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| *.jl.mem | ||
| docs/build/ | ||
| docs/site/ | ||
| docs/src/generated/ | ||
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| Manifest.toml | ||
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| [deps] | ||
| BenchmarkTools = "6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf" | ||
| CairoMakie = "13f3f980-e62b-5c42-98c6-ff1f3baf88f0" | ||
| Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4" | ||
| ImageGeoms = "9ee76f2b-840d-4475-b6d6-e485c9297852" | ||
| ImagePhantoms = "71a99df6-f52c-4da1-bd2a-69d6f37f3252" | ||
| JLArrays = "27aeb0d3-9eb9-45fb-866b-73c2ecf80fcb" | ||
| LinearOperatorCollection = "a4a2c56f-fead-462a-a3ab-85921a5f2575" | ||
| Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" | ||
| Literate = "98b081ad-f1c9-55d3-8b20-4c87d4299306" | ||
| RadonKA = "86de8297-835b-47df-b249-c04e8db91db5" | ||
| RegularizedLeastSquares = "1e9c538a-f78c-5de5-8ffb-0b6dbe892d23" | ||
| UnicodePlots = "b8865327-cd53-5732-bb35-84acbb429228" | ||
| Wavelets = "29a6e085-ba6d-5f35-a997-948ac2efa89a" | ||
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| [compat] | ||
| Documenter = "1.0, 1.1, 1.2" | ||
| Documenter = "1" |
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| # # 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. | ||
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| # ## Preparing the Inverse Problem | ||
| # To get started, let us generate a simple phantom using the ImagePhantoms and ImageGeom packages: | ||
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| using ImagePhantoms, ImageGeoms | ||
| N = 256 | ||
| image = shepp_logan(N, SheppLoganToft()) | ||
| size(image) | ||
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| # This produces a 256x256 image of a Shepp-Logan phantom. | ||
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| # 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: | ||
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| # 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)]) | ||
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| # Afterwards we build a sampling operator which samples the image at the selected indices. | ||
| A = SamplingOp(eltype(image), pattern = sampledIndices , shape = size(image)); | ||
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| # Then we apply the sampling operator to the vectorized image to obtain the sampled measurement vector | ||
| b = A*vec(image); | ||
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| # 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 | ||
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| # 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. | ||
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| # ## 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} | ||
| # ``` | ||
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| # For this purpose we build a TV regularizer with regularization parameter $λ=0.01$ | ||
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| using RegularizedLeastSquares | ||
| reg = TVRegularization(0.01; shape=size(image)); | ||
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| # We will use the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) to solve our inverse problem. Thus, we build the corresponding solver | ||
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| solver = createLinearSolver(FISTA, A; reg=reg, iterations=20); | ||
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| # and apply it to our measurement vector | ||
| img_approx = solve!(solver,b) | ||
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| # 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 |
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| # # 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. | ||
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| # 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. | ||
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| # ## Preparing the Inverse Problem | ||
| # To get started, let us generate a simple phantom using the ImagePhantoms and ImageGeom packages: | ||
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| using ImagePhantoms, ImageGeoms | ||
| N = 256 | ||
| image = shepp_logan(N, SheppLoganToft()) | ||
| size(image) | ||
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| # This produces a 64x64 image of a Shepp-Logan phantom. | ||
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| using RadonKA, LinearOperatorCollection | ||
| angles = collect(range(0, π, 256)) | ||
| sinogram = Array(RadonKA.radon(image, angles)) | ||
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| # Afterwards we build a Radon operator implementing both the forward and adjoint Radon transform | ||
| A = RadonOp(eltype(image); angles, shape = size(image)); | ||
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| # 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 | ||
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| # 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. | ||
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| # ## 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} | ||
| # ``` | ||
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| # For this purpose we build a $l^2_2$ with regularization parameter $λ=0.001$ | ||
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| using RegularizedLeastSquares | ||
| reg = L2Regularization(0.001); | ||
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| # 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); | ||
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| # and apply it to our measurement vector | ||
| img_approx = solve!(solver, vec(sinogram)) | ||
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| # 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 |
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| # # Getting Started | ||
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| # In this example we will go through a simple random inverse problem to get familiar with RegularizedLeastSquares.jl. | ||
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| # ## Installation | ||
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| # To install RegularizedLeastSquares.jl, you can use the Julia package manager. Open a Julia REPL and run the following command: | ||
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| # ```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). | ||
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| # Once the installation is complete, you can import the package with the `using` keyword: | ||
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| using RegularizedLeastSquares | ||
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| # 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. | ||
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| A = rand(32, 16) | ||
| x = rand(16) | ||
| b = A*x; | ||
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| # 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); | ||
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| # and apply it to our measurement vector | ||
| x_approx = solve!(solver, b) | ||
| isapprox(x, x_approx, rtol = 0.001) | ||
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| # 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} | ||
| # ``` | ||
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| # 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) |
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