +# The previous `solve!` call was still executed sequentially. To execute it in parallel, we have to specify a multi-threaded `scheduler` as a keyword-argument of the `solve!` call. +# RegularizedLeastSquares.jl provides a `MultiThreadingState` scheduler that can be used for this purpose. This scheduler is based on the `Threads.@threads` macro: +x_multi = solve!(solver, B; scheduler = MultiThreadingState) +x_approx == x_multi + +# ## Custom Scheduling +# It is possible to implement custom scheduling. The following code shows how to implement this for the `Threads.@spawn` macro. Usually one this to implement multi-threading with a package such as FLoop.jl or ThreadPools.jl for thread pinning: + +# Since most solver have conv. criteria, they can finish at different iteration numbers, which we track this information with flags. + mutable struct SpawnState{S, ST <: AbstractSolverState{S}} <: RegularizedLeastSquares.AbstractMatrixSolverState{S} + states::Vector{ST} + active::Vector{Bool} + SpawnState(states::Vector{ST}) where {S, ST <: AbstractSolverState{S}} = new{S, ST}(states, fill(true, length(states))) + end + +# To hook into the existing init! code we only have to supply a method that gets a copyable "vector" state. This will invoke our SpawnState constructor with copies of the given state. + prepareMultiStates(solver::AbstractLinearSolver, state::SpawnState, b::AbstractMatrix) = prepareMultiStates(solver, first(state.states), b); + +# We specialise the iterate function which is called with the idx of still active states +function Base.iterate(solver::AbstractLinearSolver, state::SpawnState, activeIdx) + @sync Threads.@spawn for i in activeIdx + res = iterate(solver, state.states[i]) + if isnothing(res) + state.active[i] = false + end + end + return state.active, state +end + +# Now we can simply use the SpawnState scheduler in the solve! call: +x_custom = solve!(solver, B; scheduler = SpawnState) +x_approx == x_multi \ No newline at end of file diff --git a/previews/PR92/literate/howto/normal_operator.jl b/previews/PR92/literate/howto/normal_operator.jl new file mode 100644 index 00000000..bded38d7 --- /dev/null +++ b/previews/PR92/literate/howto/normal_operator.jl @@ -0,0 +1,70 @@ +# # Normal Operator +# Many solvers in RegularizedLeastSquares.jl are based on the normal operator $\mathbf{A}^*\mathbf{A}$ of the linear operator $\mathbf{A}$. +# +# Solvers such as ADMM, FISTA and POGM generally solve optimization problems of the form: +# ```math +# \begin{equation} +# \underset{\mathbf{x}}{argmin} \mathbf{f(x)}+ \mathbf{R(x)} +# \end{equation} +# ``` +# and require the gradient of the function $\mathbf{f(x)}$. In this package we specialise the function $\mathbf{f(x)}$ to the least squares norm: +# ```math +# \begin{equation} +# \mathbf{f(x)} = \frac{1}{2}\vert\vert \mathbf{A}\mathbf{x}-\mathbf{b} \vert\vert^2_2 +# \end{equation} +# ``` +# The gradient of this function is: +# ```math +# \begin{equation} +# \nabla \mathbf{f(x)} = \mathbf{A}^*(\mathbf{A}\mathbf{x}-\mathbf{b}) = \mathbf{A}^*\mathbf{Ax} - \mathbf{A}^*\mathbf{b} +# \end{equation} +# ``` +# Similarily, the conjugate gradient normal residual (CGNR) algorithm applies conjugate gradient algorithm to: +# ```math +# \begin{equation} +# \mathbf{A}^*\mathbf{A}\mathbf{x} = \mathbf{A}^*\mathbf{b} +# \end{equation} +# ``` +# The normal operator can be passed directly to these solvers, otherwise it is computed internally. +using RegularizedLeastSquares +A = randn(32, 16) +x = randn(16) +b = A*x + +solver = createLinearSolver(CGNR, A; AHA = adjoint(A) * A, reg = L2Regularization(0.0001), iterations=32); +x_approx = solve!(solver, b) + +# The normal operator can also be computed using the `normalOperator` function from LinearOperatorCollection.jl. This is useful if the normal operator is not directly available or shouldn't be stored in memory. +# This function is opinionated and attempts to optimize the resulting operator for iterative applications. Specifying a custom method for a custom operator allows one to control this optimization. + +# An example of such an optimization is a matrix-free weighting of $\mathbf{A}$ as shown in the [Weighting](weighting.md) example: +using LinearOperatorCollection +weights = rand(32) +WA = ProdOp(WeightingOp(weights), A) +AHA = LinearOperatorCollection.normalOperator(WA) + +# Without an optimization a matrix-free product would apply the following operator each iteration: +# ```math +# \begin{equation} +# (\mathbf{WA})^*\mathbf{WA} = \mathbf{A}^*\mathbf{W}^*\mathbf{W}\mathbf{A} +# \end{equation} +# ``` +# This is not efficient and instead the normal operator can be optimized by initially computing the weights: +# ```math +# \begin{equation} +# \tilde{\mathbf{W}} = \mathbf{W}^*\mathbf{W} +# \end{equation} +# ``` +# and then applying the following each iteration: +# ```math +# \begin{equation} +# \mathbf{A}^*\tilde{\mathbf{W}}\mathbf{A} +# \end{equation} +# ``` + +# The optimized normal operator can then be passed to the solver: +solver = createLinearSolver(CGNR, WA; AHA = AHA, reg = L2Regularization(0.0001), iterations=32); +x_approx2 = solve!(solver, weights .* b) +# Of course it is also possible to optimize a normal operator with other means and pass it to the solver via the AHA keyword argument. + +# It is also possible to only supply the normal operator to these solvers, however on then needs to supply $\mathbf{A^*b}$ intead of $\mathbf{b}$. \ No newline at end of file diff --git a/previews/PR92/literate/howto/plug-and-play.jl b/previews/PR92/literate/howto/plug-and-play.jl new file mode 100644 index 00000000..096707e3 --- /dev/null +++ b/previews/PR92/literate/howto/plug-and-play.jl @@ -0,0 +1,30 @@ +# # Plug-and-Play Regularization +# A group of regularization terms that can not be directly written down as function are learned plug-and-play (PnP) priors. +# These are terms based on deep neural networks, which are trainted to implement the proximal map corresponding to the regularization term. +# Such a PnP prior can be used in the same way as any other regularization term. + +# The following example shows how to use a PnP prior in the context of the `Kaczmarz` solver. +using RegularizedLeastSquares +A = randn(32, 16) +x = randn(16) +b = A*x; + +# For the documentation we will just use the identity function as a placeholder for the PnP prior. +model = identity + +# In practice, you would replace this with a neural network: +# ```julia +# using Flux +# model = Flux.loadmodel!(model, ...) +# ``` + +# The model can then be used together with the `PnPRegularization` term: +reg = PnPRegularization(1.0; model = model, shape = [16]); + +# Since models often expect a specific input range, we can use the `MinMaxTransform` to normalize the input: +reg = PnPRegularization(1.0; model = model, shape = [16], input_transform = RegularizedLeastSquares.MinMaxTransform); +# Custom input transforms can be implemented by passing something callable as the `input_transform` keyword argument. For more details see the PnPRegularization documentation. + +# The regularization term can then be used in the solver: +solver = createLinearSolver(Kaczmarz, A; reg = reg, iterations = 32) +x_approx = solve!(solver, b) diff --git a/previews/PR92/literate/howto/weighting.jl b/previews/PR92/literate/howto/weighting.jl new file mode 100644 index 00000000..54fc6ebc --- /dev/null +++ b/previews/PR92/literate/howto/weighting.jl @@ -0,0 +1,37 @@ +# # Weighting +# Often time one wants to solve a weighted least squares problem of the form: +# ```math +# \begin{equation} +# \underset{\mathbf{x}}{argmin} \frac{1}{2}\vert\vert \mathbf{A}\mathbf{x}-\mathbf{b} \vert\vert^2_{\mathbf{W}} + \mathbf{R(x)} . +# \end{equation} +# ``` +# where $\mathbf{W}$ is a symmetric, positive weighting matrix and $\vert\vert\mathbf{y}\vert\vert^2_\mathbf{W}$ denotes the weighted Euclidean norm. +# An example of such a weighting matrix is a noise whitening matrix. Another example could be a scaling of the matrix rows by the reciprocal of their row energy. + +# In the following, we will solve a weighted least squares problem of the form: +# ```math +# \begin{equation} +# \underset{\mathbf{x}}{argmin} \frac{1}{2}\vert\vert \mathbf{A}\mathbf{x}-\mathbf{b} \vert\vert_\mathbf{W}^2 + \vert\vert\mathbf{x}\vert\vert^2_2 . +# \end{equation} +# ``` +using RegularizedLeastSquares, LinearOperatorCollection, LinearAlgebra +A = rand(32, 16) +x = rand(16) +b = A*x; + +# As a weighting matrix, we will use the reciprocal of the row energy of the matrix A. +weights = map(row -> 1/rownorm²(A, row), 1:size(A, 1)); + +# First, let us solve the problem with matrices we manually weighted. +WA = diagm(weights) * A +solver = createLinearSolver(Kaczmarz, WA; reg = L2Regularization(0.0001), iterations=10) +x_approx = solve!(solver, weights .* b); + +# The operator A is not always a dense matrix and the product between the operator and the weighting matrix is not always efficient or possible. +# The package [LinearOperatorCollection.jl](https://github.com/JuliaImageRecon/LinearOperatorCollection.jl) provides a matrix-free implementation of a diagonal weighting matrix, as well as a matrix free product between two matrices. +# This weighted operator has efficient implementations of the normal operator and also for the row-action operations of the Kaczmarz solver. +W = WeightingOp(weights) +P = ProdOp(W, A) +solver = createLinearSolver(Kaczmarz, P; reg = L2Regularization(0.0001), iterations=10) +x_approx2 = solve!(solver, W * b) +isapprox(x_approx, x_approx2) \ No newline at end of file diff --git a/previews/PR92/objects.inv b/previews/PR92/objects.inv new file mode 100644 index 0000000000000000000000000000000000000000..121d3d4fbba1d0fcc41e9b4582f71eb6aa329a07 GIT binary patch literal 1608 zcmV-O2DkYmAX9K?X>NERX>N99Zgg*Qc_4OWa&u{KZXhxWBOp+6Z)#;@bUGkXWoLD4 zVRC7DWn@fcVRLj-adlyGWpgfSYziYFR%LQ?X>V>iATTa5HZC&?BOp|0Wgv28ZDDC{ zWMy(7Z)PBLXlZjGW@&6?AZc?TV{dJ6a%FRKWn>_Ab7^j8AbM}{b+Bsy#lZ&iat|yYMF_q@1fk{}R&A$R*Vmp&3>pS$BvLN_RfTAd>mxNjZ z`-gA8UC?gekQcV+t?Id-kt3 z(X|vHOA1sL?9h)WVT?E%Mgk^Qw;O40FwrcRAxp}1rect6c<6^)ilg0Y2V1or(wGpU zbuJvS(ECOxE2~RICbHy1#iPi>0p@NPFq4vP>|yG@CBJnJX*ZbYzVf?dW*2_vq4}M- z+a1^DwYJDaU$e)CrXZv~_BLT8q#t-+BB+@RBWzSKTxDGtrP4)=xLB_g>71$ZKo~acN=l3i>U%JkvI!z1giY- zq{)qSt7%zWVnH3X_SGI;z4;Y&lPGKS!dt?4!*L^F(4Hug8r7V91EmXmy9cD2X|cl3 z%KE7tOUB7rW61_HqZrH%;^X?HmlASq*2@_=B?rsi5hwQgoLtHpg@(}=hPOmf|Fi4c z`+X3Vm{MqHa%{3_luqY{RT}OOJh1wJpE3VH(9YCpzN8ak@BA~O`&dz{ zmE_HDto3xyn$_L)a^D$155*n}8_vKBBwkdT)INilVxK@u5j|qa60Rs#SMr=-9z#1) zq;>jfxi1fs$)u(8-9@}Zx>G#E850^n?)G6iU#RbL$0beI{?d#sp@~Drd@-U8gZ`36 zO#vKhK@Lq5L)j&|4o(wwOHj9-E=o1(k{e_`qPE9czkdp?wHxTy%Dq^tskj-pPHk(7 z0|w2=Cr&NkC+~v4Mg-}i)$4Qva0NCP>lpa=$MgZ_Vr+Ano~o6f!VHx3E_5`8x)~XN zxNI{s{_v^A-`h~;`1r{G`?nTzg|UVTeLTAwUtTr>Z?V-Lf67Ur7iZx=rZ)+s;5RLj z{vHfKID!sn;|GsOj$F8m_jSvHnjvsEgOyMsHp#BvHTOE{${IkAtzsa92P;kTF)q?x~p*79j7vkh92TQyS z7adN@aO&M0+~E17LwI4}r~!O&K3o}yj~zWYUVLnw29EgDI+!o>zQdx+iV3*y*_STi z$sK&CD^4;`4>88dMRqZn_l A*x, hcat, xs)\nsolver = createLinearSolver(CGNR, A; iterations=32)","category":"page"},{"location":"generated/howto/multi_threading/","page":"Multi-Threading","title":"Multi-Threading","text":"We can then simply pass the measurement matrix to the solver. The result will be the same as if we passed each colument of B seperately:","category":"page"},{"location":"generated/howto/multi_threading/","page":"Multi-Threading","title":"Multi-Threading","text":"x_approx = solve!(solver, B)\nsize(x_approx)","category":"page"},{"location":"generated/howto/multi_threading/","page":"Multi-Threading","title":"Multi-Threading","text":"The previous solve! call was still executed sequentially. To execute it in parallel, we have to specify a multi-threaded scheduler as a keyword-argument of the solve! call. RegularizedLeastSquares.jl provides a MultiThreadingState scheduler that can be used for this purpose. This scheduler is based on the Threads.@threads macro:","category":"page"},{"location":"generated/howto/multi_threading/","page":"Multi-Threading","title":"Multi-Threading","text":"x_multi = solve!(solver, B; scheduler = MultiThreadingState)\nx_approx == x_multi","category":"page"},{"location":"generated/howto/multi_threading/#Custom-Scheduling","page":"Multi-Threading","title":"Custom Scheduling","text":"","category":"section"},{"location":"generated/howto/multi_threading/","page":"Multi-Threading","title":"Multi-Threading","text":"It is possible to implement custom scheduling. The following code shows how to implement this for the Threads.@spawn macro. Usually one this to implement multi-threading with a package such as FLoop.jl or ThreadPools.jl for thread pinning:","category":"page"},{"location":"generated/howto/multi_threading/","page":"Multi-Threading","title":"Multi-Threading","text":"Since most solver have conv. criteria, they can finish at different iteration numbers, which we track this information with flags.","category":"page"},{"location":"generated/howto/multi_threading/","page":"Multi-Threading","title":"Multi-Threading","text":" mutable struct SpawnState{S, ST <: AbstractSolverState{S}} <: RegularizedLeastSquares.AbstractMatrixSolverState{S}\n states::Vector{ST}\n active::Vector{Bool}\n SpawnState(states::Vector{ST}) where {S, ST <: AbstractSolverState{S}} = new{S, ST}(states, fill(true, length(states)))\n end","category":"page"},{"location":"generated/howto/multi_threading/","page":"Multi-Threading","title":"Multi-Threading","text":"To hook into the existing init! code we only have to supply a method that gets a copyable \"vector\" state. This will invoke our SpawnState constructor with copies of the given state.","category":"page"},{"location":"generated/howto/multi_threading/","page":"Multi-Threading","title":"Multi-Threading","text":" prepareMultiStates(solver::AbstractLinearSolver, state::SpawnState, b::AbstractMatrix) = prepareMultiStates(solver, first(state.states), b);\nnothing #hide","category":"page"},{"location":"generated/howto/multi_threading/","page":"Multi-Threading","title":"Multi-Threading","text":"We specialise the iterate function which is called with the idx of still active states","category":"page"},{"location":"generated/howto/multi_threading/","page":"Multi-Threading","title":"Multi-Threading","text":"function Base.iterate(solver::AbstractLinearSolver, state::SpawnState, activeIdx)\n @sync Threads.@spawn for i in activeIdx\n res = iterate(solver, state.states[i])\n if isnothing(res)\n state.active[i] = false\n end\n end\n return state.active, state\nend","category":"page"},{"location":"generated/howto/multi_threading/","page":"Multi-Threading","title":"Multi-Threading","text":"Now we can simply use the SpawnState scheduler in the solve! call:","category":"page"},{"location":"generated/howto/multi_threading/","page":"Multi-Threading","title":"Multi-Threading","text":"x_custom = solve!(solver, B; scheduler = SpawnState)\nx_approx == x_multi","category":"page"},{"location":"generated/howto/multi_threading/","page":"Multi-Threading","title":"Multi-Threading","text":"","category":"page"},{"location":"generated/howto/multi_threading/","page":"Multi-Threading","title":"Multi-Threading","text":"This page was generated using Literate.jl.","category":"page"},{"location":"solvers/#Solvers","page":"Solvers","title":"Solvers","text":"","category":"section"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"RegularizedLeastSquares.jl provides a variety of solvers, which are used in fields such as MPI and MRI. The following is a non-exhaustive list of the implemented solvers:","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"Kaczmarz algorithm (Kaczmarz, also called Algebraic reconstruction technique)\nConjugate Gradients Normal Residual method (CGNR)\nFast Iterative Shrinkage Thresholding Algorithm (FISTA)\nAlternating Direction of Multipliers Method (ADMM)","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"The solvers are organized in a type-hierarchy and inherit from:","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"abstract type AbstractLinearSolver","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"The type hierarchy is further differentiated into solver categories such as AbstractRowAtionSolver, AbstractPrimalDualSolver or AbstractProximalGradientSolver. A list of all available solvers can be returned by the linearSolverList function.","category":"page"},{"location":"solvers/#Solver-Construction","page":"Solvers","title":"Solver Construction","text":"","category":"section"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"To create a solver, one can invoke the method createLinearSolver as in","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"solver = createLinearSolver(CGNR, A; reg=reg, kwargs...)","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"Here A denotes the operator and reg are the Regularization terms to be used by the solver. All further solver parameters can be passed as keyword arguments and are solver specific. To make things more compact, it can be usefull to collect all parameters in a Dict{Symbol,Any}. In this way, the code snippet above can be written as","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"params=Dict{Symbol,Any}()\nparams[:reg] = ...\n...\n\nsolver = createLinearSolver(CGNR, A; params...)","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"This notation can be convenient when a large number of parameters are set manually.","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"It is also possible to construct a solver directly with its specific keyword arguments:","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"solver = CGNR(A, reg = reg, ...)","category":"page"},{"location":"solvers/#Solver-Usage","page":"Solvers","title":"Solver Usage","text":"","category":"section"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"Once constructed, a solver can be used to approximate a solution to a given measurement vector:","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"x_approx = solve!(solver, b; kwargs...)","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"The keyword arguments can be used to supply an inital solution x0, one or more callbacks to interact and monitor the solvers state and more. See the How-To and the API for more information.","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"It is also possible to explicitly invoke the solvers iterations using Julias iterate interface:","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"init!(solver, b; kwargs...)\nfor (iteration, x_approx) in enumerate(solver)\n println(\"Iteration $iteration\")\nend","category":"page"},{"location":"solvers/#Solver-Internals","page":"Solvers","title":"Solver Internals","text":"","category":"section"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"The fields of a solver can be divided into two groups. The first group are intended to be immutable fields that do not change during iterations, the second group are mutable fields that do change. Examples of the first group are the operator itself and examples of the second group are the current solution or the number of the current iteration.","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"The second group is usually encapsulated in its own state struct:","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"mutable struct Solver{matT, ...}\n A::matT\n # Other \"static\" fields\n state::AbstractSolverState{<:Solver}\nend\n\nmutable struct SolverState{T, tempT} <: AbstractSolverState{Solver}\n x::tempT\n rho::T\n # ...\n iteration::Int64\nend","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"States are subtypes of the parametric AbstractSolverState{S} type. The state fields of solvers can be exchanged with different state belonging to the correct solver S. This means that the states can be used to realize custom variants of an existing solver: ```julia mutable struct VariantState{T, tempT} <: AbstractSolverState{Solver} x::tempT other::tempT","category":"page"},{"location":"solvers/#...","page":"Solvers","title":"...","text":"","category":"section"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"iteration::Int64 end","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"SolverVariant(A; kwargs...) = Solver(A, VariantState(kwargs...))","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"function iterate(solver::Solver, state::VarianteState)","category":"page"},{"location":"solvers/#Custom-iteration","page":"Solvers","title":"Custom iteration","text":"","category":"section"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"end","category":"page"},{"location":"generated/howto/weighting/","page":"Weighting","title":"Weighting","text":"EditURL = \"../../literate/howto/weighting.jl\"","category":"page"},{"location":"generated/howto/weighting/#Weighting","page":"Weighting","title":"Weighting","text":"","category":"section"},{"location":"generated/howto/weighting/","page":"Weighting","title":"Weighting","text":"Often time one wants to solve a weighted least squares problem of the form:","category":"page"},{"location":"generated/howto/weighting/","page":"Weighting","title":"Weighting","text":"beginequation\n undersetmathbfxargmin frac12vertvert mathbfAmathbfx-mathbfb vertvert^2_mathbfW + mathbfR(x) \nendequation","category":"page"},{"location":"generated/howto/weighting/","page":"Weighting","title":"Weighting","text":"where mathbfW is a symmetric, positive weighting matrix and vertvertmathbfyvertvert^2_mathbfW denotes the weighted Euclidean norm. An example of such a weighting matrix is a noise whitening matrix. Another example could be a scaling of the matrix rows by the reciprocal of their row energy.","category":"page"},{"location":"generated/howto/weighting/","page":"Weighting","title":"Weighting","text":"In the following, we will solve a weighted least squares problem of the form:","category":"page"},{"location":"generated/howto/weighting/","page":"Weighting","title":"Weighting","text":"beginequation\n undersetmathbfxargmin frac12vertvert mathbfAmathbfx-mathbfb vertvert_mathbfW^2 + vertvertmathbfxvertvert^2_2 \nendequation","category":"page"},{"location":"generated/howto/weighting/","page":"Weighting","title":"Weighting","text":"using RegularizedLeastSquares, LinearOperatorCollection, LinearAlgebra\nA = rand(32, 16)\nx = rand(16)\nb = A*x;\nnothing #hide","category":"page"},{"location":"generated/howto/weighting/","page":"Weighting","title":"Weighting","text":"As a weighting matrix, we will use the reciprocal of the row energy of the matrix A.","category":"page"},{"location":"generated/howto/weighting/","page":"Weighting","title":"Weighting","text":"weights = map(row -> 1/rownorm²(A, row), 1:size(A, 1));\nnothing #hide","category":"page"},{"location":"generated/howto/weighting/","page":"Weighting","title":"Weighting","text":"First, let us solve the problem with matrices we manually weighted.","category":"page"},{"location":"generated/howto/weighting/","page":"Weighting","title":"Weighting","text":"WA = diagm(weights) * A\nsolver = createLinearSolver(Kaczmarz, WA; reg = L2Regularization(0.0001), iterations=10)\nx_approx = solve!(solver, weights .* b);\nnothing #hide","category":"page"},{"location":"generated/howto/weighting/","page":"Weighting","title":"Weighting","text":"The operator A is not always a dense matrix and the product between the operator and the weighting matrix is not always efficient or possible. The package LinearOperatorCollection.jl provides a matrix-free implementation of a diagonal weighting matrix, as well as a matrix free product between two matrices. This weighted operator has efficient implementations of the normal operator and also for the row-action operations of the Kaczmarz solver.","category":"page"},{"location":"generated/howto/weighting/","page":"Weighting","title":"Weighting","text":"W = WeightingOp(weights)\nP = ProdOp(W, A)\nsolver = createLinearSolver(Kaczmarz, P; reg = L2Regularization(0.0001), iterations=10)\nx_approx2 = solve!(solver, W * b)\nisapprox(x_approx, x_approx2)","category":"page"},{"location":"generated/howto/weighting/","page":"Weighting","title":"Weighting","text":"","category":"page"},{"location":"generated/howto/weighting/","page":"Weighting","title":"Weighting","text":"This page was generated using Literate.jl.","category":"page"},{"location":"API/regularization/#API-for-Regularizers","page":"Regularization Terms","title":"API for Regularizers","text":"","category":"section"},{"location":"API/regularization/","page":"Regularization Terms","title":"Regularization Terms","text":"This page contains documentation of the public API of the RegularizedLeastSquares. In the Julia REPL one can access this documentation by entering the help mode with ?","category":"page"},{"location":"API/regularization/","page":"Regularization Terms","title":"Regularization Terms","text":"RegularizedLeastSquares.L1Regularization\nRegularizedLeastSquares.L2Regularization\nRegularizedLeastSquares.L21Regularization\nRegularizedLeastSquares.LLRRegularization\nRegularizedLeastSquares.NuclearRegularization\nRegularizedLeastSquares.TVRegularization","category":"page"},{"location":"API/regularization/#RegularizedLeastSquares.L1Regularization","page":"Regularization Terms","title":"RegularizedLeastSquares.L1Regularization","text":"L1Regularization\n\nRegularization term implementing the proximal map for the Lasso problem.\n\n\n\n\n\n","category":"type"},{"location":"API/regularization/#RegularizedLeastSquares.L2Regularization","page":"Regularization Terms","title":"RegularizedLeastSquares.L2Regularization","text":"L2Regularization\n\nRegularization term implementing the proximal map for Tikhonov regularization.\n\n\n\n\n\n","category":"type"},{"location":"API/regularization/#RegularizedLeastSquares.L21Regularization","page":"Regularization Terms","title":"RegularizedLeastSquares.L21Regularization","text":"L21Regularization\n\nRegularization term implementing the proximal map for group-soft-thresholding.\n\nArguments\n\nλ - regularization paramter\n\nKeywords\n\nslices=1 - number of elements per group\n\n\n\n\n\n","category":"type"},{"location":"API/regularization/#RegularizedLeastSquares.LLRRegularization","page":"Regularization Terms","title":"RegularizedLeastSquares.LLRRegularization","text":"LLRRegularization\n\nRegularization term implementing the proximal map for locally low rank (LLR) regularization using singular-value-thresholding.\n\nArguments\n\nλ - regularization paramter\n\nKeywords\n\nshape::Tuple{Int} - dimensions of the image\nblockSize::Tuple{Int}=(2,2) - size of patches to perform singular value thresholding on\nrandshift::Bool=true - randomly shifts the patches to ensure translation invariance\nfullyOverlapping::Bool=false - choose between fully overlapping block or non-overlapping blocks\n\n\n\n\n\n","category":"type"},{"location":"API/regularization/#RegularizedLeastSquares.NuclearRegularization","page":"Regularization Terms","title":"RegularizedLeastSquares.NuclearRegularization","text":"NuclearRegularization\n\nRegularization term implementing the proximal map for singular value soft-thresholding.\n\nArguments:\n\nλ - regularization paramter\n\nKeywords\n\nsvtShape::NTuple - size of the underlying matrix\n\n\n\n\n\n","category":"type"},{"location":"API/regularization/#RegularizedLeastSquares.TVRegularization","page":"Regularization Terms","title":"RegularizedLeastSquares.TVRegularization","text":"TVRegularization\n\nRegularization term implementing the proximal map for TV regularization. Calculated with the Condat algorithm if the TV is calculated only along one real-valued dimension and with the Fast Gradient Projection algorithm otherwise.\n\nReference for the Condat algorithm: https://lcondat.github.io/publis/Condat-fast_TV-SPL-2013.pdf\n\nReference for the FGP algorithm: A. Beck and T. Teboulle, \"Fast Gradient-Based Algorithms for Constrained Total Variation Image Denoising and Deblurring Problems\", IEEE Trans. Image Process. 18(11), 2009\n\nArguments\n\nλ::T - regularization parameter\n\nKeywords\n\nshape::NTuple - size of the underlying image\ndims - Dimension to perform the TV along. If Integer, the Condat algorithm is called, and the FDG algorithm otherwise.\niterationsTV=20 - number of FGP iterations\n\n\n\n\n\n","category":"type"},{"location":"API/regularization/#Projection-Regularization","page":"Regularization Terms","title":"Projection Regularization","text":"","category":"section"},{"location":"API/regularization/","page":"Regularization Terms","title":"Regularization Terms","text":"RegularizedLeastSquares.PositiveRegularization\nRegularizedLeastSquares.RealRegularization","category":"page"},{"location":"API/regularization/#RegularizedLeastSquares.PositiveRegularization","page":"Regularization Terms","title":"RegularizedLeastSquares.PositiveRegularization","text":"PositiveRegularization\n\nRegularization term implementing a projection onto positive and real numbers.\n\n\n\n\n\n","category":"type"},{"location":"API/regularization/#RegularizedLeastSquares.RealRegularization","page":"Regularization Terms","title":"RegularizedLeastSquares.RealRegularization","text":"RealRegularization\n\nRegularization term implementing a projection onto real numbers.\n\n\n\n\n\n","category":"type"},{"location":"API/regularization/#Nested-Regularization","page":"Regularization Terms","title":"Nested Regularization","text":"","category":"section"},{"location":"API/regularization/","page":"Regularization Terms","title":"Regularization Terms","text":"RegularizedLeastSquares.innerreg(::AbstractNestedRegularization)\nRegularizedLeastSquares.sink(::AbstractNestedRegularization)\nRegularizedLeastSquares.sinktype(::AbstractNestedRegularization)","category":"page"},{"location":"API/regularization/#RegularizedLeastSquares.innerreg-Tuple{AbstractNestedRegularization}","page":"Regularization Terms","title":"RegularizedLeastSquares.innerreg","text":"innerreg(reg::AbstractNestedRegularization)\n\nreturn the inner regularization term of reg. Nested regularization terms also implement the iteration interface.\n\n\n\n\n\n","category":"method"},{"location":"API/regularization/#RegularizedLeastSquares.sink-Tuple{AbstractNestedRegularization}","page":"Regularization Terms","title":"RegularizedLeastSquares.sink","text":"sink(reg::AbstractNestedRegularization)\n\nreturn the innermost regularization term.\n\n\n\n\n\n","category":"method"},{"location":"API/regularization/#RegularizedLeastSquares.sinktype-Tuple{AbstractNestedRegularization}","page":"Regularization Terms","title":"RegularizedLeastSquares.sinktype","text":"sinktype(reg::AbstractNestedRegularization)\n\nreturn the type of the innermost regularization term.\n\nSee also sink.\n\n\n\n\n\n","category":"method"},{"location":"API/regularization/#Scaled-Regularization","page":"Regularization Terms","title":"Scaled Regularization","text":"","category":"section"},{"location":"API/regularization/","page":"Regularization Terms","title":"Regularization Terms","text":"RegularizedLeastSquares.AbstractScaledRegularization\nRegularizedLeastSquares.scalefactor\nRegularizedLeastSquares.NormalizedRegularization\nRegularizedLeastSquares.NoNormalization\nRegularizedLeastSquares.MeasurementBasedNormalization\nRegularizedLeastSquares.SystemMatrixBasedNormalization\nRegularizedLeastSquares.FixedParameterRegularization","category":"page"},{"location":"API/regularization/#RegularizedLeastSquares.AbstractScaledRegularization","page":"Regularization Terms","title":"RegularizedLeastSquares.AbstractScaledRegularization","text":"AbstractScaledRegularization\n\nNested regularization term that applies a scalefactor to the regularization parameter λ of its inner term.\n\nSee also scalefactor, λ, innerreg.\n\n\n\n\n\n","category":"type"},{"location":"API/regularization/#RegularizedLeastSquares.scalefactor","page":"Regularization Terms","title":"RegularizedLeastSquares.scalefactor","text":"scalescalefactor(reg::AbstractScaledRegularization)\n\nreturn the scaling scalefactor for λ\n\n\n\n\n\n","category":"function"},{"location":"API/regularization/#RegularizedLeastSquares.NormalizedRegularization","page":"Regularization Terms","title":"RegularizedLeastSquares.NormalizedRegularization","text":"NormalizedRegularization\n\nNested regularization term that scales λ according to normalization scheme. This term is commonly applied by a solver based on a given normalization keyword\n\n#See also NoNormalization, MeasurementBasedNormalization, SystemMatrixBasedNormalization.\n\n\n\n\n\n","category":"type"},{"location":"API/regularization/#RegularizedLeastSquares.NoNormalization","page":"Regularization Terms","title":"RegularizedLeastSquares.NoNormalization","text":"NoNormalization\n\nNo normalization to λ is applied.\n\n\n\n\n\n","category":"type"},{"location":"API/regularization/#RegularizedLeastSquares.MeasurementBasedNormalization","page":"Regularization Terms","title":"RegularizedLeastSquares.MeasurementBasedNormalization","text":"MeasurementBasedNormalization\n\nλ is normalized by the 1-norm of b divided by its length.\n\n\n\n\n\n","category":"type"},{"location":"API/regularization/#RegularizedLeastSquares.SystemMatrixBasedNormalization","page":"Regularization Terms","title":"RegularizedLeastSquares.SystemMatrixBasedNormalization","text":"SystemMatrixBasedNormalization\n\nλ is normalized by the energy of the system matrix rows.\n\n\n\n\n\n","category":"type"},{"location":"API/regularization/#RegularizedLeastSquares.FixedParameterRegularization","page":"Regularization Terms","title":"RegularizedLeastSquares.FixedParameterRegularization","text":"FixedParameterRegularization\n\nNested regularization term that discards any λ passed to it and instead uses λ from its inner regularization term. This can be used to selectively disallow normalization.\n\n\n\n\n\n","category":"type"},{"location":"API/regularization/#Misc.-Nested-Regularization","page":"Regularization Terms","title":"Misc. Nested Regularization","text":"","category":"section"},{"location":"API/regularization/","page":"Regularization Terms","title":"Regularization Terms","text":"RegularizedLeastSquares.MaskedRegularization\nRegularizedLeastSquares.TransformedRegularization\nRegularizedLeastSquares.PlugAndPlayRegularization","category":"page"},{"location":"API/regularization/#RegularizedLeastSquares.MaskedRegularization","page":"Regularization Terms","title":"RegularizedLeastSquares.MaskedRegularization","text":"MaskedRegularization\n\nNested regularization term that only applies prox! and norm to elements of x for which the mask is true.\n\nExamples\n\njulia> positive = PositiveRegularization();\n\njulia> masked = MaskedRegularization(reg, [true, false, true, false]);\n\njulia> prox!(masked, fill(-1, 4))\n4-element Vector{Float64}:\n 0.0\n -1.0\n 0.0\n -1.0\n\n\n\n\n\n","category":"type"},{"location":"API/regularization/#RegularizedLeastSquares.TransformedRegularization","page":"Regularization Terms","title":"RegularizedLeastSquares.TransformedRegularization","text":"TransformedRegularization(reg, trafo)\n\nNested regularization term that applies prox! or norm on z = trafo * x and returns (inplace) x = adjoint(trafo) * z.\n\nExample\n\njulia> core = L1Regularization(0.8)\nL1Regularization{Float64}(0.8)\n\njulia> wop = WaveletOp(Float32, shape = (32,32));\n\njulia> reg = TransformedRegularization(core, wop);\n\njulia> prox!(reg, randn(32*32)); # Apply soft-thresholding in Wavelet domain\n\n\n\n\n\n","category":"type"},{"location":"API/regularization/#RegularizedLeastSquares.PlugAndPlayRegularization","page":"Regularization Terms","title":"RegularizedLeastSquares.PlugAndPlayRegularization","text":" PlugAndPlayRegularization\n\nRegularization term implementing a given plug-and-play proximal mapping. The actual regularization term is indirectly defined by the learned proximal mapping and as such there is no norm implemented.\n\nArguments\n\nλ - regularization paramter\n\nKeywords\n\nmodel - model applied to the image\nshape - dimensions of the image\ninput_transform - transform of image before model\n\n\n\n\n\n","category":"type"},{"location":"API/regularization/#Miscellaneous-Functions","page":"Regularization Terms","title":"Miscellaneous Functions","text":"","category":"section"},{"location":"API/regularization/","page":"Regularization Terms","title":"Regularization Terms","text":"RegularizedLeastSquares.prox!(::AbstractParameterizedRegularization, ::AbstractArray)\nRegularizedLeastSquares.prox!(::Type{<:AbstractParameterizedRegularization}, ::Any, ::Any)\nRegularizedLeastSquares.norm(::AbstractParameterizedRegularization, ::AbstractArray)\nRegularizedLeastSquares.λ(::AbstractParameterizedRegularization)\nRegularizedLeastSquares.norm(::Type{<:AbstractParameterizedRegularization}, ::Any, ::Any)","category":"page"},{"location":"API/regularization/#RegularizedLeastSquares.prox!-Tuple{AbstractParameterizedRegularization, AbstractArray}","page":"Regularization Terms","title":"RegularizedLeastSquares.prox!","text":"prox!(reg::AbstractParameterizedRegularization, x)\n\nperform the proximal mapping defined by reg on x. Uses the regularization parameter defined for reg.\n\n\n\n\n\n","category":"method"},{"location":"API/regularization/#RegularizedLeastSquares.prox!-Tuple{Type{<:AbstractParameterizedRegularization}, Any, Any}","page":"Regularization Terms","title":"RegularizedLeastSquares.prox!","text":"prox!(regType::Type{<:AbstractParameterizedRegularization}, x, λ; kwargs...)\n\nconstruct a regularization term of type regType with given λ and kwargs and apply its prox! on x\n\n\n\n\n\n","category":"method"},{"location":"API/regularization/#LinearAlgebra.norm-Tuple{AbstractParameterizedRegularization, AbstractArray}","page":"Regularization Terms","title":"LinearAlgebra.norm","text":"norm(reg::AbstractParameterizedRegularization, x)\n\nreturns the value of the reg regularization term on x. Uses the regularization parameter defined for reg.\n\n\n\n\n\n","category":"method"},{"location":"API/regularization/#RegularizedLeastSquares.λ-Tuple{AbstractParameterizedRegularization}","page":"Regularization Terms","title":"RegularizedLeastSquares.λ","text":"λ(reg::AbstractParameterizedRegularization)\n\nreturn the regularization parameter λ of reg\n\n\n\n\n\n","category":"method"},{"location":"API/regularization/#LinearAlgebra.norm-Tuple{Type{<:AbstractParameterizedRegularization}, Any, Any}","page":"Regularization Terms","title":"LinearAlgebra.norm","text":"norm(regType::Type{<:AbstractParameterizedRegularization}, x, λ; kwargs...)\n\nconstruct a regularization term of type regType with given λ and kwargs and apply its norm on x\n\n\n\n\n\n","category":"method"},{"location":"generated/howto/callbacks/","page":"Callbacks","title":"Callbacks","text":"EditURL = \"../../literate/howto/callbacks.jl\"","category":"page"},{"location":"generated/howto/callbacks/#Callbacks","page":"Callbacks","title":"Callbacks","text":"","category":"section"},{"location":"generated/howto/callbacks/","page":"Callbacks","title":"Callbacks","text":"For certain optimization it is important to monitor the internal state of the solver. RegularizedLeastSquares.jl provides a callback mechanism to allow developres to access this state after each iteration. The package provides a variety of default callbacks, which for example store the solution after each iteration. More information can be found in the API reference for the solve! function.","category":"page"},{"location":"generated/howto/callbacks/","page":"Callbacks","title":"Callbacks","text":"In this example we will revist the compressed sensing compressed sensing example and implement a custom callback using the do-syntax of the solve! function. This allows us to implement our callback inline and access the solver state after each iteration.","category":"page"},{"location":"generated/howto/callbacks/","page":"Callbacks","title":"Callbacks","text":"We first recreate the operator A and the measurement vector b:","category":"page"},{"location":"generated/howto/callbacks/","page":"Callbacks","title":"Callbacks","text":"using ImagePhantoms, ImageGeoms\nN = 256\nimage = shepp_logan(N, SheppLoganToft())\nsize(image)\nusing Random, LinearOperatorCollection\nrandomIndices = shuffle(eachindex(image))\nsampledIndices = sort(randomIndices[1:div(end, 3)])\nA = SamplingOp(eltype(image), pattern = sampledIndices , shape = size(image));\nb = A*vec(image);\nnothing #hide","category":"page"},{"location":"generated/howto/callbacks/","page":"Callbacks","title":"Callbacks","text":"Next we prepare our visualization helper function:","category":"page"},{"location":"generated/howto/callbacks/","page":"Callbacks","title":"Callbacks","text":"using CairoMakie\nfunction plot_image(figPos, img; title = \"\", width = 150, height = 150, clim = extrema(img))\n ax = CairoMakie.Axis(figPos; yreversed=true, title, width, height)\n hidedecorations!(ax)\n heatmap!(ax, img, colorrange = clim)\nend","category":"page"},{"location":"generated/howto/callbacks/","page":"Callbacks","title":"Callbacks","text":"Now we construct the solver with the TV regularization term:","category":"page"},{"location":"generated/howto/callbacks/","page":"Callbacks","title":"Callbacks","text":"using RegularizedLeastSquares\nreg = TVRegularization(0.01; shape=size(image));\nsolver = createLinearSolver(FISTA, A; reg=reg, iterations=20);\nnothing #hide","category":"page"},{"location":"generated/howto/callbacks/","page":"Callbacks","title":"Callbacks","text":"We will now implement a callback that plots the solution every four iteration:","category":"page"},{"location":"generated/howto/callbacks/","page":"Callbacks","title":"Callbacks","text":"fig = Figure()\nidx = 1\nsolve!(solver, b) do solver, iteration\n if iteration % 4 == 0\n img_approx = copy(solversolution(solver))\n img_approx = reshape(img_approx, size(image))\n plot_image(fig[div(idx -1, 3) + 1, mod1(idx, 3)], img_approx, clim = extrema(image), title = \"$iteration\")\n global idx += 1\n end\nend","category":"page"},{"location":"generated/howto/callbacks/","page":"Callbacks","title":"Callbacks","text":"In the callback we have to copy the solution, as the solver will update it in place. As is explained in the solver section, each features fields that are intended to be immutable during a solve! call and a state that is modified in each iteration. Depending on the solvers parameters and the measurement input, the state can differ in its fields and their type. Ideally, one tries to avoid accessing the state directly and uses the provided functions to access the state.","category":"page"},{"location":"generated/howto/callbacks/","page":"Callbacks","title":"Callbacks","text":"The resulting figure shows the reconstructed image after 0, 4, 8, 12, 16 and 20 iterations:","category":"page"},{"location":"generated/howto/callbacks/","page":"Callbacks","title":"Callbacks","text":"resize_to_layout!(fig)\nfig","category":"page"},{"location":"generated/howto/callbacks/","page":"Callbacks","title":"Callbacks","text":"","category":"page"},{"location":"generated/howto/callbacks/","page":"Callbacks","title":"Callbacks","text":"This page was generated using Literate.jl.","category":"page"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"EditURL = \"../../literate/howto/normal_operator.jl\"","category":"page"},{"location":"generated/howto/normal_operator/#Normal-Operator","page":"Normal Operator","title":"Normal Operator","text":"","category":"section"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"Many solvers in RegularizedLeastSquares.jl are based on the normal operator mathbfA^*mathbfA of the linear operator mathbfA.","category":"page"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"Solvers such as ADMM, FISTA and POGM generally solve optimization problems of the form:","category":"page"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"beginequation\n undersetmathbfxargmin mathbff(x)+ mathbfR(x)\nendequation","category":"page"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"and require the gradient of the function mathbff(x). In this package we specialise the function mathbff(x) to the least squares norm:","category":"page"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"beginequation\n mathbff(x) = frac12vertvert mathbfAmathbfx-mathbfb vertvert^2_2\nendequation","category":"page"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"The gradient of this function is:","category":"page"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"beginequation\n nabla mathbff(x) = mathbfA^*(mathbfAmathbfx-mathbfb) = mathbfA^*mathbfAx - mathbfA^*mathbfb\nendequation","category":"page"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"Similarily, the conjugate gradient normal residual (CGNR) algorithm applies conjugate gradient algorithm to:","category":"page"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"beginequation\n mathbfA^*mathbfAmathbfx = mathbfA^*mathbfb\nendequation","category":"page"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"The normal operator can be passed directly to these solvers, otherwise it is computed internally.","category":"page"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"using RegularizedLeastSquares\nA = randn(32, 16)\nx = randn(16)\nb = A*x\n\nsolver = createLinearSolver(CGNR, A; AHA = adjoint(A) * A, reg = L2Regularization(0.0001), iterations=32);\nx_approx = solve!(solver, b)","category":"page"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"The normal operator can also be computed using the normalOperator function from LinearOperatorCollection.jl. This is useful if the normal operator is not directly available or shouldn't be stored in memory. This function is opinionated and attempts to optimize the resulting operator for iterative applications. Specifying a custom method for a custom operator allows one to control this optimization.","category":"page"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"An example of such an optimization is a matrix-free weighting of mathbfA as shown in the Weighting example:","category":"page"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"using LinearOperatorCollection\nweights = rand(32)\nWA = ProdOp(WeightingOp(weights), A)\nAHA = LinearOperatorCollection.normalOperator(WA)","category":"page"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"Without an optimization a matrix-free product would apply the following operator each iteration:","category":"page"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"beginequation\n (mathbfWA)^*mathbfWA = mathbfA^*mathbfW^*mathbfWmathbfA\nendequation","category":"page"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"This is not efficient and instead the normal operator can be optimized by initially computing the weights:","category":"page"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"beginequation\n tildemathbfW = mathbfW^*mathbfW\nendequation","category":"page"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"and then applying the following each iteration:","category":"page"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"beginequation\n mathbfA^*tildemathbfWmathbfA\nendequation","category":"page"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"The optimized normal operator can then be passed to the solver:","category":"page"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"solver = createLinearSolver(CGNR, WA; AHA = AHA, reg = L2Regularization(0.0001), iterations=32);\nx_approx2 = solve!(solver, weights .* b)","category":"page"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"Of course it is also possible to optimize a normal operator with other means and pass it to the solver via the AHA keyword argument.","category":"page"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"It is also possible to only supply the normal operator to these solvers, however on then needs to supply mathbfA^*b intead of mathbfb.","category":"page"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"","category":"page"},{"location":"generated/howto/normal_operator/","page":"Normal Operator","title":"Normal Operator","text":"This page was generated using Literate.jl.","category":"page"},{"location":"generated/howto/plug-and-play/","page":"Plug-and-Play Regularization","title":"Plug-and-Play Regularization","text":"EditURL = \"../../literate/howto/plug-and-play.jl\"","category":"page"},{"location":"generated/howto/plug-and-play/#Plug-and-Play-Regularization","page":"Plug-and-Play Regularization","title":"Plug-and-Play Regularization","text":"","category":"section"},{"location":"generated/howto/plug-and-play/","page":"Plug-and-Play Regularization","title":"Plug-and-Play Regularization","text":"A group of regularization terms that can not be directly written down as function are learned plug-and-play (PnP) priors. These are terms based on deep neural networks, which are trainted to implement the proximal map corresponding to the regularization term. Such a PnP prior can be used in the same way as any other regularization term.","category":"page"},{"location":"generated/howto/plug-and-play/","page":"Plug-and-Play Regularization","title":"Plug-and-Play Regularization","text":"The following example shows how to use a PnP prior in the context of the Kaczmarz solver.","category":"page"},{"location":"generated/howto/plug-and-play/","page":"Plug-and-Play Regularization","title":"Plug-and-Play Regularization","text":"using RegularizedLeastSquares\nA = randn(32, 16)\nx = randn(16)\nb = A*x;\nnothing #hide","category":"page"},{"location":"generated/howto/plug-and-play/","page":"Plug-and-Play Regularization","title":"Plug-and-Play Regularization","text":"For the documentation we will just use the identity function as a placeholder for the PnP prior.","category":"page"},{"location":"generated/howto/plug-and-play/","page":"Plug-and-Play Regularization","title":"Plug-and-Play Regularization","text":"model = identity","category":"page"},{"location":"generated/howto/plug-and-play/","page":"Plug-and-Play Regularization","title":"Plug-and-Play Regularization","text":"In practice, you would replace this with a neural network:","category":"page"},{"location":"generated/howto/plug-and-play/","page":"Plug-and-Play Regularization","title":"Plug-and-Play Regularization","text":"using Flux\nmodel = Flux.loadmodel!(model, ...)","category":"page"},{"location":"generated/howto/plug-and-play/","page":"Plug-and-Play Regularization","title":"Plug-and-Play Regularization","text":"The model can then be used together with the PnPRegularization term:","category":"page"},{"location":"generated/howto/plug-and-play/","page":"Plug-and-Play Regularization","title":"Plug-and-Play Regularization","text":"reg = PnPRegularization(1.0; model = model, shape = [16]);\nnothing #hide","category":"page"},{"location":"generated/howto/plug-and-play/","page":"Plug-and-Play Regularization","title":"Plug-and-Play Regularization","text":"Since models often expect a specific input range, we can use the MinMaxTransform to normalize the input:","category":"page"},{"location":"generated/howto/plug-and-play/","page":"Plug-and-Play Regularization","title":"Plug-and-Play Regularization","text":"reg = PnPRegularization(1.0; model = model, shape = [16], input_transform = RegularizedLeastSquares.MinMaxTransform);\nnothing #hide","category":"page"},{"location":"generated/howto/plug-and-play/","page":"Plug-and-Play Regularization","title":"Plug-and-Play Regularization","text":"Custom input transforms can be implemented by passing something callable as the input_transform keyword argument. For more details see the PnPRegularization documentation.","category":"page"},{"location":"generated/howto/plug-and-play/","page":"Plug-and-Play Regularization","title":"Plug-and-Play Regularization","text":"The regularization term can then be used in the solver:","category":"page"},{"location":"generated/howto/plug-and-play/","page":"Plug-and-Play Regularization","title":"Plug-and-Play Regularization","text":"solver = createLinearSolver(Kaczmarz, A; reg = reg, iterations = 32)\nx_approx = solve!(solver, b)","category":"page"},{"location":"generated/howto/plug-and-play/","page":"Plug-and-Play Regularization","title":"Plug-and-Play Regularization","text":"","category":"page"},{"location":"generated/howto/plug-and-play/","page":"Plug-and-Play Regularization","title":"Plug-and-Play Regularization","text":"This page was generated using Literate.jl.","category":"page"},{"location":"API/solvers/#API-for-Solvers","page":"Solvers","title":"API for Solvers","text":"","category":"section"},{"location":"API/solvers/","page":"Solvers","title":"Solvers","text":"This page contains documentation of the public API of the RegularizedLeastSquares. In the Julia REPL one can access this documentation by entering the help mode with ?","category":"page"},{"location":"API/solvers/#solve!","page":"Solvers","title":"solve!","text":"","category":"section"},{"location":"API/solvers/","page":"Solvers","title":"Solvers","text":"RegularizedLeastSquares.solve!(::AbstractLinearSolver, ::Any)\nRegularizedLeastSquares.init!(::AbstractLinearSolver, ::Any)\nRegularizedLeastSquares.init!(::AbstractLinearSolver, ::AbstractSolverState, ::AbstractMatrix)\n","category":"page"},{"location":"API/solvers/#RegularizedLeastSquares.solve!-Tuple{AbstractLinearSolver, Any}","page":"Solvers","title":"RegularizedLeastSquares.solve!","text":"solve!(solver::AbstractLinearSolver, b; x0 = 0, callbacks = (_, _) -> nothing)\n\nSolves an inverse problem for the data vector b using solver.\n\nRequired Arguments\n\nsolver::AbstractLinearSolver - linear solver (e.g., ADMM or FISTA), containing forward/normal operator and regularizer\nb::AbstractVector - data vector if A was supplied to the solver, back-projection of the data otherwise\n\nOptional Keyword Arguments\n\nx0::AbstractVector - initial guess for the solution; default is zero\ncallbacks - (optionally a vector of) function or callable struct that takes the two arguments callback(solver, iteration) and, e.g., stores, prints, or plots the intermediate solutions or convergence parameters. Be sure not to modify solver or iteration in the callback function as this would japaridze convergence. The default does nothing.\n\nExamples\n\nThe optimization problem\n\n\targmin_x Ax - b_2^2 + λ x_1\n\ncan be solved with the following lines of code:\n\njulia> using RegularizedLeastSquares\n\njulia> A = [0.831658 0.96717\n 0.383056 0.39043\n 0.820692 0.08118];\n\njulia> x = [0.5932234523399985; 0.2697534345340015];\n\njulia> b = A * x;\n\njulia> S = ADMM(A);\n\njulia> x_approx = solve!(S, b)\n2-element Vector{Float64}:\n 0.5932234523399984\n 0.26975343453400163\n\nHere, we use L1Regularization, which is default for ADMM. All regularization options can be found in API for Regularizers.\n\nThe following example solves the same problem, but stores the solution x of each interation in tr:\n\njulia> tr = Dict[]\nDict[]\n\njulia> store_trace!(tr, solver, iteration) = push!(tr, Dict(\"iteration\" => iteration, \"x\" => solver.x, \"beta\" => solver.β))\nstore_trace! (generic function with 1 method)\n\njulia> x_approx = solve!(S, b; callbacks=(solver, iteration) -> store_trace!(tr, solver, iteration))\n2-element Vector{Float64}:\n 0.5932234523399984\n 0.26975343453400163\n\njulia> tr[3]\nDict{String, Any} with 3 entries:\n \"iteration\" => 2\n \"x\" => [0.593223, 0.269753]\n \"beta\" => [1.23152, 0.927611]\n\nThe last example show demonstrates how to plot the solution at every 10th iteration and store the solvers convergence metrics:\n\njulia> using Plots\n\njulia> conv = StoreConvergenceCallback()\n\njulia> function plot_trace(solver, iteration)\n if iteration % 10 == 0\n display(scatter(solver.x))\n end\n end\nplot_trace (generic function with 1 method)\n\njulia> x_approx = solve!(S, b; callbacks = [conv, plot_trace]);\n\nThe keyword callbacks allows you to pass a (vector of) callable objects that takes the arguments solver and iteration and prints, stores, or plots intermediate result.\n\nSee also StoreSolutionCallback, StoreConvergenceCallback, CompareSolutionCallback for a number of provided callback options.\n\n\n\n\n\n","category":"method"},{"location":"API/solvers/#RegularizedLeastSquares.init!-Tuple{AbstractLinearSolver, Any}","page":"Solvers","title":"RegularizedLeastSquares.init!","text":"init!(solver::AbstractLinearSolver, b; kwargs...)\n\nPrepare the solver for iteration based on the given data vector b and kwargs.\n\n\n\n\n\n","category":"method"},{"location":"API/solvers/#RegularizedLeastSquares.init!-Tuple{AbstractLinearSolver, AbstractSolverState, AbstractMatrix}","page":"Solvers","title":"RegularizedLeastSquares.init!","text":"init!(solver::AbstractLinearSolver, state::AbstractSolverState, b::AbstractMatrix; scheduler = SequentialState, kwargs...)\n\nInitialize the solver with each column of b and pass the corresponding states to the scheduler.\n\n\n\n\n\n","category":"method"},{"location":"API/solvers/#ADMM","page":"Solvers","title":"ADMM","text":"","category":"section"},{"location":"API/solvers/","page":"Solvers","title":"Solvers","text":"RegularizedLeastSquares.ADMM","category":"page"},{"location":"API/solvers/#RegularizedLeastSquares.ADMM","page":"Solvers","title":"RegularizedLeastSquares.ADMM","text":"ADMM(A; AHA = A'*A, precon = Identity(), reg = L1Regularization(zero(real(eltype(AHA)))), regTrafo = opEye(eltype(AHA), size(AHA,1)), normalizeReg = NoNormalization(), rho = 1e-1, vary_rho = :none, iterations = 10, iterationsCG = 10, absTol = eps(real(eltype(AHA))), relTol = eps(real(eltype(AHA))), tolInner = 1e-5, verbose = false)\nADMM( ; AHA = , precon = Identity(), reg = L1Regularization(zero(real(eltype(AHA)))), regTrafo = opEye(eltype(AHA), size(AHA,1)), normalizeReg = NoNormalization(), rho = 1e-1, vary_rho = :none, iterations = 10, iterationsCG = 10, absTol = eps(real(eltype(AHA))), relTol = eps(real(eltype(AHA))), tolInner = 1e-5, verbose = false)\n\nCreates an ADMM object for the forward operator A or normal operator AHA.\n\nRequired Arguments\n\nA - forward operator\n\nOR\n\nAHA - normal operator (as a keyword argument)\n\nOptional Keyword Arguments\n\nAHA - normal operator is optional if A is supplied\nprecon - preconditionner for the internal CG algorithm\nreg::AbstractParameterizedRegularization - regularization term; can also be a vector of regularization terms\nregTrafo - transformation to a space in which reg is applied; if reg is a vector, regTrafo has to be a vector of the same length. Use opEye(eltype(AHA), size(AHA,1)) if no transformation is desired.\nnormalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()\nrho::Real - penalty of the augmented Lagrangian\nvary_rho::Symbol - vary rho to balance primal and dual feasibility; options :none, :balance, :PnP\niterations::Int - maximum number of (outer) ADMM iterations\niterationsCG::Int - maximum number of (inner) CG iterations\nabsTol::Real - absolute tolerance for stopping criterion\nrelTol::Real - relative tolerance for stopping criterion\ntolInner::Real - relative tolerance for CG stopping criterion\nverbose::Bool - print residual in each iteration\n\nADMM differs from ISTA-type algorithms in the sense that the proximal operation is applied separately from the transformation to the space in which the penalty is applied. This is reflected by the interface which has reg and regTrafo as separate arguments. E.g., for a TV penalty, you should NOT set reg=TVRegularization, but instead use reg=L1Regularization(λ), regTrafo=RegularizedLeastSquares.GradientOp(Float64; shape=(Nx,Ny,Nz)).\n\nSee also createLinearSolver, solve!.\n\n\n\n\n\n","category":"type"},{"location":"API/solvers/#CGNR","page":"Solvers","title":"CGNR","text":"","category":"section"},{"location":"API/solvers/","page":"Solvers","title":"Solvers","text":"RegularizedLeastSquares.CGNR","category":"page"},{"location":"API/solvers/#RegularizedLeastSquares.CGNR","page":"Solvers","title":"RegularizedLeastSquares.CGNR","text":"CGNR(A; AHA = A' * A, reg = L2Regularization(zero(real(eltype(AHA)))), normalizeReg = NoNormalization(), iterations = 10, relTol = eps(real(eltype(AHA))))\nCGNR( ; AHA = , reg = L2Regularization(zero(real(eltype(AHA)))), normalizeReg = NoNormalization(), iterations = 10, relTol = eps(real(eltype(AHA))))\n\ncreates an CGNR object for the forward operator A or normal operator AHA.\n\nRequired Arguments\n\nA - forward operator\n\nOR\n\nAHA - normal operator (as a keyword argument)\n\nOptional Keyword Arguments\n\nAHA - normal operator is optional if A is supplied\nreg::AbstractParameterizedRegularization - regularization term; can also be a vector of regularization terms\nnormalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()\niterations::Int - maximum number of iterations\nrelTol::Real - tolerance for stopping criterion\n\nSee also createLinearSolver, solve!.\n\n\n\n\n\n","category":"type"},{"location":"API/solvers/#Kaczmarz","page":"Solvers","title":"Kaczmarz","text":"","category":"section"},{"location":"API/solvers/","page":"Solvers","title":"Solvers","text":"RegularizedLeastSquares.Kaczmarz","category":"page"},{"location":"API/solvers/#RegularizedLeastSquares.Kaczmarz","page":"Solvers","title":"RegularizedLeastSquares.Kaczmarz","text":"Kaczmarz(A; reg = L2Regularization(0), normalizeReg = NoNormalization(), randomized=false, subMatrixFraction=0.15, shuffleRows=false, seed=1234, iterations=10)\n\nCreates a Kaczmarz object for the forward operator A.\n\nRequired Arguments\n\nA - forward operator\n\nOptional Keyword Arguments\n\nreg::AbstractParameterizedRegularization - regularization term\nnormalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()\nrandomized::Bool - randomize Kacmarz algorithm\nsubMatrixFraction::Real - fraction of rows used in randomized Kaczmarz algorithm\nshuffleRows::Bool - randomize Kacmarz algorithm\nseed::Int - seed for randomized algorithm\niterations::Int - number of iterations\n\nSee also createLinearSolver, solve!.\n\n\n\n\n\n","category":"type"},{"location":"API/solvers/#FISTA","page":"Solvers","title":"FISTA","text":"","category":"section"},{"location":"API/solvers/","page":"Solvers","title":"Solvers","text":"RegularizedLeastSquares.FISTA","category":"page"},{"location":"API/solvers/#RegularizedLeastSquares.FISTA","page":"Solvers","title":"RegularizedLeastSquares.FISTA","text":"FISTA(A; AHA=A'*A, reg=L1Regularization(zero(real(eltype(AHA)))), normalizeReg=NoNormalization(), iterations=50, verbose = false, rho = 0.95 / power_iterations(AHA), theta=1, relTol=eps(real(eltype(AHA))), restart = :none)\nFISTA( ; AHA=, reg=L1Regularization(zero(real(eltype(AHA)))), normalizeReg=NoNormalization(), iterations=50, verbose = false, rho = 0.95 / power_iterations(AHA), theta=1, relTol=eps(real(eltype(AHA))), restart = :none)\n\ncreates a FISTA object for the forward operator A or normal operator AHA.\n\nRequired Arguments\n\nA - forward operator\n\nOR\n\nAHA - normal operator (as a keyword argument)\n\nOptional Keyword Arguments\n\nAHA - normal operator is optional if A is supplied\nprecon - preconditionner for the internal CG algorithm\nreg::AbstractParameterizedRegularization - regularization term; can also be a vector of regularization terms\nnormalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()\nrho::Real - step size for gradient step; the default is 0.95 / max_eigenvalue as determined with power iterations.\ntheta::Real - parameter for predictor-corrector step\nrelTol::Real - tolerance for stopping criterion\niterations::Int - maximum number of iterations\nrestart::Symbol - :none, :gradient options for restarting\nverbose::Bool - print residual in each iteration\n\nSee also createLinearSolver, solve!.\n\n\n\n\n\n","category":"type"},{"location":"API/solvers/#OptISTA","page":"Solvers","title":"OptISTA","text":"","category":"section"},{"location":"API/solvers/","page":"Solvers","title":"Solvers","text":"RegularizedLeastSquares.OptISTA","category":"page"},{"location":"API/solvers/#RegularizedLeastSquares.OptISTA","page":"Solvers","title":"RegularizedLeastSquares.OptISTA","text":"OptISTA(A; AHA=A'*A, reg=L1Regularization(zero(real(eltype(AHA)))), normalizeReg=NoNormalization(), iterations=50, verbose = false, rho=0.95 / power_iterations(AHA), theta=1, relTol=eps(real(eltype(AHA))))\nOptISTA( ; AHA=, reg=L1Regularization(zero(real(eltype(AHA)))), normalizeReg=NoNormalization(), iterations=50, verbose = false, rho=0.95 / power_iterations(AHA), theta=1, relTol=eps(real(eltype(AHA))))\n\ncreates a OptISTA object for the forward operator A or normal operator AHA. OptISTA has a 2x better worst-case bound than FISTA, but actual performance varies by application. It stores 2 extra intermediate variables the size of the image compared to FISTA.\n\nReference:\n\nUijeong Jang, Shuvomoy Das Gupta, Ernest K. Ryu, \"Computer-Assisted Design of Accelerated Composite Optimization Methods: OptISTA,\" arXiv:2305.15704, 2023, [https://arxiv.org/abs/2305.15704]\n\nRequired Arguments\n\nA - forward operator\n\nOR\n\nAHA - normal operator (as a keyword argument)\n\nOptional Keyword Arguments\n\nAHA - normal operator is optional if A is supplied\nreg::AbstractParameterizedRegularization - regularization term\nnormalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()\nrho::Real - step size for gradient step; the default is 0.95 / max_eigenvalue as determined with power iterations.\ntheta::Real - parameter for predictor-corrector step\nrelTol::Real - tolerance for stopping criterion\niterations::Int - maximum number of iterations\nverbose::Bool - print residual in each iteration\n\nSee also createLinearSolver, solve!.\n\n\n\n\n\n","category":"type"},{"location":"API/solvers/#POGM","page":"Solvers","title":"POGM","text":"","category":"section"},{"location":"API/solvers/","page":"Solvers","title":"Solvers","text":"RegularizedLeastSquares.POGM","category":"page"},{"location":"API/solvers/#RegularizedLeastSquares.POGM","page":"Solvers","title":"RegularizedLeastSquares.POGM","text":"POGM(A; AHA = A'*A, reg = L1Regularization(zero(real(eltype(AHA)))), normalizeReg = NoNormalization(), iterations = 50, verbose = false, rho = 0.95 / power_iterations(AHA), theta = 1, sigma_fac = 1, relTol = eps(real(eltype(AHA))), restart = :none)\nPOGM( ; AHA = , reg = L1Regularization(zero(real(eltype(AHA)))), normalizeReg = NoNormalization(), iterations = 50, verbose = false, rho = 0.95 / power_iterations(AHA), theta = 1, sigma_fac = 1, relTol = eps(real(eltype(AHA))), restart = :none)\n\nCreates a POGM object for the forward operator A or normal operator AHA. POGM has a 2x better worst-case bound than FISTA, but actual performance varies by application. It stores 3 extra intermediate variables the size of the image compared to FISTA. Only gradient restart scheme is implemented for now.\n\nReferences:\n\nA.B. Taylor, J.M. Hendrickx, F. Glineur, \"Exact worst-case performance of first-order algorithms for composite convex optimization,\" Arxiv:1512.07516, 2015, SIAM J. Opt. 2017 [http://doi.org/10.1137/16m108104x]\nKim, D., & Fessler, J. A. (2018). Adaptive Restart of the Optimized Gradient Method for Convex Optimization. Journal of Optimization Theory and Applications, 178(1), 240–263. [https://doi.org/10.1007/s10957-018-1287-4]\nRequired Arguments\nA - forward operator\nOR\nAHA - normal operator (as a keyword argument)\nOptional Keyword Arguments\nAHA - normal operator is optional if A is supplied\nreg::AbstractParameterizedRegularization - regularization term\nnormalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()\nrho::Real - step size for gradient step; the default is 0.95 / max_eigenvalue as determined with power iterations.\ntheta::Real - parameter for predictor-corrector step\nsigma_fac::Real - parameter for decreasing γ-momentum ∈ [0,1]\nrelTol::Real - tolerance for stopping criterion\niterations::Int - maximum number of iterations\nrestart::Symbol - :none, :gradient options for restarting\nverbose::Bool - print residual in each iteration\n\nSee also createLinearSolver, solve!.\n\n\n\n\n\n","category":"type"},{"location":"API/solvers/#SplitBregman","page":"Solvers","title":"SplitBregman","text":"","category":"section"},{"location":"API/solvers/","page":"Solvers","title":"Solvers","text":"RegularizedLeastSquares.SplitBregman","category":"page"},{"location":"API/solvers/#RegularizedLeastSquares.SplitBregman","page":"Solvers","title":"RegularizedLeastSquares.SplitBregman","text":"SplitBregman(A; AHA = A'*A, precon = Identity(), reg = L1Regularization(zero(real(eltype(AHA)))), regTrafo = opEye(eltype(AHA), size(AHA,1)), normalizeReg = NoNormalization(), rho = 1e-1, iterations = 10, iterationsInner = 10, iterationsCG = 10, absTol = eps(real(eltype(AHA))), relTol = eps(real(eltype(AHA))), tolInner = 1e-5, verbose = false)\nSplitBregman( ; AHA = , precon = Identity(), reg = L1Regularization(zero(real(eltype(AHA)))), regTrafo = opEye(eltype(AHA), size(AHA,1)), normalizeReg = NoNormalization(), rho = 1e-1, iterations = 10, iterationsInner = 10, iterationsCG = 10, absTol = eps(real(eltype(AHA))), relTol = eps(real(eltype(AHA))), tolInner = 1e-5, verbose = false)\n\nCreates a SplitBregman object for the forward operator A or normal operator AHA.\n\nRequired Arguments\n\nA - forward operator\n\nOR\n\nAHA - normal operator (as a keyword argument)\n\nOptional Keyword Arguments\n\nAHA - normal operator is optional if A is supplied\nprecon - preconditionner for the internal CG algorithm\nreg::AbstractParameterizedRegularization - regularization term; can also be a vector of regularization terms\nregTrafo - transformation to a space in which reg is applied; if reg is a vector, regTrafo has to be a vector of the same length. Use opEye(eltype(AHA), size(AHA,1)) if no transformation is desired.\nnormalizeReg::AbstractRegularizationNormalization - regularization normalization scheme; options are NoNormalization(), MeasurementBasedNormalization(), SystemMatrixBasedNormalization()\nrho::Real - weights for condition on regularized variables; can also be a vector for multiple regularization terms\niterations::Int - maximum number of outer iterations. Set to 1 for unconstraint split Bregman (equivalent to ADMM)\niterationsInner::Int - maximum number of inner iterations\niterationsCG::Int - maximum number of (inner) CG iterations\nabsTol::Real - absolute tolerance for stopping criterion\nrelTol::Real - relative tolerance for stopping criterion\ntolInner::Real - relative tolerance for CG stopping criterion\nverbose::Bool - print residual in each iteration\n\nThis algorithm solves the constraint problem (Eq. (4.7) in Tom Goldstein and Stanley Osher), i.e. ||R(x)||₁ such that ||Ax -b||₂² < σ². In order to solve the unconstraint problem (Eq. (4.8) in Tom Goldstein and Stanley Osher), i.e. ||Ax -b||₂² + λ ||R(x)||₁, you can either set iterations=1 or use ADMM instead, which is equivalent (iterations=1 in SplitBregman in implied in ADMM and the SplitBregman variable iterationsInner is simply called iterations in ADMM)\n\nLike ADMM, SplitBregman differs from ISTA-type algorithms in the sense that the proximal operation is applied separately from the transformation to the space in which the penalty is applied. This is reflected by the interface which has reg and regTrafo as separate arguments. E.g., for a TV penalty, you should NOT set reg=TVRegularization, but instead use reg=L1Regularization(λ), regTrafo=RegularizedLeastSquares.GradientOp(Float64; shape=(Nx,Ny,Nz)).\n\nSee also createLinearSolver, solve!.\n\n\n\n\n\n","category":"type"},{"location":"API/solvers/#Miscellaneous","page":"Solvers","title":"Miscellaneous","text":"","category":"section"},{"location":"API/solvers/","page":"Solvers","title":"Solvers","text":"RegularizedLeastSquares.solverstate\nRegularizedLeastSquares.solversolution\nRegularizedLeastSquares.solverconvergence\nRegularizedLeastSquares.StoreSolutionCallback\nRegularizedLeastSquares.StoreConvergenceCallback\nRegularizedLeastSquares.CompareSolutionCallback\nRegularizedLeastSquares.linearSolverList\nRegularizedLeastSquares.createLinearSolver\nRegularizedLeastSquares.applicableSolverList\nRegularizedLeastSquares.isapplicable","category":"page"},{"location":"API/solvers/#RegularizedLeastSquares.solverstate","page":"Solvers","title":"RegularizedLeastSquares.solverstate","text":"solverstate(solver::AbstractLinearSolver)\n\nReturn the current state of the solver\n\n\n\n\n\n","category":"function"},{"location":"API/solvers/#RegularizedLeastSquares.solversolution","page":"Solvers","title":"RegularizedLeastSquares.solversolution","text":"solversolution(solver::AbstractLinearSolver)\n\nReturn the current solution of the solver\n\n\n\n\n\nsolversolution(state::AbstractSolverState)\n\nReturn the current solution of the solver's state\n\n\n\n\n\n","category":"function"},{"location":"API/solvers/#RegularizedLeastSquares.solverconvergence","page":"Solvers","title":"RegularizedLeastSquares.solverconvergence","text":"solverconvergence(solver::AbstractLinearSolver)\n\nReturn a named tuple of the solvers current convergence metrics\n\n\n\n\n\n","category":"function"},{"location":"API/solvers/#RegularizedLeastSquares.StoreSolutionCallback","page":"Solvers","title":"RegularizedLeastSquares.StoreSolutionCallback","text":"StoreSolutionCallback(T)\n\nCallback that accumlates the solvers solution per iteration. Results are stored in the solutions field.\n\n\n\n\n\n","category":"type"},{"location":"API/solvers/#RegularizedLeastSquares.StoreConvergenceCallback","page":"Solvers","title":"RegularizedLeastSquares.StoreConvergenceCallback","text":"StoreConvergenceCallback()\n\nCallback that accumlates the solvers convergence metrics per iteration. Results are stored in the convMeas field.\n\n\n\n\n\n","category":"type"},{"location":"API/solvers/#RegularizedLeastSquares.CompareSolutionCallback","page":"Solvers","title":"RegularizedLeastSquares.CompareSolutionCallback","text":"CompareSolutionCallback(ref, cmp)\n\nCallback that compares the solvers current solution with the given reference via cmp(ref, solution) per iteration. Results are stored in the results field.\n\n\n\n\n\n","category":"type"},{"location":"API/solvers/#RegularizedLeastSquares.linearSolverList","page":"Solvers","title":"RegularizedLeastSquares.linearSolverList","text":"Return a list of all available linear solvers\n\n\n\n\n\n","category":"function"},{"location":"API/solvers/#RegularizedLeastSquares.createLinearSolver","page":"Solvers","title":"RegularizedLeastSquares.createLinearSolver","text":"createLinearSolver(solver::AbstractLinearSolver, A; kargs...)\n\nThis method creates a solver. The supported solvers are methods typically used for solving regularized linear systems. All solvers return an approximate solution to Ax = b.\n\nTODO: give a hint what solvers are available\n\n\n\n\n\n","category":"function"},{"location":"API/solvers/#RegularizedLeastSquares.applicableSolverList","page":"Solvers","title":"RegularizedLeastSquares.applicableSolverList","text":"applicable(args...)\n\nlist all solvers that are applicable to the given arguments. Arguments are the same as for isapplicable without the solver type.\n\nSee also isapplicable, linearSolverList.\n\n\n\n\n\n","category":"function"},{"location":"API/solvers/#RegularizedLeastSquares.isapplicable","page":"Solvers","title":"RegularizedLeastSquares.isapplicable","text":"isapplicable(solverType::Type{<:AbstractLinearSolver}, A, x, reg)\n\nreturn true if a solver of type solverType is applicable to system matrix A, data x and regularization terms reg.\n\n\n\n\n\n","category":"function"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"EditURL = \"../../literate/howto/gpu_acceleration.jl\"","category":"page"},{"location":"generated/howto/gpu_acceleration/#GPU-Acceleration","page":"GPU Acceleration","title":"GPU Acceleration","text":"","category":"section"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"RegularizedLeastSquares.jl supports generic GPU acceleration. This means that the user can use any GPU array type that supports the GPUArrays interface. This includes CUDA.jl, AMDGPU.jl, and Metal.jl. In this example we will use the package JLArrays.jl which provides a reference implementation for GPUArrays, that can runs on CPUs.","category":"page"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"using JLArrays\ngpu = JLArray;\nnothing #hide","category":"page"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"To use the following examples on an actual GPU, load the appropraite package replace JLArray with the respective GPU array type, for example:","category":"page"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"using CUDA\ngpu = CuArray","category":"page"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"At first we will look at an example of dense GPU arrays.","category":"page"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"using RegularizedLeastSquares\nA = gpu(rand(Float32, 32, 16))\nx = gpu(rand(Float32, 16))\nb = A*x;\nnothing #hide","category":"page"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"Solvers adapt their states based on the type of the given measurement vector. This means that the solver will automatically switch to GPU acceleration if a GPU array is passed as the measurement vector.","category":"page"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"solver = createLinearSolver(CGNR, A; reg = L2Regularization(0.0001), iterations=32);\nx_approx = solve!(solver, b)","category":"page"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"This adaption does not include the operator. So if we want to compare with CPU result, we need to construct a new solver with a CPU operator.","category":"page"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"solver = createLinearSolver(CGNR, Array(A); reg = L2Regularization(0.0001), iterations=32);\nx_cpu = solve!(solver, Array(b))\nisapprox(Array(x_approx), x_cpu)","category":"page"},{"location":"generated/howto/gpu_acceleration/#Matrix-Free-Operators","page":"GPU Acceleration","title":"Matrix-Free Operators","text":"","category":"section"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"A special case is the usage of matrix-free operators. Since these operators do not have a concrete matrix representation, their GPU support depends on their implementation. Since not all multiplications within a solver approximation are in-place, the operator also needs to support the * operation and construct an appropriate result vector. For matrix-free operators based on LinearOperators.jl, this can be achieved by implementing the LinearOperators.storage_type method.","category":"page"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"In the following, we will take another look at the CS example and execute it on the GPU. Note that for the JLArray example we chose a small phantom, since the JLArray implementation is not optimized for performance:","category":"page"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"using ImagePhantoms, ImageGeoms\nN = 32\nimage = shepp_logan(N, SheppLoganToft())\n\nusing Random, LinearOperatorCollection\nrandomIndices = shuffle(eachindex(image))\nsampledIndices = sort(randomIndices[1:div(end, 3)]);\nnothing #hide","category":"page"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"To construct the operator, we need to convert the indices to a GPU array. We also need to specify the correct storage type. In both LinearOperators.jl and LinearOperatorCollection.jl this is done with the S keyword argument.","category":"page"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"gpu_indices = gpu(sampledIndices)\nA = SamplingOp(eltype(image), pattern = gpu_indices, shape = size(image), S = typeof(b));\nnothing #hide","category":"page"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"Let's inspect the storage type of the operator:","category":"page"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"using LinearOperatorCollection.LinearOperators\nLinearOperators.storage_type(A)","category":"page"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"Afterwards we can use the operator as usual:","category":"page"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"b = A*vec(gpu(image));\nnothing #hide","category":"page"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"And use it in the solver:","category":"page"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"using RegularizedLeastSquares\nreg = TVRegularization(0.01; shape=size(image))\nsolver = createLinearSolver(FISTA, A; reg=reg, iterations=20)\nimg_approx = solve!(solver,b);\nnothing #hide","category":"page"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"To visualize the reconstructed image, we need to reshape the result vector to the correct shape and convert it to a CPU array:","category":"page"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"img_approx = reshape(Array(img_approx),size(image))","category":"page"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"We will again use CairoMakie for visualization:","category":"page"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"using CairoMakie\nfunction plot_image(figPos, img; title = \"\", width = 150, height = 150)\n ax = CairoMakie.Axis(figPos; yreversed=true, title, width, height)\n hidedecorations!(ax)\n heatmap!(ax, img)\nend\nfig = Figure()\nplot_image(fig[1,1], image, title = \"Image\")\nsamplingMask = fill(false, size(image))\nsamplingMask[sampledIndices] .= true\nplot_image(fig[1,2], image .* samplingMask, title = \"Sampled Image\")\nplot_image(fig[1,3], img_approx, title = \"Reconstructed Image\")\nresize_to_layout!(fig)\nfig","category":"page"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"","category":"page"},{"location":"generated/howto/gpu_acceleration/","page":"GPU Acceleration","title":"GPU Acceleration","text":"This page was generated using Literate.jl.","category":"page"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"EditURL = \"../../literate/examples/compressed_sensing.jl\"","category":"page"},{"location":"generated/examples/compressed_sensing/#Compressed-Sensing-Example","page":"Compressed Sensing","title":"Compressed Sensing Example","text":"","category":"section"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"In this example we will go through a simple example from the field of Compressed Sensing. In addtion to RegularizedLeastSquares.jl, we will need the packages LinearOperatorCollection.jl, ImagePhantoms.jl, ImageGeoms.jl and Random.jl, as well as CairoMakie.jl for visualization. We can install them the same way we did RegularizedLeastSquares.jl.","category":"page"},{"location":"generated/examples/compressed_sensing/#Preparing-the-Inverse-Problem","page":"Compressed Sensing","title":"Preparing the Inverse Problem","text":"","category":"section"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"To get started, let us generate a simple phantom using the ImagePhantoms and ImageGeom packages:","category":"page"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"using ImagePhantoms, ImageGeoms\nN = 256\nimage = shepp_logan(N, SheppLoganToft())\nsize(image)","category":"page"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"This produces a 256x256 image of a Shepp-Logan phantom.","category":"page"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"In this example, we consider a problem in which we randomly sample a third of the pixels in the image. Such a problem and the corresponding measurement can be constructed with the packages LinearOperatorCollection and Random:","category":"page"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"We first randomly shuffle the indices of the image and then select the first third of the indices to sample.","category":"page"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"using Random, LinearOperatorCollection\nrandomIndices = shuffle(eachindex(image))\nsampledIndices = sort(randomIndices[1:div(end, 3)])","category":"page"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"Afterwards we build a sampling operator which samples the image at the selected indices.","category":"page"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"A = SamplingOp(eltype(image), pattern = sampledIndices , shape = size(image));\nnothing #hide","category":"page"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"Then we apply the sampling operator to the vectorized image to obtain the sampled measurement vector","category":"page"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"b = A*vec(image);\nnothing #hide","category":"page"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"To visualize our image we can use CairoMakie:","category":"page"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"using CairoMakie\nfunction plot_image(figPos, img; title = \"\", width = 150, height = 150)\n ax = CairoMakie.Axis(figPos; yreversed=true, title, width, height)\n hidedecorations!(ax)\n heatmap!(ax, img)\nend\nfig = Figure()\nplot_image(fig[1,1], image, title = \"Image\")\nsamplingMask = fill(false, size(image))\nsamplingMask[sampledIndices] .= true\nplot_image(fig[1,2], image .* samplingMask, title = \"Sampled Image\")\nresize_to_layout!(fig)\nfig","category":"page"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"As we can see in the right image, only a third of the pixels are sampled. The goal of the inverse problem is to recover the original image from this measurement vector.","category":"page"},{"location":"generated/examples/compressed_sensing/#Solving-the-Inverse-Problem","page":"Compressed Sensing","title":"Solving the Inverse Problem","text":"","category":"section"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"To recover the image from the measurement vector, we solve the TV-regularized least squares problem:","category":"page"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"beginequation\n undersetmathbfxargmin frac12vertvert mathbfAmathbfx-mathbfb vertvert_2^2 + vertvertmathbfxvertvert_lambdatextTV \nendequation","category":"page"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"For this purpose we build a TV regularizer with regularization parameter λ=001","category":"page"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"using RegularizedLeastSquares\nreg = TVRegularization(0.01; shape=size(image));\nnothing #hide","category":"page"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"We will use the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) to solve our inverse problem. Thus, we build the corresponding solver","category":"page"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"solver = createLinearSolver(FISTA, A; reg=reg, iterations=20);\nnothing #hide","category":"page"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"and apply it to our measurement vector","category":"page"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"img_approx = solve!(solver,b)","category":"page"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"To visualize the reconstructed image, we need to reshape the result vector to the correct shape. Afterwards we can use CairoMakie again:","category":"page"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"img_approx = reshape(img_approx,size(image));\nplot_image(fig[1,3], img_approx, title = \"Reconstructed Image\")\nresize_to_layout!(fig)\nfig","category":"page"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"","category":"page"},{"location":"generated/examples/compressed_sensing/","page":"Compressed Sensing","title":"Compressed Sensing","text":"This page was generated using Literate.jl.","category":"page"},{"location":"generated/examples/getting_started/","page":"Getting Started","title":"Getting Started","text":"EditURL = \"../../literate/examples/getting_started.jl\"","category":"page"},{"location":"generated/examples/getting_started/#Getting-Started","page":"Getting Started","title":"Getting Started","text":"","category":"section"},{"location":"generated/examples/getting_started/","page":"Getting Started","title":"Getting Started","text":"In this example we will go through a simple random inverse problem to get familiar with RegularizedLeastSquares.jl.","category":"page"},{"location":"generated/examples/getting_started/#Installation","page":"Getting Started","title":"Installation","text":"","category":"section"},{"location":"generated/examples/getting_started/","page":"Getting Started","title":"Getting Started","text":"To install RegularizedLeastSquares.jl, you can use the Julia package manager. Open a Julia REPL and run the following command:","category":"page"},{"location":"generated/examples/getting_started/","page":"Getting Started","title":"Getting Started","text":"using Pkg\nPkg.add(\"RegularizedLeastSquares\")","category":"page"},{"location":"generated/examples/getting_started/","page":"Getting Started","title":"Getting Started","text":"This will download and install the RegularizedLeastSquares.jl package and its dependencies. To install a different version, please consult the Pkg documentation.","category":"page"},{"location":"generated/examples/getting_started/","page":"Getting Started","title":"Getting Started","text":"Once the installation is complete, you can import the package with the using keyword:","category":"page"},{"location":"generated/examples/getting_started/","page":"Getting Started","title":"Getting Started","text":"using RegularizedLeastSquares","category":"page"},{"location":"generated/examples/getting_started/","page":"Getting Started","title":"Getting Started","text":"RegularizedLeastSquares aims to solve inverse problems of the form:","category":"page"},{"location":"generated/examples/getting_started/","page":"Getting Started","title":"Getting Started","text":"beginequation\n undersetmathbfxargmin frac12vertvert mathbfAmathbfx-mathbfb vertvert_2^2 + mathbfR(x) \nendequation","category":"page"},{"location":"generated/examples/getting_started/","page":"Getting Started","title":"Getting Started","text":"where mathbfA is a linear operator, mathbfy is the measurement vector, and mathbfR(x) is an (optional) regularization term. The goal is to retrieve an approximation of the unknown vector mathbfx. In this first exampel we will just work with simple random arrays. For more advanced examples, please refer to the examples.","category":"page"},{"location":"generated/examples/getting_started/","page":"Getting Started","title":"Getting Started","text":"A = rand(32, 16)\nx = rand(16)\nb = A*x;\nnothing #hide","category":"page"},{"location":"generated/examples/getting_started/","page":"Getting Started","title":"Getting Started","text":"To approximate x from b, we can use the Conjugate Gradient Normal Residual (CGNR) algorithm. We first build the corresponding solver:","category":"page"},{"location":"generated/examples/getting_started/","page":"Getting Started","title":"Getting Started","text":"solver = createLinearSolver(CGNR, A; iterations=32);\nnothing #hide","category":"page"},{"location":"generated/examples/getting_started/","page":"Getting Started","title":"Getting Started","text":"and apply it to our measurement vector","category":"page"},{"location":"generated/examples/getting_started/","page":"Getting Started","title":"Getting Started","text":"x_approx = solve!(solver, b)\nisapprox(x, x_approx, rtol = 0.001)","category":"page"},{"location":"generated/examples/getting_started/","page":"Getting Started","title":"Getting Started","text":"Usually the inverse problems are ill-posed and require regularization. RegularizedLeastSquares.jl provides a variety of regularization terms. The CGNR algorithm can solve optimzation problems of the form:","category":"page"},{"location":"generated/examples/getting_started/","page":"Getting Started","title":"Getting Started","text":"beginequation\n undersetmathbfxargmin frac12vertvert mathbfAmathbfx-mathbfb vertvert_2^2 + vertvertmathbfxvertvert^2_2 \nendequation","category":"page"},{"location":"generated/examples/getting_started/","page":"Getting Started","title":"Getting Started","text":"The corresponding solver can be built with the L2 regularization term:","category":"page"},{"location":"generated/examples/getting_started/","page":"Getting Started","title":"Getting Started","text":"solver = createLinearSolver(CGNR, A; reg = L2Regularization(0.0001), iterations=32);\nx_approx = solve!(solver, b)\nisapprox(x, x_approx, rtol = 0.001)","category":"page"},{"location":"generated/examples/getting_started/","page":"Getting Started","title":"Getting Started","text":"","category":"page"},{"location":"generated/examples/getting_started/","page":"Getting Started","title":"Getting Started","text":"This page was generated using Literate.jl.","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"EditURL = \"../../literate/explanations/regularization.jl\"","category":"page"},{"location":"generated/explanations/regularization/#Regularization","page":"Regularization","title":"Regularization","text":"","category":"section"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"When formulating inverse problems, a regulariser is formulated as an additional cost function which has to be minimised. Many algorithms deal with a regularizers g, by computing the proximal map:","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"beginequation\n prox_g (mathbfx) = undersetmathbfuargmin frac12vertvert mathbfu-mathbfx vert vert^2 + g(mathbfx)\nendequation","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"For many regularizers, the proximal map can be computed efficiently in a closed form.","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"In order to implement these proximal mappings, RegularizedLeastSquares.jl defines the following type hierarchy:","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"abstract type AbstractRegularization\nprox!(reg::AbstractRegularization, x)\nnorm(reg::AbstractRegularization, x)","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"Here prox!(reg, x) is an in-place function which computes the proximal map on the input-vector x. The function norm computes the value of the corresponding term in the inverse problem. RegularizedLeastSquares.jl provides AbstractParameterizedRegularization and AbstractProjectionRegularization as core regularization types.","category":"page"},{"location":"generated/explanations/regularization/#Parameterized-Regularization-Terms","page":"Regularization","title":"Parameterized Regularization Terms","text":"","category":"section"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"This group of regularization terms features a regularization parameter λ that is used during the prox! and normcomputations. Examples of this regulariztion group are L1, L2 or LLR (locally low rank) regularization terms.","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"These terms are constructed by supplying a λ and optionally term specific keyword arguments:","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"using RegularizedLeastSquares\nl2 = L2Regularization(0.3)","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"Parameterized regularization terms implement:","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"prox!(reg::AbstractParameterizedRegularization, x, λ)\nnorm(reg::AbstractParameterizedRegularization, x, λ)","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"where λ by default is filled with the value used during construction.","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"Invoking λ on a parameterized term retrieves its regularization parameter. This can be used in a solver to scale and overwrite the parameter as follows:","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"prox!(l2, [1.0])","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"param = λ(l2)\nprox!(l2, [1.0], param*0.2)","category":"page"},{"location":"generated/explanations/regularization/#Projection-Regularization-Terms","page":"Regularization","title":"Projection Regularization Terms","text":"","category":"section"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"This group of regularization terms implement projections, such as a positivity constraint or a projection with a given convex projection function. These are essentially proximal maps where g(mathbfx) is the indicator function of a convex set.","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"positive = PositiveRegularization()\nprox!(positive, [2.0, -0.2])","category":"page"},{"location":"generated/explanations/regularization/#Nested-Regularization-Terms","page":"Regularization","title":"Nested Regularization Terms","text":"","category":"section"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"Nested regularization terms are terms that act as decorators to the core regularization terms. These terms can be nested around other terms and add functionality to a regularization term, such as scaling λ based on the provided operator or applying a transform, such as the Wavelet, to x. As an example, we can nest a L1 regularization term around a Wavelet operator.","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"First we generate an image and apply a wavelet operator to it:","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"using Wavelets, LinearOperatorCollection, ImagePhantoms, ImageGeoms\nN = 256\nimage = shepp_logan(N, SheppLoganToft())\nwop = WaveletOp(Float32, shape = size(image))\n\nwavelet_image = reshape(wop*vec(image), size(image))\nwavelet_image = log.(abs.(wavelet_image) .+ 0.01)","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"We will use CairoMakie for visualization:","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"using CairoMakie\nfunction plot_image(figPos, img; title = \"\", width = 150, height = 150)\n ax = CairoMakie.Axis(figPos; yreversed=true, title, width, height)\n hidedecorations!(ax)\n heatmap!(ax, img)\nend\nfig = Figure()\nplot_image(fig[1,1], image, title = \"Image\")\nplot_image(fig[1,2], wavelet_image, title = \"Wavelet Image\")\nresize_to_layout!(fig)\nfig","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"To apply soft-thresholding in the wavelet domain, we can nest a L1 regularization term around the Wavelet operator:","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"core = L1Regularization(0.1)\nreg = TransformedRegularization(core, wop);\nnothing #hide","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"We can then apply the proximal map to the image or the image in the wavelet domain:","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"img_prox_image = prox!(core, copy(vec(image)));\nimg_prox_wavelet = prox!(reg, copy(vec(image)));\nnothing #hide","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"We can visualize the result:","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"img_prox_image = reshape(img_prox_image, size(image))\nimg_prox_wavelet = reshape(img_prox_wavelet, size(image))\nplot_image(fig[1,3], img_prox_image, title = \"Reg. Image Domain\")\nplot_image(fig[1,4], img_prox_wavelet, title = \"Reg. Wavelet Domain\")\nresize_to_layout!(fig)\nfig","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"Generally, regularization terms can be nested arbitrarly deep, adding new functionality with each layer. Each nested regularization term can return its inner regularization term. Furthermore, all regularization terms implement the iteration interface to iterate over the nesting. The innermost regularization term of a nested term must be a core regularization term and it can be returned by the sink function:","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"RegularizedLeastSquares.innerreg(reg) == core","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"sink(reg) == core","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"foreach(r -> println(nameof(typeof(r))), reg)","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"","category":"page"},{"location":"generated/explanations/regularization/","page":"Regularization","title":"Regularization","text":"This page was generated using Literate.jl.","category":"page"},{"location":"#RegularizedLeastSquares.jl","page":"Home","title":"RegularizedLeastSquares.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Solvers for Linear Inverse Problems using Regularization Techniques","category":"page"},{"location":"#Introduction","page":"Home","title":"Introduction","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"RegularizedLeastSquares.jl is a Julia package for solving large linear systems using various types of algorithms. Ill-conditioned problems arise in many areas of practical interest. Regularisation techniques and nonlinear problem formulations are often used to solve these problems. This package provides implementations for a variety of solvers used in areas such as MPI and MRI. In particular, this package serves as the optimization backend of the Julia packages MPIReco.jl and MRIReco.jl.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The implemented methods range from the l^2_2-regularized CGNR method to more general optimizers such as the Alternating Direction of Multipliers Method (ADMM) or the Split-Bregman method.","category":"page"},{"location":"","page":"Home","title":"Home","text":"For convenience, implementations of popular regularizers, such as l_1-regularization and TV regularization, are provided. 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    Solvers

    RegularizedLeastSquares.jl provides a variety of solvers, which are used in fields such as MPI and MRI. The following is a non-exhaustive list of the implemented solvers:

    • Kaczmarz algorithm (Kaczmarz, also called Algebraic reconstruction technique)
    • Conjugate Gradients Normal Residual method (CGNR)
    • Fast Iterative Shrinkage Thresholding Algorithm (FISTA)
    • Alternating Direction of Multipliers Method (ADMM)

    The solvers are organized in a type-hierarchy and inherit from:

    abstract type AbstractLinearSolver

    The type hierarchy is further differentiated into solver categories such as AbstractRowAtionSolver, AbstractPrimalDualSolver or AbstractProximalGradientSolver. A list of all available solvers can be returned by the linearSolverList function.

    Solver Construction

    To create a solver, one can invoke the method createLinearSolver as in

    solver = createLinearSolver(CGNR, A; reg=reg, kwargs...)

    Here A denotes the operator and reg are the Regularization terms to be used by the solver. All further solver parameters can be passed as keyword arguments and are solver specific. To make things more compact, it can be usefull to collect all parameters in a Dict{Symbol,Any}. In this way, the code snippet above can be written as

    params=Dict{Symbol,Any}()
+params[:reg] = ...
+...
+
+solver = createLinearSolver(CGNR, A; params...)

    This notation can be convenient when a large number of parameters are set manually.

    It is also possible to construct a solver directly with its specific keyword arguments:

    solver = CGNR(A, reg = reg, ...)

    Solver Usage

    Once constructed, a solver can be used to approximate a solution to a given measurement vector:

    x_approx = solve!(solver, b; kwargs...)

    The keyword arguments can be used to supply an inital solution x0, one or more callbacks to interact and monitor the solvers state and more. See the How-To and the API for more information.

    It is also possible to explicitly invoke the solvers iterations using Julias iterate interface:

    init!(solver, b; kwargs...)
+for (iteration, x_approx) in enumerate(solver)
+    println("Iteration $iteration")
+end

    Solver Internals

    The fields of a solver can be divided into two groups. The first group are intended to be immutable fields that do not change during iterations, the second group are mutable fields that do change. Examples of the first group are the operator itself and examples of the second group are the current solution or the number of the current iteration.

    The second group is usually encapsulated in its own state struct:

    mutable struct Solver{matT, ...}
+  A::matT
+  # Other "static" fields
+  state::AbstractSolverState{<:Solver}
+end
+
+mutable struct SolverState{T, tempT} <: AbstractSolverState{Solver}
+  x::tempT
+  rho::T
+  # ...
+  iteration::Int64
+end

    States are subtypes of the parametric AbstractSolverState{S} type. The state fields of solvers can be exchanged with different state belonging to the correct solver S. This means that the states can be used to realize custom variants of an existing solver: ```julia mutable struct VariantState{T, tempT} <: AbstractSolverState{Solver} x::tempT other::tempT

    ...

    iteration::Int64 end

    SolverVariant(A; kwargs...) = Solver(A, VariantState(kwargs...))

    function iterate(solver::Solver, state::VarianteState)

    Custom iteration

    end

Compressed Sensing Example

In this example we will go through a simple example from the field of Compressed Sensing. In addtion to RegularizedLeastSquares.jl, we will need the packages LinearOperatorCollection.jl, ImagePhantoms.jl, ImageGeoms.jl and Random.jl, as well as CairoMakie.jl for visualization. We can install them the same way we did RegularizedLeastSquares.jl.

Preparing the Inverse Problem

To get started, let us generate a simple phantom using the ImagePhantoms and ImageGeom packages:

using ImagePhantoms, ImageGeoms
+N = 256
+image = shepp_logan(N, SheppLoganToft())
+size(image)
(256, 256)

This produces a 256x256 image of a Shepp-Logan phantom.

In this example, we consider a problem in which we randomly sample a third of the pixels in the image. Such a problem and the corresponding measurement can be constructed with the packages LinearOperatorCollection and Random:

We first randomly shuffle the indices of the image and then select the first third of the indices to sample.

using Random, LinearOperatorCollection
+randomIndices = shuffle(eachindex(image))
+sampledIndices = sort(randomIndices[1:div(end, 3)])
21845-element Vector{Int64}:
+     1
+     3
+     5
+     7
+     8
+    13
+    16
+    19
+    21
+    22
+     ⋮
+ 65511
+ 65512
+ 65513
+ 65514
+ 65517
+ 65519
+ 65530
+ 65532
+ 65533

Afterwards we build a sampling operator which samples the image at the selected indices.

A = SamplingOp(eltype(image), pattern = sampledIndices , shape = size(image));

Then we apply the sampling operator to the vectorized image to obtain the sampled measurement vector

b = A*vec(image);

To visualize our image we can use CairoMakie:

using CairoMakie
+function plot_image(figPos, img; title = "", width = 150, height = 150)
+  ax = CairoMakie.Axis(figPos; yreversed=true, title, width, height)
+  hidedecorations!(ax)
+  heatmap!(ax, img)
+end
+fig = Figure()
+plot_image(fig[1,1], image, title = "Image")
+samplingMask = fill(false, size(image))
+samplingMask[sampledIndices] .= true
+plot_image(fig[1,2], image .* samplingMask, title = "Sampled Image")
+resize_to_layout!(fig)
+fig
Example block output

As we can see in the right image, only a third of the pixels are sampled. The goal of the inverse problem is to recover the original image from this measurement vector.

Solving the Inverse Problem

To recover the image from the measurement vector, we solve the TV-regularized least squares problem:

\[\begin{equation} + \underset{\mathbf{x}}{argmin} \frac{1}{2}\vert\vert \mathbf{A}\mathbf{x}-\mathbf{b} \vert\vert_2^2 + \vert\vert\mathbf{x}\vert\vert_{\lambda\text{TV}} . +\end{equation}\]

For this purpose we build a TV regularizer with regularization parameter $λ=0.01$

using RegularizedLeastSquares
+reg = TVRegularization(0.01; shape=size(image));

We will use the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) to solve our inverse problem. Thus, we build the corresponding solver

solver = createLinearSolver(FISTA, A; reg=reg, iterations=20);

and apply it to our measurement vector

img_approx = solve!(solver,b)
65536-element Vector{Float32}:
+ 9.888277f-19
+ 1.7648029f-18
+ 3.61334f-18
+ 7.428132f-18
+ 1.6200033f-17
+ 3.69934f-17
+ 8.16212f-17
+ 1.7291627f-16
+ 3.6464276f-16
+ 7.5773333f-16
+ ⋮
+ 4.1033114f-16
+ 1.924142f-16
+ 8.822592f-17
+ 3.88645f-17
+ 1.5847841f-17
+ 6.0871485f-18
+ 2.4316864f-18
+ 1.092523f-18
+ 6.154074f-19

To visualize the reconstructed image, we need to reshape the result vector to the correct shape. Afterwards we can use CairoMakie again:

img_approx = reshape(img_approx,size(image));
+plot_image(fig[1,3], img_approx, title = "Reconstructed Image")
+resize_to_layout!(fig)
+fig
Example block output

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