Documentation 1.6.0. > API Reference > reconst

reconst

`reconst`

`bench`([label, verbose, extra_argv])	Run benchmarks for module using nose.
`test`([label, verbose, extra_argv, doctests, ...])	Run tests for module using nose.

Module: `reconst.base`

Base-classes for reconstruction models and reconstruction fits.

All the models in the reconst module follow the same template: a Model object is used to represent the abstract properties of the model, that are independent of the specifics of the data . These properties are reused whenever fitting a particular set of data (different voxels, for example).

`ReconstFit`(model, data)	Abstract class which holds the fit result of ReconstModel
`ReconstModel`(gtab)	Abstract class for signal reconstruction models

Module: `reconst.benchmarks`

Module: `reconst.benchmarks.bench_bounding_box`

Benchmarks for bounding_box

Run all benchmarks with:

import dipy.reconst as dire
dire.bench()

With Pytest, Run this benchmark with:

pytest -svv -c bench.ini /path/to/bench_bounding_box.py

`bench_bounding_box`()
`measure`(code_str[, times, label])	Return elapsed time for executing code in the namespace of the caller.

Module: `reconst.benchmarks.bench_csd`

`ConstrainedSphericalDeconvModel`(gtab, response)	Methods
`GradientTable`(gradients[, big_delta, ...])	Diffusion gradient information
`bench_csdeconv`([center, width])
`load_nifti_data`(fname[, as_ndarray])	Load only the data array from a nifti file.
`num_grad`(gtab)
`read_stanford_labels`()	Read stanford hardi data and label map.

Module: `reconst.benchmarks.bench_peaks`

Benchmarks for peak finding

Run all benchmarks with:

import dipy.reconst as dire
dire.bench()

With Pytest, Run this benchmark with:

pytest -svv -c bench.ini /path/to/bench_peaks.py

`bench_local_maxima`()
`measure`(code_str[, times, label])	Return elapsed time for executing code in the namespace of the caller.

Module: `reconst.benchmarks.bench_squash`

Benchmarks for fast squashing

Run all benchmarks with:

import dipy.reconst as dire
dire.bench()

With Pytest, Run this benchmark with:

pytest -svv -c bench.ini /path/to/bench_squash.py

`bench_quick_squash`()
`measure`(code_str[, times, label])	Return elapsed time for executing code in the namespace of the caller.
`ndindex`(shape)	An N-dimensional iterator object to index arrays.
`old_squash`(arr[, mask, fill])	Try and make a standard array from an object array
`reduce`(function, sequence[, initial])	Apply a function of two arguments cumulatively to the items of a sequence, from left to right, so as to reduce the sequence to a single value.

Module: `reconst.benchmarks.bench_vec_val_sum`

Benchmarks for vec / val summation routine

Run benchmarks with:

import dipy.reconst as dire
dire.bench()

With Pytest, Run this benchmark with:

pytest -svv -c bench.ini /path/to/bench_vec_val_sum.py

`bench_vec_val_vect`()
`measure`(code_str[, times, label])	Return elapsed time for executing code in the namespace of the caller.

Module: `reconst.cache`

`Cache`()	Cache values based on a key object (such as a sphere or gradient table).
`auto_attr`(func)	Decorator to create OneTimeProperty attributes.

Module: `reconst.cross_validation`

Cross-validation analysis of diffusion models.

`coeff_of_determination`(data, model[, axis])	Calculate the coefficient of determination for a model prediction,
`kfold_xval`(model, data, folds, *model_args, ...)	Perform k-fold cross-validation.

Module: `reconst.csdeconv`

`AxSymShResponse`(S0, dwi_response[, bvalue])	A simple wrapper for response functions represented using only axially symmetric, even spherical harmonic functions (ie, m == 0 and n even).
`ConstrainedSDTModel`(gtab, ratio[, ...])	Methods
`ConstrainedSphericalDeconvModel`(gtab, response)	Methods
`SphHarmFit`(model, shm_coef, mask)	Diffusion data fit to a spherical harmonic model
`SphHarmModel`(gtab)	To be subclassed by all models that return a SphHarmFit when fit.
`TensorModel`(gtab[, fit_method, return_S0_hat])	Diffusion Tensor
`auto_response`(gtab, data[, roi_center, ...])	Automatic estimation of ssst response function using FA.
`auto_response_ssst`(gtab, data[, roi_center, ...])	Automatic estimation of single-shell single-tissue (ssst) response
`cart2sphere`(x, y, z)	Return angles for Cartesian 3D coordinates x, y, and z
`csdeconv`(dwsignal, X, B_reg[, tau, ...])	Constrained-regularized spherical deconvolution (CSD) [1]
`deprecate_with_version`(message[, since, ...])	Return decorator function function for deprecation warning / error.
`deprecated_params`(old_name[, new_name, ...])	Deprecate a renamed or removed function argument.
`estimate_response`(gtab, evals, S0)	Estimate single fiber response function
`fa_trace_to_lambdas`([fa, trace])
`forward_sdeconv_mat`(r_rh, n)	Build forward spherical deconvolution matrix
`forward_sdt_deconv_mat`(ratio, n[, r2_term])	Build forward sharpening deconvolution transform (SDT) matrix
`fractional_anisotropy`(evals[, axis])	Return Fractional anisotropy (FA) of a diffusion tensor.
`get_sphere`([name])	provide triangulated spheres
`lazy_index`(index)	Produces a lazy index
`lpn`(n, z)	Legendre function of the first kind.
`mask_for_response_ssst`(gtab, data[, ...])	Computation of mask for single-shell single-tissue (ssst) response
`multi_voxel_fit`(single_voxel_fit)	Method decorator to turn a single voxel model fit definition into a multi voxel model fit definition
`ndindex`(shape)	An N-dimensional iterator object to index arrays.
`odf_deconv`(odf_sh, R, B_reg[, lambda_, tau, ...])	ODF constrained-regularized spherical deconvolution using the Sharpening Deconvolution Transform (SDT) [1], [2].
`odf_sh_to_sharp`(odfs_sh, sphere[, basis, ...])	Sharpen odfs using the sharpening deconvolution transform [2]
`peaks_from_model`(model, data, sphere, ...[, ...])	Fit the model to data and computes peaks and metrics
`quad`(func, a, b[, args, full_output, ...])	Compute a definite integral.
`real_sh_descoteaux`(sh_order, theta, phi[, ...])	Compute real spherical harmonics as in Descoteaux et al. 2007 [Ra6d8f6cd2652-1], where the real harmonic \(Y^m_n\) is defined to be:.
`real_sh_descoteaux_from_index`(m, n, theta, phi)	Compute real spherical harmonics as in Descoteaux et al. 2007 [R2763a8caef3c-1], where the real harmonic \(Y^m_n\) is defined to be:.
`recursive_response`(gtab, data[, mask, ...])	Recursive calibration of response function using peak threshold
`response_from_mask`(gtab, data, mask)	Computation of single-shell single-tissue (ssst) response
`response_from_mask_ssst`(gtab, data, mask)	Computation of single-shell single-tissue (ssst) response
`sh_to_rh`(r_sh, m, n)	Spherical harmonics (SH) to rotational harmonics (RH)
`single_tensor`(gtab[, S0, evals, evecs, snr])	Simulate diffusion-weighted signals with a single tensor.
`sph_harm_ind_list`(sh_order[, full_basis])	Returns the degree (`m`) and order (`n`) of all the symmetric spherical harmonics of degree less then or equal to `sh_order`.
`vec2vec_rotmat`(u, v)	rotation matrix from 2 unit vectors

Module: `reconst.dki`

Classes and functions for fitting the diffusion kurtosis model

`DiffusionKurtosisFit`(model, model_params)	Class for fitting the Diffusion Kurtosis Model
`DiffusionKurtosisModel`(gtab[, fit_method])	Class for the Diffusion Kurtosis Model
`ReconstModel`(gtab)	Abstract class for signal reconstruction models
`TensorFit`(model, model_params[, model_S0])	Attributes:
`Wcons`(k_elements)	Construct the full 4D kurtosis tensors from its 15 independent elements
`Wrotate`(kt, Basis)	Rotate a kurtosis tensor from the standard Cartesian coordinate system to another coordinate system basis
`Wrotate_element`(kt, indi, indj, indk, indl, B)	Computes the the specified index element of a kurtosis tensor rotated to the coordinate system basis B.
`apparent_kurtosis_coef`(dki_params, sphere[, ...])	Calculates the apparent kurtosis coefficient (AKC) in each direction of a sphere [1].
`axial_kurtosis`(dki_params[, min_kurtosis, ...])	Computes axial Kurtosis (AK) from the kurtosis tensor [1], [2].
`carlson_rd`(x, y, z[, errtol])	Computes the Carlson's incomplete elliptic integral of the second kind defined as:
`carlson_rf`(x, y, z[, errtol])	Computes the Carlson's incomplete elliptic integral of the first kind defined as:
`cart2sphere`(x, y, z)	Return angles for Cartesian 3D coordinates x, y, and z
`check_multi_b`(gtab, n_bvals[, non_zero, bmag])	Check if you have enough different b-values in your gradient table
`decompose_tensor`(tensor[, min_diffusivity])	Returns eigenvalues and eigenvectors given a diffusion tensor
`design_matrix`(gtab)	Construct B design matrix for DKI.
`directional_diffusion`(dt, V[, min_diffusivity])	Calculates the apparent diffusion coefficient (adc) in each direction of a sphere for a single voxel [1].
`directional_diffusion_variance`(kt, V[, ...])	Calculates the apparent diffusion variance (adv) in each direction of a sphere for a single voxel [1].
`directional_kurtosis`(dt, md, kt, V[, ...])	Calculates the apparent kurtosis coefficient (akc) in each direction of a sphere for a single voxel [1].
`dki_prediction`(dki_params, gtab[, S0])	Predict a signal given diffusion kurtosis imaging parameters.
`from_lower_triangular`(D)	Returns a tensor given the six unique tensor elements
`get_fnames`([name])	Provide full paths to example or test datasets.
`get_sphere`([name])	provide triangulated spheres
`kurtosis_fractional_anisotropy`(dki_params)	Computes the anisotropy of the kurtosis tensor (KFA) [1].
`kurtosis_maximum`(dki_params[, sphere, gtol, ...])	Computes kurtosis maximum value
`local_maxima`	Local maxima of a function evaluated on a discrete set of points.
`lower_triangular`(tensor[, b0])	Returns the six lower triangular values of the tensor ordered as (Dxx, Dxy, Dyy, Dxz, Dyz, Dzz) and a dummy variable if b0 is not None.
`mean_diffusivity`(evals[, axis])	Mean Diffusivity (MD) of a diffusion tensor.
`mean_kurtosis`(dki_params[, min_kurtosis, ...])	Computes mean Kurtosis (MK) from the kurtosis tensor.
`mean_kurtosis_tensor`(dki_params[, ...])	Computes mean of the kurtosis tensor (MKT) [1].
`ndindex`(shape)	An N-dimensional iterator object to index arrays.
`nlls_fit_tensor`(design_matrix, data[, ...])	Fit the cumulant expansion params (e.g.
`ols_fit_dki`(design_matrix, data)	Computes the diffusion and kurtosis tensors using an ordinary linear least squares (OLS) approach [1].
`perpendicular_directions`(v[, num, half])	Computes n evenly spaced perpendicular directions relative to a given vector v
`radial_kurtosis`(dki_params[, min_kurtosis, ...])	Radial Kurtosis (RK) of a diffusion kurtosis tensor [1], [2].
`restore_fit_tensor`(design_matrix, data[, ...])	Use the RESTORE algorithm [1] to calculate a robust tensor fit
`sphere2cart`(r, theta, phi)	Spherical to Cartesian coordinates
`split_dki_param`(dki_params)	Extract the diffusion tensor eigenvalues, the diffusion tensor eigenvector matrix, and the 15 independent elements of the kurtosis tensor from the model parameters estimated from the DKI model
`vec_val_vect`	Vectorize vecs.diag(vals).`vecs`.T for last 2 dimensions of vecs
`wls_fit_dki`(design_matrix, data)	Computes the diffusion and kurtosis tensors using a weighted linear least squares (WLS) approach [1].

Module: `reconst.dki_micro`

Classes and functions for fitting the DKI-based microstructural model

`DiffusionKurtosisFit`(model, model_params)	Class for fitting the Diffusion Kurtosis Model
`DiffusionKurtosisModel`(gtab[, fit_method])	Class for the Diffusion Kurtosis Model
`KurtosisMicrostructuralFit`(model, model_params)	Class for fitting the Diffusion Kurtosis Microstructural Model
`KurtosisMicrostructureModel`(gtab[, fit_method])	Class for the Diffusion Kurtosis Microstructural Model
`axial_diffusivity`(evals[, axis])	Axial Diffusivity (AD) of a diffusion tensor.
`axonal_water_fraction`(dki_params[, sphere, ...])	Computes the axonal water fraction from DKI [1].
`decompose_tensor`(tensor[, min_diffusivity])	Returns eigenvalues and eigenvectors given a diffusion tensor
`diffusion_components`(dki_params[, sphere, ...])	Extracts the restricted and hindered diffusion tensors of well aligned fibers from diffusion kurtosis imaging parameters [1].
`directional_diffusion`(dt, V[, min_diffusivity])	Calculates the apparent diffusion coefficient (adc) in each direction of a sphere for a single voxel [1].
`directional_kurtosis`(dt, md, kt, V[, ...])	Calculates the apparent kurtosis coefficient (akc) in each direction of a sphere for a single voxel [1].
`dkimicro_prediction`(params, gtab[, S0])	Signal prediction given the DKI microstructure model parameters.
`dti_design_matrix`(gtab[, dtype])	Constructs design matrix for DTI weighted least squares or least squares fitting.
`from_lower_triangular`(D)	Returns a tensor given the six unique tensor elements
`get_sphere`([name])	provide triangulated spheres
`kurtosis_maximum`(dki_params[, sphere, gtol, ...])	Computes kurtosis maximum value
`lower_triangular`(tensor[, b0])	Returns the six lower triangular values of the tensor ordered as (Dxx, Dxy, Dyy, Dxz, Dyz, Dzz) and a dummy variable if b0 is not None.
`mean_diffusivity`(evals[, axis])	Mean Diffusivity (MD) of a diffusion tensor.
`ndindex`(shape)	An N-dimensional iterator object to index arrays.
`radial_diffusivity`(evals[, axis])	Radial Diffusivity (RD) of a diffusion tensor.
`split_dki_param`(dki_params)	Extract the diffusion tensor eigenvalues, the diffusion tensor eigenvector matrix, and the 15 independent elements of the kurtosis tensor from the model parameters estimated from the DKI model
`tortuosity`(hindered_ad, hindered_rd)	Computes the tortuosity of the hindered diffusion compartment given its axial and radial diffusivities
`trace`(evals[, axis])	Trace of a diffusion tensor.
`vec_val_vect`	Vectorize vecs.diag(vals).`vecs`.T for last 2 dimensions of vecs

Module: `reconst.dsi`

`Cache`()	Cache values based on a key object (such as a sphere or gradient table).
`DiffusionSpectrumDeconvFit`(model, data)	Methods
`DiffusionSpectrumDeconvModel`(gtab[, ...])	Methods
`DiffusionSpectrumFit`(model, data)	Methods
`DiffusionSpectrumModel`(gtab[, qgrid_size, ...])	Methods
`OdfFit`(model, data)	Methods
`OdfModel`(gtab)	An abstract class to be sub-classed by specific odf models
`LR_deconv`(prop, psf[, numit, acc_factor])	Perform Lucy-Richardson deconvolution algorithm on a 3D array.
`create_qspace`(gtab, origin)	create the 3D grid which holds the signal values (q-space)
`create_qtable`(gtab, origin)	create a normalized version of gradients
`fftn`(x[, shape, axes, overwrite_x])	Return multidimensional discrete Fourier transform.
`fftshift`(x[, axes])	Shift the zero-frequency component to the center of the spectrum.
`gen_PSF`(qgrid_sampling, siz_x, siz_y, siz_z)	Generate a PSF for DSI Deconvolution by taking the ifft of the binary q-space sampling mask and truncating it to keep only the center.
`half_to_full_qspace`(data, gtab)	Half to full Cartesian grid mapping
`hanning_filter`(gtab, filter_width, origin)	create a hanning window
`ifftshift`(x[, axes])	The inverse of fftshift.
`map_coordinates`(input, coordinates[, ...])	Map the input array to new coordinates by interpolation.
`multi_voxel_fit`(single_voxel_fit)	Method decorator to turn a single voxel model fit definition into a multi voxel model fit definition
`pdf_interp_coords`(sphere, rradius, origin)	Precompute coordinates for ODF calculation from the PDF
`pdf_odf`(Pr, rradius, interp_coords)	Calculates the real ODF from the diffusion propagator(PDF) Pr
`project_hemisph_bvecs`(gtab)	Project any near identical bvecs to the other hemisphere
`threshold_propagator`(P[, estimated_snr])	Applies hard threshold on the propagator to remove background noise for the deconvolution.

Module: `reconst.dti`

Classes and functions for fitting tensors

`ReconstModel`(gtab)	Abstract class for signal reconstruction models
`TensorFit`(model, model_params[, model_S0])	Attributes:
`TensorModel`(gtab[, fit_method, return_S0_hat])	Diffusion Tensor
`Version`(version)	This class abstracts handling of a project's versions.
`apparent_diffusion_coef`(q_form, sphere)	Calculate the apparent diffusion coefficient (ADC) in each direction of a sphere.
`auto_attr`(func)	Decorator to create OneTimeProperty attributes.
`axial_diffusivity`(evals[, axis])	Axial Diffusivity (AD) of a diffusion tensor.
`color_fa`(fa, evecs)	Color fractional anisotropy of diffusion tensor
`decompose_tensor`(tensor[, min_diffusivity])	Returns eigenvalues and eigenvectors given a diffusion tensor
`design_matrix`(gtab[, dtype])	Constructs design matrix for DTI weighted least squares or least squares fitting.
`determinant`(q_form)	The determinant of a tensor, given in quadratic form
`deviatoric`(q_form)	Calculate the deviatoric (anisotropic) part of the tensor [1].
`eig_from_lo_tri`(data[, min_diffusivity])	Calculates tensor eigenvalues/eigenvectors from an array containing the lower diagonal form of the six unique tensor elements.
`fractional_anisotropy`(evals[, axis])	Return Fractional anisotropy (FA) of a diffusion tensor.
`from_lower_triangular`(D)	Returns a tensor given the six unique tensor elements
`geodesic_anisotropy`(evals[, axis])	Geodesic anisotropy (GA) of a diffusion tensor.
`get_sphere`([name])	provide triangulated spheres
`gradient_table`(bvals[, bvecs, big_delta, ...])	A general function for creating diffusion MR gradients.
`isotropic`(q_form)	Calculate the isotropic part of the tensor [1].
`iter_fit_tensor`([step])	Wrap a fit_tensor func and iterate over chunks of data with given length
`linearity`(evals[, axis])	The linearity of the tensor [1]
`lower_triangular`(tensor[, b0])	Returns the six lower triangular values of the tensor ordered as (Dxx, Dxy, Dyy, Dxz, Dyz, Dzz) and a dummy variable if b0 is not None.
`mean_diffusivity`(evals[, axis])	Mean Diffusivity (MD) of a diffusion tensor.
`mode`(q_form)	Mode (MO) of a diffusion tensor [1].
`nlls_fit_tensor`(design_matrix, data[, ...])	Fit the cumulant expansion params (e.g.
`norm`(q_form)	Calculate the Frobenius norm of a tensor quadratic form
`ols_fit_tensor`(design_matrix, data[, ...])	Computes ordinary least squares (OLS) fit to calculate self-diffusion tensor using a linear regression model [Rd310240b4eed-1].
`pinv`(a[, rcond])	Vectorized version of numpy.linalg.pinv
`planarity`(evals[, axis])	The planarity of the tensor [1]
`quantize_evecs`(evecs[, odf_vertices])	Find the closest orientation of an evenly distributed sphere
`radial_diffusivity`(evals[, axis])	Radial Diffusivity (RD) of a diffusion tensor.
`restore_fit_tensor`(design_matrix, data[, ...])	Use the RESTORE algorithm [1] to calculate a robust tensor fit
`sphericity`(evals[, axis])	The sphericity of the tensor [1]
`tensor_prediction`(dti_params, gtab, S0)	Predict a signal given tensor parameters.
`trace`(evals[, axis])	Trace of a diffusion tensor.
`vec_val_vect`	Vectorize vecs.diag(vals).`vecs`.T for last 2 dimensions of vecs
`vector_norm`(vec[, axis, keepdims])	Return vector Euclidean (L2) norm
`wls_fit_tensor`(design_matrix, data[, ...])	Computes weighted least squares (WLS) fit to calculate self-diffusion tensor using a linear regression model [1].

Module: `reconst.eudx_direction_getter`

EuDXDirectionGetter

Deterministic Direction Getter based on peak directions.

Module: `reconst.forecast`

`Cache`()	Cache values based on a key object (such as a sphere or gradient table).
`ForecastFit`(model, data, sh_coef, d_par, d_perp)	Attributes:
`ForecastModel`(gtab[, sh_order, lambda_lb, ...])	Fiber ORientation Estimated using Continuous Axially Symmetric Tensors (FORECAST) [1,2,3]_.
`OdfFit`(model, data)	Methods
`OdfModel`(gtab)	An abstract class to be sub-classed by specific odf models
`Version`(version)	This class abstracts handling of a project's versions.
`cart2sphere`(x, y, z)	Return angles for Cartesian 3D coordinates x, y, and z
`csdeconv`(dwsignal, X, B_reg[, tau, ...])	Constrained-regularized spherical deconvolution (CSD) [1]
`find_signal_means`(b_unique, data_norm, ...)	Calculate the mean signal for each shell.
`forecast_error_func`(x, b_unique, E)	Calculates the difference between the mean signal calculated using the parameter vector x and the average signal E using FORECAST and SMT
`forecast_matrix`(sh_order, d_par, d_perp, bvals)	Compute the FORECAST radial matrix
`lb_forecast`(sh_order)	Returns the Laplace-Beltrami regularization matrix for FORECAST
`leastsq`(func, x0[, args, Dfun, full_output, ...])	Minimize the sum of squares of a set of equations.
`multi_voxel_fit`(single_voxel_fit)	Method decorator to turn a single voxel model fit definition into a multi voxel model fit definition
`optional_package`(name[, trip_msg])	Return package-like thing and module setup for package name
`psi_l`(l, b)
`real_sh_descoteaux_from_index`(m, n, theta, phi)	Compute real spherical harmonics as in Descoteaux et al. 2007 [Rded9a56186a6-1], where the real harmonic \(Y^m_n\) is defined to be:.
`rho_matrix`(sh_order, vecs)	Compute the SH matrix \(\rho\)
`warn`(/, message[, category, stacklevel, source])	Issue a warning, or maybe ignore it or raise an exception.

Module: `reconst.fwdti`

Classes and functions for fitting tensors without free water contamination

`FreeWaterTensorFit`(model, model_params)	Class for fitting the Free Water Tensor Model
`FreeWaterTensorModel`(gtab[, fit_method])	Class for the Free Water Elimination Diffusion Tensor Model
`ReconstModel`(gtab)	Abstract class for signal reconstruction models
`TensorFit`(model, model_params[, model_S0])	Attributes:
`check_multi_b`(gtab, n_bvals[, non_zero, bmag])	Check if you have enough different b-values in your gradient table
`cholesky_to_lower_triangular`(R)	Convert Cholesky decompostion elements to the diffusion tensor elements
`decompose_tensor`(tensor[, min_diffusivity])	Returns eigenvalues and eigenvectors given a diffusion tensor
`design_matrix`(gtab[, dtype])	Constructs design matrix for DTI weighted least squares or least squares fitting.
`from_lower_triangular`(D)	Returns a tensor given the six unique tensor elements
`fwdti_prediction`(params, gtab[, S0, Diso])	Signal prediction given the free water DTI model parameters.
`lower_triangular`(tensor[, b0])	Returns the six lower triangular values of the tensor ordered as (Dxx, Dxy, Dyy, Dxz, Dyz, Dzz) and a dummy variable if b0 is not None.
`lower_triangular_to_cholesky`(tensor_elements)	Performs Cholesky decomposition of the diffusion tensor
`multi_voxel_fit`(single_voxel_fit)	Method decorator to turn a single voxel model fit definition into a multi voxel model fit definition
`ndindex`(shape)	An N-dimensional iterator object to index arrays.
`nls_fit_tensor`(gtab, data[, mask, Diso, ...])	Fit the water elimination tensor model using the non-linear least-squares.
`nls_iter`(design_matrix, sig, S0[, Diso, ...])	Applies non linear least squares fit of the water free elimination model to single voxel signals.
`vec_val_vect`	Vectorize vecs.diag(vals).`vecs`.T for last 2 dimensions of vecs
`wls_fit_tensor`(gtab, data[, Diso, mask, ...])	Computes weighted least squares (WLS) fit to calculate self-diffusion tensor using a linear regression model [1].
`wls_iter`(design_matrix, sig, S0[, Diso, ...])	Applies weighted linear least squares fit of the water free elimination model to single voxel signals.

Module: `reconst.gqi`

Classes and functions for generalized q-sampling

`Cache`()	Cache values based on a key object (such as a sphere or gradient table).
`GeneralizedQSamplingFit`(model, data)	Methods
`GeneralizedQSamplingModel`(gtab[, method, ...])	Methods
`OdfFit`(model, data)	Methods
`OdfModel`(gtab)	An abstract class to be sub-classed by specific odf models
`equatorial_maximum`(vertices, odf, pole, width)
`equatorial_zone_vertices`(vertices, pole[, width])	finds the 'vertices' in the equatorial zone conjugate to 'pole' with width half 'width' degrees
`multi_voxel_fit`(single_voxel_fit)	Method decorator to turn a single voxel model fit definition into a multi voxel model fit definition
`normalize_qa`(qa[, max_qa])	Normalize quantitative anisotropy.
`npa`(self, odf[, width])	non-parametric anisotropy
`odf_sum`(odf)
`patch_maximum`(vertices, odf, pole, width)
`patch_sum`(vertices, odf, pole, width)
`patch_vertices`(vertices, pole, width)	find 'vertices' within the cone of 'width' degrees around 'pole'
`polar_zone_vertices`(vertices, pole[, width])	finds the 'vertices' in the equatorial band around the 'pole' of radius 'width' degrees
`squared_radial_component`(x[, tol])	Part of the GQI2 integral
`triple_odf_maxima`(vertices, odf, width)
`upper_hemi_map`(v)	maps a 3-vector into the z-upper hemisphere

Module: `reconst.ivim`

Classes and functions for fitting ivim model

`IvimFit`(model, model_params)	Attributes:
`IvimModelTRR`(gtab[, split_b_D, split_b_S0, ...])	Ivim model
`IvimModelVP`(gtab[, bounds, maxiter, xtol])	Methods
`ReconstModel`(gtab)	Abstract class for signal reconstruction models
`Version`(version)	This class abstracts handling of a project's versions.
`IvimModel`(gtab[, fit_method])	Selector function to switch between the 2-stage Trust-Region Reflective based NLLS fitting method (also containing the linear fit): trr and the Variable Projections based fitting method: varpro.
`differential_evolution`(func, bounds[, args, ...])	Finds the global minimum of a multivariate function.
`f_D_star_error`(params, gtab, signal, S0, D)	Error function used to fit f and D_star keeping S0 and D fixed
`f_D_star_prediction`(params, gtab, S0, D)	Function used to predict IVIM signal when S0 and D are known by considering f and D_star as the unknown parameters.
`ivim_model_selector`(gtab[, fit_method])	Selector function to switch between the 2-stage Trust-Region Reflective based NLLS fitting method (also containing the linear fit): trr and the Variable Projections based fitting method: varpro.
`ivim_prediction`(params, gtab)	The Intravoxel incoherent motion (IVIM) model function.
`least_squares`(fun, x0[, jac, bounds, ...])	Solve a nonlinear least-squares problem with bounds on the variables.
`multi_voxel_fit`(single_voxel_fit)	Method decorator to turn a single voxel model fit definition into a multi voxel model fit definition
`optional_package`(name[, trip_msg])	Return package-like thing and module setup for package name

Module: `reconst.mapmri`

`Cache`()	Cache values based on a key object (such as a sphere or gradient table).
`MapmriFit`(model, mapmri_coef, mu, R, lopt[, ...])	Attributes:
`MapmriModel`(gtab[, radial_order, ...])	Mean Apparent Propagator MRI (MAPMRI) [1] of the diffusion signal.
`Optimizer`(fun, x0[, args, method, jac, ...])	Attributes:
`PositiveDefiniteLeastSquares`(m[, A, L])	Methods
`ReconstFit`(model, data)	Abstract class which holds the fit result of ReconstModel
`ReconstModel`(gtab)	Abstract class for signal reconstruction models
`Version`(version)	This class abstracts handling of a project's versions.
`b_mat`(index_matrix)	Calculates the B coefficients from [1] Eq.
`b_mat_isotropic`(index_matrix)	Calculates the isotropic B coefficients from [1] Fig 8.
`binomialfloat`(n, k)	Custom Binomial function
`cart2sphere`(x, y, z)	Return angles for Cartesian 3D coordinates x, y, and z
`create_rspace`(gridsize, radius_max)	Create the real space table, that contains the points in which to compute the pdf.
`delta`(n, m)
`factorial2`(n[, exact])	Double factorial.
`gcv_cost_function`(weight, args)	The GCV cost function that is iterated [4].
`generalized_crossvalidation`(data, M, LR[, ...])	Generalized Cross Validation Function [1] eq.
`generalized_crossvalidation_array`(data, M, LR)	Generalized Cross Validation Function [1] eq.
`genlaguerre`(n, alpha[, monic])	Generalized (associated) Laguerre polynomial.
`gradient_table`(bvals[, bvecs, big_delta, ...])	A general function for creating diffusion MR gradients.
`hermite`(n[, monic])	Physicist's Hermite polynomial.
`isotropic_scale_factor`(mu_squared)	Estimated isotropic scaling factor _[1] Eq.
`load_sdp_constraints`(model_name[, order])	Import semidefinite programming constraint matrices for different models, generated as described for example in [1].
`map_laplace_s`(n, m)	R(m,n) static matrix for Laplacian regularization [1] eq.
`map_laplace_t`(n, m)	L(m, n) static matrix for Laplacian regularization [1] eq.
`map_laplace_u`(n, m)	S(n, m) static matrix for Laplacian regularization [1] eq.
`mapmri_STU_reg_matrices`(radial_order)	Generate the static portions of the Laplacian regularization matrix according to [1] eq.
`mapmri_index_matrix`(radial_order)	Calculates the indices for the MAPMRI [1] basis in x, y and z.
`mapmri_isotropic_K_mu_dependent`(...)	Computes mu dependent part of M.
`mapmri_isotropic_K_mu_independent`(...)	Computes mu independent part of K.
`mapmri_isotropic_M_mu_dependent`(...)	Computed the mu dependent part of the signal design matrix.
`mapmri_isotropic_M_mu_independent`(...)	Computed the mu independent part of the signal design matrix.
`mapmri_isotropic_index_matrix`(radial_order)	Calculates the indices for the isotropic MAPMRI basis [1] Fig 8.
`mapmri_isotropic_laplacian_reg_matrix`(...)	Computes the Laplacian regularization matrix for MAP-MRI's isotropic implementation [1] eq.
`mapmri_isotropic_laplacian_reg_matrix_from_index_matrix`(...)	Computes the Laplacian regularization matrix for MAP-MRI's isotropic implementation [1] eq.
`mapmri_isotropic_odf_matrix`(radial_order, ...)	Compute the isotropic MAPMRI ODF matrix [1] Eq.
`mapmri_isotropic_odf_sh_matrix`(radial_order, ...)	Compute the isotropic MAPMRI ODF matrix [1] Eq.
`mapmri_isotropic_phi_matrix`(radial_order, mu, q)	Three dimensional isotropic MAPMRI signal basis function from [1] Eq.
`mapmri_isotropic_psi_matrix`(radial_order, ...)	Three dimensional isotropic MAPMRI propagator basis function from [1] Eq.
`mapmri_isotropic_radial_pdf_basis`(j, l, mu, r)	Radial part of the isotropic 1D-SHORE propagator basis [1] eq.
`mapmri_isotropic_radial_signal_basis`(j, l, ...)	Radial part of the isotropic 1D-SHORE signal basis [1] eq.
`mapmri_laplacian_reg_matrix`(ind_mat, mu, ...)	Put the Laplacian regularization matrix together [1] eq.
`mapmri_odf_matrix`(radial_order, mu, s, vertices)	Compute the MAPMRI ODF matrix [1] Eq.
`mapmri_phi_1d`(n, q, mu)	One dimensional MAPMRI basis function from [1] Eq.
`mapmri_phi_matrix`(radial_order, mu, q_gradients)	Compute the MAPMRI phi matrix for the signal [1] eq.
`mapmri_psi_1d`(n, x, mu)	One dimensional MAPMRI propagator basis function from [1] Eq.
`mapmri_psi_matrix`(radial_order, mu, rgrad)	Compute the MAPMRI psi matrix for the propagator [1] eq.
`mfactorial`(x, /)	Find x!.
`multi_voxel_fit`(single_voxel_fit)	Method decorator to turn a single voxel model fit definition into a multi voxel model fit definition
`optional_package`(name[, trip_msg])	Return package-like thing and module setup for package name
`real_sh_descoteaux_from_index`(m, n, theta, phi)	Compute real spherical harmonics as in Descoteaux et al. 2007 [R85c991cbd5ae-1], where the real harmonic \(Y^m_n\) is defined to be:.
`sfactorial`(n[, exact])	The factorial of a number or array of numbers.
`sph_harm_ind_list`(sh_order[, full_basis])	Returns the degree (`m`) and order (`n`) of all the symmetric spherical harmonics of degree less then or equal to `sh_order`.
`warn`(/, message[, category, stacklevel, source])	Issue a warning, or maybe ignore it or raise an exception.

Module: `reconst.mcsd`

`GradientTable`(gradients[, big_delta, ...])	Diffusion gradient information
`MSDeconvFit`(model, coeff, mask)	Attributes:
`MultiShellDeconvModel`(gtab, response[, ...])	Methods
`MultiShellResponse`(response, sh_order, shells)	Attributes:
`QpFitter`(X, reg)	Methods
`TensorModel`(gtab[, fit_method, return_S0_hat])	Diffusion Tensor
`Version`(version)	This class abstracts handling of a project's versions.
`auto_response_msmt`(gtab, data[, tol, ...])	Automatic estimation of multi-shell multi-tissue (msmt) response
`fractional_anisotropy`(evals[, axis])	Return Fractional anisotropy (FA) of a diffusion tensor.
`get_bval_indices`(bvals, bval[, tol])	Get indices where the b-value is bval
`gradient_table`(bvals[, bvecs, big_delta, ...])	A general function for creating diffusion MR gradients.
`mask_for_response_msmt`(gtab, data[, ...])	Computation of masks for multi-shell multi-tissue (msmt) response
`mean_diffusivity`(evals[, axis])	Mean Diffusivity (MD) of a diffusion tensor.
`multi_shell_fiber_response`(sh_order, bvals, ...)	Fiber response function estimation for multi-shell data.
`multi_tissue_basis`(gtab, sh_order, iso_comp)	Builds a basis for multi-shell multi-tissue CSD model.
`multi_voxel_fit`(single_voxel_fit)	Method decorator to turn a single voxel model fit definition into a multi voxel model fit definition
`optional_package`(name[, trip_msg])	Return package-like thing and module setup for package name
`response_from_mask_msmt`(gtab, data, mask_wm, ...)	Computation of multi-shell multi-tissue (msmt) response
`response_from_mask_ssst`(gtab, data, mask)	Computation of single-shell single-tissue (ssst) response
`single_tensor`(gtab[, S0, evals, evecs, snr])	Simulate diffusion-weighted signals with a single tensor.
`solve_qp`(P, Q, G, H)	Helper function to set up and solve the Quadratic Program (QP) in CVXPY.
`unique_bvals_tolerance`(bvals[, tol])	Gives the unique b-values of the data, within a tolerance gap

Module: `reconst.msdki`

Classes and functions for fitting the mean signal diffusion kurtosis model

`MeanDiffusionKurtosisFit`(model, model_params)	Attributes:
`MeanDiffusionKurtosisModel`(gtab[, bmag, ...])	Mean signal Diffusion Kurtosis Model
`ReconstModel`(gtab)	Abstract class for signal reconstruction models
`auto_attr`(func)	Decorator to create OneTimeProperty attributes.
`awf_from_msk`(msk[, mask])	Computes the axonal water fraction from the mean signal kurtosis assuming the 2-compartmental spherical mean technique model [1], [2]
`check_multi_b`(gtab, n_bvals[, non_zero, bmag])	Check if you have enough different b-values in your gradient table
`design_matrix`(ubvals)	Constructs design matrix for the mean signal diffusion kurtosis model
`mean_signal_bvalue`(data, gtab[, bmag])	Computes the average signal across different diffusion directions for each unique b-value
`msdki_prediction`(msdki_params, gtab[, S0])	Predict the mean signal given the parameters of the mean signal DKI, an GradientTable object and S0 signal.
`msk_from_awf`(f)	Computes mean signal kurtosis from axonal water fraction estimates of the SMT2 model
`ndindex`(shape)	An N-dimensional iterator object to index arrays.
`round_bvals`(bvals[, bmag])	"This function rounds the b-values
`unique_bvals_magnitude`(bvals[, bmag, rbvals])	This function gives the unique rounded b-values of the data
`wls_fit_msdki`(design_matrix, msignal, ng[, ...])	Fits the mean signal diffusion kurtosis imaging based on a weighted least square solution [1].

Module: `reconst.multi_voxel`

Tools to easily make multi voxel models

`CallableArray`	An array which can be called like a function
`MultiVoxelFit`(model, fit_array, mask)	Holds an array of fits and allows access to their attributes and methods
`ReconstFit`(model, data)	Abstract class which holds the fit result of ReconstModel
`tqdm`(_, *__)	Decorate an iterable object, returning an iterator which acts exactly like the original iterable, but prints a dynamically updating progressbar every time a value is requested.
`as_strided`(x[, shape, strides, subok, writeable])	Create a view into the array with the given shape and strides.
`multi_voxel_fit`(single_voxel_fit)	Method decorator to turn a single voxel model fit definition into a multi voxel model fit definition
`ndindex`(shape)	An N-dimensional iterator object to index arrays.

Module: `reconst.odf`

`OdfFit`(model, data)	Methods
`OdfModel`(gtab)	An abstract class to be sub-classed by specific odf models
`ReconstFit`(model, data)	Abstract class which holds the fit result of ReconstModel
`ReconstModel`(gtab)	Abstract class for signal reconstruction models
`gfa`(samples)	The general fractional anisotropy of a function evaluated on the unit sphere
`minmax_normalize`(samples[, out])	Min-max normalization of a function evaluated on the unit sphere

Module: `reconst.qtdmri`

`Cache`()	Cache values based on a key object (such as a sphere or gradient table).
`QtdmriFit`(model, qtdmri_coef, us, ut, ...)	Methods
`QtdmriModel`(gtab[, radial_order, ...])	The q:math:tau-dMRI model [1] to analytically and continuously represent the q:math:tau diffusion signal attenuation over diffusion sensitization q and diffusion time \(\tau\).
`Version`(version)	This class abstracts handling of a project's versions.
`GCV_cost_function`(weight, arguments)	Generalized Cross Validation Function that is iterated [1].
`H`(value)	Step function of H(x)=1 if x>=0 and zero otherwise.
`angular_basis_EAP_opt`(j, l, m, r, theta, phi)
`angular_basis_opt`(l, m, q, theta, phi)	Angular basis independent of spatial scaling factor us.
`cart2sphere`(x, y, z)	Return angles for Cartesian 3D coordinates x, y, and z
`create_rt_space_grid`(grid_size_r, ...)	Generates EAP grid (for potential positivity constraint).
`design_matrix_spatial`(bvecs, qvals)	Constructs design matrix for DTI weighted least squares or least squares fitting.
`elastic_crossvalidation`(b0s_mask, E, M, L, lopt)	cross-validation function to find the optimal weight of alpha for sparsity regularization when also Laplacian regularization is used.
`factorial`(n[, exact])	The factorial of a number or array of numbers.
`factorial2`(n[, exact])	Double factorial.
`fmin_l_bfgs_b`(func, x0[, fprime, args, ...])	Minimize a function func using the L-BFGS-B algorithm.
`generalized_crossvalidation`(data, M, LR[, ...])	Generalized Cross Validation Function [1].
`genlaguerre`(n, alpha[, monic])	Generalized (associated) Laguerre polynomial.
`gradient_table_from_gradient_strength_bvecs`(...)	A general function for creating diffusion MR gradients.
`l1_crossvalidation`(b0s_mask, E, M[, ...])	cross-validation function to find the optimal weight of alpha for sparsity regularization
`multi_voxel_fit`(single_voxel_fit)	Method decorator to turn a single voxel model fit definition into a multi voxel model fit definition
`optional_package`(name[, trip_msg])	Return package-like thing and module setup for package name
`part1_reg_matrix_tau`(ind_mat, ut)	Partial temporal Laplacian regularization matrix following Appendix B in [1].
`part23_iso_reg_matrix_q`(ind_mat, us)	Partial spherical spatial Laplacian regularization matrix following the equation below Eq.
`part23_reg_matrix_q`(ind_mat, U_mat, T_mat, us)	Partial cartesian spatial Laplacian regularization matrix following second line of Eq.
`part23_reg_matrix_tau`(ind_mat, ut)	Partial temporal Laplacian regularization matrix following Appendix B in [1].
`part4_iso_reg_matrix_q`(ind_mat, us)	Partial spherical spatial Laplacian regularization matrix following the equation below Eq.
`part4_reg_matrix_q`(ind_mat, U_mat, us)	Partial cartesian spatial Laplacian regularization matrix following equation Eq.
`part4_reg_matrix_tau`(ind_mat, ut)	Partial temporal Laplacian regularization matrix following Appendix B in [1].
`qtdmri_anisotropic_scaling`(data, q, bvecs, tau)	Constructs design matrix for fitting an exponential to the diffusion time points.
`qtdmri_eap_matrix`(radial_order, time_order, ...)	Constructs the design matrix as a product of 3 separated radial, angular and temporal design matrices.
`qtdmri_eap_matrix_`(radial_order, time_order, ...)
`qtdmri_index_matrix`(radial_order, time_order)	Computes the SHORE basis order indices according to [1].
`qtdmri_isotropic_eap_matrix`(radial_order, ...)	Constructs the design matrix as a product of 3 separated radial, angular and temporal design matrices.
`qtdmri_isotropic_eap_matrix_`(radial_order, ...)
`qtdmri_isotropic_index_matrix`(radial_order, ...)	Computes the SHORE basis order indices according to [1].
`qtdmri_isotropic_laplacian_reg_matrix`(...[, ...])	Computes the spherical qt-dMRI Laplacian regularization matrix.
`qtdmri_isotropic_scaling`(data, q, tau)	Constructs design matrix for fitting an exponential to the diffusion time points.
`qtdmri_isotropic_signal_matrix`(radial_order, ...)
`qtdmri_isotropic_signal_matrix_`(...[, ...])
`qtdmri_isotropic_to_mapmri_matrix`(...)	Generates the matrix that maps the spherical qtdmri coefficients to MAP-MRI coefficients.
`qtdmri_laplacian_reg_matrix`(ind_mat, us, ut)	Computes the cartesian qt-dMRI Laplacian regularization matrix.
`qtdmri_mapmri_isotropic_normalization`(j, l, u0)	Normalization factor for Spherical MAP-MRI basis.
`qtdmri_mapmri_normalization`(mu)	Normalization factor for Cartesian MAP-MRI basis.
`qtdmri_number_of_coefficients`(radial_order, ...)	Computes the total number of coefficients of the qtdmri basis given a radial and temporal order.
`qtdmri_signal_matrix`(radial_order, ...)	Constructs the design matrix as a product of 3 separated radial, angular and temporal design matrices.
`qtdmri_signal_matrix_`(radial_order, ...[, ...])	Function to generate the qtdmri signal basis.
`qtdmri_temporal_normalization`(ut)	Normalization factor for the temporal basis
`qtdmri_to_mapmri_matrix`(radial_order, ...)	Generates the matrix that maps the qtdmri coefficients to MAP-MRI coefficients.
`radial_basis_EAP_opt`(j, l, us, r)
`radial_basis_opt`(j, l, us, q)	Spatial basis dependent on spatial scaling factor us
`real_sh_descoteaux_from_index`(m, n, theta, phi)	Compute real spherical harmonics as in Descoteaux et al. 2007 [R6e399bd6d05a-1], where the real harmonic \(Y^m_n\) is defined to be:.
`temporal_basis`(o, ut, tau)	Temporal basis dependent on temporal scaling factor ut
`visualise_gradient_table_G_Delta_rainbow`(gtab)	This function visualizes a q-tau acquisition scheme as a function of gradient strength and pulse separation (big_delta).
`warn`(/, message[, category, stacklevel, source])	Issue a warning, or maybe ignore it or raise an exception.

Module: `reconst.qti`

Classes and functions for fitting the covariance tensor model of q-space trajectory imaging (QTI) by Westin et al. as presented in “Q-space trajectory imaging for multidimensional diffusion MRI of the human brain” NeuroImage vol. 135 (2016): 345-62. https://doi.org/10.1016/j.neuroimage.2016.02.039

`QtiFit`(params)	Methods
`QtiModel`(gtab[, fit_method, cvxpy_solver])	Methods
`ReconstModel`(gtab)	Abstract class for signal reconstruction models
`Version`(version)	This class abstracts handling of a project's versions.
`auto_attr`(func)	Decorator to create OneTimeProperty attributes.
`cvxpy_1x21_to_6x6`(V)	Convert 1 x 21 vector into a symmetric 6 x 6 matrix.
`cvxpy_1x6_to_3x3`(V)	Convert a 1 x 6 vector into a symmetric 3 x 3 matrix.
`design_matrix`(btens)	Calculate the design matrix from the b-tensors.
`dtd_covariance`(DTD)	Calculate covariance of a diffusion tensor distribution (DTD).
`from_21x1_to_6x6`(V)	Convert 21 x 1 vectors into symmetric 6 x 6 matrices.
`from_3x3_to_6x1`(T)	Convert symmetric 3 x 3 matrices into 6 x 1 vectors.
`from_6x1_to_3x3`(V)	Convert 6 x 1 vectors into symmetric 3 x 3 matrices.
`from_6x6_to_21x1`(T)	Convert symmetric 6 x 6 matrices into 21 x 1 vectors.
`ndindex`(shape)	An N-dimensional iterator object to index arrays.
`optional_package`(name[, trip_msg])	Return package-like thing and module setup for package name
`qti_signal`(gtab, D, C[, S0])	Generate signals using the covariance tensor signal representation.
`warn`(/, message[, category, stacklevel, source])	Issue a warning, or maybe ignore it or raise an exception.

Module: `reconst.quick_squash`

Detect common dtype across object array

`quick_squash`	Try and make a standard array from an object array
`reduce`(function, sequence[, initial])	Apply a function of two arguments cumulatively to the items of a sequence, from left to right, so as to reduce the sequence to a single value.

Module: `reconst.recspeed`

Optimized routines for creating voxel diffusion models

`adj_to_countarrs`	Convert adjacency sequence to counts and flattened indices
`argmax_from_adj`	Indices of local maxima from vals given adjacent points
`argmax_from_countarrs`	Indices of local maxima from vals from count, array neighbors
`le_to_odf`	odf for interpolated Laplacian normalized signal
`local_maxima`	Local maxima of a function evaluated on a discrete set of points.
`proc_reco_args`
`remove_similar_vertices`(vertices, theta[, ...])	Remove vertices that are less than theta degrees from any other
`search_descending`(a, relative_threshold)	i in descending array a so a[i] < a[0] * relative_threshold
`sum_on_blocks_1d`	Summations on blocks of 1d array

Module: `reconst.rumba`

Robust and Unbiased Model-BAsed Spherical Deconvolution (RUMBA-SD)

`AxSymShResponse`(S0, dwi_response[, bvalue])	A simple wrapper for response functions represented using only axially symmetric, even spherical harmonic functions (ie, m == 0 and n even).
`OdfFit`(model, data)	Methods
`OdfModel`(gtab)	An abstract class to be sub-classed by specific odf models
`RumbaFit`(model, model_params)	Methods
`RumbaSDModel`(gtab[, wm_response, ...])	Methods
`Sphere`([x, y, z, theta, phi, xyz, faces, edges])	Points on the unit sphere.
`all_tensor_evecs`(e0)	Given the principle tensor axis, return the array of all eigenvectors column-wise (or, the rotation matrix that orientates the tensor).
`auto_attr`(func)	Decorator to create OneTimeProperty attributes.
`bounding_box`(vol)	Compute the bounding box of nonzero intensity voxels in the volume.
`crop`(vol, mins, maxs)	Crops the input volume.
`generate_kernel`(gtab, sphere, wm_response, ...)	Generate deconvolution kernel
`get_bval_indices`(bvals, bval[, tol])	Get indices where the b-value is bval
`get_sphere`([name])	provide triangulated spheres
`gradient_table`(bvals[, bvecs, big_delta, ...])	A general function for creating diffusion MR gradients.
`lazy_index`(index)	Produces a lazy index
`mbessel_ratio`(n, x)	Fast computation of modified Bessel function ratio (first kind).
`normalize_data`(data, where_b0[, min_signal, out])	Normalizes the data with respect to the mean b0
`rumba_deconv`(data, kernel[, n_iter, ...])	Fit fODF and GM/CSF volume fractions for a voxel using RUMBA-SD [1].
`rumba_deconv_global`(data, kernel, mask[, ...])	Fit fODF for all voxels simultaneously using RUMBA-SD.
`single_tensor`(gtab[, S0, evals, evecs, snr])	Simulate diffusion-weighted signals with a single tensor.
`unique_bvals_tolerance`(bvals[, tol])	Gives the unique b-values of the data, within a tolerance gap
`vec2vec_rotmat`(u, v)	rotation matrix from 2 unit vectors

Module: `reconst.sfm`

The Sparse Fascicle Model.

This is an implementation of the sparse fascicle model described in [Rokem2015]. The multi b-value version of this model is described in [Rokem2014].

[Rokem2015]

Ariel Rokem, Jason D. Yeatman, Franco Pestilli, Kendrick N. Kay, Aviv Mezer, Stefan van der Walt, Brian A. Wandell (2015). Evaluating the accuracy of diffusion MRI models in white matter. PLoS ONE 10(4): e0123272. doi:10.1371/journal.pone.0123272

[Rokem2014]

Ariel Rokem, Kimberly L. Chan, Jason D. Yeatman, Franco Pestilli, Brian A. Wandell (2014). Evaluating the accuracy of diffusion models at multiple b-values with cross-validation. ISMRM 2014.

`Cache`()	Cache values based on a key object (such as a sphere or gradient table).
`ExponentialIsotropicFit`(model, params)	A fit to the ExponentialIsotropicModel object, based on data.
`ExponentialIsotropicModel`(gtab)	Representing the isotropic signal as a fit to an exponential decay function with b-values
`IsotropicFit`(model, params)	A fit object for representing the isotropic signal as the mean of the diffusion-weighted signal.
`IsotropicModel`(gtab)	A base-class for the representation of isotropic signals.
`OrderedDict`	Dictionary that remembers insertion order
`ReconstFit`(model, data)	Abstract class which holds the fit result of ReconstModel
`ReconstModel`(gtab)	Abstract class for signal reconstruction models
`SparseFascicleFit`(model, beta, S0, iso)	Methods
`SparseFascicleModel`(gtab[, sphere, ...])	Methods
`auto_attr`(func)	Decorator to create OneTimeProperty attributes.
`determine_num_processes`(num_processes)	Determine the effective number of processes for parallelization.
`nanmean`(a[, axis, dtype, out, keepdims, where])	Compute the arithmetic mean along the specified axis, ignoring NaNs.
`optional_package`(name[, trip_msg])	Return package-like thing and module setup for package name
`sfm_design_matrix`(gtab, sphere, response[, mode])	Construct the SFM design matrix

Module: `reconst.shm`

Tools for using spherical harmonic models to fit diffusion data.

References

[1]

Aganj, I., et al. 2009. ODF Reconstruction in Q-Ball Imaging With Solid Angle Consideration.

[2]

Descoteaux, M., et al. 2007. Regularized, fast, and robust analytical Q-ball imaging.

[3]

Tristan-Vega, A., et al. 2010. A new methodology for estimation of fiber populations in white matter of the brain with Funk-Radon transform.

[4]

Tristan-Vega, A., et al. 2009. Estimation of fiber orientation probability density functions in high angular resolution diffusion imaging.

Note about the Transpose: In the literature the matrix representation of these methods is often written as Y = Bx where B is some design matrix and Y and x are column vectors. In our case the input data, a dwi stored as a nifti file for example, is stored as row vectors (ndarrays) of the form (x, y, z, n), where n is the number of diffusion directions. We could transpose and reshape the data to be (n, x*y*z), so that we could directly plug it into the above equation. However, I have chosen to keep the data as is and implement the relevant equations rewritten in the following form: Y.T = x.T B.T, or in python syntax data = np.dot(sh_coef, B.T) where data is Y.T and sh_coef is x.T.

`Cache`()	Cache values based on a key object (such as a sphere or gradient table).
`CsaOdfModel`(gtab, sh_order[, smooth, ...])	Implementation of Constant Solid Angle reconstruction method.
`OdfFit`(model, data)	Methods
`OdfModel`(gtab)	An abstract class to be sub-classed by specific odf models
`OpdtModel`(gtab, sh_order[, smooth, ...])	Implementation of Orientation Probability Density Transform reconstruction method.
`QballBaseModel`(gtab, sh_order[, smooth, ...])	To be subclassed by Qball type models.
`QballModel`(gtab, sh_order[, smooth, ...])	Implementation of regularized Qball reconstruction method.
`ResidualBootstrapWrapper`(signal_object, B, ...)	Returns a residual bootstrap sample of the signal_object when indexed
`SphHarmFit`(model, shm_coef, mask)	Diffusion data fit to a spherical harmonic model
`SphHarmModel`(gtab)	To be subclassed by all models that return a SphHarmFit when fit.
`anisotropic_power`(sh_coeffs[, norm_factor, ...])	Calculate anisotropic power map with a given SH coefficient matrix.
`auto_attr`(func)	Decorator to create OneTimeProperty attributes.
`bootstrap_data_array`(data, H, R[, permute])	Applies the Residual Bootstraps to the data given H and R
`bootstrap_data_voxel`(data, H, R[, permute])	Like bootstrap_data_array but faster when for a single voxel
`calculate_max_order`(n_coeffs[, full_basis])	Calculate the maximal harmonic order, given that you know the number of parameters that were estimated.
`cart2sphere`(x, y, z)	Return angles for Cartesian 3D coordinates x, y, and z
`convert_sh_from_legacy`(sh_coeffs, sh_basis)	Convert SH coefficients in legacy SH basis to SH coefficients of the new SH basis for `descoteaux07` [1] or `tournier07` [R8020d68d5dcd-2]_[R8020d68d5dcd-3]_ bases.
`convert_sh_to_full_basis`(sh_coeffs)	Given an array of SH coeffs from a symmetric basis, returns the coefficients for the full SH basis by filling odd order SH coefficients with zeros
`convert_sh_to_legacy`(sh_coeffs, sh_basis[, ...])	Convert SH coefficients in new SH basis to SH coefficients for the legacy SH basis for `descoteaux07` [1] or `tournier07` [R2032a14b59d6-2]_[R2032a14b59d6-3]_ bases.
`deprecate_with_version`(message[, since, ...])	Return decorator function function for deprecation warning / error.
`forward_sdeconv_mat`(r_rh, n)	Build forward spherical deconvolution matrix
`gen_dirac`(m, n, theta, phi[, legacy])	Generate Dirac delta function orientated in (theta, phi) on the sphere
`hat`(B)	Returns the hat matrix for the design matrix B
`lazy_index`(index)	Produces a lazy index
`lcr_matrix`(H)	Returns a matrix for computing leveraged, centered residuals from data
`normalize_data`(data, where_b0[, min_signal, out])	Normalizes the data with respect to the mean b0
`order_from_ncoef`(ncoef[, full_basis])	Given a number `n` of coefficients, calculate back the `sh_order`
`randint`(low[, high, size, dtype])	Return random integers from low (inclusive) to high (exclusive).
`real_sh_descoteaux`(sh_order, theta, phi[, ...])	Compute real spherical harmonics as in Descoteaux et al. 2007 [R191e35d27b5b-1], where the real harmonic \(Y^m_n\) is defined to be:.
`real_sh_descoteaux_from_index`(m, n, theta, phi)	Compute real spherical harmonics as in Descoteaux et al. 2007 [R700bd48b1688-1], where the real harmonic \(Y^m_n\) is defined to be:.
`real_sh_tournier`(sh_order, theta, phi[, ...])	Compute real spherical harmonics as initially defined in Tournier 2007 [1] then updated in MRtrix3 [2], where the real harmonic \(Y^m_n\) is defined to be:
`real_sh_tournier_from_index`(m, n, theta, phi)	Compute real spherical harmonics as initially defined in Tournier 2007 [1] then updated in MRtrix3 [2], where the real harmonic \(Y^m_n\) is defined to be:
`real_sph_harm`(m, n, theta, phi)	Compute real spherical harmonics.
`real_sym_sh_basis`(sh_order, theta, phi)	Samples a real symmetric spherical harmonic basis at point on the sphere
`real_sym_sh_mrtrix`(sh_order, theta, phi)	dipy.reconst.shm.real_sym_sh_mrtrix is deprecated, Please use dipy.reconst.shm.real_sh_tournier instead
`sf_to_sh`(sf, sphere[, sh_order, basis_type, ...])	Spherical function to spherical harmonics (SH).
`sh_to_rh`(r_sh, m, n)	Spherical harmonics (SH) to rotational harmonics (RH)
`sh_to_sf`(sh, sphere[, sh_order, basis_type, ...])	Spherical harmonics (SH) to spherical function (SF).
`sh_to_sf_matrix`(sphere[, sh_order, ...])	Matrix that transforms Spherical harmonics (SH) to spherical function (SF).
`smooth_pinv`(B, L)	Regularized pseudo-inverse
`sph_harm_ind_list`(sh_order[, full_basis])	Returns the degree (`m`) and order (`n`) of all the symmetric spherical harmonics of degree less then or equal to `sh_order`.
`spherical_harmonics`(m, n, theta, phi[, ...])	Compute spherical harmonics.
`warn`(/, message[, category, stacklevel, source])	Issue a warning, or maybe ignore it or raise an exception.

Module: `reconst.shore`

`Cache`()	Cache values based on a key object (such as a sphere or gradient table).
`ShoreFit`(model, shore_coef)	Attributes:
`ShoreModel`(gtab[, radial_order, zeta, ...])	Simple Harmonic Oscillator based Reconstruction and Estimation (SHORE) [1] of the diffusion signal.
`Version`(version)	This class abstracts handling of a project's versions.
`cart2sphere`(x, y, z)	Return angles for Cartesian 3D coordinates x, y, and z
`create_rspace`(gridsize, radius_max)	Create the real space table, that contains the points in which
`factorial`(x, /)	Find x!.
`genlaguerre`(n, alpha[, monic])	Generalized (associated) Laguerre polynomial.
`l_shore`(radial_order)	Returns the angular regularisation matrix for SHORE basis
`multi_voxel_fit`(single_voxel_fit)	Method decorator to turn a single voxel model fit definition into a multi voxel model fit definition
`n_shore`(radial_order)	Returns the angular regularisation matrix for SHORE basis
`optional_package`(name[, trip_msg])	Return package-like thing and module setup for package name
`real_sh_descoteaux_from_index`(m, n, theta, phi)	Compute real spherical harmonics as in Descoteaux et al. 2007 [Rf0cd5775fa0d-1], where the real harmonic \(Y^m_n\) is defined to be:.
`shore_indices`(radial_order, index)	Given the basis order and the index, return the shore indices n, l, m for modified Merlet's 3D-SHORE ..math:: :nowrap: begin{equation} textbf{E}(qtextbf{u})=sum_{l=0, even}^{N_{max}} sum_{n=l}^{(N_{max}+l)/2} sum_{m=-l}^l c_{nlm} phi_{nlm}(qtextbf{u}) end{equation}
`shore_matrix`(radial_order, zeta, gtab[, tau])	Compute the SHORE matrix for modified Merlet's 3D-SHORE [1]
`shore_matrix_odf`(radial_order, zeta, ...)	Compute the SHORE ODF matrix [1]"
`shore_matrix_pdf`(radial_order, zeta, rtab)	Compute the SHORE propagator matrix [1]"
`shore_order`(n, l, m)	Given the indices (n,l,m) of the basis, return the minimum order for those indices and their index for modified Merlet's 3D-SHORE.
`warn`(/, message[, category, stacklevel, source])	Issue a warning, or maybe ignore it or raise an exception.

Module: `reconst.utils`

dki_design_matrix(gtab)

Construct B design matrix for DKI.

Module: `reconst.vec_val_sum`

vec_val_vect

Vectorize vecs.diag(vals).`vecs`.T for last 2 dimensions of vecs

bench

dipy.reconst.bench(label='fast', verbose=1, extra_argv=None)

Run benchmarks for module using nose.

Parameters:

label{‘fast’, ‘full’, ‘’, attribute identifier}, optional

Identifies the benchmarks to run. This can be a string to pass to the nosetests executable with the ‘-A’ option, or one of several special values. Special values are:

‘fast’ - the default - which corresponds to the nosetests -A option of ‘not slow’.
‘full’ - fast (as above) and slow benchmarks as in the ‘no -A’ option to nosetests - this is the same as ‘’.
None or ‘’ - run all tests.
attribute_identifier - string passed directly to nosetests as ‘-A’.

verboseint, optional

Verbosity value for benchmark outputs, in the range 1-10. Default is 1.

extra_argvlist, optional

List with any extra arguments to pass to nosetests.

Returns:

successbool: Returns True if running the benchmarks works, False if an error occurred.

Notes

Benchmarks are like tests, but have names starting with “bench” instead of “test”, and can be found under the “benchmarks” sub-directory of the module.

Each NumPy module exposes bench in its namespace to run all benchmarks for it.

Examples

>>> success = np.lib.bench() 
Running benchmarks for numpy.lib
...
using 562341 items:
unique:
0.11
unique1d:
0.11
ratio: 1.0
nUnique: 56230 == 56230
...
OK

>>> success 
True

test

dipy.reconst.test(label='fast', verbose=1, extra_argv=None, doctests=False, coverage=False, raise_warnings=None, timer=False)

Run tests for module using nose.

Parameters:

label{‘fast’, ‘full’, ‘’, attribute identifier}, optional

Identifies the tests to run. This can be a string to pass to the nosetests executable with the ‘-A’ option, or one of several special values. Special values are:

‘fast’ - the default - which corresponds to the nosetests -A option of ‘not slow’.
‘full’ - fast (as above) and slow tests as in the ‘no -A’ option to nosetests - this is the same as ‘’.
None or ‘’ - run all tests.
attribute_identifier - string passed directly to nosetests as ‘-A’.

verboseint, optional

Verbosity value for test outputs, in the range 1-10. Default is 1.

extra_argvlist, optional

List with any extra arguments to pass to nosetests.

doctestsbool, optional

If True, run doctests in module. Default is False.

coveragebool, optional

If True, report coverage of NumPy code. Default is False. (This requires the coverage module).

raise_warningsNone, str or sequence of warnings, optional

This specifies which warnings to configure as ‘raise’ instead of being shown once during the test execution. Valid strings are:

“develop” : equals (Warning,)
“release” : equals (), do not raise on any warnings.

timerbool or int, optional

Timing of individual tests with nose-timer (which needs to be installed). If True, time tests and report on all of them. If an integer (say N), report timing results for N slowest tests.

Returns:

resultobject: Returns the result of running the tests as a nose.result.TextTestResult object.

Notes

Each NumPy module exposes test in its namespace to run all tests for it. For example, to run all tests for numpy.lib:

>>> np.lib.test() 

Examples

>>> result = np.lib.test() 
Running unit tests for numpy.lib
...
Ran 976 tests in 3.933s

>>> result.errors 
[]
>>> result.knownfail 
[]

`ReconstFit`

class dipy.reconst.base.ReconstFit(model, data)

Bases: object

Abstract class which holds the fit result of ReconstModel

For example that could be holding FA or GFA etc.

__init__(model, data)

`ReconstModel`

class dipy.reconst.base.ReconstModel(gtab)

Bases: object

Abstract class for signal reconstruction models

Methods

fit

__init__(gtab)

Initialization of the abstract class for signal reconstruction models

Parameters:

gtabGradientTable class instance

fit(data, mask=None, **kwargs)

bench_bounding_box

dipy.reconst.benchmarks.bench_bounding_box.bench_bounding_box()

measure

dipy.reconst.benchmarks.bench_bounding_box.measure(code_str, times=1, label=None)

Return elapsed time for executing code in the namespace of the caller.

The supplied code string is compiled with the Python builtin compile. The precision of the timing is 10 milli-seconds. If the code will execute fast on this timescale, it can be executed many times to get reasonable timing accuracy.

Parameters:

code_strstr: The code to be timed.
timesint, optional: The number of times the code is executed. Default is 1. The code is only compiled once.
labelstr, optional: A label to identify code_str with. This is passed into compile as the second argument (for run-time error messages).

Returns:

elapsedfloat: Total elapsed time in seconds for executing code_str times times.

Examples

>>> times = 10
>>> etime = np.testing.measure('for i in range(1000): np.sqrt(i**2)', times=times)
>>> print("Time for a single execution : ", etime / times, "s")  
Time for a single execution :  0.005 s

`ConstrainedSphericalDeconvModel`

class dipy.reconst.benchmarks.bench_csd.ConstrainedSphericalDeconvModel(gtab, response, reg_sphere=None, sh_order=8, lambda_=1, tau=0.1, convergence=50)

Bases: SphHarmModel

Methods

`cache_clear`()	Clear the cache.
`cache_get`(tag, key[, default])	Retrieve a value from the cache.
`cache_set`(tag, key, value)	Store a value in the cache.
`fit`(data[, mask])	Fit method for every voxel in data
`predict`(sh_coeff[, gtab, S0])	Compute a signal prediction given spherical harmonic coefficients for the provided GradientTable class instance.
`sampling_matrix`(sphere)	The matrix needed to sample ODFs from coefficients of the model.

__init__(gtab, response, reg_sphere=None, sh_order=8, lambda_=1, tau=0.1, convergence=50)

Constrained Spherical Deconvolution (CSD) [1].

Spherical deconvolution computes a fiber orientation distribution (FOD), also called fiber ODF (fODF) [2], as opposed to a diffusion ODF as the QballModel or the CsaOdfModel. This results in a sharper angular profile with better angular resolution that is the best object to be used for later deterministic and probabilistic tractography [3].

A sharp fODF is obtained because a single fiber response function is injected as a priori knowledge. The response function is often data-driven and is thus provided as input to the ConstrainedSphericalDeconvModel. It will be used as deconvolution kernel, as described in [1].

Parameters:

gtabGradientTable
responsetuple or AxSymShResponse object: A tuple with two elements. The first is the eigen-values as an (3,) ndarray and the second is the signal value for the response function without diffusion weighting (i.e. S0). This is to be able to generate a single fiber synthetic signal. The response function will be used as deconvolution kernel ([1]).
reg_sphereSphere (optional): sphere used to build the regularization B matrix. Default: ‘symmetric362’.
sh_orderint (optional): maximal spherical harmonics order. Default: 8
lambda_float (optional): weight given to the constrained-positivity regularization part of the deconvolution equation (see [1]). Default: 1
taufloat (optional): threshold controlling the amplitude below which the corresponding fODF is assumed to be zero. Ideally, tau should be set to zero. However, to improve the stability of the algorithm, tau is set to tau*100 % of the mean fODF amplitude (here, 10% by default) (see [1]). Default: 0.1
convergenceint: Maximum number of iterations to allow the deconvolution to converge.

References

[1] (1,2,3,4,5)

Tournier, J.D., et al. NeuroImage 2007. Robust determination of the fibre orientation distribution in diffusion MRI: Non-negativity constrained super-resolved spherical deconvolution

[2]

Descoteaux, M., et al. IEEE TMI 2009. Deterministic and Probabilistic Tractography Based on Complex Fibre Orientation Distributions

[3]

Côté, M-A., et al. Medical Image Analysis 2013. Tractometer: Towards validation of tractography pipelines

[4]

Tournier, J.D, et al. Imaging Systems and Technology 2012. MRtrix: Diffusion Tractography in Crossing Fiber Regions

fit(data, mask=None): Fit method for every voxel in data

predict(sh_coeff, gtab=None, S0=1.0)

Compute a signal prediction given spherical harmonic coefficients for the provided GradientTable class instance.

Parameters:

sh_coeffndarray: The spherical harmonic representation of the FOD from which to make the signal prediction.
gtabGradientTable: The gradients for which the signal will be predicted. Uses the model’s gradient table by default.
S0ndarray or float: The non diffusion-weighted signal value.

Returns:

pred_signdarray: The predicted signal.

`GradientTable`

class dipy.reconst.benchmarks.bench_csd.GradientTable(gradients, big_delta=None, small_delta=None, b0_threshold=50, btens=None)

Bases: object

Diffusion gradient information

Parameters:

gradientsarray_like (N, 3): Diffusion gradients. The direction of each of these vectors corresponds to the b-vector, and the length corresponds to the b-value.
b0_thresholdfloat: Gradients with b-value less than or equal to b0_threshold are considered as b0s i.e. without diffusion weighting.

See also

gradient_table

Notes

The GradientTable object is immutable. Do NOT assign attributes. If you have your gradient table in a bval & bvec format, we recommend using the factory function gradient_table

Attributes:

gradients(N,3) ndarray: diffusion gradients
bvals(N,) ndarray: The b-value, or magnitude, of each gradient direction.
qvals: (N,) ndarray: The q-value for each gradient direction. Needs big and small delta.
bvecs(N,3) ndarray: The direction, represented as a unit vector, of each gradient.
b0s_mask(N,) ndarray: Boolean array indicating which gradients have no diffusion weighting, ie b-value is close to 0.
b0_thresholdfloat: Gradients with b-value less than or equal to b0_threshold are considered to not have diffusion weighting.
btens(N,3,3) ndarray: The b-tensor of each gradient direction.

Methods

b0s_mask
bvals
bvecs
gradient_strength
qvals
tau

__init__(gradients, big_delta=None, small_delta=None, b0_threshold=50, btens=None): Constructor for GradientTable class

b0s_mask()

bvals()

bvecs()

gradient_strength()

property info

qvals()

tau()

bench_csdeconv

dipy.reconst.benchmarks.bench_csd.bench_csdeconv(center=(50, 40, 40), width=12)

load_nifti_data

dipy.reconst.benchmarks.bench_csd.load_nifti_data(fname, as_ndarray=True)

Load only the data array from a nifti file.

Parameters:

fnamestr: Full path to the file.
as_ndarray: bool, optional: convert nibabel ArrayProxy to a numpy.ndarray. If you want to save memory and delay this casting, just turn this option to False (default: True)

Returns:

data: np.ndarray or nib.ArrayProxy

See also

load_nifti

num_grad

dipy.reconst.benchmarks.bench_csd.num_grad(gtab)

read_stanford_labels

dipy.reconst.benchmarks.bench_csd.read_stanford_labels(): Read stanford hardi data and label map.

bench_local_maxima

dipy.reconst.benchmarks.bench_peaks.bench_local_maxima()

measure

dipy.reconst.benchmarks.bench_peaks.measure(code_str, times=1, label=None)

Return elapsed time for executing code in the namespace of the caller.

Parameters:

code_strstr: The code to be timed.
timesint, optional: The number of times the code is executed. Default is 1. The code is only compiled once.
labelstr, optional: A label to identify code_str with. This is passed into compile as the second argument (for run-time error messages).

Returns:

elapsedfloat: Total elapsed time in seconds for executing code_str times times.

Examples

>>> times = 10
>>> etime = np.testing.measure('for i in range(1000): np.sqrt(i**2)', times=times)
>>> print("Time for a single execution : ", etime / times, "s")  
Time for a single execution :  0.005 s

bench_quick_squash

dipy.reconst.benchmarks.bench_squash.bench_quick_squash()

measure

dipy.reconst.benchmarks.bench_squash.measure(code_str, times=1, label=None)

Return elapsed time for executing code in the namespace of the caller.

Parameters:

code_strstr: The code to be timed.
timesint, optional: The number of times the code is executed. Default is 1. The code is only compiled once.
labelstr, optional: A label to identify code_str with. This is passed into compile as the second argument (for run-time error messages).

Returns:

elapsedfloat: Total elapsed time in seconds for executing code_str times times.

Examples

>>> times = 10
>>> etime = np.testing.measure('for i in range(1000): np.sqrt(i**2)', times=times)
>>> print("Time for a single execution : ", etime / times, "s")  
Time for a single execution :  0.005 s

ndindex

dipy.reconst.benchmarks.bench_squash.ndindex(shape)

An N-dimensional iterator object to index arrays.

Given the shape of an array, an ndindex instance iterates over the N-dimensional index of the array. At each iteration a tuple of indices is returned; the last dimension is iterated over first.

Parameters:

shapetuple of ints: The dimensions of the array.

Examples

>>> from dipy.core.ndindex import ndindex
>>> shape = (3, 2, 1)
>>> for index in ndindex(shape):
...     print(index)
(0, 0, 0)
(0, 1, 0)
(1, 0, 0)
(1, 1, 0)
(2, 0, 0)
(2, 1, 0)

old_squash

dipy.reconst.benchmarks.bench_squash.old_squash(arr, mask=None, fill=0)

Try and make a standard array from an object array

This function takes an object array and attempts to convert it to a more useful dtype. If array can be converted to a better dtype, Nones are replaced by fill. To make the behaviour of this function more clear, here are the most common cases:

arr is an array of scalars of type T. Returns an array like arr.astype(T)
arr is an array of arrays. All items in arr have the same shape S. Returns an array with shape arr.shape + S.
arr is an array of arrays of different shapes. Returns arr.
Items in arr are not ndarrys or scalars. Returns arr.

Parameters:

arrarray, dtype=object: The array to be converted.
maskarray, dtype=bool, optional: Where arr has Nones.
fillnumber, optional: Nones are replaced by fill.

Returns:

resultarray

Examples

>>> arr = np.empty(3, dtype=object)
>>> arr.fill(2)
>>> old_squash(arr)
array([2, 2, 2])
>>> arr[0] = None
>>> old_squash(arr)
array([0, 2, 2])
>>> arr.fill(np.ones(2))
>>> r = old_squash(arr)
>>> r.shape == (3, 2)
True
>>> r.dtype
dtype('float64')

reduce

dipy.reconst.benchmarks.bench_squash.reduce(function, sequence[, initial]) → value: Apply a function of two arguments cumulatively to the items of a sequence, from left to right, so as to reduce the sequence to a single value. For example, reduce(lambda x, y: x+y, [1, 2, 3, 4, 5]) calculates ((((1+2)+3)+4)+5). If initial is present, it is placed before the items of the sequence in the calculation, and serves as a default when the sequence is empty.

bench_vec_val_vect

dipy.reconst.benchmarks.bench_vec_val_sum.bench_vec_val_vect()

measure

dipy.reconst.benchmarks.bench_vec_val_sum.measure(code_str, times=1, label=None)

Return elapsed time for executing code in the namespace of the caller.

Parameters:

code_strstr: The code to be timed.
timesint, optional: The number of times the code is executed. Default is 1. The code is only compiled once.
labelstr, optional: A label to identify code_str with. This is passed into compile as the second argument (for run-time error messages).

Returns:

elapsedfloat: Total elapsed time in seconds for executing code_str times times.

Examples

>>> times = 10
>>> etime = np.testing.measure('for i in range(1000): np.sqrt(i**2)', times=times)
>>> print("Time for a single execution : ", etime / times, "s")  
Time for a single execution :  0.005 s

`Cache`

class dipy.reconst.cache.Cache

Bases: object

Cache values based on a key object (such as a sphere or gradient table).

Notes

This class is meant to be used as a mix-in:

class MyModel(Model, Cache):
    pass

class MyModelFit(Fit):
    pass

Inside a method on the fit, typical usage would be:

def odf(sphere):
    M = self.model.cache_get('odf_basis_matrix', key=sphere)

    if M is None:
        M = self._compute_basis_matrix(sphere)
        self.model.cache_set('odf_basis_matrix', key=sphere, value=M)

Methods

`cache_clear`()	Clear the cache.
`cache_get`(tag, key[, default])	Retrieve a value from the cache.
`cache_set`(tag, key, value)	Store a value in the cache.

__init__(*args, **kwargs)

cache_clear(): Clear the cache.

cache_get(tag, key, default=None)

Retrieve a value from the cache.

Parameters:

tagstr: Description of the cached value.
keyobject: Key object used to look up the cached value.
defaultobject: Value to be returned if no cached entry is found.

Returns:

vobject: Value from the cache associated with (tag, key). Returns default if no cached entry is found.

cache_set(tag, key, value)

Store a value in the cache.

Parameters:

tagstr: Description of the cached value.
keyobject: Key object used to look up the cached value.
valueobject: Value stored in the cache for each unique combination of (tag, key).

Examples

>>> def compute_expensive_matrix(parameters):
...     # Imagine the following computation is very expensive
...     return (p**2 for p in parameters)

>>> c = Cache()

>>> parameters = (1, 2, 3)
>>> X1 = compute_expensive_matrix(parameters)

>>> c.cache_set('expensive_matrix', parameters, X1)
>>> X2 = c.cache_get('expensive_matrix', parameters)

>>> X1 is X2
True

auto_attr

dipy.reconst.cache.auto_attr(func)

Decorator to create OneTimeProperty attributes.

Parameters:

funcmethod: The method that will be called the first time to compute a value. Afterwards, the method’s name will be a standard attribute holding the value of this computation.

Examples

>>> class MagicProp(object):
...     @auto_attr
...     def a(self):
...         return 99
...
>>> x = MagicProp()
>>> 'a' in x.__dict__
False
>>> x.a
99
>>> 'a' in x.__dict__
True

coeff_of_determination

dipy.reconst.cross_validation.coeff_of_determination(data, model, axis=-1)

Calculate the coefficient of determination for a model prediction,: relative to data.

Parameters:

datandarray: The data
modelndarray: The predictions of a model for this data. Same shape as the data.
axis: int, optional: The axis along which different samples are laid out (default: -1).

Returns:

CODndarray: The coefficient of determination. This has shape data.shape[:-1]

rac{SSE}{SSD})

where SSE is the sum of the squared error between the model and the data (sum of the squared residuals) and SSD is the sum of the squares of the deviations of the data from the mean of the data (variance * N).

kfold_xval

dipy.reconst.cross_validation.kfold_xval(model, data, folds, *model_args, **model_kwargs)

Perform k-fold cross-validation.

It generate out-of-sample predictions for each measurement.

Parameters:

modelModel class instance: The type of the model to use for prediction. The corresponding Fit object must have a predict function implementd One of the following: reconst.dti.TensorModel or reconst.csdeconv.ConstrainedSphericalDeconvModel.
datandarray: Diffusion MRI data acquired with the GradientTable of the model. Shape will typically be (x, y, z, b) where xyz are spatial dimensions and b is the number of bvals/bvecs in the GradientTable.
foldsint: The number of divisions to apply to the data
model_argslist: Additional arguments to the model initialization
model_kwargsdict: Additional key-word arguments to the model initialization. If contains the kwarg mask, this will be used as a key-word argument to the fit method of the model object, rather than being used in the initialization of the model object

Notes

This function assumes that a prediction API is implemented in the Model class for which prediction is conducted. That is, the Fit object that gets generated upon fitting the model needs to have a predict method, which receives a GradientTable class instance as input and produces a predicted signal as output.

It also assumes that the model object has bval and bvec attributes holding b-values and corresponding unit vectors.

References

[1]

Rokem, A., Chan, K.L. Yeatman, J.D., Pestilli, F., Mezer, A., Wandell, B.A., 2014. Evaluating the accuracy of diffusion models at multiple b-values with cross-validation. ISMRM 2014.

`AxSymShResponse`

class dipy.reconst.csdeconv.AxSymShResponse(S0, dwi_response, bvalue=None)

Bases: object

A simple wrapper for response functions represented using only axially symmetric, even spherical harmonic functions (ie, m == 0 and n even).

Parameters:

S0float: Signal with no diffusion weighting.
dwi_responsearray: Response function signal as coefficients to axially symmetric, even spherical harmonic.

Methods

`basis`(sphere)	A basis that maps the response coefficients onto a sphere.
`on_sphere`(sphere)	Evaluates the response function on sphere.

__init__(S0, dwi_response, bvalue=None)

basis(sphere): A basis that maps the response coefficients onto a sphere.

on_sphere(sphere): Evaluates the response function on sphere.

`ConstrainedSDTModel`

class dipy.reconst.csdeconv.ConstrainedSDTModel(gtab, ratio, reg_sphere=None, sh_order=8, lambda_=1.0, tau=0.1)

Bases: SphHarmModel

Methods

`cache_clear`()	Clear the cache.
`cache_get`(tag, key[, default])	Retrieve a value from the cache.
`cache_set`(tag, key, value)	Store a value in the cache.
`fit`(data[, mask])	Fit method for every voxel in data
`sampling_matrix`(sphere)	The matrix needed to sample ODFs from coefficients of the model.

__init__(gtab, ratio, reg_sphere=None, sh_order=8, lambda_=1.0, tau=0.1)

Spherical Deconvolution Transform (SDT) [1].

The SDT computes a fiber orientation distribution (FOD) as opposed to a diffusion ODF as the QballModel or the CsaOdfModel. This results in a sharper angular profile with better angular resolution. The Constrained SDTModel is similar to the Constrained CSDModel but mathematically it deconvolves the q-ball ODF as oppposed to the HARDI signal (see [1] for a comparison and a through discussion).

A sharp fODF is obtained because a single fiber response function is injected as a priori knowledge. In the SDTModel, this response is a single fiber q-ball ODF as opposed to a single fiber signal function for the CSDModel. The response function will be used as deconvolution kernel.

Parameters:

gtabGradientTable
ratiofloat: ratio of the smallest vs the largest eigenvalue of the single prolate tensor response function
reg_sphereSphere: sphere used to build the regularization B matrix
sh_orderint: maximal spherical harmonics order
lambda_float: weight given to the constrained-positivity regularization part of the deconvolution equation
taufloat: threshold (tau *mean(fODF)) controlling the amplitude below which the corresponding fODF is assumed to be zero.

References

[1] (1,2)

Descoteaux, M., et al. IEEE TMI 2009. Deterministic and Probabilistic Tractography Based on Complex Fibre Orientation Distributions.

fit(data, mask=None): Fit method for every voxel in data

`ConstrainedSphericalDeconvModel`

class dipy.reconst.csdeconv.ConstrainedSphericalDeconvModel(gtab, response, reg_sphere=None, sh_order=8, lambda_=1, tau=0.1, convergence=50)

Bases: SphHarmModel

Methods

`cache_clear`()	Clear the cache.
`cache_get`(tag, key[, default])	Retrieve a value from the cache.
`cache_set`(tag, key, value)	Store a value in the cache.
`fit`(data[, mask])	Fit method for every voxel in data
`predict`(sh_coeff[, gtab, S0])	Compute a signal prediction given spherical harmonic coefficients for the provided GradientTable class instance.
`sampling_matrix`(sphere)	The matrix needed to sample ODFs from coefficients of the model.

__init__(gtab, response, reg_sphere=None, sh_order=8, lambda_=1, tau=0.1, convergence=50)

Constrained Spherical Deconvolution (CSD) [1].

Parameters:

gtabGradientTable
responsetuple or AxSymShResponse object: A tuple with two elements. The first is the eigen-values as an (3,) ndarray and the second is the signal value for the response function without diffusion weighting (i.e. S0). This is to be able to generate a single fiber synthetic signal. The response function will be used as deconvolution kernel ([1]).
reg_sphereSphere (optional): sphere used to build the regularization B matrix. Default: ‘symmetric362’.
sh_orderint (optional): maximal spherical harmonics order. Default: 8
lambda_float (optional): weight given to the constrained-positivity regularization part of the deconvolution equation (see [1]). Default: 1
taufloat (optional): threshold controlling the amplitude below which the corresponding fODF is assumed to be zero. Ideally, tau should be set to zero. However, to improve the stability of the algorithm, tau is set to tau*100 % of the mean fODF amplitude (here, 10% by default) (see [1]). Default: 0.1
convergenceint: Maximum number of iterations to allow the deconvolution to converge.

References

[1] (1,2,3,4,5)

Tournier, J.D., et al. NeuroImage 2007. Robust determination of the fibre orientation distribution in diffusion MRI: Non-negativity constrained super-resolved spherical deconvolution

[2]

Descoteaux, M., et al. IEEE TMI 2009. Deterministic and Probabilistic Tractography Based on Complex Fibre Orientation Distributions

[3]

Côté, M-A., et al. Medical Image Analysis 2013. Tractometer: Towards validation of tractography pipelines

[4]

Tournier, J.D, et al. Imaging Systems and Technology 2012. MRtrix: Diffusion Tractography in Crossing Fiber Regions

fit(data, mask=None): Fit method for every voxel in data

predict(sh_coeff, gtab=None, S0=1.0)

Compute a signal prediction given spherical harmonic coefficients for the provided GradientTable class instance.

Parameters:

sh_coeffndarray: The spherical harmonic representation of the FOD from which to make the signal prediction.
gtabGradientTable: The gradients for which the signal will be predicted. Uses the model’s gradient table by default.
S0ndarray or float: The non diffusion-weighted signal value.

Returns:

pred_signdarray: The predicted signal.

`SphHarmFit`

class dipy.reconst.csdeconv.SphHarmFit(model, shm_coef, mask)

Bases: OdfFit

Diffusion data fit to a spherical harmonic model

Attributes:

shape
shm_coeff: The spherical harmonic coefficients of the odf

Methods

`odf`(sphere)	Samples the odf function on the points of a sphere
`predict`([gtab, S0])	Predict the diffusion signal from the model coefficients.

gfa

__init__(model, shm_coef, mask)

gfa()

odf(sphere)

Samples the odf function on the points of a sphere

Parameters:

sphereSphere: The points on which to sample the odf.

Returns:

valuesndarray: The value of the odf on each point of sphere.

predict(gtab=None, S0=1.0)

Predict the diffusion signal from the model coefficients.

Parameters:

gtaba GradientTable class instance: The directions and bvalues on which prediction is desired
S0float array: The mean non-diffusion-weighted signal in each voxel.

property shape

property shm_coeff

The spherical harmonic coefficients of the odf

Make this a property for now, if there is a use case for modifying the coefficients we can add a setter or expose the coefficients more directly

`SphHarmModel`

class dipy.reconst.csdeconv.SphHarmModel(gtab)

Bases: OdfModel, Cache

To be subclassed by all models that return a SphHarmFit when fit.

Methods

`cache_clear`()	Clear the cache.
`cache_get`(tag, key[, default])	Retrieve a value from the cache.
`cache_set`(tag, key, value)	Store a value in the cache.
`fit`(data)	To be implemented by specific odf models
`sampling_matrix`(sphere)	The matrix needed to sample ODFs from coefficients of the model.

__init__(gtab)

Initialization of the abstract class for signal reconstruction models

Parameters:

gtabGradientTable class instance

sampling_matrix(sphere)

The matrix needed to sample ODFs from coefficients of the model.

Parameters:

sphereSphere: Points used to sample ODF.

Returns:

sampling_matrixarray: The size of the matrix will be (N, M) where N is the number of vertices on sphere and M is the number of coefficients needed by the model.

`TensorModel`

class dipy.reconst.csdeconv.TensorModel(gtab, fit_method='WLS', return_S0_hat=False, *args, **kwargs)

Bases: ReconstModel

Diffusion Tensor

Methods

`fit`(data[, mask])	Fit method of the DTI model class
`predict`(dti_params[, S0])	Predict a signal for this TensorModel class instance given parameters.

__init__(gtab, fit_method='WLS', return_S0_hat=False, *args, **kwargs)

A Diffusion Tensor Model [1], [2].

Parameters:

gtabGradientTable class instance

fit_methodstr or callable

str can be one of the following:

‘WLS’ for weighted least squares: dti.wls_fit_tensor()
‘LS’ or ‘OLS’ for ordinary least squares: dti.ols_fit_tensor()
‘NLLS’ for non-linear least-squares: dti.nlls_fit_tensor()
‘RT’ or ‘restore’ or ‘RESTORE’ for RESTORE robust tensor: fitting [3] dti.restore_fit_tensor()
callable has to have the signature:: fit_method(design_matrix, data, *args, **kwargs)

return_S0_hatbool

Boolean to return (True) or not (False) the S0 values for the fit.

args, kwargsarguments and key-word arguments passed to the

fit_method. See dti.wls_fit_tensor, dti.ols_fit_tensor for details

min_signalfloat

The minimum signal value. Needs to be a strictly positive number. Default: minimal signal in the data provided to fit.

Notes

In order to increase speed of processing, tensor fitting is done simultaneously over many voxels. Many fit_methods use the ‘step’ parameter to set the number of voxels that will be fit at once in each iteration. This is the chunk size as a number of voxels. A larger step value should speed things up, but it will also take up more memory. It is advisable to keep an eye on memory consumption as this value is increased.

E.g., in iter_fit_tensor() we have a default step value of 1e4

References

[1]

Basser, P.J., Mattiello, J., LeBihan, D., 1994. Estimation of the effective self-diffusion tensor from the NMR spin echo. J Magn Reson B 103, 247-254.

[2]

Basser, P., Pierpaoli, C., 1996. Microstructural and physiological features of tissues elucidated by quantitative diffusion-tensor MRI. Journal of Magnetic Resonance 111, 209-219.

[3]

Lin-Ching C., Jones D.K., Pierpaoli, C. 2005. RESTORE: Robust estimation of tensors by outlier rejection. MRM 53: 1088-1095

fit(data, mask=None)

Fit method of the DTI model class

Parameters:

dataarray: The measured signal from one voxel.
maskarray: A boolean array used to mark the coordinates in the data that should be analyzed that has the shape data.shape[:-1]

predict(dti_params, S0=1.0)

Predict a signal for this TensorModel class instance given parameters.

Parameters:

dti_paramsndarray: The last dimension should have 12 tensor parameters: 3 eigenvalues, followed by the 3 eigenvectors
S0float or ndarray: The non diffusion-weighted signal in every voxel, or across all voxels. Default: 1

auto_response

dipy.reconst.csdeconv.auto_response(gtab, data, roi_center=None, roi_radius=10, fa_thr=0.7, fa_callable=None, return_number_of_voxels=None)

Automatic estimation of ssst response function using FA.

dipy.reconst.csdeconv.auto_response is deprecated, Please use dipy.reconst.csdeconv.auto_response_ssst instead

deprecated from version: 1.2
Raises <class ‘dipy.utils.deprecator.ExpiredDeprecationError’> as of version: 1.4

Parameters:

gtabGradientTable
datandarray: diffusion data
roi_centerarray-like, (3,): Center of ROI in data. If center is None, it is assumed that it is the center of the volume with shape data.shape[:3].
roi_radiusint: radius of cubic ROI
fa_thrfloat: FA threshold
fa_callablecallable: A callable that defines an operation that compares FA with the fa_thr. The operator should have two positional arguments (e.g., fa_operator(FA, fa_thr)) and it should return a bool array.
return_number_of_voxelsbool: If True, returns the number of voxels used for estimating the response function

Returns:

responsetuple, (2,): (evals, S0)
ratiofloat: The ratio between smallest versus largest eigenvalue of the response.

Notes

In CSD there is an important pre-processing step: the estimation of the fiber response function. In order to do this, we look for voxels with very anisotropic configurations. We get this information from csdeconv.mask_for_response_ssst(), which returns a mask of selected voxels (more details are available in the description of the function).

With the mask, we compute the response function by using csdeconv.response_from_mask_ssst(), which returns the response and the ratio (more details are available in the description of the function).

auto_response_ssst

dipy.reconst.csdeconv.auto_response_ssst(gtab, data, roi_center=None, roi_radii=10, fa_thr=0.7)

Automatic estimation of single-shell single-tissue (ssst) response: function using FA.

Parameters:

gtabGradientTable
datandarray: diffusion data
roi_centerarray-like, (3,): Center of ROI in data. If center is None, it is assumed that it is the center of the volume with shape data.shape[:3].
roi_radiiint or array-like, (3,): radii of cuboid ROI
fa_thrfloat: FA threshold

Returns:

responsetuple, (2,): (evals, S0)
ratiofloat: The ratio between smallest versus largest eigenvalue of the response.

Notes

With the mask, we compute the response function by using csdeconv.response_from_mask_ssst(), which returns the response and the ratio (more details are available in the description of the function).

cart2sphere

dipy.reconst.csdeconv.cart2sphere(x, y, z)

Return angles for Cartesian 3D coordinates x, y, and z

See doc for sphere2cart for angle conventions and derivation of the formulae.

\(0\le\theta\mathrm{(theta)}\le\pi\) and \(-\pi\le\phi\mathrm{(phi)}\le\pi\)

Parameters:

xarray_like: x coordinate in Cartesian space
yarray_like: y coordinate in Cartesian space
zarray_like: z coordinate

Returns:

rarray: radius
thetaarray: inclination (polar) angle
phiarray: azimuth angle

csdeconv

dipy.reconst.csdeconv.csdeconv(dwsignal, X, B_reg, tau=0.1, convergence=50, P=None)

Constrained-regularized spherical deconvolution (CSD) [1]

Deconvolves the axially symmetric single fiber response function r_rh in rotational harmonics coefficients from the diffusion weighted signal in dwsignal.

Parameters:

dwsignalarray: Diffusion weighted signals to be deconvolved.
Xarray: Prediction matrix which estimates diffusion weighted signals from FOD coefficients.
B_regarray (N, B): SH basis matrix which maps FOD coefficients to FOD values on the surface of the sphere. B_reg should be scaled to account for lambda.
taufloat: Threshold controlling the amplitude below which the corresponding fODF is assumed to be zero. Ideally, tau should be set to zero. However, to improve the stability of the algorithm, tau is set to tau*100 % of the max fODF amplitude (here, 10% by default). This is similar to peak detection where peaks below 0.1 amplitude are usually considered noise peaks. Because SDT is based on a q-ball ODF deconvolution, and not signal deconvolution, using the max instead of mean (as in CSD), is more stable.
convergenceint: Maximum number of iterations to allow the deconvolution to converge.
Pndarray: This is an optimization to avoid computing dot(X.T, X) many times. If the same X is used many times, P can be precomputed and passed to this function.

Returns:

fodf_shndarray ((sh_order + 1)*(sh_order + 2)/2,): Spherical harmonics coefficients of the constrained-regularized fiber ODF.
num_itint: Number of iterations in the constrained-regularization used for convergence.

Notes

This section describes how the fitting of the SH coefficients is done. Problem is to minimise per iteration:

\(F(f_n) = ||Xf_n - S||^2 + \lambda^2 ||H_{n-1} f_n||^2\)

Where \(X\) maps current FOD SH coefficients \(f_n\) to DW signals \(s\) and \(H_{n-1}\) maps FOD SH coefficients \(f_n\) to amplitudes along set of negative directions identified in previous iteration, i.e. the matrix formed by the rows of \(B_{reg}\) for which \(Hf_{n-1}<0\) where \(B_{reg}\) maps \(f_n\) to FOD amplitude on a sphere.

Solve by differentiating and setting to zero:

\(\Rightarrow \frac{\delta F}{\delta f_n} = 2X^T(Xf_n - S) + 2 \lambda^2 H_{n-1}^TH_{n-1}f_n=0\)

Or:

\((X^TX + \lambda^2 H_{n-1}^TH_{n-1})f_n = X^Ts\)

Define \(Q = X^TX + \lambda^2 H_{n-1}^TH_{n-1}\) , which by construction is a square positive definite symmetric matrix of size \(n_{SH} by n_{SH}\). If needed, positive definiteness can be enforced with a small minimum norm regulariser (helps a lot with poorly conditioned direction sets and/or superresolution):

\(Q = X^TX + (\lambda H_{n-1}^T) (\lambda H_{n-1}) + \mu I\)

Solve \(Qf_n = X^Ts\) using Cholesky decomposition:

\(Q = LL^T\)

where \(L\) is lower triangular. Then problem can be solved by back-substitution:

\(L_y = X^Ts\)

\(L^Tf_n = y\)

To speeds things up further, form \(P = X^TX + \mu I\), and update to form \(Q\) by rankn update with \(H_{n-1}\). The dipy implementation looks like:

form initially \(P = X^T X + \mu I\) and \(\lambda B_{reg}\)

for each voxel: form \(z = X^Ts\)

estimate \(f_0\) by solving \(Pf_0=z\). We use a simplified \(l_{max}=4\) solution here, but it might not make a big difference.

Then iterate until no change in rows of \(H\) used in \(H_n\)

form \(H_{n}\) given \(f_{n-1}\)

form \(Q = P + (\lambda H_{n-1}^T) (\lambda H_{n-1}\)) (this can be done by rankn update, but we currently do not use rankn update).

solve \(Qf_n = z\) using Cholesky decomposition

We’d like to thanks Donald Tournier for his help with describing and implementing this algorithm.

References

[1] (1,2)

Tournier, J.D., et al. NeuroImage 2007. Robust determination of the fibre orientation distribution in diffusion MRI: Non-negativity constrained super-resolved spherical deconvolution.

deprecate_with_version

dipy.reconst.csdeconv.deprecate_with_version(message, since='', until='', version_comparator=<function cmp_pkg_version>, warn_class=<class 'DeprecationWarning'>, error_class=<class 'dipy.utils.deprecator.ExpiredDeprecationError'>)

Return decorator function function for deprecation warning / error.

The decorated function / method will:

Raise the given warning_class warning when the function / method gets called, up to (and including) version until (if specified);
Raise the given error_class error when the function / method gets called, when the package version is greater than version until (if specified).

Parameters:

messagestr: Message explaining deprecation, giving possible alternatives.
sincestr, optional: Released version at which object was first deprecated.
untilstr, optional: Last released version at which this function will still raise a deprecation warning. Versions higher than this will raise an error.
version_comparatorcallable: Callable accepting string as argument, and return 1 if string represents a higher version than encoded in the version_comparator, 0 if the version is equal, and -1 if the version is lower. For example, the version_comparator may compare the input version string to the current package version string.
warn_classclass, optional: Class of warning to generate for deprecation.
error_classclass, optional: Class of error to generate when version_comparator returns 1 for a given argument of until.

Returns:

deprecatorfunc: Function returning a decorator.

deprecated_params

dipy.reconst.csdeconv.deprecated_params(old_name, new_name=None, since='', until='', version_comparator=<function cmp_pkg_version>, arg_in_kwargs=False, warn_class=<class 'dipy.utils.deprecator.ArgsDeprecationWarning'>, error_class=<class 'dipy.utils.deprecator.ExpiredDeprecationError'>, alternative='')

Deprecate a renamed or removed function argument.

The decorator assumes that the argument with the old_name was removed from the function signature and the new_name replaced it at the same position in the signature. If the old_name argument is given when calling the decorated function the decorator will catch it and issue a deprecation warning and pass it on as new_name argument.

Parameters:

old_namestr or list/tuple thereof: The old name of the argument.
new_namestr or list/tuple thereof or None, optional: The new name of the argument. Set this to None to remove the argument old_name instead of renaming it.
sincestr or number or list/tuple thereof, optional: The release at which the old argument became deprecated.
untilstr or number or list/tuple thereof, optional: Last released version at which this function will still raise a deprecation warning. Versions higher than this will raise an error.
version_comparatorcallable: Callable accepting string as argument, and return 1 if string represents a higher version than encoded in the version_comparator, 0 if the version is equal, and -1 if the version is lower. For example, the version_comparator may compare the input version string to the current package version string.
arg_in_kwargsbool or list/tuple thereof, optional: If the argument is not a named argument (for example it was meant to be consumed by **kwargs) set this to True. Otherwise the decorator will throw an Exception if the new_name cannot be found in the signature of the decorated function. Default is False.
warn_classwarning, optional: Warning to be issued.
error_classException, optional: Error to be issued
alternativestr, optional: An alternative function or class name that the user may use in place of the deprecated object if new_name is None. The deprecation warning will tell the user about this alternative if provided.

Raises:

TypeError: If the new argument name cannot be found in the function signature and arg_in_kwargs was False or if it is used to deprecate the name of the *args-, **kwargs-like arguments. At runtime such an Error is raised if both the new_name and old_name were specified when calling the function and “relax=False”.

Notes

This function is based on the Astropy (major modification). https://github.com/astropy/astropy. See COPYING file distributed along with the astropy package for the copyright and license terms.

Examples

The deprecation warnings are not shown in the following examples. To deprecate a positional or keyword argument:: >>> from dipy.utils.deprecator import deprecated_params >>> @deprecated_params(‘sig’, ‘sigma’, ‘0.3’) … def test(sigma): … return sigma >>> test(2) 2 >>> test(sigma=2) 2 >>> test(sig=2) # doctest: +SKIP 2

It is also possible to replace multiple arguments. The old_name, new_name and since have to be tuple or list and contain the same number of entries:: >>> @deprecated_params([‘a’, ‘b’], [‘alpha’, ‘beta’], … [‘0.2’, 0.4]) … def test(alpha, beta): … return alpha, beta >>> test(a=2, b=3) # doctest: +SKIP (2, 3)

estimate_response

dipy.reconst.csdeconv.estimate_response(gtab, evals, S0)

Estimate single fiber response function

Parameters:

gtabGradientTable
evalsndarray
S0float: non diffusion weighted

Returns:

Sestimated signal

fa_trace_to_lambdas

dipy.reconst.csdeconv.fa_trace_to_lambdas(fa=0.08, trace=0.0021)

forward_sdeconv_mat

dipy.reconst.csdeconv.forward_sdeconv_mat(r_rh, n)

Build forward spherical deconvolution matrix

Parameters:

r_rhndarray: Rotational harmonics coefficients for the single fiber response function. Each element rh[i] is associated with spherical harmonics of degree 2*i.
nndarray: The order of spherical harmonic function associated with each row of the deconvolution matrix. Only even orders are allowed

Returns:

Rndarray (N, N): Deconvolution matrix with shape (N, N)

forward_sdt_deconv_mat

dipy.reconst.csdeconv.forward_sdt_deconv_mat(ratio, n, r2_term=False)

Build forward sharpening deconvolution transform (SDT) matrix

Parameters:

ratiofloat: ratio = \(\frac{\lambda_2}{\lambda_1}\) of the single fiber response function
nndarray (N,): The degree of spherical harmonic function associated with each row of the deconvolution matrix. Only even degrees are allowed.
r2_termbool: True if ODF comes from an ODF computed from a model using the \(r^2\) term in the integral. For example, DSI, GQI, SHORE, CSA, Tensor, Multi-tensor ODFs. This results in using the proper analytical response function solution solving from the single-fiber ODF with the r^2 term. This derivation is not published anywhere but is very similar to [1].

Returns:

Rndarray (N, N): SDT deconvolution matrix
Pndarray (N, N): Funk-Radon Transform (FRT) matrix

References

[1]

Descoteaux, M. PhD Thesis. INRIA Sophia-Antipolis. 2008.

fractional_anisotropy

dipy.reconst.csdeconv.fractional_anisotropy(evals, axis=-1)

Return Fractional anisotropy (FA) of a diffusion tensor.

Parameters:

evalsarray-like: Eigenvalues of a diffusion tensor.
axisint: Axis of evals which contains 3 eigenvalues.

Returns:

faarray: Calculated FA. Range is 0 <= FA <= 1.

Notes

FA is calculated using the following equation:

\[FA = \sqrt{\frac{1}{2}\frac{(\lambda_1-\lambda_2)^2+(\lambda_1- \lambda_3)^2+(\lambda_2-\lambda_3)^2}{\lambda_1^2+ \lambda_2^2+\lambda_3^2}}\]

get_sphere

dipy.reconst.csdeconv.get_sphere(name='symmetric362')

provide triangulated spheres

Parameters:

namestr: which sphere - one of: * ‘symmetric362’ * ‘symmetric642’ * ‘symmetric724’ * ‘repulsion724’ * ‘repulsion100’ * ‘repulsion200’

Returns:

spherea dipy.core.sphere.Sphere class instance

Examples

>>> import numpy as np
>>> from dipy.data import get_sphere
>>> sphere = get_sphere('symmetric362')
>>> verts, faces = sphere.vertices, sphere.faces
>>> verts.shape == (362, 3)
True
>>> faces.shape == (720, 3)
True
>>> verts, faces = get_sphere('not a sphere name') 
Traceback (most recent call last):
    ...
DataError: No sphere called "not a sphere name"

lazy_index

dipy.reconst.csdeconv.lazy_index(index)

Produces a lazy index

Returns a slice that can be used for indexing an array, if no slice can be made index is returned as is.

lpn

dipy.reconst.csdeconv.lpn(n, z)

Legendre function of the first kind.

Compute sequence of Legendre functions of the first kind (polynomials), Pn(z) and derivatives for all degrees from 0 to n (inclusive).

See also special.legendre for polynomial class.

References

[1]

Zhang, Shanjie and Jin, Jianming. “Computation of Special Functions”, John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

mask_for_response_ssst

dipy.reconst.csdeconv.mask_for_response_ssst(gtab, data, roi_center=None, roi_radii=10, fa_thr=0.7)

Computation of mask for single-shell single-tissue (ssst) response: function using FA.

Parameters:

gtabGradientTable
datandarray: diffusion data (4D)
roi_centerarray-like, (3,): Center of ROI in data. If center is None, it is assumed that it is the center of the volume with shape data.shape[:3].
roi_radiiint or array-like, (3,): radii of cuboid ROI
fa_thrfloat: FA threshold

Returns:

maskndarray: Mask of voxels within the ROI and with FA above the FA threshold.

Notes

In CSD there is an important pre-processing step: the estimation of the fiber response function. In order to do this, we look for voxels with very anisotropic configurations. This function aims to accomplish that by returning a mask of voxels within a ROI, that have a FA value above a given threshold. For example we can use a ROI (20x20x20) at the center of the volume and store the signal values for the voxels with FA values higher than 0.7 (see [1]).

References

[1]

Tournier, J.D., et al. NeuroImage 2004. Direct estimation of the

fiber orientation density function from diffusion-weighted MRI data using spherical deconvolution

multi_voxel_fit

dipy.reconst.csdeconv.multi_voxel_fit(single_voxel_fit): Method decorator to turn a single voxel model fit definition into a multi voxel model fit definition

ndindex

dipy.reconst.csdeconv.ndindex(shape)

An N-dimensional iterator object to index arrays.

Given the shape of an array, an ndindex instance iterates over the N-dimensional index of the array. At each iteration a tuple of indices is returned; the last dimension is iterated over first.

Parameters:

shapetuple of ints: The dimensions of the array.

Examples

>>> from dipy.core.ndindex import ndindex
>>> shape = (3, 2, 1)
>>> for index in ndindex(shape):
...     print(index)
(0, 0, 0)
(0, 1, 0)
(1, 0, 0)
(1, 1, 0)
(2, 0, 0)
(2, 1, 0)

odf_deconv

dipy.reconst.csdeconv.odf_deconv(odf_sh, R, B_reg, lambda_=1.0, tau=0.1, r2_term=False)

ODF constrained-regularized spherical deconvolution using the Sharpening Deconvolution Transform (SDT) [1], [2].

Parameters:

odf_shndarray ((sh_order + 1)*(sh_order + 2)/2,): ndarray of SH coefficients for the ODF spherical function to be deconvolved
Rndarray ((sh_order + 1)(sh_order + 2)/2,: (sh_order + 1)(sh_order + 2)/2) SDT matrix in SH basis
B_regndarray ((sh_order + 1)(sh_order + 2)/2,: (sh_order + 1)(sh_order + 2)/2) SH basis matrix used for deconvolution
lambda_float: lambda parameter in minimization equation (default 1.0)
taufloat: threshold (tau *max(fODF)) controlling the amplitude below which the corresponding fODF is assumed to be zero.
r2_termbool: True if ODF is computed from model that uses the \(r^2\) term in the integral. Recall that Tuch’s ODF (used in Q-ball Imaging [1]) and the true normalized ODF definition differ from a \(r^2\) term in the ODF integral. The original Sharpening Deconvolution Transform (SDT) technique [2] is expecting Tuch’s ODF without the \(r^2\) (see [3] for the mathematical details). Now, this function supports ODF that have been computed using the \(r^2\) term because the proper analytical response function has be derived. For example, models such as DSI, GQI, SHORE, CSA, Tensor, Multi-tensor ODFs, should now be deconvolved with the r2_term=True.

Returns:

fodf_shndarray ((sh_order + 1)(sh_order + 2)/2,): Spherical harmonics coefficients of the constrained-regularized fiber ODF
num_itint: Number of iterations in the constrained-regularization used for convergence

References

[1] (1,2,3)

Tuch, D. MRM 2004. Q-Ball Imaging.

[2] (1,2,3)

Descoteaux, M., et al. IEEE TMI 2009. Deterministic and Probabilistic Tractography Based on Complex Fibre Orientation Distributions

[3]

Descoteaux, M, PhD thesis, INRIA Sophia-Antipolis, 2008.

odf_sh_to_sharp

dipy.reconst.csdeconv.odf_sh_to_sharp(odfs_sh, sphere, basis=None, ratio=0.2, sh_order=8, lambda_=1.0, tau=0.1, r2_term=False)

Sharpen odfs using the sharpening deconvolution transform [2]

This function can be used to sharpen any smooth ODF spherical function. In theory, this should only be used to sharpen QballModel ODFs, but in practice, one can play with the deconvolution ratio and sharpen almost any ODF-like spherical function. The constrained-regularization is stable and will not only sharpen the ODF peaks but also regularize the noisy peaks.

Parameters:

odfs_shndarray ((sh_order + 1)*(sh_order + 2)/2, ): array of odfs expressed as spherical harmonics coefficients
sphereSphere: sphere used to build the regularization matrix
basis{None, ‘tournier07’, ‘descoteaux07’}: different spherical harmonic basis: None for the default DIPY basis, tournier07 for the Tournier 2007 [4] basis, and descoteaux07 for the Descoteaux 2007 [3] basis (None defaults to descoteaux07).
ratiofloat,: ratio of the smallest vs the largest eigenvalue of the single prolate tensor response function (\(\frac{\lambda_2}{\lambda_1}\))
sh_orderint: maximal SH order of the SH representation
lambda_float: lambda parameter (see odfdeconv) (default 1.0)
taufloat: tau parameter in the L matrix construction (see odfdeconv) (default 0.1)
r2_termbool: True if ODF is computed from model that uses the \(r^2\) term in the integral. Recall that Tuch’s ODF (used in Q-ball Imaging [1]) and the true normalized ODF definition differ from a \(r^2\) term in the ODF integral. The original Sharpening Deconvolution Transform (SDT) technique [2] is expecting Tuch’s ODF without the \(r^2\) (see [3] for the mathematical details). Now, this function supports ODF that have been computed using the \(r^2\) term because the proper analytical response function has be derived. For example, models such as DSI, GQI, SHORE, CSA, Tensor, Multi-tensor ODFs, should now be deconvolved with the r2_term=True.

Returns:

fodf_shndarray: sharpened odf expressed as spherical harmonics coefficients

References

[1]

Tuch, D. MRM 2004. Q-Ball Imaging.

[2] (1,2,3)

Descoteaux, M., et al. IEEE TMI 2009. Deterministic and Probabilistic Tractography Based on Complex Fibre Orientation Distributions

[3] (1,2)

Descoteaux, M., Angelino, E., Fitzgibbons, S. and Deriche, R. Regularized, Fast, and Robust Analytical Q-ball Imaging. Magn. Reson. Med. 2007;58:497-510.

[4]

Tournier J.D., Calamante F. and Connelly A. Robust determination of the fibre orientation distribution in diffusion MRI: Non-negativity constrained super-resolved spherical deconvolution. NeuroImage. 2007;35(4):1459-1472.

peaks_from_model

dipy.reconst.csdeconv.peaks_from_model(model, data, sphere, relative_peak_threshold, min_separation_angle, mask=None, return_odf=False, return_sh=True, gfa_thr=0, normalize_peaks=False, sh_order=8, sh_basis_type=None, npeaks=5, B=None, invB=None, parallel=False, num_processes=None)

Fit the model to data and computes peaks and metrics

Parameters:

modela model instance: model will be used to fit the data.
datandarray: Diffusion data.
sphereSphere: The Sphere providing discrete directions for evaluation.
relative_peak_thresholdfloat: Only return peaks greater than relative_peak_threshold * m where m is the largest peak.
min_separation_anglefloat in [0, 90] The minimum distance between: directions. If two peaks are too close only the larger of the two is returned.
maskarray, optional: If mask is provided, voxels that are False in mask are skipped and no peaks are returned.
return_odfbool: If True, the odfs are returned.
return_shbool: If True, the odf as spherical harmonics coefficients is returned
gfa_thrfloat: Voxels with gfa less than gfa_thr are skipped, no peaks are returned.
normalize_peaksbool: If true, all peak values are calculated relative to max(odf).
sh_orderint, optional: Maximum SH order in the SH fit. For sh_order, there will be (sh_order + 1) * (sh_order + 2) / 2 SH coefficients (default 8).
sh_basis_type{None, ‘tournier07’, ‘descoteaux07’}: None for the default DIPY basis, tournier07 for the Tournier 2007 [2] basis, and descoteaux07 for the Descoteaux 2007 [1] basis (None defaults to descoteaux07).
npeaksint: Maximum number of peaks found (default 5 peaks).
Bndarray, optional: Matrix that transforms spherical harmonics to spherical function sf = np.dot(sh, B).
invBndarray, optional: Inverse of B.
parallel: bool: If True, use multiprocessing to compute peaks and metric (default False). Temporary files are saved in the default temporary directory of the system. It can be changed using import tempfile and tempfile.tempdir = '/path/to/tempdir'.
num_processes: int, optional: If parallel is True, the number of subprocesses to use (default multiprocessing.cpu_count()). If < 0 the maximal number of cores minus num_processes + 1 is used (enter -1 to use as many cores as possible). 0 raises an error.

Returns:

pamPeaksAndMetrics: An object with gfa, peak_directions, peak_values, peak_indices, odf, shm_coeffs as attributes

References

[1]

Descoteaux, M., Angelino, E., Fitzgibbons, S. and Deriche, R. Regularized, Fast, and Robust Analytical Q-ball Imaging. Magn. Reson. Med. 2007;58:497-510.

[2]

quad

dipy.reconst.csdeconv.quad(func, a, b, args=(), full_output=0, epsabs=1.49e-08, epsrel=1.49e-08, limit=50, points=None, weight=None, wvar=None, wopts=None, maxp1=50, limlst=50, complex_func=False)

Compute a definite integral.

Integrate func from a to b (possibly infinite interval) using a technique from the Fortran library QUADPACK.

Parameters:

func{function, scipy.LowLevelCallable}

A Python function or method to integrate. If func takes many arguments, it is integrated along the axis corresponding to the first argument.

If the user desires improved integration performance, then f may be a scipy.LowLevelCallable with one of the signatures:

double func(double x)
double func(double x, void *user_data)
double func(int n, double *xx)
double func(int n, double *xx, void *user_data)

The user_data is the data contained in the scipy.LowLevelCallable. In the call forms with xx, n is the length of the xx array which contains xx[0] == x and the rest of the items are numbers contained in the args argument of quad.

In addition, certain ctypes call signatures are supported for backward compatibility, but those should not be used in new code.

afloat

Lower limit of integration (use -numpy.inf for -infinity).

bfloat

Upper limit of integration (use numpy.inf for +infinity).

argstuple, optional

Extra arguments to pass to func.

full_outputint, optional

Non-zero to return a dictionary of integration information. If non-zero, warning messages are also suppressed and the message is appended to the output tuple.

complex_funcbool, optional

Indicate if the function’s (func) return type is real (complex_func=False: default) or complex (complex_func=True). In both cases, the function’s argument is real. If full_output is also non-zero, the infodict, message, and explain for the real and complex components are returned in a dictionary with keys “real output” and “imag output”.

Returns:

yfloat: The integral of func from a to b.
abserrfloat: An estimate of the absolute error in the result.
infodictdict: A dictionary containing additional information.
message: A convergence message.
explain: Appended only with ‘cos’ or ‘sin’ weighting and infinite integration limits, it contains an explanation of the codes in infodict[‘ierlst’]

Other Parameters:

epsabsfloat or int, optional: Absolute error tolerance. Default is 1.49e-8. quad tries to obtain an accuracy of abs(i-result) <= max(epsabs, epsrel*abs(i)) where i = integral of func from a to b, and result is the numerical approximation. See epsrel below.
epsrelfloat or int, optional: Relative error tolerance. Default is 1.49e-8. If epsabs <= 0, epsrel must be greater than both 5e-29 and 50 * (machine epsilon). See epsabs above.
limitfloat or int, optional: An upper bound on the number of subintervals used in the adaptive algorithm.
points(sequence of floats,ints), optional: A sequence of break points in the bounded integration interval where local difficulties of the integrand may occur (e.g., singularities, discontinuities). The sequence does not have to be sorted. Note that this option cannot be used in conjunction with weight.
weightfloat or int, optional: String indicating weighting function. Full explanation for this and the remaining arguments can be found below.
wvaroptional: Variables for use with weighting functions.
woptsoptional: Optional input for reusing Chebyshev moments.
maxp1float or int, optional: An upper bound on the number of Chebyshev moments.
limlstint, optional: Upper bound on the number of cycles (>=3) for use with a sinusoidal weighting and an infinite end-point.

See also

dblquad: double integral
tplquad: triple integral
nquad: n-dimensional integrals (uses quad recursively)
fixed_quad: fixed-order Gaussian quadrature
quadrature: adaptive Gaussian quadrature
odeint: ODE integrator
ode: ODE integrator
simpson: integrator for sampled data
romb: integrator for sampled data
scipy.special: for coefficients and roots of orthogonal polynomials

Notes

Extra information for quad() inputs and outputs

If full_output is non-zero, then the third output argument (infodict) is a dictionary with entries as tabulated below. For infinite limits, the range is transformed to (0,1) and the optional outputs are given with respect to this transformed range. Let M be the input argument limit and let K be infodict[‘last’]. The entries are:

‘neval’: The number of function evaluations.
‘last’: The number, K, of subintervals produced in the subdivision process.
‘alist’: A rank-1 array of length M, the first K elements of which are the left end points of the subintervals in the partition of the integration range.
‘blist’: A rank-1 array of length M, the first K elements of which are the right end points of the subintervals.
‘rlist’: A rank-1 array of length M, the first K elements of which are the integral approximations on the subintervals.
‘elist’: A rank-1 array of length M, the first K elements of which are the moduli of the absolute error estimates on the subintervals.
‘iord’: A rank-1 integer array of length M, the first L elements of which are pointers to the error estimates over the subintervals with L=K if K<=M/2+2 or L=M+1-K otherwise. Let I be the sequence infodict['iord'] and let E be the sequence infodict['elist']. Then E[I[1]], ..., E[I[L]] forms a decreasing sequence.

If the input argument points is provided (i.e., it is not None), the following additional outputs are placed in the output dictionary. Assume the points sequence is of length P.

‘pts’: A rank-1 array of length P+2 containing the integration limits and the break points of the intervals in ascending order. This is an array giving the subintervals over which integration will occur.
‘level’: A rank-1 integer array of length M (=limit), containing the subdivision levels of the subintervals, i.e., if (aa,bb) is a subinterval of (pts[1], pts[2]) where pts[0] and pts[2] are adjacent elements of infodict['pts'], then (aa,bb) has level l if |bb-aa| = |pts[2]-pts[1]| * 2**(-l).
‘ndin’: A rank-1 integer array of length P+2. After the first integration over the intervals (pts[1], pts[2]), the error estimates over some of the intervals may have been increased artificially in order to put their subdivision forward. This array has ones in slots corresponding to the subintervals for which this happens.

Weighting the integrand

The input variables, weight and wvar, are used to weight the integrand by a select list of functions. Different integration methods are used to compute the integral with these weighting functions, and these do not support specifying break points. The possible values of weight and the corresponding weighting functions are.

`weight`	Weight function used	`wvar`
‘cos’	cos(w*x)	wvar = w
‘sin’	sin(w*x)	wvar = w
‘alg’	g(x) = ((x-a)*alpha)((b-x)**beta)	wvar = (alpha, beta)
‘alg-loga’	g(x)*log(x-a)	wvar = (alpha, beta)
‘alg-logb’	g(x)*log(b-x)	wvar = (alpha, beta)
‘alg-log’	g(x)log(x-a)log(b-x)	wvar = (alpha, beta)
‘cauchy’	1/(x-c)	wvar = c

wvar holds the parameter w, (alpha, beta), or c depending on the weight selected. In these expressions, a and b are the integration limits.

For the ‘cos’ and ‘sin’ weighting, additional inputs and outputs are available.

For finite integration limits, the integration is performed using a Clenshaw-Curtis method which uses Chebyshev moments. For repeated calculations, these moments are saved in the output dictionary:

‘momcom’: The maximum level of Chebyshev moments that have been computed, i.e., if M_c is infodict['momcom'] then the moments have been computed for intervals of length |b-a| * 2**(-l), l=0,1,...,M_c.
‘nnlog’: A rank-1 integer array of length M(=limit), containing the subdivision levels of the subintervals, i.e., an element of this array is equal to l if the corresponding subinterval is |b-a|* 2**(-l).
‘chebmo’: A rank-2 array of shape (25, maxp1) containing the computed Chebyshev moments. These can be passed on to an integration over the same interval by passing this array as the second element of the sequence wopts and passing infodict[‘momcom’] as the first element.

If one of the integration limits is infinite, then a Fourier integral is computed (assuming w neq 0). If full_output is 1 and a numerical error is encountered, besides the error message attached to the output tuple, a dictionary is also appended to the output tuple which translates the error codes in the array info['ierlst'] to English messages. The output information dictionary contains the following entries instead of ‘last’, ‘alist’, ‘blist’, ‘rlist’, and ‘elist’:

‘lst’: The number of subintervals needed for the integration (call it K_f).
‘rslst’: A rank-1 array of length M_f=limlst, whose first K_f elements contain the integral contribution over the interval (a+(k-1)c, a+kc) where c = (2*floor(|w|) + 1) * pi / |w| and k=1,2,...,K_f.
‘erlst’: A rank-1 array of length M_f containing the error estimate corresponding to the interval in the same position in infodict['rslist'].
‘ierlst’: A rank-1 integer array of length M_f containing an error flag corresponding to the interval in the same position in infodict['rslist']. See the explanation dictionary (last entry in the output tuple) for the meaning of the codes.

Details of QUADPACK level routines

quad calls routines from the FORTRAN library QUADPACK. This section provides details on the conditions for each routine to be called and a short description of each routine. The routine called depends on weight, points and the integration limits a and b.

QUADPACK routine	weight	points	infinite bounds
qagse	None	No	No
qagie	None	No	Yes
qagpe	None	Yes	No
qawoe	‘sin’, ‘cos’	No	No
qawfe	‘sin’, ‘cos’	No	either a or b
qawse	‘alg*’	No	No
qawce	‘cauchy’	No	No

The following provides a short desciption from [1] for each routine.

qagse

is an integrator based on globally adaptive interval subdivision in connection with extrapolation, which will eliminate the effects of integrand singularities of several types.

qagie

handles integration over infinite intervals. The infinite range is mapped onto a finite interval and subsequently the same strategy as in QAGS is applied.

qagpe

serves the same purposes as QAGS, but also allows the user to provide explicit information about the location and type of trouble-spots i.e. the abscissae of internal singularities, discontinuities and other difficulties of the integrand function.

qawoe

is an integrator for the evaluation of \(\int^b_a \cos(\omega x)f(x)dx\) or \(\int^b_a \sin(\omega x)f(x)dx\) over a finite interval [a,b], where \(\omega\) and \(f\) are specified by the user. The rule evaluation component is based on the modified Clenshaw-Curtis technique

An adaptive subdivision scheme is used in connection with an extrapolation procedure, which is a modification of that in QAGS and allows the algorithm to deal with singularities in \(f(x)\).

qawfe

calculates the Fourier transform \(\int^\infty_a \cos(\omega x)f(x)dx\) or \(\int^\infty_a \sin(\omega x)f(x)dx\) for user-provided \(\omega\) and \(f\). The procedure of QAWO is applied on successive finite intervals, and convergence acceleration by means of the \(\varepsilon\)-algorithm is applied to the series of integral approximations.

qawse

approximate \(\int^b_a w(x)f(x)dx\), with \(a < b\) where \(w(x) = (x-a)^{\alpha}(b-x)^{\beta}v(x)\) with \(\alpha,\beta > -1\), where \(v(x)\) may be one of the following functions: \(1\), \(\log(x-a)\), \(\log(b-x)\), \(\log(x-a)\log(b-x)\).

The user specifies \(\alpha\), \(\beta\) and the type of the function \(v\). A globally adaptive subdivision strategy is applied, with modified Clenshaw-Curtis integration on those subintervals which contain a or b.

qawce

compute \(\int^b_a f(x) / (x-c)dx\) where the integral must be interpreted as a Cauchy principal value integral, for user specified \(c\) and \(f\). The strategy is globally adaptive. Modified Clenshaw-Curtis integration is used on those intervals containing the point \(x = c\).

Integration of Complex Function of a Real Variable

A complex valued function, \(f\), of a real variable can be written as \(f = g + ih\). Similarly, the integral of \(f\) can be written as

\[\int_a^b f(x) dx = \int_a^b g(x) dx + i\int_a^b h(x) dx\]

assuming that the integrals of \(g\) and \(h\) exist over the inteval \([a,b]\) [2]. Therefore, quad integrates complex-valued functions by integrating the real and imaginary components separately.

References

[1]

Piessens, Robert; de Doncker-Kapenga, Elise; Überhuber, Christoph W.; Kahaner, David (1983). QUADPACK: A subroutine package for automatic integration. Springer-Verlag. ISBN 978-3-540-12553-2.

[2]

McCullough, Thomas; Phillips, Keith (1973). Foundations of Analysis in the Complex Plane. Holt Rinehart Winston. ISBN 0-03-086370-8

Examples

Calculate \(\int^4_0 x^2 dx\) and compare with an analytic result

>>> from scipy import integrate
>>> import numpy as np
>>> x2 = lambda x: x**2
>>> integrate.quad(x2, 0, 4)
(21.333333333333332, 2.3684757858670003e-13)
>>> print(4**3 / 3.)  # analytical result
21.3333333333

Calculate \(\int^\infty_0 e^{-x} dx\)

>>> invexp = lambda x: np.exp(-x)
>>> integrate.quad(invexp, 0, np.inf)
(1.0, 5.842605999138044e-11)

Calculate \(\int^1_0 a x \,dx\) for \(a = 1, 3\)

>>> f = lambda x, a: a*x
>>> y, err = integrate.quad(f, 0, 1, args=(1,))
>>> y
0.5
>>> y, err = integrate.quad(f, 0, 1, args=(3,))
>>> y
1.5

Calculate \(\int^1_0 x^2 + y^2 dx\) with ctypes, holding y parameter as 1:

testlib.c =>
    double func(int n, double args[n]){
        return args[0]*args[0] + args[1]*args[1];}
compile to library testlib.*

from scipy import integrate
import ctypes
lib = ctypes.CDLL('/home/.../testlib.*') #use absolute path
lib.func.restype = ctypes.c_double
lib.func.argtypes = (ctypes.c_int,ctypes.c_double)
integrate.quad(lib.func,0,1,(1))
#(1.3333333333333333, 1.4802973661668752e-14)
print((1.0**3/3.0 + 1.0) - (0.0**3/3.0 + 0.0)) #Analytic result
# 1.3333333333333333

Be aware that pulse shapes and other sharp features as compared to the size of the integration interval may not be integrated correctly using this method. A simplified example of this limitation is integrating a y-axis reflected step function with many zero values within the integrals bounds.

>>> y = lambda x: 1 if x<=0 else 0
>>> integrate.quad(y, -1, 1)
(1.0, 1.1102230246251565e-14)
>>> integrate.quad(y, -1, 100)
(1.0000000002199108, 1.0189464580163188e-08)
>>> integrate.quad(y, -1, 10000)
(0.0, 0.0)

real_sh_descoteaux

dipy.reconst.csdeconv.real_sh_descoteaux(sh_order, theta, phi, full_basis=False, legacy=True)

Compute real spherical harmonics as in Descoteaux et al. 2007 [Ra6d8f6cd2652-1], where the real harmonic \(Y^m_n\) is defined to be:

Imag(\(Y^m_n\)) * sqrt(2) if m > 0 \(Y^0_n\) if m = 0 Real(\(Y^m_n\)) * sqrt(2) if m < 0

This may take scalar or array arguments. The inputs will be broadcast against each other.

Parameters:

sh_orderint: The maximum degree or the spherical harmonic basis.
thetafloat [0, pi]: The polar (colatitudinal) coordinate.
phifloat [0, 2*pi]: The azimuthal (longitudinal) coordinate.
full_basis: bool, optional: If true, returns a basis including odd order SH functions as well as even order SH functions. Otherwise returns only even order SH functions.
legacy: bool, optional: If true, uses DIPY’s legacy descoteaux07 implementation (where |m| for m < 0). Else, implements the basis as defined in Descoteaux et al. 2007.

Returns:

real_shreal float: The real harmonic \(Y^m_n\) sampled at theta and phi.
marray: The degree of the harmonics.
narray: The order of the harmonics.

real_sh_descoteaux_from_index

dipy.reconst.csdeconv.real_sh_descoteaux_from_index(m, n, theta, phi, legacy=True)

Compute real spherical harmonics as in Descoteaux et al. 2007 [R2763a8caef3c-1], where the real harmonic \(Y^m_n\) is defined to be:

Imag(\(Y^m_n\)) * sqrt(2) if m > 0 \(Y^0_n\) if m = 0 Real(\(Y^m_n\)) * sqrt(2) if m < 0

This may take scalar or array arguments. The inputs will be broadcast against each other.

Parameters:

mint |m| <= n: The degree of the harmonic.
nint >= 0: The order of the harmonic.
thetafloat [0, pi]: The polar (colatitudinal) coordinate.
phifloat [0, 2*pi]: The azimuthal (longitudinal) coordinate.
legacy: bool, optional: If true, uses DIPY’s legacy descoteaux07 implementation (where |m| is used for m < 0). Else, implements the basis as defined in Descoteaux et al. 2007 (without the absolute value).

Returns:

real_shreal float: The real harmonic \(Y^m_n\) sampled at theta and phi.

recursive_response

dipy.reconst.csdeconv.recursive_response(gtab, data, mask=None, sh_order=8, peak_thr=0.01, init_fa=0.08, init_trace=0.0021, iter=8, convergence=0.001, parallel=False, num_processes=None, sphere=<dipy.core.sphere.HemiSphere object>)

Recursive calibration of response function using peak threshold

Parameters:

gtabGradientTable
datandarray: diffusion data
maskndarray, optional: mask for recursive calibration, for example a white matter mask. It has shape data.shape[0:3] and dtype=bool. Default: use the entire data array.
sh_orderint, optional: maximal spherical harmonics order. Default: 8
peak_thrfloat, optional: peak threshold, how large the second peak can be relative to the first peak in order to call it a single fiber population [1]. Default: 0.01
init_fafloat, optional: FA of the initial ‘fat’ response function (tensor). Default: 0.08
init_tracefloat, optional: trace of the initial ‘fat’ response function (tensor). Default: 0.0021
iterint, optional: maximum number of iterations for calibration. Default: 8.
convergencefloat, optional: convergence criterion, maximum relative change of SH coefficients. Default: 0.001.
parallelbool, optional: Whether to use parallelization in peak-finding during the calibration procedure. Default: True
num_processesint, optional: If parallel is True, the number of subprocesses to use (default multiprocessing.cpu_count()). If < 0 the maximal number of cores minus num_processes + 1 is used (enter -1 to use as many cores as possible). 0 raises an error.
sphereSphere, optional.: The sphere used for peak finding. Default: default_sphere.

Returns:

responsendarray: response function in SH coefficients

Notes

In CSD there is an important pre-processing step: the estimation of the fiber response function. Using an FA threshold is not a very robust method. It is dependent on the dataset (non-informed used subjectivity), and still depends on the diffusion tensor (FA and first eigenvector), which has low accuracy at high b-value. This function recursively calibrates the response function, for more information see [1].

References

[1]

Tax, C.M.W., et al. NeuroImage 2014. Recursive calibration of the fiber response function for spherical deconvolution of diffusion MRI data.

response_from_mask

dipy.reconst.csdeconv.response_from_mask(gtab, data, mask)

Computation of single-shell single-tissue (ssst) response: function from a given mask.

dipy.reconst.csdeconv.response_from_mask is deprecated, Please use dipy.reconst.csdeconv.response_from_mask_ssst instead

deprecated from version: 1.2
Raises <class ‘dipy.utils.deprecator.ExpiredDeprecationError’> as of version: 1.4

Parameters:

gtabGradientTable
datandarray: diffusion data
maskndarray: mask from where to compute the response function

Returns:

responsetuple, (2,): (evals, S0)
ratiofloat: The ratio between smallest versus largest eigenvalue of the response.

Notes

In CSD there is an important pre-processing step: the estimation of the fiber response function. In order to do this, we look for voxels with very anisotropic configurations. This information can be obtained by using csdeconv.mask_for_response_ssst() through a mask of selected voxels (see[R64bc28ca561d-1]_). The present function uses such a mask to compute the ssst response function.

For the response we also need to find the average S0 in the ROI. This is possible using gtab.b0s_mask() we can find all the S0 volumes (which correspond to b-values equal 0) in the dataset.

The response consists always of a prolate tensor created by averaging the highest and second highest eigenvalues in the ROI with FA higher than threshold. We also include the average S0s.

We also return the ratio which is used for the SDT models.

References

[1]

Tournier, J.D., et al. NeuroImage 2004. Direct estimation of the

fiber orientation density function from diffusion-weighted MRI data using spherical deconvolution

response_from_mask_ssst

dipy.reconst.csdeconv.response_from_mask_ssst(gtab, data, mask)

Computation of single-shell single-tissue (ssst) response: function from a given mask.

Parameters:

gtabGradientTable
datandarray: diffusion data
maskndarray: mask from where to compute the response function

Returns:

responsetuple, (2,): (evals, S0)
ratiofloat: The ratio between smallest versus largest eigenvalue of the response.

Notes

In CSD there is an important pre-processing step: the estimation of the fiber response function. In order to do this, we look for voxels with very anisotropic configurations. This information can be obtained by using csdeconv.mask_for_response_ssst() through a mask of selected voxels (see[Rb280cc58c30e-1]_). The present function uses such a mask to compute the ssst response function.

For the response we also need to find the average S0 in the ROI. This is possible using gtab.b0s_mask() we can find all the S0 volumes (which correspond to b-values equal 0) in the dataset.

The response consists always of a prolate tensor created by averaging the highest and second highest eigenvalues in the ROI with FA higher than threshold. We also include the average S0s.

We also return the ratio which is used for the SDT models.

References

[1]

Tournier, J.D., et al. NeuroImage 2004. Direct estimation of the

fiber orientation density function from diffusion-weighted MRI data using spherical deconvolution

sh_to_rh

dipy.reconst.csdeconv.sh_to_rh(r_sh, m, n)

Spherical harmonics (SH) to rotational harmonics (RH)

Calculate the rotational harmonic decomposition up to harmonic order n, degree m for an axially and antipodally symmetric function. Note that all m != 0 coefficients will be ignored as axial symmetry is assumed. Hence, there will be (sh_order/2 + 1) non-zero coefficients.

Parameters:

r_shndarray (N,): ndarray of SH coefficients for the single fiber response function. These coefficients must correspond to the real spherical harmonic functions produced by shm.real_sh_descoteaux_from_index.
mndarray (N,): The degree of the spherical harmonic function associated with each coefficient.
nndarray (N,): The order of the spherical harmonic function associated with each coefficient.

Returns:

r_rhndarray ((sh_order + 1)*(sh_order + 2)/2,): Rotational harmonics coefficients representing the input r_sh

See also

shm.real_sh_descoteaux_from_index, shm.real_sh_descoteaux

References

[1]

Tournier, J.D., et al. NeuroImage 2007. Robust determination of the fibre orientation distribution in diffusion MRI: Non-negativity constrained super-resolved spherical deconvolution

single_tensor

dipy.reconst.csdeconv.single_tensor(gtab, S0=1, evals=None, evecs=None, snr=None)

Simulate diffusion-weighted signals with a single tensor.

Parameters:

gtabGradientTable: Table with information of b-values and gradient directions g. Note that if gtab has a btens attribute, simulations will be performed according to the given b-tensor B information.
S0double, optional: Strength of signal in the presence of no diffusion gradient (also called the b=0 value).
evals(3,) ndarray, optional: Eigenvalues of the diffusion tensor. By default, values typical for prolate white matter are used.
evecs(3, 3) ndarray, optional: Eigenvectors of the tensor. You can also think of this as a rotation matrix that transforms the direction of the tensor. The eigenvectors need to be column wise.
snrfloat, optional: Signal to noise ratio, assuming Rician noise. None implies no noise.

Returns:

S(N,) ndarray

Simulated signal:: S(b, g) = S_0 e^(-b g^T R D R.T g), if gtab.tens=None S(B) = S_0 e^(-B:D), if gtab.tens information is given

References

[1]

M. Descoteaux, “High Angular Resolution Diffusion MRI: from Local Estimation to Segmentation and Tractography”, PhD thesis, University of Nice-Sophia Antipolis, p. 42, 2008.

[2]

E. Stejskal and J. Tanner, “Spin diffusion measurements: spin echos in the presence of a time-dependent field gradient”, Journal of Chemical Physics, nr. 42, pp. 288–292, 1965.

sph_harm_ind_list

dipy.reconst.csdeconv.sph_harm_ind_list(sh_order, full_basis=False)

Returns the degree (m) and order (n) of all the symmetric spherical harmonics of degree less then or equal to sh_order. The results, m_list and n_list are kx1 arrays, where k depends on sh_order. They can be passed to real_sh_descoteaux_from_index() and :func:real_sh_tournier_from_index.

Parameters:

sh_orderint: even int > 0, max order to return
full_basis: bool, optional: True for SH basis with even and odd order terms

Returns:

m_listarray: degrees of even spherical harmonics
n_listarray: orders of even spherical harmonics

See also

shm.real_sh_descoteaux_from_index, shm.real_sh_tournier_from_index

vec2vec_rotmat

dipy.reconst.csdeconv.vec2vec_rotmat(u, v)

rotation matrix from 2 unit vectors

u, v being unit 3d vectors return a 3x3 rotation matrix R than aligns u to v.

In general there are many rotations that will map u to v. If S is any rotation using v as an axis then R.S will also map u to v since (S.R)u = S(Ru) = Sv = v. The rotation R returned by vec2vec_rotmat leaves fixed the perpendicular to the plane spanned by u and v.

The transpose of R will align v to u.

Parameters:

uarray, shape(3,)
varray, shape(3,)

Returns:

Rarray, shape(3,3)

Examples

>>> import numpy as np
>>> from dipy.core.geometry import vec2vec_rotmat
>>> u=np.array([1,0,0])
>>> v=np.array([0,1,0])
>>> R=vec2vec_rotmat(u,v)
>>> np.dot(R,u)
array([ 0.,  1.,  0.])
>>> np.dot(R.T,v)
array([ 1.,  0.,  0.])

`DiffusionKurtosisFit`

class dipy.reconst.dki.DiffusionKurtosisFit(model, model_params)

Bases: TensorFit

Class for fitting the Diffusion Kurtosis Model

Attributes:

S0_hat
directions: For tracking - return the primary direction in each voxel
evals: Returns the eigenvalues of the tensor as an array
evecs: Returns the eigenvectors of the tensor as an array, columnwise
kfa: Returns the kurtosis tensor (KFA) [1].
kt: Returns the 15 independent elements of the kurtosis tensor as an array
quadratic_form: Calculates the 3x3 diffusion tensor for each voxel
shape

Methods

`ad`()	Axial diffusivity (AD) calculated from cached eigenvalues.
`adc`(sphere)	Calculate the apparent diffusion coefficient (ADC) in each direction on
`ak`([min_kurtosis, max_kurtosis, analytical])	Axial Kurtosis (AK) of a diffusion kurtosis tensor [1].
`akc`(sphere)	Calculates the apparent kurtosis coefficient (AKC) in each direction on the sphere for each voxel in the data
`color_fa`()	Color fractional anisotropy of diffusion tensor
`fa`()	Fractional anisotropy (FA) calculated from cached eigenvalues.
`ga`()	Geodesic anisotropy (GA) calculated from cached eigenvalues.
`kmax`([sphere, gtol, mask])	Computes the maximum value of a single voxel kurtosis tensor
`linearity`()	Returns:
`md`()	Mean diffusivity (MD) calculated from cached eigenvalues.
`mk`([min_kurtosis, max_kurtosis, analytical])	Computes mean Kurtosis (MK) from the kurtosis tensor.
`mkt`([min_kurtosis, max_kurtosis])	Computes mean of the kurtosis tensor (MKT) [1].
`mode`()	Tensor mode calculated from cached eigenvalues.
`odf`(sphere)	The diffusion orientation distribution function (dODF).
`planarity`()	Returns:
`predict`(gtab[, S0])	Given a DKI model fit, predict the signal on the vertices of a gradient table
`rd`()	Radial diffusivity (RD) calculated from cached eigenvalues.
`rk`([min_kurtosis, max_kurtosis, analytical])	Radial Kurtosis (RK) of a diffusion kurtosis tensor [1].
`sphericity`()	Returns:
`trace`()	Trace of the tensor calculated from cached eigenvalues.

lower_triangular

__init__(model, model_params)

Initialize a DiffusionKurtosisFit class instance.

Since DKI is an extension of DTI, class instance is defined as subclass of the TensorFit from dti.py

Parameters:

modelDiffusionKurtosisModel Class instance

Class instance containing the Diffusion Kurtosis Model for the fit

model_paramsndarray (x, y, z, 27) or (n, 27)

All parameters estimated from the diffusion kurtosis model. Parameters are ordered as follows:

Three diffusion tensor’s eigenvalues

Three lines of the eigenvector matrix each containing the first, second and third coordinates of the eigenvector

Fifteen elements of the kurtosis tensor

ak(min_kurtosis=-0.42857142857142855, max_kurtosis=10, analytical=True)

Axial Kurtosis (AK) of a diffusion kurtosis tensor [1].

Parameters:

min_kurtosisfloat (optional): To keep kurtosis values within a plausible biophysical range, axial kurtosis values that are smaller than min_kurtosis are replaced with -3./7 (theoretical kurtosis limit for regions that consist of water confined to spherical pores [2])
max_kurtosisfloat (optional): To keep kurtosis values within a plausible biophysical range, axial kurtosis values that are larger than max_kurtosis are replaced with max_kurtosis. Default = 10
analyticalbool (optional): If True, AK is calculated from rotated diffusion kurtosis tensor, otherwise it will be computed from the apparent diffusion kurtosis values along the principal axis of the diffusion tensor (see notes). Default is set to True.

Returns:

akarray: Calculated AK.

Notes

AK is defined as the directional kurtosis parallel to the fiber’s main direction e1 [1], [2]. You can compute AK using to approaches:

AK is calculated from rotated diffusion kurtosis tensor [2], i.e.:

\[AK = \hat{W}_{1111} \frac{(\lambda_{1}+\lambda_{2}+\lambda_{3})^2}{(9 \lambda_{1}^2)}\]

AK can be sampled from the principal axis of the diffusion tensor:

\[AK = K(\mathbf{\mathbf{e}_1)\]

Although both approaches leads to an exact calculation of AK, the first approach will be referred to as the analytical method while the second approach will be referred to as the numerical method based on their analogy to the estimation strategies for MK and RK.

References

[1] (1,2,3)

Jensen, J.H., Helpern, J.A., 2010. MRI quantification of non-Gaussian water diffusion by kurtosis analysis. NMR in Biomedicine 23(7): 698-710

[2] (1,2,3)

Tabesh, A., Jensen, J.H., Ardekani, B.A., Helpern, J.A., 2011. Estimation of tensors and tensor-derived measures in diffusional kurtosis imaging. Magn Reson Med. 65(3), 823-836

[3]

Barmpoutis, A., & Zhuo, J., 2011. Diffusion kurtosis imaging: Robust estimation from DW-MRI using homogeneous polynomials. Proceedings of the 8th {IEEE} International Symposium on Biomedical Imaging: From Nano to Macro, ISBI 2011, 262-265. doi: 10.1109/ISBI.2011.5872402

akc(sphere)

Calculates the apparent kurtosis coefficient (AKC) in each direction on the sphere for each voxel in the data

Parameters:

sphereSphere class instance

Returns:

akcndarray: The estimates of the apparent kurtosis coefficient in every direction on the input sphere

Notes

For each sphere direction with coordinates \((n_{1}, n_{2}, n_{3})\), the calculation of AKC is done using formula:

\[AKC(n)=\frac{MD^{2}}{ADC(n)^{2}}\sum_{i=1}^{3}\sum_{j=1}^{3} \sum_{k=1}^{3}\sum_{l=1}^{3}n_{i}n_{j}n_{k}n_{l}W_{ijkl}\]

where \(W_{ijkl}\) are the elements of the kurtosis tensor, MD the mean diffusivity and ADC the apparent diffusion coefficent computed as:

\[ADC(n)=\sum_{i=1}^{3}\sum_{j=1}^{3}n_{i}n_{j}D_{ij}\]

where \(D_{ij}\) are the elements of the diffusion tensor.

property kfa

Returns the kurtosis tensor (KFA) [1].

Notes

The KFA is defined as [1]:

\[KFA \equiv \frac{||\mathbf{W} - MKT \mathbf{I}^{(4)}||_F}{||\mathbf{W}||_F}\]

where \(W\) is the kurtosis tensor, MKT the kurtosis tensor mean, \(I^(4)\) is the fully symmetric rank 2 isotropic tensor and \(||...||_F\) is the tensor’s Frobenius norm [1].

References

[1] (1,2,3)

Glenn, G. R., Helpern, J. A., Tabesh, A., and Jensen, J. H. (2015). Quantitative assessment of diffusional kurtosis anisotropy. NMR in Biomedicine 28, 448–459. doi:10.1002/nbm.3271

kmax(sphere='repulsion100', gtol=1e-05, mask=None)

Computes the maximum value of a single voxel kurtosis tensor

Parameters:

sphereSphere class instance, optional: The sphere providing sample directions for the initial search of the maximum value of kurtosis.
gtolfloat, optional: This input is to refine kurtosis maximum under the precision of the directions sampled on the sphere class instance. The gradient of the convergence procedure must be less than gtol before successful termination. If gtol is None, fiber direction is directly taken from the initial sampled directions of the given sphere object

Returns:

max_valuefloat: kurtosis tensor maximum value

property kt: Returns the 15 independent elements of the kurtosis tensor as an array

mk(min_kurtosis=-0.42857142857142855, max_kurtosis=10, analytical=True)

Computes mean Kurtosis (MK) from the kurtosis tensor.

Parameters:

min_kurtosisfloat (optional): To keep kurtosis values within a plausible biophysical range, mean kurtosis values that are smaller than min_kurtosis are replaced with min_kurtosis. Default = -3./7 (theoretical kurtosis limit for regions that consist of water confined to spherical pores [4])
max_kurtosisfloat (optional): To keep kurtosis values within a plausible biophysical range, mean kurtosis values that are larger than max_kurtosis are replaced with max_kurtosis. Default = 10
analyticalbool (optional): If True, MK is calculated using its analytical solution, otherwise an exact numerical estimator is used (see Notes). Default is set to True.

Returns:

mkarray: Calculated MK.

Notes

The MK is defined as the average of directional kurtosis coefficients across all spatial directions, which can be formulated by the following surface integral[R1a4c5980fd18-1]_:

\[MK \equiv \frac{1}{4\pi} \int d\Omega_\mathbf{n} K(\mathbf{n})\]

This integral can be numerically solved by averaging directional kurtosis values sampled for directions of a spherical t-design [2].

Alternatively, MK can be solved from the analytical solution derived by Tabesh et al. [3]. This solution is given by:

\[\begin{split}MK=F_1(\lambda_1,\lambda_2,\lambda_3)\hat{W}_{1111}+ F_1(\lambda_2,\lambda_1,\lambda_3)\hat{W}_{2222}+ F_1(\lambda_3,\lambda_2,\lambda_1)\hat{W}_{3333}+ \\ F_2(\lambda_1,\lambda_2,\lambda_3)\hat{W}_{2233}+ F_2(\lambda_2,\lambda_1,\lambda_3)\hat{W}_{1133}+ F_2(\lambda_3,\lambda_2,\lambda_1)\hat{W}_{1122}\end{split}\]

where \(\hat{W}_{ijkl}\) are the components of the \(W\) tensor in the coordinates system defined by the eigenvectors of the diffusion tensor \(\mathbf{D}\) and

\[ \begin{align}\begin{aligned}\begin{split}F_1(\lambda_1,\lambda_2,\lambda_3)= \frac{(\lambda_1+\lambda_2+\lambda_3)^2} {18(\lambda_1-\lambda_2)(\lambda_1-\lambda_3)} [\frac{\sqrt{\lambda_2\lambda_3}}{\lambda_1} R_F(\frac{\lambda_1}{\lambda_2},\frac{\lambda_1}{\lambda_3},1)+\\ \frac{3\lambda_1^2-\lambda_1\lambda_2-\lambda_2\lambda_3- \lambda_1\lambda_3} {3\lambda_1 \sqrt{\lambda_2 \lambda_3}} R_D(\frac{\lambda_1}{\lambda_2},\frac{\lambda_1}{\lambda_3},1)-1 ]\end{split}\\\begin{split}F_2(\lambda_1,\lambda_2,\lambda_3)= \frac{(\lambda_1+\lambda_2+\lambda_3)^2} {3(\lambda_2-\lambda_3)^2} [\frac{\lambda_2+\lambda_3}{\sqrt{\lambda_2\lambda_3}} R_F(\frac{\lambda_1}{\lambda_2},\frac{\lambda_1}{\lambda_3},1)+\\ \frac{2\lambda_1-\lambda_2-\lambda_3}{3\sqrt{\lambda_2 \lambda_3}} R_D(\frac{\lambda_1}{\lambda_2},\frac{\lambda_1}{\lambda_3},1)-2]\end{split}\end{aligned}\end{align} \]

where \(R_f\) and \(R_d\) are the Carlson’s elliptic integrals.

References

[1]

Jensen, J.H., Helpern, J.A., 2010. MRI quantification of non-Gaussian water diffusion by kurtosis analysis. NMR in Biomedicine 23(7): 698-710

[2]

Hardin, R.H., Sloane, N.J.A., 1996. McLaren’s Improved Snub Cube and Other New Spherical Designs in Three Dimensions. Discrete and Computational Geometry 15, 429-441.

[3]

Tabesh, A., Jensen, J.H., Ardekani, B.A., Helpern, J.A., 2011. Estimation of tensors and tensor-derived measures in diffusional kurtosis imaging. Magn Reson Med. 65(3), 823-836

[4]

mkt(min_kurtosis=-0.42857142857142855, max_kurtosis=10)

Computes mean of the kurtosis tensor (MKT) [1].

Parameters:

min_kurtosisfloat (optional): To keep kurtosis values within a plausible biophysical range, mean kurtosis values that are smaller than min_kurtosis are replaced with min_kurtosis. Default = -3./7 (theoretical kurtosis limit for regions that consist of water confined to spherical pores [2])
max_kurtosisfloat (optional): To keep kurtosis values within a plausible biophysical range, mean kurtosis values that are larger than max_kurtosis are replaced with max_kurtosis. Default = 10

Returns:

mktarray: Calculated mean kurtosis tensor.

Notes

The MKT is defined as [1]:

\[MKT \equiv \frac{1}{4\pi} \int d \Omega_{\mathnbf{n}} n_i n_j n_k n_l W_{ijkl}\]

which can be directly computed from the trace of the kurtosis tensor:

\[\]

MKT = frac{1}{5} Tr(mathbf{W}) = frac{1}{5} (W_{1111} + W_{2222} + W_{3333} + 2W_{1122} + 2W_{1133} + 2W_{2233})

References

[1] (1,2,3)

Hansen, B., Lund, T. E., Sangill, R., and Jespersen, S. N. 2013. Experimentally and computationally fast method for estimation of a mean kurtosis. Magnetic Resonance in Medicine69, 1754–1760. 388. doi:10.1002/mrm.24743

[2]

predict(gtab, S0=1.0)

Given a DKI model fit, predict the signal on the vertices of a gradient table

Parameters:

gtaba GradientTable class instance: The gradient table for this prediction
S0float or ndarray (optional): The non diffusion-weighted signal in every voxel, or across all voxels. Default: 1

Notes

The predicted signal is given by:

\[S(n,b)=S_{0}e^{-bD(n)+\frac{1}{6}b^{2}D(n)^{2}K(n)}\]

\(\mathbf{D(n)}\) and \(\mathbf{K(n)}\) can be computed from the DT and KT using the following equations:

\[D(n)=\sum_{i=1}^{3}\sum_{j=1}^{3}n_{i}n_{j}D_{ij}\]

and

\[K(n)=\frac{MD^{2}}{D(n)^{2}}\sum_{i=1}^{3}\sum_{j=1}^{3} \sum_{k=1}^{3}\sum_{l=1}^{3}n_{i}n_{j}n_{k}n_{l}W_{ijkl}\]

where \(D_{ij}\) and \(W_{ijkl}\) are the elements of the second-order DT and the fourth-order KT tensors, respectively, and \(MD\) is the mean diffusivity.

rk(min_kurtosis=-0.42857142857142855, max_kurtosis=10, analytical=True)

Radial Kurtosis (RK) of a diffusion kurtosis tensor [1].

Parameters:

min_kurtosisfloat (optional): To keep kurtosis values within a plausible biophysical range, radial kurtosis values that are smaller than min_kurtosis are replaced with min_kurtosis. Default = -3./7 (theoretical kurtosis limit for regions that consist of water confined to spherical pores [3])
max_kurtosisfloat (optional): To keep kurtosis values within a plausible biophysical range, radial kurtosis values that are larger than max_kurtosis are replaced with max_kurtosis. Default = 10
analyticalbool (optional): If True, RK is calculated using its analytical solution, otherwise an exact numerical estimator is used (see Notes). Default is set to True

Returns:

rkarray: Calculated RK.

Notes

RK is defined as the average of the directional kurtosis perpendicular to the fiber’s main direction e1 [1], [2]:

\[\]

RK equiv frac{1}{2pi} int dOmega _mathbf{theta}: K(mathbf{theta}) delta (mathbf{theta}cdot mathbf{e}_1)

This equation can be numerically computed by averaging apparent directional kurtosis samples for directions perpendicular to e1.

Otherwise, RK can be calculated from its analytical solution [2]:

\[K_{\bot} = G_1(\lambda_1,\lambda_2,\lambda_3)\hat{W}_{2222} + G_1(\lambda_1,\lambda_3,\lambda_2)\hat{W}_{3333} + G_2(\lambda_1,\lambda_2,\lambda_3)\hat{W}_{2233}\]

where:

\[G_1(\lambda_1,\lambda_2,\lambda_3)= \frac{(\lambda_1+\lambda_2+\lambda_3)^2}{18\lambda_2(\lambda_2- \lambda_3)} \left (2\lambda_2 + \frac{\lambda_3^2-3\lambda_2\lambda_3}{\sqrt{\lambda_2\lambda_3}} \right)\]

and

\[ G_2(\lambda_1,\lambda_2,\lambda_3)= \frac{(\lambda_1+\lambda_2+\lambda_3)^2}{(\lambda_2-\lambda_3)^2} \left ( \frac{\lambda_2+\lambda_3}{\sqrt{\lambda_2\lambda_3}}- 2\right )\]

References

[1] (1,2,3)

Jensen, J.H., Helpern, J.A., 2010. MRI quantification of non-Gaussian water diffusion by kurtosis analysis. NMR in Biomedicine 23(7): 698-710

[2] (1,2)

Tabesh, A., Jensen, J.H., Ardekani, B.A., Helpern, J.A., 2011. Estimation of tensors and tensor-derived measures in diffusional kurtosis imaging. Magn Reson Med. 65(3), 823-836

[3]

`DiffusionKurtosisModel`

class dipy.reconst.dki.DiffusionKurtosisModel(gtab, fit_method='WLS', *args, **kwargs)

Bases: ReconstModel

Class for the Diffusion Kurtosis Model

Methods

`fit`(data[, mask])	Fit method of the DKI model class
`predict`(dki_params[, S0])	Predict a signal for this DKI model class instance given parameters.

__init__(gtab, fit_method='WLS', *args, **kwargs)

Diffusion Kurtosis Tensor Model [1]

Parameters:

gtabGradientTable class instance

fit_methodstr or callable

str can be one of the following: ‘OLS’ or ‘ULLS’ for ordinary least squares

dki.ols_fit_dki

‘WLS’ or ‘UWLLS’ for weighted ordinary least squares: dki.wls_fit_dki
callable has to have the signature:: fit_method(design_matrix, data, *args, **kwargs)

args, kwargsarguments and key-word arguments passed to the

fit_method. See dki.ols_fit_dki, dki.wls_fit_dki for details

References

[1]

Tabesh, A., Jensen, J.H., Ardekani, B.A., Helpern, J.A., 2011.

Estimation of tensors and tensor-derived measures in diffusional kurtosis imaging. Magn Reson Med. 65(3), 823-836

fit(data, mask=None)

Fit method of the DKI model class

Parameters:

dataarray: The measured signal from one voxel.
maskarray: A boolean array used to mark the coordinates in the data that should be analyzed that has the shape data.shape[-1]

predict(dki_params, S0=1.0)

Predict a signal for this DKI model class instance given parameters.

Parameters:

dki_paramsndarray (x, y, z, 27) or (n, 27)

All parameters estimated from the diffusion kurtosis model. Parameters are ordered as follows:

Three diffusion tensor’s eigenvalues

2. Three lines of the eigenvector matrix each containing the first, second and third coordinates of the eigenvector 3. Fifteen elements of the kurtosis tensor

S0float or ndarray (optional)

The non diffusion-weighted signal in every voxel, or across all voxels. Default: 1

`ReconstModel`

class dipy.reconst.dki.ReconstModel(gtab)

Bases: object

Abstract class for signal reconstruction models

Methods

fit

__init__(gtab)

Initialization of the abstract class for signal reconstruction models

Parameters:

gtabGradientTable class instance

fit(data, mask=None, **kwargs)

`TensorFit`

class dipy.reconst.dki.TensorFit(model, model_params, model_S0=None)

Bases: object

Attributes:

S0_hat
directions: For tracking - return the primary direction in each voxel
evals: Returns the eigenvalues of the tensor as an array
evecs: Returns the eigenvectors of the tensor as an array, columnwise
quadratic_form: Calculates the 3x3 diffusion tensor for each voxel
shape

Methods

`ad`()	Axial diffusivity (AD) calculated from cached eigenvalues.
`adc`(sphere)	Calculate the apparent diffusion coefficient (ADC) in each direction on
`color_fa`()	Color fractional anisotropy of diffusion tensor
`fa`()	Fractional anisotropy (FA) calculated from cached eigenvalues.
`ga`()	Geodesic anisotropy (GA) calculated from cached eigenvalues.
`linearity`()	Returns:
`md`()	Mean diffusivity (MD) calculated from cached eigenvalues.
`mode`()	Tensor mode calculated from cached eigenvalues.
`odf`(sphere)	The diffusion orientation distribution function (dODF).
`planarity`()	Returns:
`predict`(gtab[, S0, step])	Given a model fit, predict the signal on the vertices of a sphere
`rd`()	Radial diffusivity (RD) calculated from cached eigenvalues.
`sphericity`()	Returns:
`trace`()	Trace of the tensor calculated from cached eigenvalues.

lower_triangular

__init__(model, model_params, model_S0=None): Initialize a TensorFit class instance.

property S0_hat

ad()

Axial diffusivity (AD) calculated from cached eigenvalues.

Returns:

adarray (V, 1): Calculated AD.

Notes

RD is calculated with the following equation:

\[AD = \lambda_1\]

adc(sphere)

Calculate the apparent diffusion coefficient (ADC) in each direction on the sphere for each voxel in the data

Parameters:

sphereSphere class instance

Returns:

adcndarray: The estimates of the apparent diffusion coefficient in every direction on the input sphere

ec{b} Q ec{b}^T

Where Q is the quadratic form of the tensor.

color_fa(): Color fractional anisotropy of diffusion tensor

property directions: For tracking - return the primary direction in each voxel

property evals: Returns the eigenvalues of the tensor as an array

property evecs: Returns the eigenvectors of the tensor as an array, columnwise

fa(): Fractional anisotropy (FA) calculated from cached eigenvalues.

ga(): Geodesic anisotropy (GA) calculated from cached eigenvalues.

linearity()

Returns:

linearityarray: Calculated linearity of the diffusion tensor [1].

Notes

Linearity is calculated with the following equation:

\[Linearity = \frac{\lambda_1-\lambda_2}{\lambda_1+\lambda_2+\lambda_3}\]

References

[1]

Westin C.-F., Peled S., Gubjartsson H., Kikinis R., Jolesz F., “Geometrical diffusion measures for MRI from tensor basis analysis” in Proc. 5th Annual ISMRM, 1997.

lower_triangular(b0=None)

md()

Mean diffusivity (MD) calculated from cached eigenvalues.

Returns:

mdarray (V, 1): Calculated MD.

Notes

MD is calculated with the following equation:

\[MD = \frac{\lambda_1+\lambda_2+\lambda_3}{3}\]

mode(): Tensor mode calculated from cached eigenvalues.

odf(sphere)

The diffusion orientation distribution function (dODF). This is an estimate of the diffusion distance in each direction

Parameters:

sphereSphere class instance.: The dODF is calculated in the vertices of this input.

Returns:

odfndarray: The diffusion distance in every direction of the sphere in every voxel in the input data.

Notes

This is based on equation 3 in [1]. To re-derive it from scratch, follow steps in [2], Section 7.9 Equation 7.24 but with an \(r^2\) term in the integral.

References

[1]

Aganj, I., Lenglet, C., Sapiro, G., Yacoub, E., Ugurbil, K., & Harel, N. (2010). Reconstruction of the orientation distribution function in single- and multiple-shell q-ball imaging within constant solid angle. Magnetic Resonance in Medicine, 64(2), 554-566. doi:DOI: 10.1002/mrm.22365

[2]

Descoteaux, M. (2008). PhD Thesis: High Angular Resolution Diffusion MRI: from Local Estimation to Segmentation and Tractography. ftp://ftp-sop.inria.fr/athena/Publications/PhDs/descoteaux_thesis.pdf

planarity()

Returns:

sphericityarray: Calculated sphericity of the diffusion tensor [1].

Notes

Sphericity is calculated with the following equation:

\[Sphericity = \frac{2 (\lambda_2 - \lambda_3)}{\lambda_1+\lambda_2+\lambda_3}\]

References

[1]

Westin C.-F., Peled S., Gubjartsson H., Kikinis R., Jolesz F., “Geometrical diffusion measures for MRI from tensor basis analysis” in Proc. 5th Annual ISMRM, 1997.

predict(gtab, S0=None, step=None)

Given a model fit, predict the signal on the vertices of a sphere

Parameters:

gtaba GradientTable class instance

This encodes the directions for which a prediction is made

S0float array

The mean non-diffusion weighted signal in each voxel. Default: The fitted S0 value in all voxels if it was fitted. Otherwise 1 in all voxels.

stepint

The chunk size as a number of voxels. Optional parameter with default value 10,000.

In order to increase speed of processing, tensor fitting is done simultaneously over many voxels. This parameter sets the number of voxels that will be fit at once in each iteration. A larger step value should speed things up, but it will also take up more memory. It is advisable to keep an eye on memory consumption as this value is increased.

Notes

The predicted signal is given by:

\[S( heta, b) = S_0 * e^{-b ADC}\]

Where: .. math

ADC =       heta Q  heta^T

:math:` heta` is a unit vector pointing at any direction on the sphere for which a signal is to be predicted and \(b\) is the b value provided in the GradientTable input for that direction

property quadratic_form: Calculates the 3x3 diffusion tensor for each voxel

rd()

Radial diffusivity (RD) calculated from cached eigenvalues.

Returns:

rdarray (V, 1): Calculated RD.

Notes

RD is calculated with the following equation:

\[RD = \frac{\lambda_2 + \lambda_3}{2}\]

property shape

sphericity()

Returns:

sphericityarray: Calculated sphericity of the diffusion tensor [1].

Notes

Sphericity is calculated with the following equation:

\[Sphericity = \frac{3 \lambda_3}{\lambda_1+\lambda_2+\lambda_3}\]

References

[1]

Westin C.-F., Peled S., Gubjartsson H., Kikinis R., Jolesz F., “Geometrical diffusion measures for MRI from tensor basis analysis” in Proc. 5th Annual ISMRM, 1997.

trace()

Trace of the tensor calculated from cached eigenvalues.

Returns:

tracearray (V, 1): Calculated trace.

Notes

The trace is calculated with the following equation:

\[trace = \lambda_1 + \lambda_2 + \lambda_3\]

Wcons

dipy.reconst.dki.Wcons(k_elements)

Construct the full 4D kurtosis tensors from its 15 independent elements

Parameters:

k_elements(15,): elements of the kurtosis tensor in the following order:
.. math::
begin{matrix} ( & W_{xxxx} & W_{yyyy} & W_{zzzz} & W_{xxxy} & W_{xxxz}: & … \ & W_{xyyy} & W_{yyyz} & W_{xzzz} & W_{yzzz} & W_{xxyy} & … \ & W_{xxzz} & W_{yyzz} & W_{xxyz} & W_{xyyz} & W_{xyzz} & & )end{matrix}

Returns:

Warray(3, 3, 3, 3): Full 4D kurtosis tensor

Wrotate

dipy.reconst.dki.Wrotate(kt, Basis)

Rotate a kurtosis tensor from the standard Cartesian coordinate system to another coordinate system basis

Parameters:

kt(15,): Vector with the 15 independent elements of the kurtosis tensor
Basisarray (3, 3): Vectors of the basis column-wise oriented
indsarray(m, 4) (optional): Array of vectors containing the four indexes of m specific elements of the rotated kurtosis tensor. If not specified all 15 elements of the rotated kurtosis tensor are computed.

Returns:

Wrotarray (m,) or (15,): Vector with the m independent elements of the rotated kurtosis tensor. If ‘indices’ is not specified all 15 elements of the rotated kurtosis tensor are computed.

Notes

KT elements are assumed to be ordered as follows:

\[\]

begin{matrix} ( & W_{xxxx} & W_{yyyy} & W_{zzzz} & W_{xxxy} & W_{xxxz}: & … \ & W_{xyyy} & W_{yyyz} & W_{xzzz} & W_{yzzz} & W_{xxyy} & … \ & W_{xxzz} & W_{yyzz} & W_{xxyz} & W_{xyyz} & W_{xyzz} & & )end{matrix}

References

[1] Hui ES, Cheung MM, Qi L, Wu EX, 2008. Towards better MR characterization of neural tissues using directional diffusion kurtosis analysis. Neuroimage 42(1): 122-34

Wrotate_element

dipy.reconst.dki.Wrotate_element(kt, indi, indj, indk, indl, B)

Computes the the specified index element of a kurtosis tensor rotated to the coordinate system basis B.

Parameters:

ktndarray (x, y, z, 15) or (n, 15): Array containing the 15 independent elements of the kurtosis tensor
indiint: Rotated kurtosis tensor element index i (0 for x, 1 for y, 2 for z)
indjint: Rotated kurtosis tensor element index j (0 for x, 1 for y, 2 for z)
indkint: Rotated kurtosis tensor element index k (0 for x, 1 for y, 2 for z)
indl: int: Rotated kurtosis tensor element index l (0 for x, 1 for y, 2 for z)
B: array (x, y, z, 3, 3) or (n, 15): Vectors of the basis column-wise oriented

Returns:

Wrefloat: rotated kurtosis tensor element of index ind_i, ind_j, ind_k, ind_l

Notes

It is assumed that initial kurtosis tensor elementes are defined on the Cartesian coordinate system.

References

[1] Hui ES, Cheung MM, Qi L, Wu EX, 2008. Towards better MR characterization of neural tissues using directional diffusion kurtosis analysis. Neuroimage 42(1): 122-34

apparent_kurtosis_coef

dipy.reconst.dki.apparent_kurtosis_coef(dki_params, sphere, min_diffusivity=0, min_kurtosis=-0.42857142857142855)

Calculates the apparent kurtosis coefficient (AKC) in each direction of a sphere [1].

Parameters:

dki_paramsndarray (x, y, z, 27) or (n, 27)

All parameters estimated from the diffusion kurtosis model. Parameters are ordered as follows:

Three diffusion tensor’s eigenvalues

Three lines of the eigenvector matrix each containing the first, second and third coordinates of the eigenvectors respectively

Fifteen elements of the kurtosis tensor

spherea Sphere class instance

The AKC will be calculated for each of the vertices in the sphere

min_diffusivityfloat (optional)

Because negative eigenvalues are not physical and small eigenvalues cause quite a lot of noise in diffusion-based metrics, diffusivity values smaller than min_diffusivity are replaced with min_diffusivity. Default = 0

min_kurtosisfloat (optional)

Because high-amplitude negative values of kurtosis are not physicaly and biologicaly pluasible, and these cause artefacts in kurtosis-based measures, directional kurtosis values smaller than min_kurtosis are replaced with min_kurtosis. Default = -3./7 (theoretical kurtosis limit for regions that consist of water confined to spherical pores [2])

Returns:

akcndarray (x, y, z, g) or (n, g): Apparent kurtosis coefficient (AKC) for all g directions of a sphere.

Notes

For each sphere direction with coordinates \((n_{1}, n_{2}, n_{3})\), the calculation of AKC is done using formula [1]:

\[AKC(n)=\frac{MD^{2}}{ADC(n)^{2}}\sum_{i=1}^{3}\sum_{j=1}^{3} \sum_{k=1}^{3}\sum_{l=1}^{3}n_{i}n_{j}n_{k}n_{l}W_{ijkl}\]

where \(W_{ijkl}\) are the elements of the kurtosis tensor, MD the mean diffusivity and ADC the apparent diffusion coefficent computed as:

\[ADC(n)=\sum_{i=1}^{3}\sum_{j=1}^{3}n_{i}n_{j}D_{ij}\]

where \(D_{ij}\) are the elements of the diffusion tensor.

References

[1] (1,2,3)

Neto Henriques R, Correia MM, Nunes RG, Ferreira HA (2015). Exploring the 3D geometry of the diffusion kurtosis tensor - Impact on the development of robust tractography procedures and novel biomarkers, NeuroImage 111: 85-99

[2]

axial_kurtosis

dipy.reconst.dki.axial_kurtosis(dki_params, min_kurtosis=-0.42857142857142855, max_kurtosis=10, analytical=True)

Computes axial Kurtosis (AK) from the kurtosis tensor [1], [2].

Parameters:

dki_paramsndarray (x, y, z, 27) or (n, 27)

All parameters estimated from the diffusion kurtosis model. Parameters are ordered as follows:

Three diffusion tensor’s eigenvalues

Three lines of the eigenvector matrix each containing the first, second and third coordinates of the eigenvector

Fifteen elements of the kurtosis tensor

min_kurtosisfloat (optional)

To keep kurtosis values within a plausible biophysical range, axial kurtosis values that are smaller than min_kurtosis are replaced with min_kurtosis. Default = -3./7 (theoretical kurtosis limit for regions that consist of water confined to spherical pores [3])

max_kurtosisfloat (optional)

To keep kurtosis values within a plausible biophysical range, axial kurtosis values that are larger than max_kurtosis are replaced with max_kurtosis. Default = 10

analyticalbool (optional)

If True, AK is calculated from rotated diffusion kurtosis tensor, otherwise it will be computed from the apparent diffusion kurtosis values along the principal axis of the diffusion tensor (see notes). Default is set to True.

Returns:

akarray: Calculated AK.

Notes

AK is defined as the directional kurtosis parallel to the fiber’s main direction e1 [1], [2]. You can compute AK using to approaches:

AK is calculated from rotated diffusion kurtosis tensor [2], i.e.:

\[AK = \hat{W}_{1111} \frac{(\lambda_{1}+\lambda_{2}+\lambda_{3})^2}{(9 \lambda_{1}^2)}\]

AK can be sampled from the principal axis of the diffusion tensor:

\[AK = K(\mathbf{\mathbf{e}_1)\]

References

[1] (1,2,3)

Jensen, J.H., Helpern, J.A., 2010. MRI quantification of non-Gaussian water diffusion by kurtosis analysis. NMR in Biomedicine 23(7): 698-710

[2] (1,2,3,4)

Tabesh, A., Jensen, J.H., Ardekani, B.A., Helpern, J.A., 2011. Estimation of tensors and tensor-derived measures in diffusional kurtosis imaging. Magn Reson Med. 65(3), 823-836

[3]

carlson_rd

dipy.reconst.dki.carlson_rd(x, y, z, errtol=0.0001)

Computes the Carlson’s incomplete elliptic integral of the second kind defined as:

\[R_D = \frac{3}{2} \int_{0}^{\infty} (t+x)^{-\frac{1}{2}} (t+y)^{-\frac{1}{2}}(t+z) ^{-\frac{3}{2}}\]

Parameters:

xndarray: First independent variable of the integral.
yndarray: Second independent variable of the integral.
zndarray: Third independent variable of the integral.
errtolfloat: Error tolerance. Integral is computed with relative error less in magnitude than the defined value

Returns:

RDndarray: Value of the incomplete second order elliptic integral

Notes

x, y, and z have to be nonnegative and at most x or y is zero.

carlson_rf

dipy.reconst.dki.carlson_rf(x, y, z, errtol=0.0003)

Computes the Carlson’s incomplete elliptic integral of the first kind defined as:

\[R_F = \frac{1}{2} \int_{0}^{\infty} \left [(t+x)(t+y)(t+z) \right ] ^{-\frac{1}{2}}dt\]

Parameters:

xndarray: First independent variable of the integral.
yndarray: Second independent variable of the integral.
zndarray: Third independent variable of the integral.
errtolfloat: Error tolerance. Integral is computed with relative error less in magnitude than the defined value

Returns:

RFndarray: Value of the incomplete first order elliptic integral

Notes

x, y, and z have to be nonnegative and at most one of them is zero.

References

[1]

Carlson, B.C., 1994. Numerical computation of real or complex elliptic integrals. arXiv:math/9409227 [math.CA]

cart2sphere

dipy.reconst.dki.cart2sphere(x, y, z)

Return angles for Cartesian 3D coordinates x, y, and z

See doc for sphere2cart for angle conventions and derivation of the formulae.

\(0\le\theta\mathrm{(theta)}\le\pi\) and \(-\pi\le\phi\mathrm{(phi)}\le\pi\)

Parameters:

xarray_like: x coordinate in Cartesian space
yarray_like: y coordinate in Cartesian space
zarray_like: z coordinate

Returns:

rarray: radius
thetaarray: inclination (polar) angle
phiarray: azimuth angle

check_multi_b

dipy.reconst.dki.check_multi_b(gtab, n_bvals, non_zero=True, bmag=None)

Check if you have enough different b-values in your gradient table

Parameters:

gtabGradientTable class instance.
n_bvalsint: The number of different b-values you are checking for.
non_zerobool: Whether to check only non-zero bvalues. In this case, we will require at least n_bvals non-zero b-values (where non-zero is defined depending on the gtab object’s b0_threshold attribute)
bmagint: The order of magnitude of the b-values used. The function will normalize the b-values relative \(10^{bmag}\). Default: derive this value from the maximal b-value provided: \(bmag=log_{10}(max(bvals)) - 1\).

Returns:

boolWhether there are at least n_bvals different b-values in the
gradient table used.

decompose_tensor

dipy.reconst.dki.decompose_tensor(tensor, min_diffusivity=0)

Returns eigenvalues and eigenvectors given a diffusion tensor

Computes tensor eigen decomposition to calculate eigenvalues and eigenvectors (Basser et al., 1994a).

Parameters:

tensorarray (…, 3, 3): Hermitian matrix representing a diffusion tensor.
min_diffusivityfloat: Because negative eigenvalues are not physical and small eigenvalues, much smaller than the diffusion weighting, cause quite a lot of noise in metrics such as fa, diffusivity values smaller than min_diffusivity are replaced with min_diffusivity.

Returns:

eigvalsarray (…, 3): Eigenvalues from eigen decomposition of the tensor. Negative eigenvalues are replaced by zero. Sorted from largest to smallest.
eigvecsarray (…, 3, 3): Associated eigenvectors from eigen decomposition of the tensor. Eigenvectors are columnar (e.g. eigvecs[…, :, j] is associated with eigvals[…, j])

design_matrix

dipy.reconst.dki.design_matrix(gtab)

Construct B design matrix for DKI.

Parameters:

gtabGradientTable: Measurement directions.

Returns:

Barray (N, 22): Design matrix or B matrix for the DKI model B[j, :] = (Bxx, Bxy, Bzz, Bxz, Byz, Bzz,

Bxxxx, Byyyy, Bzzzz, Bxxxy, Bxxxz, Bxyyy, Byyyz, Bxzzz, Byzzz, Bxxyy, Bxxzz, Byyzz, Bxxyz, Bxyyz, Bxyzz, BlogS0)

directional_diffusion

dipy.reconst.dki.directional_diffusion(dt, V, min_diffusivity=0)

Calculates the apparent diffusion coefficient (adc) in each direction of a sphere for a single voxel [1].

Parameters:

dtarray (6,): elements of the diffusion tensor of the voxel.
Varray (g, 3): g directions of a Sphere in Cartesian coordinates
min_diffusivityfloat (optional): Because negative eigenvalues are not physical and small eigenvalues cause quite a lot of noise in diffusion-based metrics, diffusivity values smaller than min_diffusivity are replaced with min_diffusivity. Default = 0

Returns:

adcndarray (g,): Apparent diffusion coefficient (adc) in all g directions of a sphere for a single voxel.

References

[1] (1,2)

directional_diffusion_variance

dipy.reconst.dki.directional_diffusion_variance(kt, V, min_kurtosis=-0.42857142857142855)

Calculates the apparent diffusion variance (adv) in each direction of a sphere for a single voxel [1].

Parameters:

dtarray (6,): elements of the diffusion tensor of the voxel.
ktarray (15,): elements of the kurtosis tensor of the voxel.
Varray (g, 3): g directions of a Sphere in Cartesian coordinates
min_kurtosisfloat (optional): Because high-amplitude negative values of kurtosis are not physicaly and biologicaly pluasible, and these cause artefacts in kurtosis-based measures, directional kurtosis values smaller than min_kurtosis are replaced with min_kurtosis. Default = -3./7 (theoretical kurtosis limit for regions that consist of water confined to spherical pores [2]_)
adcndarray(g,) (optional): Apparent diffusion coefficient (adc) in all g directions of a sphere for a single voxel.
advndarray(g,) (optional): Apparent diffusion variance coefficient (advc) in all g directions of a sphere for a single voxel.

Returns:

advndarray (g,): Apparent diffusion variance (adv) in all g directions of a sphere for a single voxel.

References

[1] (1,2)

directional_kurtosis

dipy.reconst.dki.directional_kurtosis(dt, md, kt, V, min_diffusivity=0, min_kurtosis=-0.42857142857142855, adc=None, adv=None)

Calculates the apparent kurtosis coefficient (akc) in each direction of a sphere for a single voxel [1].

Parameters:

dtarray (6,): elements of the diffusion tensor of the voxel.
mdfloat: mean diffusivity of the voxel
ktarray (15,): elements of the kurtosis tensor of the voxel.
Varray (g, 3): g directions of a Sphere in Cartesian coordinates
min_diffusivityfloat (optional): Because negative eigenvalues are not physical and small eigenvalues cause quite a lot of noise in diffusion-based metrics, diffusivity values smaller than min_diffusivity are replaced with min_diffusivity. Default = 0
min_kurtosisfloat (optional): Because high-amplitude negative values of kurtosis are not physicaly and biologicaly pluasible, and these cause artefacts in kurtosis-based measures, directional kurtosis values smaller than min_kurtosis are replaced with min_kurtosis. Default = -3./7 (theoretical kurtosis limit for regions that consist of water confined to spherical pores [2])
adcndarray(g,) (optional): Apparent diffusion coefficient (adc) in all g directions of a sphere for a single voxel.
advndarray(g,) (optional): Apparent diffusion variance (advc) in all g directions of a sphere for a single voxel.

Returns:

akcndarray (g,): Apparent kurtosis coefficient (AKC) in all g directions of a sphere for a single voxel.

References

[1] (1,2)

[2]

dki_prediction

dipy.reconst.dki.dki_prediction(dki_params, gtab, S0=1.0)

Predict a signal given diffusion kurtosis imaging parameters.

Parameters:

dki_paramsndarray (x, y, z, 27) or (n, 27)

All parameters estimated from the diffusion kurtosis model. Parameters are ordered as follows:

Three diffusion tensor’s eigenvalues

Three lines of the eigenvector matrix each containing the first, second and third coordinates of the eigenvector

Fifteen elements of the kurtosis tensor

gtaba GradientTable class instance

The gradient table for this prediction

S0float or ndarray (optional)

The non diffusion-weighted signal in every voxel, or across all voxels. Default: 1

Returns:

S(…, N) ndarray: Simulated signal based on the DKI model:

\[S=S_{0}e^{-bD+\]
rac{1}{6}b^{2}D^{2}K}

from_lower_triangular

dipy.reconst.dki.from_lower_triangular(D)

Returns a tensor given the six unique tensor elements

Given the six unique tensor elements (in the order: Dxx, Dxy, Dyy, Dxz, Dyz, Dzz) returns a 3 by 3 tensor. All elements after the sixth are ignored.

Parameters:

Darray_like, (…, >6): Unique elements of the tensors

Returns:

tensorndarray (…, 3, 3): 3 by 3 tensors

get_fnames

dipy.reconst.dki.get_fnames(name='small_64D')

Provide full paths to example or test datasets.

Parameters:

namestr

the filename/s of which dataset to return, one of:

‘small_64D’ small region of interest nifti,bvecs,bvals 64 directions
‘small_101D’ small region of interest nifti, bvecs, bvals 101 directions
‘aniso_vox’ volume with anisotropic voxel size as Nifti
‘fornix’ 300 tracks in Trackvis format (from Pittsburgh Brain Competition)
‘gqi_vectors’ the scanner wave vectors needed for a GQI acquisitions of 101 directions tested on Siemens 3T Trio
‘small_25’ small ROI (10x8x2) DTI data (b value 2000, 25 directions)
‘test_piesno’ slice of N=8, K=14 diffusion data
‘reg_c’ small 2D image used for validating registration
‘reg_o’ small 2D image used for validation registration
‘cb_2’ two vectorized cingulum bundles

Returns:

fnamestuple: filenames for dataset

Examples

>>> import numpy as np
>>> from dipy.io.image import load_nifti
>>> from dipy.data import get_fnames
>>> fimg, fbvals, fbvecs = get_fnames('small_101D')
>>> bvals=np.loadtxt(fbvals)
>>> bvecs=np.loadtxt(fbvecs).T
>>> data, affine = load_nifti(fimg)
>>> data.shape == (6, 10, 10, 102)
True
>>> bvals.shape == (102,)
True
>>> bvecs.shape == (102, 3)
True

get_sphere

dipy.reconst.dki.get_sphere(name='symmetric362')

provide triangulated spheres

Parameters:

namestr: which sphere - one of: * ‘symmetric362’ * ‘symmetric642’ * ‘symmetric724’ * ‘repulsion724’ * ‘repulsion100’ * ‘repulsion200’

Returns:

spherea dipy.core.sphere.Sphere class instance

Examples

>>> import numpy as np
>>> from dipy.data import get_sphere
>>> sphere = get_sphere('symmetric362')
>>> verts, faces = sphere.vertices, sphere.faces
>>> verts.shape == (362, 3)
True
>>> faces.shape == (720, 3)
True
>>> verts, faces = get_sphere('not a sphere name') 
Traceback (most recent call last):
    ...
DataError: No sphere called "not a sphere name"

kurtosis_fractional_anisotropy

dipy.reconst.dki.kurtosis_fractional_anisotropy(dki_params)

Computes the anisotropy of the kurtosis tensor (KFA) [1].

Parameters:

dki_paramsndarray (x, y, z, 27) or (n, 27)

All parameters estimated from the diffusion kurtosis model. Parameters are ordered as follows:

Three diffusion tensor’s eigenvalues

Three lines of the eigenvector matrix each containing the first, second and third coordinates of the eigenvector

Fifteen elements of the kurtosis tensor

Returns

——-

kfaarray

Calculated mean kurtosis tensor.

Notes

The KFA is defined as [1]:

\[KFA \equiv \frac{||\mathbf{W} - MKT \mathbf{I}^{(4)}||_F}{||\mathbf{W}||_F}\]

where \(W\) is the kurtosis tensor, MKT the kurtosis tensor mean, \(I^(4)\) is the fully symmetric rank 2 isotropic tensor and \(||...||_F\) is the tensor’s Frobenius norm [1].

References

[1] (1,2,3,4)

Glenn, G. R., Helpern, J. A., Tabesh, A., and Jensen, J. H. (2015). Quantitative assessment of diffusional kurtosis anisotropy. NMR in Biomedicine 28, 448–459. doi:10.1002/nbm.3271

kurtosis_maximum

dipy.reconst.dki.kurtosis_maximum(dki_params, sphere='repulsion100', gtol=0.01, mask=None)

Computes kurtosis maximum value

Parameters:

dki_paramsndarray (x, y, z, 27) or (n, 27)

All parameters estimated from the diffusion kurtosis model. Parameters are ordered as follows:

Three diffusion tensor’s eingenvalues

Three lines of the eigenvector matrix each containing the first, second and third coordinates of the eigenvector

Fifteen elements of the kurtosis tensor

sphereSphere class instance, optional

The sphere providing sample directions for the initial search of the maximal value of kurtosis.

gtolfloat, optional

This input is to refine kurtosis maximum under the precision of the directions sampled on the sphere class instance. The gradient of the convergence procedure must be less than gtol before successful termination. If gtol is None, fiber direction is directly taken from the initial sampled directions of the given sphere object

maskndarray

A boolean array used to mark the coordinates in the data that should be analyzed that has the shape dki_params.shape[:-1]

Returns:

max_valuefloat: kurtosis tensor maximum value
max_dirarray (3,): Cartesian coordinates of the direction of the maximal kurtosis value

local_maxima

dipy.reconst.dki.local_maxima()

Local maxima of a function evaluated on a discrete set of points.

If a function is evaluated on some set of points where each pair of neighboring points is an edge in edges, find the local maxima.

Parameters:

odfarray, 1d, dtype=double: The function evaluated on a set of discrete points.
edgesarray (N, 2): The set of neighbor relations between the points. Every edge, ie edges[i, :], is a pair of neighboring points.

Returns:

peak_valuesndarray: Value of odf at a maximum point. Peak values is sorted in descending order.
peak_indicesndarray: Indices of maximum points. Sorted in the same order as peak_values so odf[peak_indices[i]] == peak_values[i].

See also

dipy.core.sphere

Notes

A point is a local maximum if it is > at least one neighbor and >= all neighbors. If no points meet the above criteria, 1 maximum is returned such that odf[maximum] == max(odf).

lower_triangular

dipy.reconst.dki.lower_triangular(tensor, b0=None)

Returns the six lower triangular values of the tensor ordered as (Dxx, Dxy, Dyy, Dxz, Dyz, Dzz) and a dummy variable if b0 is not None.

Parameters:

tensorarray_like (…, 3, 3): a collection of 3, 3 diffusion tensors
b0float: if b0 is not none log(b0) is returned as the dummy variable

Returns:

Dndarray: If b0 is none, then the shape will be (…, 6) otherwise (…, 7)

mean_diffusivity

dipy.reconst.dki.mean_diffusivity(evals, axis=-1)

Mean Diffusivity (MD) of a diffusion tensor.

Parameters:

evalsarray-like: Eigenvalues of a diffusion tensor.
axisint: Axis of evals which contains 3 eigenvalues.

Returns:

mdarray: Calculated MD.

Notes

MD is calculated with the following equation:

\[MD = \frac{\lambda_1 + \lambda_2 + \lambda_3}{3}\]

mean_kurtosis

dipy.reconst.dki.mean_kurtosis(dki_params, min_kurtosis=-0.42857142857142855, max_kurtosis=3, analytical=True)

Computes mean Kurtosis (MK) from the kurtosis tensor.

Parameters:

dki_paramsndarray (x, y, z, 27) or (n, 27)

All parameters estimated from the diffusion kurtosis model. Parameters are ordered as follows:

Three diffusion tensor’s eigenvalues

Three lines of the eigenvector matrix each containing the first, second and third coordinates of the eigenvector

Fifteen elements of the kurtosis tensor

min_kurtosisfloat (optional)

To keep kurtosis values within a plausible biophysical range, mean kurtosis values that are smaller than min_kurtosis are replaced with min_kurtosis. Default = -3./7 (theoretical kurtosis limit for regions that consist of water confined to spherical pores [4])

max_kurtosisfloat (optional)

To keep kurtosis values within a plausible biophysical range, mean kurtosis values that are larger than max_kurtosis are replaced with max_kurtosis. Default = 10

analyticalbool (optional)

If True, MK is calculated using its analytical solution, otherwise an exact numerical estimator is used (see Notes). Default is set to True

Returns:

mkarray: Calculated MK.

Notes

The MK is defined as the average of directional kurtosis coefficients across all spatial directions, which can be formulated by the following surface integral[R953e26c55b6a-1]_:

\[MK \equiv \frac{1}{4\pi} \int d\Omega_\mathbf{n} K(\mathbf{n})\]

This integral can be numerically solved by averaging directional kurtosis values sampled for directions of a spherical t-design [2].

Alternatively, MK can be solved from the analytical solution derived by Tabesh et al. [3]. This solution is given by:

where \(\hat{W}_{ijkl}\) are the components of the \(W\) tensor in the coordinates system defined by the eigenvectors of the diffusion tensor \(\mathbf{D}\) and

where \(R_f\) and \(R_d\) are the Carlson’s elliptic integrals.

References

[1]

Jensen, J.H., Helpern, J.A., 2010. MRI quantification of non-Gaussian water diffusion by kurtosis analysis. NMR in Biomedicine 23(7): 698-710

[2]

Hardin, R.H., Sloane, N.J.A., 1996. McLaren’s Improved Snub Cube and Other New Spherical Designs in Three Dimensions. Discrete and Computational Geometry 15, 429-441.

[3]

Tabesh, A., Jensen, J.H., Ardekani, B.A., Helpern, J.A., 2011. Estimation of tensors and tensor-derived measures in diffusional kurtosis imaging. Magn Reson Med. 65(3), 823-836

[4]

mean_kurtosis_tensor

dipy.reconst.dki.mean_kurtosis_tensor(dki_params, min_kurtosis=-0.42857142857142855, max_kurtosis=10)

Computes mean of the kurtosis tensor (MKT) [1].

Parameters:

dki_paramsndarray (x, y, z, 27) or (n, 27)

All parameters estimated from the diffusion kurtosis model. Parameters are ordered as follows:

Three diffusion tensor’s eigenvalues

Three lines of the eigenvector matrix each containing the first, second and third coordinates of the eigenvector

Fifteen elements of the kurtosis tensor

min_kurtosisfloat (optional)

max_kurtosisfloat (optional)

To keep kurtosis values within a plausible biophysical range, mean kurtosis values that are larger than max_kurtosis are replaced with max_kurtosis. Default = 10

Returns

——-

mktarray

Calculated mean kurtosis tensor.

Notes

The MKT is defined as [1]:

\[MKT \equiv \frac{1}{4\pi} \int d \Omega_{\mathnbf{n}} n_i n_j n_k n_l W_{ijkl}\]

which can be directly computed from the trace of the kurtosis tensor:

\[\]

MKT = frac{1}{5} Tr(mathbf{W}) = frac{1}{5} (W_{1111} + W_{2222} + W_{3333} + 2W_{1122} + 2W_{1133} + 2W_{2233})

References

[1] (1,2,3)

Hansen, B., Lund, T. E., Sangill, R., and Jespersen, S. N. (2013). Experimentally and computationally fast method for estimation of a mean kurtosis.Magnetic Resonance in Medicine69, 1754–1760.388 doi:10.1002/mrm.24743

[2]

ndindex

dipy.reconst.dki.ndindex(shape)

An N-dimensional iterator object to index arrays.

Given the shape of an array, an ndindex instance iterates over the N-dimensional index of the array. At each iteration a tuple of indices is returned; the last dimension is iterated over first.

Parameters:

shapetuple of ints: The dimensions of the array.

Examples

>>> from dipy.core.ndindex import ndindex
>>> shape = (3, 2, 1)
>>> for index in ndindex(shape):
...     print(index)
(0, 0, 0)
(0, 1, 0)
(1, 0, 0)
(1, 1, 0)
(2, 0, 0)
(2, 1, 0)

nlls_fit_tensor

dipy.reconst.dki.nlls_fit_tensor(design_matrix, data, weighting=None, sigma=None, jac=True, return_S0_hat=False, fail_is_nan=False)

Fit the cumulant expansion params (e.g. DTI, DKI) using non-linear least-squares.

Parameters:

design_matrixarray (g, Npar): Design matrix holding the covariants used to solve for the regression coefficients. First six parameters of design matrix should correspond to the six unique diffusion tensor elements in the lower triangular order (Dxx, Dxy, Dyy, Dxz, Dyz, Dzz), while last parameter to -log(S0)
dataarray ([X, Y, Z, …], g): Data or response variables holding the data. Note that the last dimension should contain the data. It makes no copies of data.
weighting: str: the weighting scheme to use in considering the squared-error. Default behavior is to use uniform weighting. Other options: ‘sigma’ ‘gmm’
sigma: float: If the ‘sigma’ weighting scheme is used, a value of sigma needs to be provided here. According to [Chang2005], a good value to use is 1.5267 * std(background_noise), where background_noise is estimated from some part of the image known to contain no signal (only noise).
jacbool: Use the Jacobian? Default: True
return_S0_hatbool: Boolean to return (True) or not (False) the S0 values for the fit.
fail_is_nanbool: Boolean to set failed NL fitting to NaN (True) or LS (False, default).

Returns:

nlls_params: the eigen-values and eigen-vectors of the tensor in each: voxel.

ols_fit_dki

dipy.reconst.dki.ols_fit_dki(design_matrix, data)

Computes the diffusion and kurtosis tensors using an ordinary linear least squares (OLS) approach [1].

Parameters:

design_matrixarray (g, 22): Design matrix holding the covariants used to solve for the regression coefficients.
dataarray (N, g): Data or response variables holding the data. Note that the last dimension should contain the data. It makes no copies of data.

Returns:

dki_paramsarray (N, 27)

All parameters estimated from the diffusion kurtosis model. Parameters are ordered as follows:

Three diffusion tensor’s eigenvalues

Three lines of the eigenvector matrix each containing the first, second and third coordinates of the eigenvector

Fifteen elements of the kurtosis tensor

See also

wls_fit_dki, nls_fit_dki

References

[1] Lu, H., Jensen, J. H., Ramani, A., and Helpern, J. A. (2006).: Three-dimensional characterization of non-gaussian water diffusion in humans using diffusion kurtosis imaging. NMR in Biomedicine 19, 236–247. doi:10.1002/nbm.1020

perpendicular_directions

dipy.reconst.dki.perpendicular_directions(v, num=30, half=False)

Computes n evenly spaced perpendicular directions relative to a given vector v

Parameters:

varray (3,): Array containing the three cartesian coordinates of vector v
numint, optional: Number of perpendicular directions to generate
halfbool, optional: If half is True, perpendicular directions are sampled on half of the unit circumference perpendicular to v, otherwive perpendicular directions are sampled on the full circumference. Default of half is False

Returns:

psamplesarray (n, 3): array of vectors perpendicular to v

Notes

Perpendicular directions are estimated using the following two step procedure:

1) the perpendicular directions are first sampled in a unit circumference parallel to the plane normal to the x-axis.

2) Samples are then rotated and aligned to the plane normal to vector v. The rotational matrix for this rotation is constructed as reference frame basis which axis are the following:

The first axis is vector v

The second axis is defined as the normalized vector given by the

cross product between vector v and the unit vector aligned to the x-axis - The third axis is defined as the cross product between the previous computed vector and vector v.

Following this two steps, coordinates of the final perpendicular directions are given as:

\[\left [ -\sin(a_{i}) \sqrt{{v_{y}}^{2}+{v_{z}}^{2}} \; , \; \frac{v_{x}v_{y}\sin(a_{i})-v_{z}\cos(a_{i})} {\sqrt{{v_{y}}^{2}+{v_{z}}^{2}}} \; , \; \frac{v_{x}v_{z}\sin(a_{i})-v_{y}\cos(a_{i})} {\sqrt{{v_{y}}^{2}+{v_{z}}^{2}}} \right ]\]

This procedure has a singularity when vector v is aligned to the x-axis. To solve this singularity, perpendicular directions in procedure’s step 1 are defined in the plane normal to y-axis and the second axis of the rotated frame of reference is computed as the normalized vector given by the cross product between vector v and the unit vector aligned to the y-axis. Following this, the coordinates of the perpendicular directions are given as:

left [ -frac{left (v_{x}v_{y}sin(a_{i})+v_{z}cos(a_{i}) right )} {sqrt{{v_{x}}^{2}+{v_{z}}^{2}}} ; , ; sin(a_{i}) sqrt{{v_{x}}^{2}+{v_{z}}^{2}} ; , ; frac{v_{y}v_{z}sin(a_{i})+v_{x}cos(a_{i})} {sqrt{{v_{x}}^{2}+{v_{z}}^{2}}} right ]

For more details on this calculation, see ` here <http://gsoc2015dipydki.blogspot.it/2015/07/rnh-post-8-computing-perpendicular.html>`_.

radial_kurtosis

dipy.reconst.dki.radial_kurtosis(dki_params, min_kurtosis=-0.42857142857142855, max_kurtosis=10, analytical=True)

Radial Kurtosis (RK) of a diffusion kurtosis tensor [1], [2].

Parameters:

dki_paramsndarray (x, y, z, 27) or (n, 27)

All parameters estimated from the diffusion kurtosis model. Parameters are ordered as follows:

Three diffusion tensor’s eigenvalues

Three lines of the eigenvector matrix each containing the first, second and third coordinates of the eigenvector

Fifteen elements of the kurtosis tensor

min_kurtosisfloat (optional)

To keep kurtosis values within a plausible biophysical range, radial kurtosis values that are smaller than min_kurtosis are replaced with min_kurtosis. Default = -3./7 (theoretical kurtosis limit for regions that consist of water confined to spherical pores [3])

max_kurtosisfloat (optional)

To keep kurtosis values within a plausible biophysical range, radial kurtosis values that are larger than max_kurtosis are replaced with max_kurtosis. Default = 10

analyticalbool (optional)

If True, RK is calculated using its analytical solution, otherwise an exact numerical estimator is used (see Notes). Default is set to True.

Returns:

rkarray: Calculated RK.

Notes

RK is defined as the average of the directional kurtosis perpendicular to the fiber’s main direction e1 [1], [2]:

\[\]

RK equiv frac{1}{2pi} int dOmega _mathbf{theta} K(mathbf{theta}): delta (mathbf{theta}cdot mathbf{e}_1)

This equation can be numerically computed by averaging apparent directional kurtosis samples for directions perpendicular to e1.

Otherwise, RK can be calculated from its analytical solution [2]:

\[K_{\bot} = G_1(\lambda_1,\lambda_2,\lambda_3)\hat{W}_{2222} + G_1(\lambda_1,\lambda_3,\lambda_2)\hat{W}_{3333} + G_2(\lambda_1,\lambda_2,\lambda_3)\hat{W}_{2233}\]

where:

and

\[G_2(\lambda_1,\lambda_2,\lambda_3)= \frac{(\lambda_1+\lambda_2+\lambda_3)^2}{(\lambda_2-\lambda_3)^2} \left ( \frac{\lambda_2+\lambda_3}{\sqrt{\lambda_2\lambda_3}}-2\right )\]

References

[1] (1,2,3)

Jensen, J.H., Helpern, J.A., 2010. MRI quantification of non-Gaussian water diffusion by kurtosis analysis. NMR in Biomedicine 23(7): 698-710

[2] (1,2,3,4)

Tabesh, A., Jensen, J.H., Ardekani, B.A., Helpern, J.A., 2011. Estimation of tensors and tensor-derived measures in diffusional kurtosis imaging. Magn Reson Med. 65(3), 823-836

[3]

restore_fit_tensor

dipy.reconst.dki.restore_fit_tensor(design_matrix, data, sigma=None, jac=True, return_S0_hat=False, fail_is_nan=False)

Use the RESTORE algorithm [1] to calculate a robust tensor fit

Parameters:

design_matrixarray of shape (g, 7): Design matrix holding the covariants used to solve for the regression coefficients.
dataarray of shape ([X, Y, Z, n_directions], g): Data or response variables holding the data. Note that the last dimension should contain the data. It makes no copies of data.
sigmafloat, array of shape [n_directions], array of shape [X, Y, Z]: An estimate of the variance. [1] recommend to use 1.5267 * std(background_noise), where background_noise is estimated from some part of the image known to contain no signal (only noise). Array with ndim > 1 corresponds to spatially varying sigma, so if providing spatially-flattened data and spatially-varying sigma, provide array with shape [num_vox, 1].
jacbool, optional: Whether to use the Jacobian of the tensor to speed the non-linear optimization procedure used to fit the tensor parameters (see also nlls_fit_tensor()). Default: True
return_S0_hatbool: Boolean to return (True) or not (False) the S0 values for the fit.
fail_is_nanbool: Boolean to set failed NL fitting to NaN (True) or LS (False, default).

Returns:

restore_paramsan estimate of the tensor parameters in each voxel.

References

[1] (1,2,3)

Chang, L-C, Jones, DK and Pierpaoli, C (2005). RESTORE: robust

estimation of tensors by outlier rejection. MRM, 53: 1088-95.

sphere2cart

dipy.reconst.dki.sphere2cart(r, theta, phi)

Spherical to Cartesian coordinates

This is the standard physics convention where theta is the inclination (polar) angle, and phi is the azimuth angle.

Imagine a sphere with center (0,0,0). Orient it with the z axis running south-north, the y axis running west-east and the x axis from posterior to anterior. theta (the inclination angle) is the angle to rotate from the z-axis (the zenith) around the y-axis, towards the x axis. Thus the rotation is counter-clockwise from the point of view of positive y. phi (azimuth) gives the angle of rotation around the z-axis towards the y axis. The rotation is counter-clockwise from the point of view of positive z.

Equivalently, given a point P on the sphere, with coordinates x, y, z, theta is the angle between P and the z-axis, and phi is the angle between the projection of P onto the XY plane, and the X axis.

Geographical nomenclature designates theta as ‘co-latitude’, and phi as ‘longitude’

Parameters:

rarray_like: radius
thetaarray_like: inclination or polar angle
phiarray_like: azimuth angle

Returns:

xarray: x coordinate(s) in Cartesion space
yarray: y coordinate(s) in Cartesian space
zarray: z coordinate

Notes

See these pages:

for excellent discussion of the many different conventions possible. Here we use the physics conventions, used in the wikipedia page.

Derivations of the formulae are simple. Consider a vector x, y, z of length r (norm of x, y, z). The inclination angle (theta) can be found from: cos(theta) == z / r -> z == r * cos(theta). This gives the hypotenuse of the projection onto the XY plane, which we will call Q. Q == r*sin(theta). Now x / Q == cos(phi) -> x == r * sin(theta) * cos(phi) and so on.

We have deliberately named this function sphere2cart rather than sph2cart to distinguish it from the Matlab function of that name, because the Matlab function uses an unusual convention for the angles that we did not want to replicate. The Matlab function is trivial to implement with the formulae given in the Matlab help.

split_dki_param

dipy.reconst.dki.split_dki_param(dki_params)

Extract the diffusion tensor eigenvalues, the diffusion tensor eigenvector matrix, and the 15 independent elements of the kurtosis tensor from the model parameters estimated from the DKI model

Parameters:

dki_paramsndarray (x, y, z, 27) or (n, 27)

All parameters estimated from the diffusion kurtosis model. Parameters are ordered as follows:

Three diffusion tensor’s eigenvalues

Three lines of the eigenvector matrix each containing the first, second and third coordinates of the eigenvector

Fifteen elements of the kurtosis tensor

Returns:

eigvalsarray (x, y, z, 3) or (n, 3): Eigenvalues from eigen decomposition of the tensor.
eigvecsarray (x, y, z, 3, 3) or (n, 3, 3): Associated eigenvectors from eigen decomposition of the tensor. Eigenvectors are columnar (e.g. eigvecs[:,j] is associated with eigvals[j])
ktarray (x, y, z, 15) or (n, 15): Fifteen elements of the kurtosis tensor

vec_val_vect

dipy.reconst.dki.vec_val_vect()

Vectorize vecs.diag(vals).`vecs`.T for last 2 dimensions of vecs

Parameters:

vecsshape (…, M, N) array: containing tensor in last two dimensions; M, N usually equal to (3, 3)
valsshape (…, N) array: diagonal values carried in last dimension, ... shape above must match that for vecs

Returns:

resshape (…, M, M) array: For all the dimensions ellided by ..., loops to get (M, N) vec matrix, and (N,) vals vector, and calculates vec.dot(np.diag(val).dot(vec.T).

Raises:

ValueErrornon-matching ... dimensions of vecs, vals
ValueErrornon-matching N dimensions of vecs, vals

Examples

Make a 3D array where the first dimension is only 1

>>> vecs = np.arange(9).reshape((1, 3, 3))
>>> vals = np.arange(3).reshape((1, 3))
>>> vec_val_vect(vecs, vals)
array([[[   9.,   24.,   39.],
        [  24.,   66.,  108.],
        [  39.,  108.,  177.]]])

That’s the same as the 2D case (apart from the float casting):

>>> vecs = np.arange(9).reshape((3, 3))
>>> vals = np.arange(3)
>>> np.dot(vecs, np.dot(np.diag(vals), vecs.T))
array([[  9,  24,  39],
       [ 24,  66, 108],
       [ 39, 108, 177]])

wls_fit_dki

dipy.reconst.dki.wls_fit_dki(design_matrix, data)

Computes the diffusion and kurtosis tensors using a weighted linear least squares (WLS) approach [1].

Parameters:

design_matrixarray (g, 22): Design matrix holding the covariants used to solve for the regression coefficients.
dataarray (N, g): Data or response variables holding the data. Note that the last dimension should contain the data. It makes no copies of data.

Returns:

dki_paramsarray (N, 27)

All parameters estimated from the diffusion kurtosis model for all N voxels. Parameters are ordered as follows:

Three diffusion tensor’s eigenvalues

Three lines of the eigenvector matrix each containing the first second and third coordinates of the eigenvector

Fifteen elements of the kurtosis tensor

References

[1] Veraart, J., Sijbers, J., Sunaert, S., Leemans, A., Jeurissen, B.,: 2013. Weighted linear least squares estimation of diffusion MRI parameters: Strengths, limitations, and pitfalls. Magn Reson Med 81, 335-346.

`DiffusionKurtosisFit`

class dipy.reconst.dki_micro.DiffusionKurtosisFit(model, model_params)

Bases: TensorFit

Class for fitting the Diffusion Kurtosis Model

Attributes:

S0_hat
directions: For tracking - return the primary direction in each voxel
evals: Returns the eigenvalues of the tensor as an array
evecs: Returns the eigenvectors of the tensor as an array, columnwise
kfa: Returns the kurtosis tensor (KFA) [1].
kt: Returns the 15 independent elements of the kurtosis tensor as an array
quadratic_form: Calculates the 3x3 diffusion tensor for each voxel
shape

Methods

`ad`()	Axial diffusivity (AD) calculated from cached eigenvalues.
`adc`(sphere)	Calculate the apparent diffusion coefficient (ADC) in each direction on
`ak`([min_kurtosis, max_kurtosis, analytical])	Axial Kurtosis (AK) of a diffusion kurtosis tensor [1].
`akc`(sphere)	Calculates the apparent kurtosis coefficient (AKC) in each direction on the sphere for each voxel in the data
`color_fa`()	Color fractional anisotropy of diffusion tensor
`fa`()	Fractional anisotropy (FA) calculated from cached eigenvalues.
`ga`()	Geodesic anisotropy (GA) calculated from cached eigenvalues.
`kmax`([sphere, gtol, mask])	Computes the maximum value of a single voxel kurtosis tensor
`linearity`()	Returns:
`md`()	Mean diffusivity (MD) calculated from cached eigenvalues.
`mk`([min_kurtosis, max_kurtosis, analytical])	Computes mean Kurtosis (MK) from the kurtosis tensor.
`mkt`([min_kurtosis, max_kurtosis])	Computes mean of the kurtosis tensor (MKT) [1].
`mode`()	Tensor mode calculated from cached eigenvalues.
`odf`(sphere)	The diffusion orientation distribution function (dODF).
`planarity`()	Returns:
`predict`(gtab[, S0])	Given a DKI model fit, predict the signal on the vertices of a gradient table
`rd`()	Radial diffusivity (RD) calculated from cached eigenvalues.
`rk`([min_kurtosis, max_kurtosis, analytical])	Radial Kurtosis (RK) of a diffusion kurtosis tensor [1].
`sphericity`()	Returns:
`trace`()	Trace of the tensor calculated from cached eigenvalues.

lower_triangular

__init__(model, model_params)

Initialize a DiffusionKurtosisFit class instance.

Since DKI is an extension of DTI, class instance is defined as subclass of the TensorFit from dti.py

Parameters:

modelDiffusionKurtosisModel Class instance

Class instance containing the Diffusion Kurtosis Model for the fit

model_paramsndarray (x, y, z, 27) or (n, 27)

All parameters estimated from the diffusion kurtosis model. Parameters are ordered as follows:

Three diffusion tensor’s eigenvalues

Three lines of the eigenvector matrix each containing the first, second and third coordinates of the eigenvector

Fifteen elements of the kurtosis tensor

ak(min_kurtosis=-0.42857142857142855, max_kurtosis=10, analytical=True)

Axial Kurtosis (AK) of a diffusion kurtosis tensor [1].

Parameters:

min_kurtosisfloat (optional): To keep kurtosis values within a plausible biophysical range, axial kurtosis values that are smaller than min_kurtosis are replaced with -3./7 (theoretical kurtosis limit for regions that consist of water confined to spherical pores [2])
max_kurtosisfloat (optional): To keep kurtosis values within a plausible biophysical range, axial kurtosis values that are larger than max_kurtosis are replaced with max_kurtosis. Default = 10
analyticalbool (optional): If True, AK is calculated from rotated diffusion kurtosis tensor, otherwise it will be computed from the apparent diffusion kurtosis values along the principal axis of the diffusion tensor (see notes). Default is set to True.

Returns:

akarray: Calculated AK.

Notes

AK is defined as the directional kurtosis parallel to the fiber’s main direction e1 [1], [2]. You can compute AK using to approaches:

AK is calculated from rotated diffusion kurtosis tensor [2], i.e.:

\[AK = \hat{W}_{1111} \frac{(\lambda_{1}+\lambda_{2}+\lambda_{3})^2}{(9 \lambda_{1}^2)}\]

AK can be sampled from the principal axis of the diffusion tensor:

\[AK = K(\mathbf{\mathbf{e}_1)\]

References

[1] (1,2,3)

Jensen, J.H., Helpern, J.A., 2010. MRI quantification of non-Gaussian water diffusion by kurtosis analysis. NMR in Biomedicine 23(7): 698-710

[2] (1,2,3)

Tabesh, A., Jensen, J.H., Ardekani, B.A., Helpern, J.A., 2011. Estimation of tensors and tensor-derived measures in diffusional kurtosis imaging. Magn Reson Med. 65(3), 823-836

[3]

akc(sphere)

Calculates the apparent kurtosis coefficient (AKC) in each direction on the sphere for each voxel in the data

Parameters:

sphereSphere class instance

Returns:

akcndarray: The estimates of the apparent kurtosis coefficient in every direction on the input sphere

Notes

For each sphere direction with coordinates \((n_{1}, n_{2}, n_{3})\), the calculation of AKC is done using formula:

\[AKC(n)=\frac{MD^{2}}{ADC(n)^{2}}\sum_{i=1}^{3}\sum_{j=1}^{3} \sum_{k=1}^{3}\sum_{l=1}^{3}n_{i}n_{j}n_{k}n_{l}W_{ijkl}\]

where \(W_{ijkl}\) are the elements of the kurtosis tensor, MD the mean diffusivity and ADC the apparent diffusion coefficent computed as:

\[ADC(n)=\sum_{i=1}^{3}\sum_{j=1}^{3}n_{i}n_{j}D_{ij}\]

where \(D_{ij}\) are the elements of the diffusion tensor.

property kfa

Returns the kurtosis tensor (KFA) [1].

Notes

The KFA is defined as [1]:

\[KFA \equiv \frac{||\mathbf{W} - MKT \mathbf{I}^{(4)}||_F}{||\mathbf{W}||_F}\]

where \(W\) is the kurtosis tensor, MKT the kurtosis tensor mean, \(I^(4)\) is the fully symmetric rank 2 isotropic tensor and \(||...||_F\) is the tensor’s Frobenius norm [1].

References

[1] (1,2,3)

Glenn, G. R., Helpern, J. A., Tabesh, A., and Jensen, J. H. (2015). Quantitative assessment of diffusional kurtosis anisotropy. NMR in Biomedicine 28, 448–459. doi:10.1002/nbm.3271

kmax(sphere='repulsion100', gtol=1e-05, mask=None)

Computes the maximum value of a single voxel kurtosis tensor

Parameters:

sphereSphere class instance, optional: The sphere providing sample directions for the initial search of the maximum value of kurtosis.
gtolfloat, optional: This input is to refine kurtosis maximum under the precision of the directions sampled on the sphere class instance. The gradient of the convergence procedure must be less than gtol before successful termination. If gtol is None, fiber direction is directly taken from the initial sampled directions of the given sphere object

Returns:

max_valuefloat: kurtosis tensor maximum value

property kt: Returns the 15 independent elements of the kurtosis tensor as an array

mk(min_kurtosis=-0.42857142857142855, max_kurtosis=10, analytical=True)

Computes mean Kurtosis (MK) from the kurtosis tensor.

Parameters:

min_kurtosisfloat (optional): To keep kurtosis values within a plausible biophysical range, mean kurtosis values that are smaller than min_kurtosis are replaced with min_kurtosis. Default = -3./7 (theoretical kurtosis limit for regions that consist of water confined to spherical pores [4])
max_kurtosisfloat (optional): To keep kurtosis values within a plausible biophysical range, mean kurtosis values that are larger than max_kurtosis are replaced with max_kurtosis. Default = 10
analyticalbool (optional): If True, MK is calculated using its analytical solution, otherwise an exact numerical estimator is used (see Notes). Default is set to True.

Returns:

mkarray: Calculated MK.

Notes

The MK is defined as the average of directional kurtosis coefficients across all spatial directions, which can be formulated by the following surface integral[Rb657f27beb9e-1]_:

\[MK \equiv \frac{1}{4\pi} \int d\Omega_\mathbf{n} K(\mathbf{n})\]

This integral can be numerically solved by averaging directional kurtosis values sampled for directions of a spherical t-design [2].

Alternatively, MK can be solved from the analytical solution derived by Tabesh et al. [3]. This solution is given by:

where \(\hat{W}_{ijkl}\) are the components of the \(W\) tensor in the coordinates system defined by the eigenvectors of the diffusion tensor \(\mathbf{D}\) and

where \(R_f\) and \(R_d\) are the Carlson’s elliptic integrals.

References

[1]

Jensen, J.H., Helpern, J.A., 2010. MRI quantification of non-Gaussian water diffusion by kurtosis analysis. NMR in Biomedicine 23(7): 698-710

[2]

Hardin, R.H., Sloane, N.J.A., 1996. McLaren’s Improved Snub Cube and Other New Spherical Designs in Three Dimensions. Discrete and Computational Geometry 15, 429-441.

[3]

Tabesh, A., Jensen, J.H., Ardekani, B.A., Helpern, J.A., 2011. Estimation of tensors and tensor-derived measures in diffusional kurtosis imaging. Magn Reson Med. 65(3), 823-836

[4]

mkt(min_kurtosis=-0.42857142857142855, max_kurtosis=10)

Computes mean of the kurtosis tensor (MKT) [1].

Parameters:

min_kurtosisfloat (optional): To keep kurtosis values within a plausible biophysical range, mean kurtosis values that are smaller than min_kurtosis are replaced with min_kurtosis. Default = -3./7 (theoretical kurtosis limit for regions that consist of water confined to spherical pores [2])
max_kurtosisfloat (optional): To keep kurtosis values within a plausible biophysical range, mean kurtosis values that are larger than max_kurtosis are replaced with max_kurtosis. Default = 10

Returns:

mktarray: Calculated mean kurtosis tensor.

Notes

The MKT is defined as [1]:

\[MKT \equiv \frac{1}{4\pi} \int d \Omega_{\mathnbf{n}} n_i n_j n_k n_l W_{ijkl}\]

which can be directly computed from the trace of the kurtosis tensor:

\[\]

MKT = frac{1}{5} Tr(mathbf{W}) = frac{1}{5} (W_{1111} + W_{2222} + W_{3333} + 2W_{1122} + 2W_{1133} + 2W_{2233})

References

[1] (1,2,3)

[2]

predict(gtab, S0=1.0)

Given a DKI model fit, predict the signal on the vertices of a gradient table

Parameters:

gtaba GradientTable class instance: The gradient table for this prediction
S0float or ndarray (optional): The non diffusion-weighted signal in every voxel, or across all voxels. Default: 1

Notes

The predicted signal is given by:

\[S(n,b)=S_{0}e^{-bD(n)+\frac{1}{6}b^{2}D(n)^{2}K(n)}\]

\(\mathbf{D(n)}\) and \(\mathbf{K(n)}\) can be computed from the DT and KT using the following equations:

\[D(n)=\sum_{i=1}^{3}\sum_{j=1}^{3}n_{i}n_{j}D_{ij}\]

and

\[K(n)=\frac{MD^{2}}{D(n)^{2}}\sum_{i=1}^{3}\sum_{j=1}^{3} \sum_{k=1}^{3}\sum_{l=1}^{3}n_{i}n_{j}n_{k}n_{l}W_{ijkl}\]

where \(D_{ij}\) and \(W_{ijkl}\) are the elements of the second-order DT and the fourth-order KT tensors, respectively, and \(MD\) is the mean diffusivity.

rk(min_kurtosis=-0.42857142857142855, max_kurtosis=10, analytical=True)

Radial Kurtosis (RK) of a diffusion kurtosis tensor [1].

Parameters:

min_kurtosisfloat (optional): To keep kurtosis values within a plausible biophysical range, radial kurtosis values that are smaller than min_kurtosis are replaced with min_kurtosis. Default = -3./7 (theoretical kurtosis limit for regions that consist of water confined to spherical pores [3])
max_kurtosisfloat (optional): To keep kurtosis values within a plausible biophysical range, radial kurtosis values that are larger than max_kurtosis are replaced with max_kurtosis. Default = 10
analyticalbool (optional): If True, RK is calculated using its analytical solution, otherwise an exact numerical estimator is used (see Notes). Default is set to True

Returns:

rkarray: Calculated RK.

Notes

RK is defined as the average of the directional kurtosis perpendicular to the fiber’s main direction e1 [1], [2]:

\[\]

RK equiv frac{1}{2pi} int dOmega _mathbf{theta}: K(mathbf{theta}) delta (mathbf{theta}cdot mathbf{e}_1)

This equation can be numerically computed by averaging apparent directional kurtosis samples for directions perpendicular to e1.

Otherwise, RK can be calculated from its analytical solution [2]:

\[K_{\bot} = G_1(\lambda_1,\lambda_2,\lambda_3)\hat{W}_{2222} + G_1(\lambda_1,\lambda_3,\lambda_2)\hat{W}_{3333} + G_2(\lambda_1,\lambda_2,\lambda_3)\hat{W}_{2233}\]

where:

and

\[ G_2(\lambda_1,\lambda_2,\lambda_3)= \frac{(\lambda_1+\lambda_2+\lambda_3)^2}{(\lambda_2-\lambda_3)^2} \left ( \frac{\lambda_2+\lambda_3}{\sqrt{\lambda_2\lambda_3}}- 2\right )\]

References

[1] (1,2,3)

Jensen, J.H., Helpern, J.A., 2010. MRI quantification of non-Gaussian water diffusion by kurtosis analysis. NMR in Biomedicine 23(7): 698-710

[2] (1,2)

Tabesh, A., Jensen, J.H., Ardekani, B.A., Helpern, J.A., 2011. Estimation of tensors and tensor-derived measures in diffusional kurtosis imaging. Magn Reson Med. 65(3), 823-836

[3]

`DiffusionKurtosisModel`

class dipy.reconst.dki_micro.DiffusionKurtosisModel(gtab, fit_method='WLS', *args, **kwargs)

Bases: ReconstModel

Class for the Diffusion Kurtosis Model

Methods

`fit`(data[, mask])	Fit method of the DKI model class
`predict`(dki_params[, S0])	Predict a signal for this DKI model class instance given parameters.

__init__(gtab, fit_method='WLS', *args, **kwargs)

Diffusion Kurtosis Tensor Model [1]

Parameters:

gtabGradientTable class instance

fit_methodstr or callable

str can be one of the following: ‘OLS’ or ‘ULLS’ for ordinary least squares

dki.ols_fit_dki

‘WLS’ or ‘UWLLS’ for weighted ordinary least squares: dki.wls_fit_dki
callable has to have the signature:: fit_method(design_matrix, data, *args, **kwargs)

args, kwargsarguments and key-word arguments passed to the

fit_method. See dki.ols_fit_dki, dki.wls_fit_dki for details

References

[1]

Tabesh, A., Jensen, J.H., Ardekani, B.A., Helpern, J.A., 2011.

Estimation of tensors and tensor-derived measures in diffusional kurtosis imaging. Magn Reson Med. 65(3), 823-836

fit(data, mask=None)

Fit method of the DKI model class

Parameters:

dataarray: The measured signal from one voxel.
maskarray: A boolean array used to mark the coordinates in the data that should be analyzed that has the shape data.shape[-1]

predict(dki_params, S0=1.0)

Predict a signal for this DKI model class instance given parameters.

Parameters:

dki_paramsndarray (x, y, z, 27) or (n, 27)

All parameters estimated from the diffusion kurtosis model. Parameters are ordered as follows:

Three diffusion tensor’s eigenvalues

2. Three lines of the eigenvector matrix each containing the first, second and third coordinates of the eigenvector 3. Fifteen elements of the kurtosis tensor

S0float or ndarray (optional)

The non diffusion-weighted signal in every voxel, or across all voxels. Default: 1

`KurtosisMicrostructuralFit`

class dipy.reconst.dki_micro.KurtosisMicrostructuralFit(model, model_params)

Bases: DiffusionKurtosisFit

Class for fitting the Diffusion Kurtosis Microstructural Model

Attributes:

S0_hat
awf: Returns the volume fraction of the restricted diffusion compartment also known as axonal water fraction.
axonal_diffusivity: Returns the axonal diffusivity defined as the restricted diffusion tensor trace [1].
directions: For tracking - return the primary direction in each voxel
evals: Returns the eigenvalues of the tensor as an array
evecs: Returns the eigenvectors of the tensor as an array, columnwise
hindered_ad: Returns the axial diffusivity of the hindered compartment.

reconst

Module: reconst.base

Module: reconst.benchmarks

Module: reconst.benchmarks.bench_bounding_box

Module: reconst.benchmarks.bench_csd

Module: reconst.benchmarks.bench_peaks

Module: reconst.benchmarks.bench_squash

Module: reconst.benchmarks.bench_vec_val_sum

Module: reconst.cache

Module: reconst.cross_validation

Module: reconst.csdeconv

Module: reconst.dki

Module: reconst.dki_micro

Module: reconst.dsi

Module: reconst.dti

Module: reconst.eudx_direction_getter

Module: reconst.forecast

Module: reconst.fwdti

Module: reconst.gqi

Module: reconst.ivim

Module: reconst.mapmri

Module: reconst.mcsd

Module: reconst.msdki

Module: reconst.multi_voxel

Module: reconst.odf

Module: reconst.qtdmri

Module: reconst.qti

Module: reconst.quick_squash

Module: reconst.recspeed

Module: reconst.rumba

Module: reconst.sfm

Module: reconst.shm

References

Module: reconst.shore

Module: reconst.utils

Module: reconst.vec_val_sum

bench

test

bench_bounding_box

measure

bench_csdeconv

load_nifti_data

num_grad

read_stanford_labels

bench_local_maxima

measure

bench_quick_squash

measure

ndindex

old_squash

reduce

bench_vec_val_vect

measure

auto_attr

coeff_of_determination

kfold_xval

auto_response

auto_response_ssst

cart2sphere

csdeconv

deprecate_with_version

deprecated_params

estimate_response

fa_trace_to_lambdas

forward_sdeconv_mat

forward_sdt_deconv_mat

fractional_anisotropy

get_sphere

lazy_index

lpn

mask_for_response_ssst

multi_voxel_fit

ndindex

odf_deconv

odf_sh_to_sharp

peaks_from_model

quad

real_sh_descoteaux

real_sh_descoteaux_from_index

recursive_response

`reconst`

Module: `reconst.base`

Module: `reconst.benchmarks`

Module: `reconst.benchmarks.bench_bounding_box`

Module: `reconst.benchmarks.bench_csd`

Module: `reconst.benchmarks.bench_peaks`

Module: `reconst.benchmarks.bench_squash`

Module: `reconst.benchmarks.bench_vec_val_sum`

Module: `reconst.cache`

Module: `reconst.cross_validation`

Module: `reconst.csdeconv`

Module: `reconst.dki`

Module: `reconst.dki_micro`

Module: `reconst.dsi`

Module: `reconst.dti`

Module: `reconst.eudx_direction_getter`

Module: `reconst.forecast`

Module: `reconst.fwdti`

Module: `reconst.gqi`

Module: `reconst.ivim`

Module: `reconst.mapmri`

Module: `reconst.mcsd`

Module: `reconst.msdki`

Module: `reconst.multi_voxel`

Module: `reconst.odf`

Module: `reconst.qtdmri`

Module: `reconst.qti`

Module: `reconst.quick_squash`

Module: `reconst.recspeed`

Module: `reconst.rumba`

Module: `reconst.sfm`

Module: `reconst.shm`

Module: `reconst.shore`

Module: `reconst.utils`

Module: `reconst.vec_val_sum`