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+# Diffusion derivatives
+
+## Preprocessed diffusion-weighted images
+
+-   As with [raw diffusion imaging data](../04-modality-specific-files/01-magnetic-resonance-imaging-data.md#required-gradient-orientation-information),
+    inclusion of gradient orientation information is REQUIRED. While
+    [the inheritance principle](../02-common-principles.md#the-inheritance-principle)
+    applies, it is common for DWI preprocessing to include rotation of
+    gradient orientations according to subject motion, so series-specific
+    gradient information is typically expected.
+
+-   As per [file naming conventions for BIDS Derivatives](01-introduction.md#file-naming-conventions),
+    preprocessed DWI data must not possess the same name as that of the raw
+    DWI data. It is RECOMMENDED to disambiguate through use of the key-value
+    "`_desc-preproc`".
+
+-   As per [common data types](02-common-data-types.md) for derivative data, a
+    JSON sidecar file is REQUIRED due to the REQUIRED `SkullStripped` field.
+
+```Text
+<pipeline_name>/
+    sub-<participant_label>/
+        dwi/
+            <source_keywords>[_space-<space>]_desc-preproc_dwi.nii[.gz]
+            <source_keywords>[_space-<space>]_desc-preproc_dwi.bvals
+            <source_keywords>[_space-<space>]_desc-preproc_dwi.bvecs
+            <source_keywords>[_space-<space>]_desc-preproc_dwi.json
+```
+
+## Diffusion models
+
+Diffusion MRI can be modeled using various paradigms
+to extract more informative representations of the diffusion process
+and the underlying biological structure.
+There are two key attributes of diffusion models
+that warrant explicit mention due to their consequence in how they are represented:
+
+-   The encoding of *anisotropic* information;
+    that is, quantities for which the value depends on the orientation in which it is sampled.
+    Much of the detail in diffusion derivatives is therefore dedicated
+    to the definition of how such data are serialized into corresponding NIfTI images,
+    and how corresponding metadata is used to facilitate correct interpretation of those data.
+
+-   A diffusion model may represent the contents of each image voxel as the sum of *multiple compartments*;
+    further, each of these different compartments may have different intrinsic properties,
+    such as units or anisotropy (as above).
+    An individual model fit may therefore yield data for its different parameters
+    distributed across multiple NIfTI images,
+    and each image may require unique metadata fields to facilitate interpretation.
+
+### Basic filesystem structure
+
+```Text
+<pipeline_name>/
+    sub-<participant_label>/
+        dwi/
+            <source_keywords>[_space-<space>]_model-<label>[_desc-<desc>]_param-<label1>_dwimap.nii[.gz]
+            <source_keywords>[_space-<space>]_model-<label>[_desc-<desc>]_param-<label1>_dwimap.json
+            <source_keywords>[_space-<space>]_model-<label>[_desc-<desc>]_param-<label2>_dwimap.nii[.gz]
+            <source_keywords>[_space-<space>]_model-<label>[_desc-<desc>]_param-<label2>_dwimap.json
+            <source_keywords>[_space-<space>]_model-<label>[_desc-<desc>]_param-<label3>_dwimap.nii[.gz]
+            <source_keywords>[_space-<space>]_model-<label>[_desc-<desc>]_param-<label3>_dwimap.json
+```
+
+-   Files "`<source_keywords>_model-<label>_param-<label*>_dwimap.nii[.gz]`"
+    provide image data encoding the different parameters that may be estimated by the model.
+    If the image is a three-dimensional volume,
+    then in the absence of any metadata indicating to the contrary,
+    the image should be interpreted as yielding a single scalar parameter per voxel.
+    If the image is of a higher dimensionality,
+    then relevant metadata fields MUST be specified in the corresponding sidecar JSON file
+    indicating how data across dimensions beyond the three spatial dimensions
+    should be interpreted.
+
+-   Files "`<source_keywords>_model-<label>_param-<label*>_dwimap.json`"
+    MUST provide information about the model,
+    and SHOULD provide information about how it was fit to the empirical image data.
+    In circumstances where the dimensionality of the corresponding NIfTI image is greater than three,
+    they MUST also specify requisite metadata fields regarding
+    how data across dimensions beyond the three spatial dimensions are to be interpreted.
+
+### Orientation encoding types
+
+There are many mathematical bases that may be used
+in the encoding of parameters that vary as a function of orientation.
+A list of such functions supported by the specification is enumerated below,
+accompanied by requisite information specific to each representation.
+
+For image data that encode orientation information,
+there are fields that MUST be specified in the sidecar JSON file
+in order to ensure appropriate interpretation of that information;
+see [parameter metadata](#parameter-metadata).
+
+1.  <a name="encoding-scalar">*Scalar image*</a>:
+
+    Image data that do not encode orientation information
+    are referred to henceforth here as "scalar" parameters.
+
+1.  <a name="encoding-dec">*Directionally-Encoded Colors (DEC)*</a>:
+
+    An image with three volumes,
+    intended to be interpreted as red, green and blue color intensities for visualization.
+    Image data MUST NOT contain negative values.
+
+1.  <a name="encoding-spherical">*Spherical coordinates*</a>:
+
+    An image where data across volumes within each voxel encode
+    one or more discrete orientations using angles on the 2-sphere,
+    optionally encoding some parameter as the distance from origin.
+
+    This may take one of two forms:
+
+    1.  Value per direction
+
+        Each consecutive triplet of image volumes encodes a 3-tuple spherical coordinate,
+        using ISO convention for both the order of parameters
+        and reference frame for angles:
+
+        1.  Distance from origin,
+            encoding some non-negative parameter of interest.
+
+        1.  Inclination / polar angle in radians,
+            relative to the zenith direction being the positive direction of the *third* reference axis
+            (see [parameter metadata](#parameter-metadata));
+
+        1.  Azimuth angle, in radians,
+            orthogonal to the zenith direction,
+            with value of 0.0 corresponding to the *first* reference axis
+            (see [parameter metadata](#parameter-metadata)),
+            increasing toward the positive direction of the *second* reference axis.
+
+        Number of image volumes is equal to (3x*N*),
+        where *N* is the maximum number of discrete orientations in any voxel in the image.
+
+    1.  Orientations only
+
+        Each consecutive pair of image volumes encodes an inclination / azimuth pair,
+        with order & convention identical to that above
+        (equivalent to spherical coordinate with assumed unity distance from origin).
+
+        Number of image volumes is equal to (2x*N*), where *N* is the maximum
+        number of discrete orientations in any voxel in the image.
+
+1.  <a name="encoding-3vector">*3-Vectors*</a>:
+
+    An image where data across volumes within each voxel encode
+    one or more discrete orientations
+    using triplets of axis dot products.
+
+    This may take one of two forms:
+
+    1.  Value per orientation
+
+        The *norm* of the 3-vector encodes some non-negative parameter of interest,
+        while its normalized form encodes an orientation on the unit sphere.
+
+    1.  Orientations only
+
+        Each triplet of values encodes an orientation on the unit sphere
+        (that is, the vector norm MUST be 1.0);
+        no quantitative value is associated with each orientation.
+
+    Number of image volumes is equal to (3x*N*),
+    where *N* is the maximum number of discrete orientations in any voxel in the image.
+
+1.  <a name="encoding-tensor">*Tensor*</a>:
+
+    An image where volumes encode coefficients of a tensor.
+
+    -   For *Rank 2* tensors:
+
+        If antipodal symmetry is set (or implicitly assumed),
+        then the image MUST contain six volumes, in the order:
+        *D<sub>11</sub>*, *D<sub>12</sub>*, *D<sub>13</sub>*, *D<sub>22</sub>*, *D<sub>23</sub>*, *D<sub>33</sub>,
+        where *1*, *2* and *3* index the three spatial dimensions according to their reference
+        (see field `"OrientationEncoding"["Reference"]` in [parameter metadata](#parameter-metadata)).
+        If the data are not antipodally symmetric,
+        then the image MUST contain nine volumes, in the order:
+        *D<sub>11</sub>*, *D<sub>12</sub>*, *D<sub>13</sub>*, *D<sub>21</sub>*, *D<sub>22</sub>*, *D<sub>23</sub>, *D<sub>31</sub>*, *D<sub>32</sub>*, *D<sub>33</sub>,
+        with subscripts indexing row then column.
+
+1.  <a name="encoding-sh">*Spherical Harmonics (SH)*</a>:
+
+    Image where data across volumes within each voxel encode
+    a continuous function spanning the 2-sphere
+    using coefficients within a spherical harmonics basis.
+
+    Number of image volumes depends on the spherical harmonic basis employed,
+    and the maximal spherical harmonic degree *l<sub>max</sub>*
+    (see [spherical harmonics bases](#spherical-harmonics-bases)).
+
+1.  <a name="encoding-amp">*Amplitudes*</a>:
+
+    Image where data across volumes within each voxel encode
+    amplitudes of a discrete function spanning the 2-sphere.
+
+    Number of image volumes corresponds to the number of discrete orientations
+    on the unit sphere along which samples for the spherical function in
+    each voxel are provided;
+    these orientations MUST themselves be provided in the associated sidecar JSON file
+    (see [parameter metadata](#parameter-metadata)).
+
+### Bootstrap encoding
+
+For some models,
+it is common to export not only the best fit of the model to the empirical data,
+but multiple realizations of that fit taking into account image noise,
+typically through some form of bootstrapping procedure.
+Where this occurs,
+the image data for each parameter possess an additional dimension,
+along which those multiple realisations are stored.
+
+For some models,
+it is common to explicitly store *both* the multiple realisations of the model
+*and* either the parameters corresponding to the maximum *a posteriori* fit
+or the mean of each parameter computed across those realisations.
+In these circumstances,
+it is RECOMMENDED to use the same label for the "`_model-`" entity,
+and use the "`_desc-`" entity to disambiguate between these two versions
+at the filesystem level.
+
+### Metadata fields
+
+#### Model vs. parameter metadata
+
+For a NIfTI image that encodes some parameter of some model,
+there are are range of metadata fields that may be relevant.
+These can be broadly separated into two categories:
+
+1.  Metadata that apply to *the model as a whole*
+    are not specific to any individual parameter estimated by that model.
+    It is therefore RECOMMENDED that any such metadata
+    be *equivalent* across all sidecar JSON files for all parameters of that model.
+
+1.  Metadata that apply *only to that specific parameter*
+    may include attributes such as units and orientation encoding
+    essential for correct interpretation of that parameter image *only*.
+    it is therefore possible that these metadata fields may *differ*
+    across the multiple parameters estimated by that model.
+
+#### Model metadata
+
+At the root of the metadata dictionary,
+REQUIRED field `"Model"` defines a dictionary that contains relevant information
+about what the model is and how it was fit to empirical image data.
+The following table defines reserved fields within the `"Model"` sub-dictionary.
+
+| **Key name**        | **Description**                                                                                                                 |
+| ------------------- | ------------------------------------------------------------------------------------------------------------------------------- |
+| BootstrapParameters | OPTIONAL. Dictionary. Parameters relating to the generation of multiple realisations of the model fit using bootstrapping.      |
+| Description         | OPTIONAL. String. Extended text-based information to describe the model.                                                        |
+| Parameters          | OPTIONAL. Dictionary. Parameters that influenced the process of fitting the model to empirical image data (see examples below). |
+| URL                 | OPTIONAL. String. URL to the specific implementation of the model utilized.                                                     |
+
+Dictionary `"Model["Parameters"]"` has the following reserved keywords that may be applicable to a broad range of models:
+
+| **Key name**           | **Description**                                                                                                                                                                                                                                                                                               |
+| ---------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
+| FitMethod              | OPTIONAL. String. The optimization procedure used to fit the intrinsic model parameters to the empirical diffusion-weighted signal. Reserved values are: "`ols`" (Ordinary Least Squares); "`wls`" (Weighted Least Squares); "`iwls`" (Iterative Weighted Least Squares); "`nlls`" (Non-Linear Least Squares). |
+| Iterations             | OPTIONAL. Integer. The number of iterations used for any form of model fitting procedure where the number of iterations is a fixed input parameter.                                                                                                                                                           |
+| OutlierRejectionMethod | OPTIONAL. String. Text describing any form of rejection of outlier values that was performed during fitting of the model.                                                                                                                                                                                     |
+| Samples                | OPTIONAL. Integer. The number of realisations of a diffusion model from which statistical summaries (such as mean, standard deviation) of those parameters were computed.                                                                                                                                        |
+
+#### Parameter metadata
+
+The following tables define reserved fields relevant to individual model parameters.
+
+Some fields are relevant only to specific [orientation encoding types](#orientation-encoding-types):
+
+-   Where a field *is* relevant for the corresponding image,
+    designators OPTIONAL and REQUIRED within the "Description" column apply.
+
+-   Where a field is *not* relevant for the corresponding image,
+    that metadata field MUST NOT be specified.
+
+| **Key name**        | Relevant [orientation encoding types](#orientation-encoding-types)                         | **Description**                                                                                                                                                                                                                       |
+| ------------------- | ------------------------------------------------------------------------------------------ | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
+| BootstrapAxis       | Any                                                                                        | OPTIONAL. Integer. If multiple realisations of a given parameter are stored in a NIfTI image, this field nominates the image axis (indexed from zero) along which those multiple realisations are stored.                             |
+| Description         | Any                                                                                        | OPTIONAL. String. Text description of what model parameter is encoded in the corresponding data file.                                                                                                                                 |
+| NonNegativity       | All except [spherical coordinates](#encoding-spherical) and [3-vectors](#encoding-3vector) | OPTIONAL. String. Options are: { `regularized`, `constrained` }. Specifies whether, during model fitting, the parameter was regularized to not take extreme negative values, or was explicitly forbidden from taking negative values. |
+| OrientationEncoding | Any                                                                                        | REQUIRED if dimensionality of NIfTI image is greater than three. Dictionary. Provides information requisite to the interpretation of orientation information encoded in each voxel; more details below.                               |
+| ParameterURL        | Any                                                                                        | OPTIONAL. String. URL to documentation that describes the specific model parameter that is encoded within the data file.                                                                                                              |
+| ResponseFunction    | [Spherical harmonics](#encoding-sh)                                                        | OPTIONAL. Dictionary. Specifies a response function that was utilized by a deconvolution algorithm; more details below.                                                                                                               |
+
+Dictionary `"OrientationEncoding"` has the following reserved keywords:
+
+| **Key name**            | Relevant [orientation encoding types](#orientation-encoding-types)                                                                                                         | **Description**                                                                                                                                                                                                                                                                                                                                                                                                              |
+| ----------------------- | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
+| AmplitudesDirections    | [Amplitudes](#encoding-amp)                                                                                                                                                | REQUIRED for `"Type": "amplitudes"`; MUST NOT be specified otherwise. List of lists of floats. Data are either [spherical coordinates (directions only)](#encoding-spherical) or [3-vectors](#encoding-3vector) with unit norm. Defines the dense directional basis set on which samples of a spherical function within each voxel are provided. The length of the list must be equal to the number of volumes in the image. |
+| AntipodalSymmetry       | [spherical coordinates](#encoding-spherical), [3-vectors](#encoding-3vector), [tensor](#encoding-tensor), [amplitudes](#encoding-amp), [spherical harmonics](#encoding-sh) | OPTIONAL. Boolean. Indicates whether orientation information should be interpreted as being antipodally symmetric. Assumed to be True if omitted.                                                                                                                                                                                                                                                                            |
+| EncodingAxis            | All except [scalar](#encoding-scalar)                                                                                                                                      | REQUIRED. Integer. Indicates the image axis (indexed from zero) along which image intensities should be interpreted as corresponding to orientation encoding.                                                                                                                                                                                                                                                                |
+| FillValue               | [Scalar](#encoding-scalar), [spherical coordinates](#encoding-spherical), [3-vectors](#encoding-3vector)                                                                   | OPTIONAL. Float; allowed values: { 0.0, NaN }. Value stored in image when the number of discrete orientations in a given voxel is fewer than the maximal number for that image.                                                                                                                                                                                                                                              |
+| Reference               | All except [scalar](#encoding-scalar)                                                                                                                                      | REQUIRED. String; allowed values: { `bvec`, `ijk`, `xyz` }. Defines the reference coordinate system against which orientation information is encoded (more below).                                                                                                                                                                                                                                                           |
+| SphericalHarmonicBasis  | [Spherical harmonics](#encoding-sh)                                                                                                                                        | REQUIRED for `"Type": "sh"`; MUST NOT be specified otherwise. String. Options are: { `mrtrix3`, `descoteaux` }. Details are provided in the [spherical harmonics bases](#spherical-harmonics-bases) section.                                                                                                                                                                                                                 |
+| SphericalHarmonicDegree | [Spherical harmonics](#encoding-sh)                                                                                                                                        | OPTIONAL for `"Type": "sh"`; MUST NOT be specified otherwise. Integer. The maximal spherical harmonic order *l<sub>max</sub>*; the number of volumes in the associated NIfTI image must correspond to this value as per the relationship described in [spherical harmonics bases](#spherical-harmonics-bases) section.                                                                                                       |
+| TensorRank              | [Tensor](#encoding-tensor)                                                                                                                                                 | REQUIRED for `"Type": "tensor"; MUST NOT be specified otherwise. Integer. Rank of tensor reporesentation. Specification currently only supports a value of 2.                                                                                                                                                                                                                                                                |
+| Type                    | Any                                                                                                                                                                        | REQUIRED. String. Specifies the type of orientation information (if any) encoded in the NIfTI image. Permitted values: { `scalar`, `dec`, `unitspherical`, `spherical`, `unit3vector`, `3vector`, `tensor`, `sh`, `amplitudes` }.                                                                                                                                                                                            |
+
+Field `"OrientationEncoding"["Reference"]` MUST contain one of the following values:
+
+| **Value** | **Interpretation**                                                                                                                                                                                                                                                            |
+| --------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
+| bvec      | The three spatial image axes; **unless** those axes form a right-handed coordinate system (that is, the 3x3 linear component of the NIfTI header transformation has a positive determinant), in which case the negative of the first axis orientation is the first reference. |
+| ijk       | The three spatial image axes define the orientation reference.                                                                                                                                                                                                                |
+| xyz       | The "real" / "scanner" space axes, which are independent of the NIfTI image header transform, define the orientation reference.                                                                                                                                               |
+
+Dictionary `"ResponseFunction"` has the following reserved keywords:
+
+| **Key name** | **Description**                                                                                                                                                                            |
+| ------------ | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ |
+| Coefficients | REQUIRED. Either a list of floats, or a list of lists of floats, depending on the mathematical form of the response function and possibly the data to which it applies; see further below. |
+| Type         | REQUIRED. String. Options are: { `eigen`, `zsh` }. The mathematical form in which the response function coefficients are provided; see further below.                                      |
+
+-   If `"ResponseFunction"["Type"]: "eigen"`,
+    then a list of 4 floating-point values must be specified;
+    these are interpreted as three ordered eigenvalues of a rank 2 tensor,
+    followed by a reference *b*=0 intensity.
+
+-   If `"ResponseFunction"["Type"]": "zsh`",
+    then the values provided can be one of the following:
+
+    -   List of floating-point values.
+        Values correspond to the response function coefficient for each consecutive even zonal spherical harmonic degree starting from zero.
+
+    -   List of lists of floating-point values.
+        One list per unique *b*-value.
+        Each individual list contains a coefficient per even zonal spherical harmonic degree starting from zero.
+        If the response function utilized has a different number of non-zero zonal spherical harmonic coefficients for different *b*-values,
+        these must be padded with zeroes such that all lists contain the same number of floating-point values.
+
+### Demonstrative examples
+
+#### A basic Diffusion Tensor fit
+
+<!-- This block generates a file tree.
+A guide for using macros can be found at
+ https://github.com/bids-standard/bids-specification/blob/master/macros_doc.md
+-->
+{{ MACROS___make_filetree_example(
+    {
+    "dti_pipeline": {
+        "sub-01": {
+        "dwi": {
+            "sub-01_model-tensor_param-diffusivity_dwimap.nii.gz": "",
+            "sub-01_model-tensor_param-diffusivity_dwimap.json": "",
+            "sub-01_model-tensor_param-s0_dwimap.nii.gz": "",
+            "sub-01_model-tensor_param-s0_dwimap.json": "",
+            "sub-01_model-tensor_param-fa_dwimap.nii.gz": "",
+            "sub-01_model-tensor_param-fa_dwimap.json": "",
+        },
+        },
+    },
+    }
+) }}
+
+Dimensions of NIfTI image "`sub-01_model-tensor_param-diffusivity_dwimap.nii.gz`": *I*x*J*x*K*x6 ([symmetric rank 2 tensor image](#encoding-tensor))
+Dimensions of NIfTI image "`sub-01_model-tensor_param-s0_dwimap.nii.gz`": *I*x*J*x*K* ([scalar](#encoding-scalar))
+Dimensions of NIfTI image "`sub-01_model-tensor_param-fa_dwimap.nii.gz`": *I*x*J*x*K* ([scalar](#encoding-scalar))
+
+Contents of file `sub-01_model-tensor_param-diffusivity_dwimap.json`:
+
+```JSON
+{
+    "Description": "Diffusion Coefficient, encoded as a tensor representation",
+    "Model": {
+        "Description": "Diffusion Tensor",
+        "Parameters": {
+            "FitMethod": "ols",
+            "OutlierRejectionMethod": "None"
+        }
+    },
+    "OrientationEncoding": {
+        "AntipodalSymmetry": true,
+        "EncodingAxis": 3,
+        "Reference": "xyz",
+        "TensorRank": 2,
+        "Type": "tensor"
+    },
+    "Units": "mm^2/s"
+}
+```
+
+Contents of file `sub-01_model-tensor_param-s0_dwimap.json`:
+
+```JSON
+{
+    "Description": "Estimated signal intensity with no diffusion weighting, ie. S0",
+    "Model": {
+        "Description": "Diffusion Tensor",
+        "Parameters": {
+            "FitMethod": "ols",
+            "OutlierRejectionmethod": "None"
+        }
+    }
+}
+```
+
+Contents of file `sub-01_model-tensor_param-fa_dwimap.json`:
+
+```JSON
+{
+    "Description": "Fractional Anisotropy",
+    "Model": {
+        "Description": "Diffusion Tensor",
+        "Parameters": {
+            "FitMethod": "ols",
+            "OutlierRejectionmethod": "None"
+        }
+    },
+    "ParameterURL": "https://doi.org/10.1002/nbm.1940080707"
+}
+```
+
+Notes:
+
+-   "The diffusion tensor" intrinsically is a mathematical model of how
+    the diffusivity is estimated to vary as a function of orientation.
+    As such, within this model,
+    it is not the *parameter* that is a tensor;
+    rather, it is the *diffusivity* that is the estimated parameter,
+    and the *way in which the anisotropy of that parameter is encoded* is a tensor.
+
+-   Even if image `sub-01_model-tensor_param-diffusivity_dwimap.nii.gz`
+    were to be the only image yielded by the pipeline,
+    it is nevertheless REQUIRED to include entity "`_param-`" in those file names.
+
+-   Metadata fields relevant to the interpretation of the anisotropy of the tensor
+    are *not relevant* to the scalar measures
+    "`s0`" (estimated signal intensity with no diffusion weighting)
+    or "`fa`" (Fractional Anisotropy).
+    Those fields therefore MUST be omitted from the corresponding sidecar JSONs.
+
+#### A multi-shell, multi-tissue Constrained Spherical Deconvolution fit
+
+<!-- This block generates a file tree.
+A guide for using macros can be found at
+ https://github.com/bids-standard/bids-specification/blob/master/macros_doc.md
+-->
+{{ MACROS___make_filetree_example(
+    {
+    "msmtcsd_pipeline": {
+        "sub-01": {
+        "dwi": {
+            "sub-01_model-csd_param-wm_dwimap.nii.gz": "",
+            "sub-01_model-csd_param-wm_dwimap.json": "",
+            "sub-01_model-csd_param-gm_dwimap.nii.gz": "",
+            "sub-01_model-csd_param-gm_dwimap.json": "",
+            "sub-01_model-csd_param-csf_dwimap.nii.gz": "",
+            "sub-01_model-csd_param-csf_dwimap.nii.gz": "",
+        },
+        },
+    },
+    }
+) }}
+
+Dimensions of NIfTI image "`sub-01_model-csd_param-wm_model.nii.gz`": *I*x*J*x*K*x45 ([spherical harmonics](#encoding-sh))
+Dimensions of NIfTI image "`sub-01_model-csd_param-gm_model.nii.gz`": *I*x*J*x*K*x1 ([spherical harmonics](#encoding-sh))
+Dimensions of NIfTI image "`sub-01_model-csd_param-csf_model.nii.gz`": *I*x*J*x*K*x1 ([spherical harmonics](#encoding-sh))
+
+Contents of JSON file "`sub-01_model-csd_param-wm_model.json`":
+
+```JSON
+{
+    "Model": {
+        "Description": "Multi-Shell Multi-Tissue (MSMT) Constrained Spherical Deconvolution (CSD)",
+        "URL": "https://mrtrix.readthedocs.io/en/latest/constrained_spherical_deconvolution/multi_shell_multi_tissue_csd.html",
+    },
+    "Description": "White matter",
+    "NonNegativity": "constrained",
+    "OrientationEncoding": {
+        "EncodingAxis": 3,
+        "Reference": "xyz",
+        "SphericalHarmonicBasis": "MRtrix3",
+        "SphericalHarmonicDegree": 8,
+        "Type": "sh",
+    },
+    "ParameterURL": "http://www.sciencedirect.com/science/article/pii/S1053811911012092",
+    "ResponseFunction": {
+        "Coefficients": [ [ 600.2 0.0 0.0 0.0 0.0 0.0 ],
+                            [ 296.3 -115.2 24.7 -4.4 -0.5 1.8 ],
+                            [ 199.8 -111.3 41.8 -10.2 2.1 -0.7 ],
+                            [ 158.3 -98.7 48.4 -17.1 4.5 -1.4 ] ],
+        "Type": "zsh"
+    }
+}
+```
+
+Contents of JSON file "`sub-01_model-csd_param-gm_dwimap.json`":
+
+```JSON
+{
+    "Model": {
+        "Description": "Multi-Shell Multi-Tissue (MSMT) Constrained Spherical Deconvolution (CSD)",
+        "URL": "https://mrtrix.readthedocs.io/en/latest/constrained_spherical_deconvolution/multi_shell_multi_tissue_csd.html",
+    },
+    "Description": "Gray matter",
+    "NonNegativity": "constrained",
+    "OrientationEncoding": {
+        "EncodingAxis": 3,
+        "Reference": "xyz",
+        "SphericalHarmonicBasis": "MRtrix3",
+        "SphericalHarmonicDegree": 0,
+        "Type": "sh",
+    },
+    "ResponseFunction": {
+        "Coefficients": [ [ 1041.0 ],
+                            [ 436.6 ],
+                            [ 224.9 ],
+                            [ 128.8 ] ],
+        "Type": "zsh"
+    }
+}
+```
+
+Contents of JSON file "`sub-01_model-csd_param-csf_dwimap.json`":
+
+```JSON
+{
+    "Model": {
+        "Description": "Multi-Shell Multi-Tissue (MSMT) Constrained Spherical Deconvolution (CSD)",
+        "URL": "https://mrtrix.readthedocs.io/en/latest/constrained_spherical_deconvolution/multi_shell_multi_tissue_csd.html",
+    },
+    "Description": "Cerebro-spinal fluid",
+    "NonNegativity": "constrained",
+    "OrientationEncoding": {
+        "EncodingAxis": 3,
+        "Reference": "xyz",
+        "SphericalHarmonicBasis": "MRtrix3",
+        "SphericalHarmonicDegree": 0,
+        "Type": "sh"
+    },
+    "ResponseFunction": {
+        "Coefficients": [ [ 3544.90770181 ],
+                            [ 134.441453035 ],
+                            [ 32.0826839826 ],
+                            [ 29.3674604452 ] ],
+        "Type": "zsh"
+    }
+}
+```
+
+Notes:
+
+-   In this example,
+    the gray matter and CSF compartments are specified in the spherical harmonics basis
+    with maximal spherical harmonic degrees of zero,
+    even though each image only contains a single volume
+    and could therefore be interpreted as simply scalar parameters.
+    This is recommended in this instance
+    given that the spherical harmonic basis imposes a $\sqrt(4\pi)$ scaling
+    that should be taken into account if comparing the values of these parameters
+    with the *l*=0 term of the white matter ODF.
+
+-   The response functions for GM and CSF have a maximal zonal spherical harmonic degree of zero,
+    such that only one coefficient is required for each unique *b*-value shell.
+    It is however nevertheless vital that these data be provided as a list of lists of floats,
+    where the length of each list is one;
+    storing these values as a list of floats would be erroneously interpreted
+    as coefficients of different zonal spherical harmonic degrees for a single *b*-value shell.
+
+#### An FSL `bedpostx` Ball-And-Sticks fit
+
+This example includes both bootstrap realisations of the model fit,
+and the aggregated means of those parameters across realisations.
+
+<!-- This block generates a file tree.
+A guide for using macros can be found at
+ https://github.com/bids-standard/bids-specification/blob/master/macros_doc.md
+-->
+{{ MACROS___make_filetree_example(
+    {
+    "bedpostx_pipeline": {
+        "sub-01": {
+        "dwi": {
+            "sub-01_model-bs_desc-mean_param-s0_dwimap.nii.gz": "",
+            "sub-01_model-bs_desc-mean_param-s0_dwimap.json": "",
+            "sub-01_model-bs_desc-mean_param-polar_dwimap.nii.gz": "",
+            "sub-01_model-bs_desc-mean_param-polar_dwimap.json": "",
+            "sub-01_model-bs_desc-mean_param-vector_dwimap.nii.gz": "",
+            "sub-01_model-bs_desc-mean_param-vector_dwimap.json": "",
+            "sub-01_model-bs_desc-mean_param-vf_dwimap.nii.gz": "",
+            "sub-01_model-bs_desc-mean_param-vf_dwimap.json": "",
+            "sub-01_model-bs_desc-mean_param-vfsum_dwimap.nii.gz": "",
+            "sub-01_model-bs_desc-mean_param-vfsum_dwimap.json": "",
+            "sub-01_model-bs_desc-mean_param-diffusivity_dwimap.nii.gz": "",
+            "sub-01_model-bs_desc-mean_param-diffusivity_dwimap.json": "",
+            "sub-01_model-bs_desc-mean_param-dstd_dwimap.nii.gz": "",
+            "sub-01_model-bs_desc-mean_param-dstd_dwimap.json": "",
+            "sub-01_model-bs_desc-merged_param-polar_dwimap.nii.gz": "",
+            "sub-01_model-bs_desc-merged_param-polar_dwimap.json": "",
+            "sub-01_model-bs_desc-merged_param-vf_dwimap.nii.gz": "",
+            "sub-01_model-bs_desc-merged_param-vf_dwimap.json": "",
+        },
+        },
+    },
+    }
+) }}
+
+Dimensions of NIfTI image "`sub-01_model-bs_desc-mean_param-s0_dwimap.nii.gz`": *I*x*J*x*K* ([scalar](#encoding-scalar))
+Dimensions of NIfTI image "`sub-01_model-bs_desc-mean_param-polar_dwimap.nii.gz`": *I*x*J*x*K*x(*2*x*N*) ([spherical coordinates](#encoding-spherical), orientations only; *N* orientations per voxel)
+Dimensions of NIfTI image "`sub-01_model-bs_desc-mean_param-vector_dwimap.nii.gz`": *I*x*J*x*K*x(*3*x*N*) ([3-vectors](#encoding-3vectors), unit norm; *N* orientations per voxel)
+Dimensions of NIfTI image "`sub-01_model-bs_desc-mean_param-vf_dwimap.nii.gz`": *I*x*J*x*K*x*N* ([scalar](#encoding-scalar); *N* values per voxel)
+Dimensions of NIfTI image "`sub-01_model-bs_desc-mean_param-vfsum_dwimap.nii.gz`": *I*x*J*x*K* ([scalar](#encoding-scalar))
+Dimensions of NIfTI image "`sub-01_model-bs_desc-mean_param-diffusivity_dwimap.nii.gz`": *I*x*J*x*K* ([scalar](#encoding-scalar))
+Dimensions of NIfTI image "`sub-01_model-bs_desc-mean_param-dstd_dwimap.nii.gz`": *I*x*J*x*K* ([scalar](#encoding-scalar))
+Dimensions of NIfTI image "`sub-01_model-bs_desc-merged_param-polar_dwimap.nii.gz`": *I*x*J*x*K*x(*2*x*N*)x*R* ([spherical coordinates](#encoding-spherical), orientations only; *N* orientations per voxel; *R* bootstrap realisations)
+Dimensions of NIfTI image "`sub-01_model-bs_desc-merged_param-vf_dwimap.nii.gz`": *I*x*J*x*K*x*N*x*R* ([scalar](#encoding-scalar); *N* values per voxel; *R* bootstrap realisations)
+
+Contents of JSON file "`sub-01_model-bs_desc-mean_param-s0_dwimap.json`":
+
+```JSON
+{
+    "Description": "Estimated signal intensity with no diffusion weighting, ie. S0; mean across bootstrap realisations",
+    "Model": {
+        "Description": "Ball-And-Sticks model using FSL bedpostx",
+        "URL": "https://fsl.fmrib.ox.ac.uk/fsl/fslwiki/FDT",
+        "Parameters": {
+            "ARDFudgeFactor": 1.0,
+            "Fibers": 3
+        },
+        "BootstrapParameters": {
+            "Burnin": 1000,
+            "Jumps": 1250,
+            "SampleEvery": 25
+        }
+    }
+}
+```
+
+Contents of JSON file "`sub-01_model-bs_desc-mean_param-polar_dwimap.json`":
+
+```JSON
+{
+    "Description": "Fibre orientations encoded using polar angles; mean across bootstrap realisations",
+    "Model": {
+        "Description": "Ball-And-Sticks model using FSL bedpostx",
+        "URL": "https://fsl.fmrib.ox.ac.uk/fsl/fslwiki/FDT",
+        "Parameters": {
+            "ARDFudgeFactor": 1.0,
+            "Fibers": 3
+        },
+        "BootstrapParameters": {
+            "Burnin": 1000,
+            "Jumps": 1250,
+            "SampleEvery": 25
+        }
+    },
+    "OrientationEncoding": {
+        "EncodingAxis": 3,
+        "Reference": "bvec",
+        "Type": "unitspherical"
+    }
+}
+```
+
+Contents of JSON file "`sub-01_model-bs_desc-mean_param-vector_dwimap.json`":
+
+```JSON
+{
+    "Description": "Fibre orientations encoded using 3-vectors; mean across bootstrap realisations",
+    "Model": {
+        "Description": "Ball-And-Sticks model using FSL bedpostx",
+        "URL": "https://fsl.fmrib.ox.ac.uk/fsl/fslwiki/FDT",
+        "Parameters": {
+            "ARDFudgeFactor": 1.0,
+            "Fibers": 3
+        },
+        "BootstrapParameters": {
+            "Burnin": 1000,
+            "Jumps": 1250,
+            "SampleEvery": 25
+        }
+    },
+    "OrientationEncoding": {
+        "EncodingAxis": 3,
+        "Reference": "bvec",
+        "Type": "unit3vector"
+    }
+}
+```
+
+Contents of JSON file "`sub-01_model-bs_desc-mean_param-vf_dwimap.json`":
+
+```JSON
+{
+    "Description": "Volume fractions of stick components; mean across bootstrap realisations",
+    "Model": {
+        "Description": "Ball-And-Sticks model using FSL bedpostx",
+        "URL": "https://fsl.fmrib.ox.ac.uk/fsl/fslwiki/FDT",
+        "Parameters": {
+            "ARDFudgeFactor": 1.0,
+            "Fibers": 3
+        },
+        "BootstrapParameters": {
+            "Burnin": 1000,
+            "Jumps": 1250,
+            "SampleEvery": 25
+        }
+    },
+    "OrientationEncoding": {
+        "Type": "scalar"
+    }
+}
+```
+
+Contents of JSON file "`sub-01_model-bs_desc-mean_param-vfsum_dwimap.json`":
+
+```JSON
+{
+    "Description": "Sum of volume fractions of stick components; mean across bootstrap realisations",
+    "Model": {
+        "Description": "Ball-And-Sticks model using FSL bedpostx",
+        "URL": "https://fsl.fmrib.ox.ac.uk/fsl/fslwiki/FDT",
+        "Parameters": {
+            "ARDFudgeFactor": 1.0,
+            "Fibers": 3
+        },
+        "BootstrapParameters": {
+            "Burnin": 1000,
+            "Jumps": 1250,
+            "SampleEvery": 25
+        }
+    }
+}
+```
+
+Contents of JSON file "`sub-01_model-bs_desc-mean_param-diffusivity_dwimap.json`":
+
+```JSON
+{
+    "Description": "Diffusivity; mean across bootstrap realisations",
+    "Model": {
+        "Description": "Ball-And-Sticks model using FSL bedpostx",
+        "URL": "https://fsl.fmrib.ox.ac.uk/fsl/fslwiki/FDT",
+        "Parameters": {
+            "ARDFudgeFactor": 1.0,
+            "Fibers": 3
+        },
+        "BootstrapParameters": {
+            "Burnin": 1000,
+            "Jumps": 1250,
+            "SampleEvery": 25
+        }
+    },
+    "Units": "mm^2/s"
+}
+```
+
+Contents of JSON file "`sub-01_model-bs_desc-mean_param-dstd_dwimap.json`":
+
+```JSON
+{
+    "Description": "Diffusivity variance parameter; mean across bootstrap realisations",
+    "Model": {
+        "Description": "Ball-And-Sticks model using FSL bedpostx",
+        "URL": "https://fsl.fmrib.ox.ac.uk/fsl/fslwiki/FDT",
+        "Parameters": {
+            "ARDFudgeFactor": 1.0,
+            "Fibers": 3
+        },
+        "BootstrapParameters": {
+            "Burnin": 1000,
+            "Jumps": 1250,
+            "SampleEvery": 25
+        }
+    },
+    "Units": "TODO"
+}
+
+Contents of JSON file "`sub-01_model-bs_desc-merged_param-polar_dwimap.json`":
+
+```JSON
+{
+    "BootstrapAxis": 4,
+    "Description": "Fibre orientations encoded using polar angles; concatenated bootstrap realisations",
+    "Model": {
+        "Description": "Ball-And-Sticks model using FSL bedpostx",
+        "URL": "https://fsl.fmrib.ox.ac.uk/fsl/fslwiki/FDT",
+        "Parameters": {
+            "ARDFudgeFactor": 1.0,
+            "Fibers": 3
+        },
+        "BootstrapParameters": {
+            "Burnin": 1000,
+            "Jumps": 1250,
+            "SampleEvery": 25
+        }
+    },
+    "OrientationEncoding": {
+        "EncodingAxis": 3,
+        "ReferenceAxes": "bvec",
+        "Type": "unitspherical"
+    }
+}
+
+Contents of JSON file "`sub-01_model-bs_desc-merged_param-vf_dwimap.json`":
+
+```JSON
+{
+    "BootstrapAxis": 4,
+    "Description": "Volume fractions of stick components; concatenated bootstrap realisations",
+    "Model": {
+        "Description": "Ball-And-Sticks model using FSL bedpostx",
+        "URL": "https://fsl.fmrib.ox.ac.uk/fsl/fslwiki/FDT",
+        "Parameters": {
+            "ARDFudgeFactor": 1.0,
+            "Fibers": 3
+        },
+        "BootstrapParameters": {
+            "Burnin": 1000,
+            "Jumps": 1250,
+            "SampleEvery": 25
+        }
+    },
+    "OrientationEncoding": {
+        "Type": "scalar"
+    }
+}
+
+Notes:
+
+-   Here the descriptors "`merged`" and "`mean`" have been utilized
+    to distinguish between the comprehensive set of all bootstrap realisations
+    and the computed mean statistics of parameters across all realisations respectively,
+    as this is the terminology utilized by the FSL software itself.
+    These labels are not however a part of the specification.
+
+-   Care must be taken for images of greater than three dimensions
+    where additional dimensions do *not* encode anisotropy information:
+
+    -   In image `"*_desc-mean*_param-vf_*"`,
+        the fourth image axis encodes scalar information across stick components.
+        Since this is *not* coefficients in some orientation encoding,
+        but the image possesses more than three axes,
+        field `"OrientationEncoding"["Type"]` MUST be specified as `"scalar"`.
+
+    -   Image `"*_param-vfsum_*"`is a three-dimensional image,
+        and therefore the fact that it encodes a scalar parameter
+        can be robustly inferred without reference to metadata information.
+
+    -   In image `"sub-01_model-bs_desc-merged_param-vf_dwimap.json"`,
+        there are two extra image dimensions beyond the three spatial dimensions:
+        the fourth image axis encodes across the multiple stick components per voxel,
+        and the fifth axis encodes realisations across bootstraps.
+        it is therefore necessary to explicitly specify
+        that the parameter being encoded in the data,
+        being the volume fractions of individual stick components,
+        do not have any anisotropy,
+        and therefore field `"OrientationEncoding"["Type"]` MUST be specified as `"scalar"`.
+
+-   TODO CONFIRM WHETHER WE WANT TO ENFORCE THIS OR INCLUDE EXPLICIT METADATA FIELDS
+    THAT INDICATE THE AXIS / AXES OF DIRECTION ENCODING VS. BOOTSTRAP REALISATIONS
+
+-   TODO DISCUSS
+    The example presented here is not necessarily a unique solution
+    for the storage of the outcomes of the FSL `bedpostx` command as BIDS Derivatives:
+
+    -   WOULD SPLITTING STICK COMPONENTS ACROSS NIFTIS
+        REQUIRE A NEW ENTITY BY WHICH TO INDEX THEM?
+        OR JUST GIVE THEM EG. `_param-spherical1`, `_param-spherical2`?
+
+    -   While the FSL `bedpostx` command yields the fibre orientation for each individual stick
+        as polar angles within separate NIfTI images,
+        for BIDS it is RECOMMENDED that such orientation information be encoded
+        either as [spherical coordinates](#encoding-spherical) or [3-vectors](#encoding-3vector).
+
+    -   Given that it is possible to encode a scalar parameter
+        into either [spherical coordinates](#encoding-spherical) or [3-vectors](#encoding-3vector) encodings,
+        it is possible to store an image that encodes,
+        for each stick component,
+        both volume fraction and orientation.
+        It is however RECOMMENDED to encode this information in separate files,
+        given that in the more general case there may be multiple scalar parameters
+        individually attributed to each component.
+
+### Appendix
+
+#### Spherical Harmonics
+
+-   Concepts shared across all spherical harmonics bases:
+
+    -   Basis functions:
+
+        ![SH basis functions](https://latex.codecogs.com/gif.latex?Y_l^m(\theta,\phi)&space;=&space;\sqrt{\frac{(2l&plus;1)}{4\pi}\frac{(l-m)!}{(l&plus;m)!}}&space;P_l^m(\cos&space;\theta)&space;e^{im\phi}")
+
+        for integer *order* *l*, *phase* *m*, associated Legendre polynomials *P*.
+
+    -   (Truncated) basis coefficients:
+
+        ![SH basis coefficients](https://latex.codecogs.com/gif.latex?f(\theta,\phi)&space;=&space;\sum_{l=0}^{l_\text{max}}&space;\sum_{m=-l}^{l}&space;c_l^m&space;Y_l^m(\theta,\phi)")
+
+        for *maximum* spherical harmonic order *l<sub>max</sub>*.
+
+    -   Functions assumed to be real: conjugate symmetry is assumed, that is,
+        *Y*(*l*,-*m*) = *Y*(*l*,*m*)\*, where \* denotes the complex
+        conjugate.
+
+    -   Antipodally symmetric: all basis functions with odd degree are
+        assumed zero; `AntipodalSymmetry` MUST NOT be set to `False`.
+
+    -   Utilized basis functions:
+
+        -   `mrtrix3`
+
+        ![MRtrix3 SH basis functions](https://latex.codecogs.com/gif.latex?Y_{lm}(\theta,\phi)=\begin{Bmatrix}&space;0&\text{if&space;}l\text{&space;is&space;odd},\\&space;\sqrt{2}\times\text{Im}\left[Y_l^{-m}(\theta,\phi)\right]&\text{if&space;}m<0,\\&space;Y_l^0(\theta,\phi)&\text{if&space;}m=0,\\&space;\sqrt{2}\times\text{Re}\left[Y_l^m(\theta,\phi)\right]&\text{if&space;}m>0\\&space;\end{Bmatrix})
+
+        -   `descoteaux`
+
+        ![Descoteaux SH basis functions](https://latex.codecogs.com/gif.latex?Y_{lm}(\theta,\phi)=\begin{Bmatrix}&space;0&\text{if&space;}l\text{&space;is&space;odd},\\&space;\sqrt{2}\times\text{Re}\left[Y_l^{-m}(\theta,\phi)\right]&\text{if&space;}m<0,\\&space;Y_l^0(\theta,\phi)&\text{if&space;}m=0,\\&space;\sqrt{2}\times\text{Im}\left[Y_l^m(\theta,\phi)\right]&\text{if&space;}m>0\\&space;\end{Bmatrix})
+
+    -   Mapping between image volume *V* and spherical harmonic basis
+        function coefficient *Y<sub>l,m</sub>*:
+
+        *V<sub>l,m</sub>* = (*l*(*l*+1) / 2) + *m*
+
+        | ***V*** | **Coefficient**    |
+        | ------- | ------------------ |
+        | 0       | *Y<sub>0,0</sub>*  |
+        | 1       | *Y<sub>2,-2</sub>* |
+        | 2       | *Y<sub>2,-1</sub>* |
+        | 3       | *Y<sub>2,0</sub>*  |
+        | 4       | *Y<sub>2,1</sub>*  |
+        | 5       | *Y<sub>2,2</sub>*  |
+        | 6       | *Y<sub>4,-4</sub>* |
+        | 7       | *Y<sub>4,-3</sub>* |
+        | ...     | ...                |
+
+    -   Relationship between maximal spherical harmonic degree *l<sub>max</sub>*
+        and number of image volumes *N*:
+
+        *N* = ((*l<sub>max</sub>*+1) x (*l<sub>max</sub>*+2)) / 2
+
+        | ***l<sub>max</sub>*** | 0 | 2 | 4  | 6  | 8  | 10 | ... |
+        | --------------------- |--:|--:|--: |--: |--: |--: | :--: |
+        | ***N***               | 1 | 6 | 15 | 28 | 45 | 66 | ... |
+
+    -   Relationship between maximal degree of *zonal* spherical harmonic
+        function (spherical harmonics function where all *m* != 0 terms are
+        assumed to be zero; used for response function definition and similar) and
+        number of coefficients *N*:
+
+        *N* = 1 + (*l<sub>max</sub>* / 2)
+
+        | ***l<sub>max</sub>*** | 0 | 2 | 4 | 6 | 8 | 10 | ... |
+        | --------------------- |--:|--:|--:|--:|--:|--: | :--: |
+        | ***N***               | 1 | 2 | 3 | 4 | 5 | 6  | ... |