# 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 whenver 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.bench_bounding_box

 Compute the bounding box of nonzero intensity voxels in the volume. 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 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

 Local maxima of a function evaluated on a discrete set of points. measure(code_str[, times, label]) Return elapsed time for executing code in the namespace of the caller. unique_edges(faces[, return_mapping]) Extract all unique edges from given triangular faces.

## 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

 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 Try and make a standard array from an object array 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

 measure(code_str[, times, label]) Return elapsed time for executing code in the namespace of the caller. randn(d0, d1, …, dn) Return a sample (or samples) from the “standard normal” distribution. Vectorize vecs.diag(vals).vecs.T for last 2 dimensions of vecs

## 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, relative kfold_xval(model, data, folds, \*model_args, …) Perform k-fold cross-validation to generate out-of-sample predictions for each measurement.

## 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 response function using FA. 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] estimate_response(gtab, evals, S0) Estimate single fiber response function fa_inferior(FA, fa_thr) Check that the FA is lower than the FA threshold fa_superior(FA, fa_thr) Check that the FA is greater than the FA threshold 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]) 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. 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_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 recursive_response(gtab, data[, mask, …]) Recursive calibration of response function using peak threshold response_from_mask(gtab, data, mask) Estimate the response function from a given mask. sh_to_rh(r_sh, m, n) Spherical harmonics (SH) to rotational harmonics (RH) single_tensor(gtab[, S0, evals, evecs, snr]) Simulated Q-space signal with a single tensor. sph_harm_ind_list(sh_order) Returns the degree (n) and order (m) 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. 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 of a function evaluated on a discrete set of points. lower_triangular(tensor[, b0]) Returns the six lower triangular values of the tensor 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 [Chang2005] 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 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. 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 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. 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 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 apparent_diffusion_coef(q_form, sphere) Calculate the apparent diffusion coefficient (ADC) in each direction of a 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]) Fractional anisotropy (FA) of a diffusion tensor. 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 [Rd0568a744381-1]. 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 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 [Chang2005] 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. 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.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 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_sph_harm(m, n, theta, phi) Compute real spherical harmonics. 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 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. 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 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. 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 gfa(samples) The general fractional anisotropy of a function evaluated on the unit sphere Local maxima of a function evaluated on a discrete set of points. 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 Remove vertices that are less than theta degrees from any other squared_radial_component(x[, tol]) Part of the GQI2 integral triple_odf_maxima(vertices, odf, width) 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 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) [Rb71d87403e8f-1] of the diffusion signal. Optimizer(fun, x0[, args, method, jac, …]) Attributes ReconstFit(model, data) Abstract class which holds the fit result of ReconstModel ReconstModel(gtab) Abstract class for signal reconstruction models 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 [Rb690cd738504-1] 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. map_laplace_s(n, m) R(m,n) static matrix for Laplacian regularization [R932dd40ca52e-1] eq. map_laplace_t(n, m) L(m, n) static matrix for Laplacian regularization [Reb78d789d6c4-1] eq. map_laplace_u(n, m) S(n, m) static matrix for Laplacian regularization [Rb93dd9dab8c9-1] eq. mapmri_STU_reg_matrices(radial_order) Generate the static portions of the Laplacian regularization matrix according to [R1d585103467a-1] eq. mapmri_index_matrix(radial_order) Calculates the indices for the MAPMRI [1] basis in x, y and z. Computes mu dependent part of M. Computes mu independent part of K. Computed the mu dependent part of the signal design matrix. 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. Computes the Laplacian regularization matrix for MAP-MRI’s isotropic implementation [R156f27ca005f-1] eq. Computes the Laplacian regularization matrix for MAP-MRI’s isotropic implementation [Rdcc29394f577-1] 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. Radial part of the isotropic 1D-SHORE signal basis [1] eq. mapmri_laplacian_reg_matrix(ind_mat, mu, …) Put the Laplacian regularization matrix together [Rc66aaccd07c1-1] 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_sph_harm(m, n, theta, phi) Compute real spherical harmonics. sfactorial(n[, exact]) The factorial of a number or array of numbers. sph_harm_ind_list(sh_order) Returns the degree (n) and order (m) 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 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 solve_qp(P, Q, G, H) Helper function to set up and solve the Quadratic Program (QP) in CVXPY.

## 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. 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. ndindex(shape) An N-dimensional iterator object to index arrays. round_bvals(bvals[, bmag]) “This function rounds the b-values unique_bvals(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 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 $$\tau$$. 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[, dtype]) Constructs design matrix for DTI weighted least squares or least squares fitting. elastic_crossvalidation()) 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. A general function for creating diffusion MR gradients. l1_crossvalidation()) 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]. 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, …) 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. Normalization factor for Spherical MAP-MRI basis. 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. 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_sph_harm(m, n, theta, phi) Compute real spherical harmonics. temporal_basis(o, ut, tau) Temporal basis dependent on temporal scaling factor ut 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.sfm

The Sparse Fascicle Model.

This is an implementation of the sparse fascicle model described in [R204bb22f26e5-Rokem2015]. The multi b-value version of this model is described in [R204bb22f26e5-Rokem2014].

R204bb22f26e5-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

R204bb22f26e5-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. 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. 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. nanmean(a[, axis, dtype, out, keepdims]) 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

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

Angle Consideration.

Descoteaux, M., et al. 2007. Regularized, fast, and robust analytical

Q-ball imaging.

Tristan-Vega, A., et al. 2010. A new methodology for estimation of fiber

populations in white matter of the brain with Funk-Radon transform.

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. LooseVersion([vstring]) Version numbering for anarchists and software realists. 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, …]) Calculates 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) Calculate the maximal harmonic order, given that you know the cart2sphere(x, y, z) Return angles for Cartesian 3D coordinates x, y, and z concatenate([axis, out]) Join a sequence of arrays along an existing axis. diag(v[, k]) Extract a diagonal or construct a diagonal array. diff(a[, n, axis, prepend, append]) Calculate the n-th discrete difference along the given axis. dot(a, b[, out]) Dot product of two arrays. empty(shape[, dtype, order]) Return a new array of given shape and type, without initializing entries. eye(N[, M, k, dtype, order]) Return a 2-D array with ones on the diagonal and zeros elsewhere. forward_sdeconv_mat(r_rh, n) Build forward spherical deconvolution matrix gen_dirac(m, n, theta, phi) Generate Dirac delta function orientated in (theta, phi) on the sphere Returns the hat matrix for the design matrix B lazy_index(index) Produces a lazy index Returns a matrix for computing leveraged, centered residuals from data lpn(n, z) Legendre function of the first kind. normalize_data(data, where_b0[, min_signal, out]) Normalizes the data with respect to the mean b0 order_from_ncoef(ncoef) Given a number n of coefficients, calculate back the sh_order pinv(a[, rcond, hermitian]) Compute the (Moore-Penrose) pseudo-inverse of a matrix. randint(low[, high, size, dtype]) Return random integers from low (inclusive) to high (exclusive). 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) Compute real spherical harmonics as in Tournier 2007 [2], where the real harmonic $$Y^m_n$$ is defined to be. 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) Returns the degree (n) and order (m) of all the symmetric spherical harmonics of degree less then or equal to sh_order. spherical_harmonics(m, n, theta, phi) Compute spherical harmonics svd(a[, full_matrices, compute_uv, hermitian]) Singular Value Decomposition. unique(ar[, return_index, return_inverse, …]) Find the unique elements of an array.

## 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) [Rd47fc6332129-1] of the diffusion signal. 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_sph_harm(m, n, theta, phi) Compute real spherical harmonics. 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

 Construct B design matrix for DKI.

### 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


OK

>>> 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__(self, model, data)

Initialize self. See help(type(self)) for accurate signature.

### ReconstModel

class dipy.reconst.base.ReconstModel(gtab)

Bases: object

Abstract class for signal reconstruction models

Methods

 fit
__init__(self, gtab)

Initialization of the abstract class for signal reconstruction models

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

### bench_bounding_box

dipy.reconst.benchmarks.bench_bounding_box.bench_bounding_box()

### bounding_box

dipy.reconst.benchmarks.bench_bounding_box.bounding_box(vol)

Compute the bounding box of nonzero intensity voxels in the volume.

Parameters
volndarray

Volume to compute bounding box on.

Returns
npminslist

Array containg minimum index of each dimension

npmaxslist

Array containg maximum index of each dimension

### 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)

Methods

 cache_clear(self) Clear the cache. cache_get(self, tag, key[, default]) Retrieve a value from the cache. cache_set(self, tag, key, value) Store a value in the cache. fit(self, data[, mask]) Fit method for every voxel in data predict(self, sh_coeff[, gtab, S0]) Compute a signal prediction given spherical harmonic coefficients for the provided GradientTable class instance. sampling_matrix(self, sphere) The matrix needed to sample ODFs from coefficients of the model.
__init__(self, 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
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(self, data, mask=None)

Fit method for every voxel in data

predict(self, 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.

The gradients for which the signal will be predicted. Use 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)

Bases: object

Parameters

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.

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

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.

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.

Methods

__init__(self, gradients, big_delta=None, small_delta=None, b0_threshold=50)

b0s_mask(self)
bvals(self)
bvecs(self)
gradient_strength(self)
property info
qvals(self)
tau(self)

### bench_csdeconv

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

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

load_nifti

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

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()

### local_maxima

dipy.reconst.benchmarks.bench_peaks.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].

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).

### 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.

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


### unique_edges

dipy.reconst.benchmarks.bench_peaks.unique_edges(faces, return_mapping=False)

Extract all unique edges from given triangular faces.

Parameters
faces(N, 3) ndarray

Vertex indices forming triangular faces.

return_mappingbool

If true, a mapping to the edges of each face is returned.

Returns
edges(N, 2) ndarray

Unique edges.

mapping(N, 3)

For each face, [x, y, z], a mapping to it’s edges [a, b, c].

   y
/               /               a/    
/                  /                   /__________          x      c     z


### 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.

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


### 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:

1. arr is an array of scalars of type T. Returns an array like arr.astype(T)

2. arr is an array of arrays. All items in arr have the same shape S. Returns an array with shape arr.shape + S.

3. arr is an array of arrays of different shapes. Returns arr.

4. Items in arr are not ndarrys or scalars. Returns arr.

Parameters
arrarray, dtype=object

The array to be converted.

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')


### quick_squash

dipy.reconst.benchmarks.bench_squash.quick_squash()

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:

1. obj_arr is an array of scalars of type T. Returns an array like obj_arr.astype(T)

2. obj_arr is an array of arrays. All items in obj_arr have the same shape S. Returns an array with shape obj_arr.shape + S

3. obj_arr is an array of arrays of different shapes. Returns obj_arr.

4. Items in obj_arr are not ndarrays or scalars. Returns obj_arr.

Parameters
obj_arrarray, dtype=object

The array to be converted.

mask is nonzero where obj_arr has Nones.

fillnumber, optional

Nones are replaced by fill.

Returns
resultarray

Examples

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


### reduce

dipy.reconst.benchmarks.bench_squash.reduce()

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.

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


### randn

dipy.reconst.benchmarks.bench_vec_val_sum.randn(d0, d1, ..., dn)

Return a sample (or samples) from the “standard normal” distribution.

Note

This is a convenience function for users porting code from Matlab, and wraps numpy.random.standard_normal. That function takes a tuple to specify the size of the output, which is consistent with other NumPy functions like numpy.zeros and numpy.ones.

If positive int_like arguments are provided, randn generates an array of shape (d0, d1, ..., dn), filled with random floats sampled from a univariate “normal” (Gaussian) distribution of mean 0 and variance 1. A single float randomly sampled from the distribution is returned if no argument is provided.

Parameters
d0, d1, …, dnint, optional

The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned.

Returns
Zndarray or float

A (d0, d1, ..., dn)-shaped array of floating-point samples from the standard normal distribution, or a single such float if no parameters were supplied.

standard_normal

Similar, but takes a tuple as its argument.

normal

Also accepts mu and sigma arguments.

Notes

For random samples from $$N(\mu, \sigma^2)$$, use:

sigma * np.random.randn(...) + mu

Examples

>>> np.random.randn()
2.1923875335537315  # random


Two-by-four array of samples from N(3, 6.25):

>>> 3 + 2.5 * np.random.randn(2, 4)
array([[-4.49401501,  4.00950034, -1.81814867,  7.29718677],   # random
[ 0.39924804,  4.68456316,  4.99394529,  4.84057254]])  # random


### vec_val_vect

dipy.reconst.benchmarks.bench_vec_val_sum.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]])


### 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(self) Clear the cache. cache_get(self, tag, key[, default]) Retrieve a value from the cache. cache_set(self, tag, key, value) Store a value in the cache.
__init__(self, /, *args, **kwargs)

Initialize self. See help(type(self)) for accurate signature.

cache_clear(self)

Clear the cache.

cache_get(self, 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(self, 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 to 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(self, sphere) A basis that maps the response coefficients onto a sphere. on_sphere(self, sphere) Evaluates the response function on sphere.
__init__(self, S0, dwi_response, bvalue=None)

Initialize self. See help(type(self)) for accurate signature.

basis(self, sphere)

A basis that maps the response coefficients onto a sphere.

on_sphere(self, 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)

Methods

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

Methods

 cache_clear(self) Clear the cache. cache_get(self, tag, key[, default]) Retrieve a value from the cache. cache_set(self, tag, key, value) Store a value in the cache. fit(self, data[, mask]) Fit method for every voxel in data predict(self, sh_coeff[, gtab, S0]) Compute a signal prediction given spherical harmonic coefficients for the provided GradientTable class instance. sampling_matrix(self, sphere) The matrix needed to sample ODFs from coefficients of the model.
__init__(self, 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
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(self, data, mask=None)

Fit method for every voxel in data

predict(self, 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.

The gradients for which the signal will be predicted. Use 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)

Diffusion data fit to a spherical harmonic model

Attributes
shape
shm_coeff

The spherical harmonic coefficients of the odf

Methods

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

Initialize self. See help(type(self)) for accurate signature.

gfa(self)
odf(self, 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(self, gtab=None, S0=1.0)

Predict the diffusion signal from the model coefficients.

Parameters

The directions and bvalues on which prediction is desired

S0float array

The mean non-diffusion-weighted signal in each voxel. Default: 1.0 in all voxels

property shape
property shm_coeff

The spherical harmonic coefficients of the odf

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

### SphHarmModel

class dipy.reconst.csdeconv.SphHarmModel(gtab)

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

Methods

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

Initialization of the abstract class for signal reconstruction models

Parameters
sampling_matrix(self, 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)

Diffusion Tensor

Methods

 fit(self, data[, mask]) Fit method of the DTI model class predict(self, dti_params[, S0]) Predict a signal for this TensorModel class instance given parameters.
__init__(self, gtab, fit_method='WLS', return_S0_hat=False, *args, **kwargs)

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

Parameters
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(self, data, mask=None)

Fit method of the DTI model class

Parameters
dataarray

The measured signal from one voxel.

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

predict(self, 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=<function fa_superior at 0x128719b90>, return_number_of_voxels=False)

Automatic estimation of response function using FA.

Parameters
datandarray

diffusion data

roi_centertuple, (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].

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.

number of voxelsint (optional)

The number of voxels used for estimating the response function.

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. For example we can use an ROI (20x20x20) at the center of the volume and store the signal values for the voxels with FA values higher than 0.7. Of course, if we haven’t precalculated FA we need to fit a Tensor model to the datasets. Which is what we do in this 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. If requested, the number of voxels used for estimating the response function is also returned, which can be used to judge the fidelity of the response function. As a rule of thumb, at least 300 voxels should be used to estimate a good response function (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

### 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

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.

### estimate_response

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

Estimate single fiber response function

Parameters
evalsndarray
S0float

non diffusion weighted

Returns
Sestimated signal

### fa_inferior

dipy.reconst.csdeconv.fa_inferior(FA, fa_thr)

Check that the FA is lower than the FA threshold

Parameters
FAarray

Fractional Anisotropy

fa_thrint

FA threshold

Returns
True when the FA value is lower than the FA threshold, otherwise False.

### fa_superior

dipy.reconst.csdeconv.fa_superior(FA, fa_thr)

Check that the FA is greater than the FA threshold

Parameters
FAarray

Fractional Anisotropy

fa_thrint

FA threshold

Returns
True when the FA value is greater than the FA threshold, otherwise
False.

### 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 degree of spherical harmonic function associated with each row of the deconvolution matrix. Only even degrees 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)

References

1

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

### fractional_anisotropy

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

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).

References

1

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

### 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, nbr_processes=None)

Fit the model to data and computes peaks and metrics

Parameters
modela model instance

model will be used to fit the 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.

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).

sh_smoothfloat, optional

Lambda-regularization in the SH fit (default 0.0).

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'.

nbr_processes: int

If parallel is True, the number of subprocesses to use (default multiprocessing.cpu_count()).

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

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.

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)

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.

Returns
yfloat

The integral of func from a to b.

abserrfloat

An estimate of the absolute error in the result.

infodictdict

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.

epsrelfloat or int, optional

Relative error tolerance.

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.

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.

dblquad

double integral

tplquad

triple integral

nquad

fixed_quad

quadrature

odeint

ODE integrator

ode

ODE integrator

simps

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. 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.

Examples

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

>>> from scipy import integrate
>>> x2 = lambda x: x**2
(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)
(1.0, 5.842605999138044e-11)

>>> 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)
#(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
(1.0, 1.1102230246251565e-14)
(1.0000000002199108, 1.0189464580163188e-08)
(0.0, 0.0)


### real_sph_harm

dipy.reconst.csdeconv.real_sph_harm(m, n, theta, phi)

Compute real spherical harmonics.

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 broadcasted against each other.

Parameters
mint |m| <= n

The order of the harmonic.

nint >= 0

The degree of the harmonic.

thetafloat [0, 2*pi]

The azimuthal (longitudinal) coordinate.

phifloat [0, pi]

The polar (colatitudinal) coordinate.

Returns
y_mnreal float

The real harmonic $$Y^m_n$$ sampled at theta and phi.

scipy.special.sph_harm

### real_sym_sh_basis

dipy.reconst.csdeconv.real_sym_sh_basis(sh_order, theta, phi)

Samples a real symmetric spherical harmonic basis at point on the sphere

Samples the basis functions up to order sh_order at points on the sphere given by theta and phi. The basis functions are defined here the same way as in Descoteaux et al. 2007 [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 broadcasted against each other.

Parameters
sh_orderint

even int > 0, max spherical harmonic degree

thetafloat [0, 2*pi]

The azimuthal (longitudinal) coordinate.

phifloat [0, pi]

The polar (colatitudinal) coordinate.

Returns
y_mnreal float

The real harmonic $$Y^m_n$$ sampled at theta and phi

marray

The order of the harmonics.

narray

The degree of the harmonics.

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.

### 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=True, nbr_processes=None, sphere=<dipy.core.sphere.HemiSphere object at 0x11fcc2710>)

Recursive calibration of response function using peak threshold

Parameters
datandarray

diffusion data

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

nbr_processes: int

If parallel is True, the number of subprocesses to use (default multiprocessing.cpu_count()).

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.

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

Estimate the response function from a given mask.

Parameters
datandarray

Diffusion data

Mask to use for the estimation of the response function. For example a mask of the white matter voxels with FA values higher than 0.7 (see [1]).

Returns
responsetuple, (2,)

(evals, S0)

ratiofloat

The ratio between smallest versus largest eigenvalue of the response.

Notes

See csdeconv.auto_response() or csdeconv.recursive_response() if you don’t have a computed mask for the response function estimation.

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 m, degree n 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_sph_harm.

mndarray (N,)

The order of the spherical harmonic function associated with each coefficient.

nndarray (N,)

The degree 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

shm.real_sph_harm, shm.real_sym_sh_basis

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)

Simulated Q-space signal with a single tensor.

Parameters

Measurement directions.

S0double,

Strength of signal in the presence of no diffusion gradient (also called the b=0 value).

evals(3,) ndarray

Eigenvalues of the diffusion tensor. By default, values typical for prolate white matter are used.

evecs(3, 3) ndarray

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

Signal to noise ratio, assuming Rician noise. None implies no noise.

Returns
S(N,) ndarray

Simulated signal: S(q, tau) = S_0 e^(-b g^T R D R.T g).

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)

Returns the degree (n) and order (m) 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_sph_harm().

Parameters
sh_orderint

even int > 0, max degree to return

Returns
m_listarray

orders of even spherical harmonics

n_listarray

degrees of even spherical harmonics

### 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)

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

1. 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

ak(self, 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:

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

$AK = \hat{W}_{1111} \frac{(\lambda_{1}+\lambda_{2}+\lambda_{3})^2}{(9 \lambda_{1}^2)}$
1. 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(self, 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(self, 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(self, 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

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

mkt(self, 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

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

predict(self, gtab, S0=1.0)

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

Parameters

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(self, 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

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

### DiffusionKurtosisModel

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

Class for the Diffusion Kurtosis Model

Methods

 fit(self, data[, mask]) Fit method of the DKI model class predict(self, dki_params[, S0]) Predict a signal for this DKI model class instance given parameters.
__init__(self, gtab, fit_method='WLS', *args, **kwargs)

Diffusion Kurtosis Tensor Model [1]

Parameters
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(self, data, mask=None)

Fit method of the DKI model class

Parameters
dataarray

The measured signal from one voxel.

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

predict(self, 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:

1. 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__(self, gtab)

Initialization of the abstract class for signal reconstruction models

Parameters
fit(self, 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(self) Axial diffusivity (AD) calculated from cached eigenvalues. adc(self, sphere) Calculate the apparent diffusion coefficient (ADC) in each direction on color_fa(self) Color fractional anisotropy of diffusion tensor fa(self) Fractional anisotropy (FA) calculated from cached eigenvalues. ga(self) Geodesic anisotropy (GA) calculated from cached eigenvalues. linearity(self) Returns md(self) Mean diffusivity (MD) calculated from cached eigenvalues. mode(self) Tensor mode calculated from cached eigenvalues. odf(self, sphere) The diffusion orientation distribution function (dODF). planarity(self) Returns predict(self, gtab[, S0, step]) Given a model fit, predict the signal on the vertices of a sphere rd(self) Radial diffusivity (RD) calculated from cached eigenvalues. sphericity(self) Returns trace(self) Trace of the tensor calculated from cached eigenvalues.
 lower_triangular
__init__(self, model, model_params, model_S0=None)

Initialize a TensorFit class instance.

property S0_hat
ad(self)

Axial diffusivity (AD) calculated from cached eigenvalues.

Returns

Notes

RD is calculated with the following equation:

$AD = \lambda_1$
adc(self, sphere)

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

Parameters
sphereSphere class instance
Returns

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(self)

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(self)

Fractional anisotropy (FA) calculated from cached eigenvalues.

ga(self)

Geodesic anisotropy (GA) calculated from cached eigenvalues.

linearity(self)
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(self, b0=None)
md(self)

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(self)

Tensor mode calculated from cached eigenvalues.

odf(self, 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 [Aganj2010]. To re-derive it from scratch, follow steps in [Descoteaux2008], Section 7.9 Equation 7.24 but with an $$r^2$$ term in the integral.

References

Aganj2010

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

Descoteaux2008

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

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

Parameters

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(self)

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(self)
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(self)

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:

1. Three diffusion tensor’s eigenvalues

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

3. 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

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

### 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:

1. 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

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:

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

$AK = \hat{W}_{1111} \frac{(\lambda_{1}+\lambda_{2}+\lambda_{3})^2}{(9 \lambda_{1}^2)}$
1. 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(1,2,3,4)

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

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

### 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

thetaarray

inclination (polar) angle

phiarray

azimuth angle

### check_multi_b

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

Parameters
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

### 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

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

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

References

1(1,2)

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

### 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]_)

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

Apparent diffusion variance coefficient (advc) in all g directions of a sphere for a single voxel.

Returns

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

References

1(1,2)

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

### 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])

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

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)

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

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

### 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:

1. 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

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.data import get_fnames
>>> fimg, fbvals, fbvecs = get_fnames('small_101D')
>>> 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:

1. 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

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:

1. Three diffusion tensor’s eingenvalues

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

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

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].

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 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:

1. 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

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:

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

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

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

### 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:

1. 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

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

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

### 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)

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.

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:

1. 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

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>_.

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:

1. 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

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(1,2,3,4)

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

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

### restore_fit_tensor

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

Use the RESTORE algorithm [Chang2005] 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

An estimate of the variance. [Chang2005] 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).

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.

Returns
restore_paramsan estimate of the tensor parameters in each voxel.

References

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

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:

1. 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

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:

1. 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

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)

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

1. 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

ak(self, 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:

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

$AK = \hat{W}_{1111} \frac{(\lambda_{1}+\lambda_{2}+\lambda_{3})^2}{(9 \lambda_{1}^2)}$
1. 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(self, 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(self, 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(self, 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:

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

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

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

mkt(self, 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

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

predict(self, gtab, S0=1.0)

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

Parameters

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(self, 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

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

### DiffusionKurtosisModel

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

Class for the Diffusion Kurtosis Model

Methods

 fit(self, data[, mask]) Fit method of the DKI model class predict(self, dki_params[, S0]) Predict a signal for this DKI model class instance given parameters.
__init__(self, gtab, fit_method='WLS', *args, **kwargs)

Diffusion Kurtosis Tensor Model [1]

Parameters
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(self, data, mask=None)

Fit method of the DKI model class

Parameters
dataarray

The measured signal from one voxel.

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

predict(self, 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:

1. 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)

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.

hindered_evals

Returns the eigenvalues of the hindered diffusion compartment.

hindered_rd

Returns the radial diffusivity of the hindered compartment.

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

restricted_evals

Returns the eigenvalues of the restricted diffusion compartment.

shape
tortuosity

Returns the tortuosity of the hindered diffusion which is defined by ADe / RDe, where ADe and RDe are the axial and radial diffusivities of the hindered compartment 1.

Methods

 ad(self) Axial diffusivity (AD) calculated from cached eigenvalues. adc(self, sphere) Calculate the apparent diffusion coefficient (ADC) in each direction on ak(self[, min_kurtosis, max_kurtosis, …]) Axial Kurtosis (AK) of a diffusion kurtosis tensor [R0b1a747e81c9-1]. akc(self, sphere) Calculates the apparent kurtosis coefficient (AKC) in each direction on the sphere for each voxel in the data color_fa(self) Color fractional anisotropy of diffusion tensor fa(self) Fractional anisotropy (FA) calculated from cached eigenvalues. ga(self) Geodesic anisotropy (GA) calculated from cached eigenvalues. kmax(self[, sphere, gtol, mask]) Computes the maximum value of a single voxel kurtosis tensor linearity(self) Returns md(self) Mean diffusivity (MD) calculated from cached eigenvalues. mk(self[, min_kurtosis, max_kurtosis, …]) Computes mean Kurtosis (MK) from the kurtosis tensor. mkt(self[, min_kurtosis, max_kurtosis]) Computes mean of the kurtosis tensor (MKT) [R8b3dd90f2e0d-1]. mode(self) Tensor mode calculated from cached eigenvalues. odf(self, sphere) The diffusion orientation distribution function (dODF). planarity(self) Returns predict(self, gtab[, S0]) Given a DKI microstructural model fit, predict the signal on the vertices of a gradient table rd(self) Radial diffusivity (RD) calculated from cached eigenvalues. rk(self[, min_kurtosis, max_kurtosis, …]) Radial Kurtosis (RK) of a diffusion kurtosis tensor [Rc4101656d30e-1]. sphericity(self) Returns trace(self) Trace of the tensor calculated from cached eigenvalues.
 lower_triangular
__init__(self, model, model_params)

Initialize a KurtosisMicrostructural Fit class instance.

Parameters
modelDiffusionKurtosisModel Class instance

Class instance containing the Diffusion Kurtosis Model for the fit

model_paramsndarray (x, y, z, 40) or (n, 40)

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

1. 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

4. Six elements of the hindered diffusion tensor

5. Six elements of the restricted diffusion tensor

6. Axonal water fraction

Notes

In the original article of DKI microstructural model [1], the hindered and restricted tensors were definde as the intra-cellular and extra-cellular diffusion compartments respectively.

References

1

Fieremans, E., Jensen, J.H., Helpern, J.A., 2011. White Matter Characterization with Diffusion Kurtosis Imaging. Neuroimage 58(1): 177-188. doi:10.1016/j.neuroimage.2011.06.006

property awf

Returns the volume fraction of the restricted diffusion compartment also known as axonal water fraction.

Notes

The volume fraction of the restricted diffusion compartment can be seem as the volume fraction of the intra-cellular compartment [1].

References

1

Fieremans, E., Jensen, J.H., Helpern, J.A., 2011. White Matter Characterization with Diffusion Kurtosis Imaging. Neuroimage 58(1): 177-188. doi:10.1016/j.neuroimage.2011.06.006

property axonal_diffusivity

Returns the axonal diffusivity defined as the restricted diffusion tensor trace [1].

References

1

Fieremans, E., Jensen, J.H., Helpern, J.A., 2011. White Matter Characterization with Diffusion Kurtosis Imaging. Neuroimage 58(1): 177-188. doi:10.1016/j.neuroimage.2011.06.006

property hindered_ad

Returns the axial diffusivity of the hindered compartment.

Notes

The hindered diffusion tensor can be seem as the tissue’s extra-cellular diffusion compartment [1].

References

1

Fieremans, E., Jensen, J.H., Helpern, J.A., 2011. White Matter Characterization with Diffusion Kurtosis Imaging. Neuroimage 58(1): 177-188. doi:10.1016/j.neuroimage.2011.06.006

property hindered_evals

Returns the eigenvalues of the hindered diffusion compartment.

Notes

The hindered diffusion tensor can be seem as the tissue’s extra-cellular diffusion compartment [1].

References

1

Fieremans, E., Jensen, J.H., Helpern, J.A., 2011. White Matter Characterization with Diffusion Kurtosis Imaging. Neuroimage 58(1): 177-188. doi:10.1016/j.neuroimage.2011.06.006

property hindered_rd

Returns the radial diffusivity of the hindered compartment.

Notes

The hindered diffusion tensor can be seem as the tissue’s extra-cellular diffusion compartment [1].

References

1

Fieremans, E., Jensen, J.H., Helpern, J.A., 2011. White Matter Characterization with Diffusion Kurtosis Imaging. Neuroimage 58(1): 177-188. doi:10.1016/j.neuroimage.2011.06.006

predict(self, gtab, S0=1.0)

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

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:

$$S(\theta, b) = S_0 * [f * e^{-b ADC_{r}} + (1-f) * e^{-b ADC_{h}]$$, where $$ADC_{r}$$ and $$ADC_{h}$$ are the apparent diffusion coefficients of the diffusion hindered and restricted compartment for a given direction $$\theta$$, $$b$$ is the b value provided in the GradientTable input for that direction, $$f$$ is the volume fraction of the restricted diffusion compartment (also known as the axonal water fraction).

property restricted_evals

Returns the eigenvalues of the restricted diffusion compartment.

Notes

The restricted diffusion tensor can be seem as the tissue’s intra-cellular diffusion compartment [1].

References

1

Fieremans, E., Jensen, J.H., Helpern, J.A., 2011. White Matter Characterization with Diffusion Kurtosis Imaging. Neuroimage 58(1): 177-188. doi:10.1016/j.neuroimage.2011.06.006

property tortuosity

Returns the tortuosity of the hindered diffusion which is defined by ADe / RDe, where ADe and RDe are the axial and radial diffusivities of the hindered compartment [1].

Notes

The hindered diffusion tensor can be seem as the tissue’s extra-cellular diffusion compartment [1].

References

1(1,2)

Fieremans, E., Jensen, J.H., Helpern, J.A., 2011. White Matter Characterization with Diffusion Kurtosis Imaging. Neuroimage 58(1): 177-188. doi:10.1016/j.neuroimage.2011.06.006

### KurtosisMicrostructureModel

class dipy.reconst.dki_micro.KurtosisMicrostructureModel(gtab, fit_method='WLS', *args, **kwargs)

Class for the Diffusion Kurtosis Microstructural Model

Methods

 fit(self, data[, mask, sphere, gtol, awf_only]) Fit method of the Diffusion Kurtosis Microstructural Model predict(self, params[, S0]) Predict a signal for the DKI microstructural model class instance given parameters.
__init__(self, gtab, fit_method='WLS', *args, **kwargs)

Initialize a KurtosisMicrostrutureModel class instance [1].

Parameters
fit_methodstr or callable

str can be one of the following: ‘OLS’ or ‘ULLS’ to fit the diffusion tensor and kurtosis tensor using the ordinary linear least squares solution

dki.ols_fit_dki

‘WLS’ or ‘UWLLS’ to fit the diffusion tensor and kurtosis tensor using the ordinary linear least squares solution

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

Fieremans, E., Jensen, J.H., Helpern, J.A., 2011. White Matter Characterization with Diffusion Kurtosis Imaging. Neuroimage 58(1): 177-188. doi:10.1016/j.neuroimage.2011.06.006

fit(self, data, mask=None, sphere='repulsion100', gtol=0.01, awf_only=False)

Fit method of the Diffusion Kurtosis Microstructural Model

Parameters
dataarray

An 4D matrix containing the diffusion-weighted data.

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

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 maxima 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

awf_onlybool, optiomal

If set to true only the axonal volume fraction is computed from the kurtosis tensor. Default = False

predict(self, params, S0=1.0)

Predict a signal for the DKI microstructural model class instance given parameters.

Parameters
paramsndarray (x, y, z, 40) or (n, 40)

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

1. 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

4. Six elements of the hindered diffusion tensor

5. Six elements of the restricted diffusion tensor

6. Axonal water fraction

S0float or ndarray (optional)

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

Notes

In the original article of DKI microstructural model [1], the hindered and restricted tensors were definde as the intra-cellular and extra-cellular diffusion compartments respectively.

References

1

Fieremans, E., Jensen, J.H., Helpern, J.A., 2011. White Matter Characterization with Diffusion Kurtosis Imaging. Neuroimage 58(1): 177-188. doi:10.1016/j.neuroimage.2011.06.006

### axial_diffusivity

dipy.reconst.dki_micro.axial_diffusivity(evals, axis=-1)

Axial Diffusivity (AD) of a diffusion tensor. Also called parallel diffusivity.

Parameters
evalsarray-like

Eigenvalues of a diffusion tensor, must be sorted in descending order along axis.

axisint

Axis of evals which contains 3 eigenvalues.

Returns

Notes

AD is calculated with the following equation:

$AD = \lambda_1$

### axonal_water_fraction

dipy.reconst.dki_micro.axonal_water_fraction(dki_params, sphere='repulsion100', gtol=0.01, mask=None)

Computes the axonal water fraction from DKI [1].

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

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

1. 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

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 maxima 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

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

Returns
awfndarray (x, y, z) or (n)

Axonal Water Fraction

References

1(1,2)

Fieremans E, Jensen JH, Helpern JA, 2011. White matter characterization with diffusional kurtosis imaging. Neuroimage 58(1):177-88. doi: 10.1016/j.neuroimage.2011.06.006

### decompose_tensor

dipy.reconst.dki_micro.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])

### diffusion_components

dipy.reconst.dki_micro.diffusion_components(dki_params, sphere='repulsion100', awf=None, mask=None)

Extracts the restricted and hindered diffusion tensors of well aligned fibers from diffusion kurtosis imaging parameters [1].

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

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

1. 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

sphereSphere class instance, optional

The sphere providing sample directions to sample the restricted and hindered cellular diffusion tensors. For more details see Fieremans et al., 2011.

awfndarray (optional)

Array containing values of the axonal water fraction that has the shape dki_params.shape[:-1]. If not given this will be automatically computed using axonal_water_fraction()” with function’s default precision.

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

Returns
edtndarray (x, y, z, 6) or (n, 6)

Parameters of the hindered diffusion tensor.

idtndarray (x, y, z, 6) or (n, 6)

Parameters of the restricted diffusion tensor.

Notes

In the original article of DKI microstructural model [1], the hindered and restricted tensors were definde as the intra-cellular and extra-cellular diffusion compartments respectively.

References

1(1,2,3)

Fieremans E, Jensen JH, Helpern JA, 2011. White matter characterization with diffusional kurtosis imaging. Neuroimage 58(1):177-88. doi: 10.1016/j.neuroimage.2011.06.006

### directional_diffusion

dipy.reconst.dki_micro.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

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

References

1(1,2)

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

### directional_kurtosis

dipy.reconst.dki_micro.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])

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

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)

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

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

### dkimicro_prediction

dipy.reconst.dki_micro.dkimicro_prediction(params, gtab, S0=1)

Signal prediction given the DKI microstructure model parameters.

Parameters
paramsndarray (x, y, z, 40) or (n, 40)
All parameters estimated from the diffusion kurtosis microstructure model.
Parameters are ordered as follows:
1. 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

4. Six elements of the hindered diffusion tensor

5. Six elements of the restricted diffusion tensor

6. Axonal water fraction

The gradient table for this prediction

S0float or ndarray

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

Returns
S(…, N) ndarray

Simulated signal based on the DKI microstructure model

Notes

1) The predicted signal is given by: $$S(\theta, b) = S_0 * [f * e^{-b ADC_{r}} + (1-f) * e^{-b ADC_{h}]$$, where :math: ADC_{r} and ADC_{h} are the apparent diffusion coefficients of the diffusion hindered and restricted compartment for a given direction theta:math:, b:math: is the b value provided in the GradientTable input for that direction, f\$ is the volume fraction of the restricted diffusion compartment (also known as the axonal water fraction).

2) In the original article of DKI microstructural model 1, the hindered and restricted tensors were definde as the intra-cellular and extra-cellular diffusion compartments respectively.

### dti_design_matrix

dipy.reconst.dki_micro.dti_design_matrix(gtab, dtype=None)

Constructs design matrix for DTI weighted least squares or least squares fitting. (Basser et al., 1994a)

Parameters
dtypestring

Parameter to control the dtype of returned designed matrix

Returns
design_matrixarray (g,7)

Design matrix or B matrix assuming Gaussian distributed tensor model design_matrix[j, :] = (Bxx, Byy, Bzz, Bxy, Bxz, Byz, dummy)

### from_lower_triangular

dipy.reconst.dki_micro.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_sphere

dipy.reconst.dki_micro.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_maximum

dipy.reconst.dki_micro.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:

1. Three diffusion tensor’s eingenvalues

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

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

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

### lower_triangular

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

Returns the six lower triangular values of the tensor 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_micro.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:

$MD = \frac{\lambda_1 + \lambda_2 + \lambda_3}{3}$

### ndindex

dipy.reconst.dki_micro.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)


dipy.reconst.dki_micro.radial_diffusivity(evals, axis=-1)

Radial Diffusivity (RD) of a diffusion tensor. Also called perpendicular diffusivity.

Parameters
evalsarray-like

Eigenvalues of a diffusion tensor, must be sorted in descending order along axis.

axisint

Axis of evals which contains 3 eigenvalues.

Returns
rdarray

Calculated RD.

Notes

RD is calculated with the following equation:

$RD = \frac{\lambda_2 + \lambda_3}{2}$

### split_dki_param

dipy.reconst.dki_micro.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:

1. 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

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

### tortuosity

dipy.reconst.dki_micro.tortuosity(hindered_ad, hindered_rd)

Computes the tortuosity of the hindered diffusion compartment given its axial and radial diffusivities

Parameters

Array containing the values of the hindered axial diffusivity.

hindered_rd: ndarray

Array containing the values of the hindered radial diffusivity.

Returns
Tortuosity of the hindered diffusion compartment

### trace

dipy.reconst.dki_micro.trace(evals, axis=-1)

Trace of a diffusion tensor.

Parameters
evalsarray-like

Eigenvalues of a diffusion tensor.

axisint

Axis of evals which contains 3 eigenvalues.

Returns
tracearray

Calculated trace of the diffusion tensor.

Notes

Trace is calculated with the following equation:

$Trace = \lambda_1 + \lambda_2 + \lambda_3$

### vec_val_vect

dipy.reconst.dki_micro.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]])


### Cache

class dipy.reconst.dsi.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(self) Clear the cache. cache_get(self, tag, key[, default]) Retrieve a value from the cache. cache_set(self, tag, key, value) Store a value in the cache.
__init__(self, /, *args, **kwargs)

Initialize self. See help(type(self)) for accurate signature.

cache_clear(self)

Clear the cache.

cache_get(self, 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(self, 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


### DiffusionSpectrumDeconvFit

class dipy.reconst.dsi.DiffusionSpectrumDeconvFit(model, data)

Methods

 msd_discrete(self[, normalized]) Calculates the mean squared displacement on the discrete propagator odf(self, sphere) Calculates the real discrete odf for a given discrete sphere pdf(self) Applies the 3D FFT in the q-space grid to generate the DSI diffusion propagator, remove the background noise with a hard threshold and then deconvolve the propagator with the Lucy-Richardson deconvolution algorithm rtop_pdf(self[, normalized]) Calculates the return to origin probability from the propagator, which is the propagator evaluated at zero (see Descoteaux et Al. rtop_signal(self[, filtering]) Calculates the return to origin probability (rtop) from the signal rtop equals to the sum of all signal values
__init__(self, model, data)

Calculates PDF and ODF and other properties for a single voxel

Parameters
modelobject,

DiffusionSpectrumModel

data1d ndarray,

signal values

pdf(self)

Applies the 3D FFT in the q-space grid to generate the DSI diffusion propagator, remove the background noise with a hard threshold and then deconvolve the propagator with the Lucy-Richardson deconvolution algorithm

### DiffusionSpectrumDeconvModel

class dipy.reconst.dsi.DiffusionSpectrumDeconvModel(gtab, qgrid_size=35, r_start=4.1, r_end=13.0, r_step=0.4, filter_width=inf, normalize_peaks=False)

Methods

 cache_clear(self) Clear the cache. cache_get(self, tag, key[, default]) Retrieve a value from the cache. cache_set(self, tag, key, value) Store a value in the cache. fit(self, data[, mask]) Fit method for every voxel in data
__init__(self, gtab, qgrid_size=35, r_start=4.1, r_end=13.0, r_step=0.4, filter_width=inf, normalize_peaks=False)

Diffusion Spectrum Deconvolution

The idea is to remove the convolution on the DSI propagator that is caused by the truncation of the q-space in the DSI sampling.

..math::
nowrap
begin{eqnarray*}

P_{dsi}(mathbf{r}) & = & S_{0}^{-1}iiintlimits_{| mathbf{q} | le mathbf{q_{max}}} S(mathbf{q})exp(-i2pimathbf{q}cdotmathbf{r})dmathbf{q} \ & = & S_{0}^{-1}iiintlimits_{mathbf{q}} left( S(mathbf{q}) cdot M(mathbf{q}) right) exp(-i2pimathbf{q}cdotmathbf{r})dmathbf{q} \ & = & P(mathbf{r}) otimes left( S_{0}^{-1}iiintlimits_{mathbf{q}} M(mathbf{q}) exp(-i2pimathbf{q}cdotmathbf{r})dmathbf{q} right) \

end{eqnarray*}

where $$\mathbf{r}$$ is the displacement vector and $$\mathbf{q}$$ is the wave vector which corresponds to different gradient directions, $$M(\mathbf{q})$$ is a mask corresponding to your q-space sampling and $$\otimes$$ is the convolution operator [1].

Parameters

Gradient directions and bvalues container class

qgrid_sizeint,

has to be an odd number. Sets the size of the q_space grid. For example if qgrid_size is 35 then the shape of the grid will be (35, 35, 35).

r_startfloat,

ODF is sampled radially in the PDF. This parameters shows where the sampling should start.

r_endfloat,

r_stepfloat,

Step size of the ODf sampling from r_start to r_end

filter_widthfloat,

Strength of the hanning filter

References

1

Canales-Rodriguez E.J et al., “Deconvolution in Diffusion

Spectrum Imaging”, Neuroimage, 2010.

2

Biggs David S.C. et al., “Acceleration of Iterative Image

Restoration Algorithms”, Applied Optics, vol. 36, No. 8, p. 1766-1775, 1997.

fit(self, data, mask=None)

Fit method for every voxel in data

### DiffusionSpectrumFit

class dipy.reconst.dsi.DiffusionSpectrumFit(model, data)

Methods

 msd_discrete(self[, normalized]) Calculates the mean squared displacement on the discrete propagator odf(self, sphere) Calculates the real discrete odf for a given discrete sphere pdf(self[, normalized]) Applies the 3D FFT in the q-space grid to generate the diffusion propagator rtop_pdf(self[, normalized]) Calculates the return to origin probability from the propagator, which is the propagator evaluated at zero (see Descoteaux et Al. rtop_signal(self[, filtering]) Calculates the return to origin probability (rtop) from the signal rtop equals to the sum of all signal values
__init__(self, model, data)

Calculates PDF and ODF and other properties for a single voxel

Parameters
modelobject,

DiffusionSpectrumModel

data1d ndarray,

signal values

msd_discrete(self, normalized=True)

Calculates the mean squared displacement on the discrete propagator

..math::
nowrap
begin{equation}

MSD:{DSI}=int_{-infty}^{infty}int_{-infty}^{infty}int_{-infty}^{infty} P(hat{mathbf{r}}) cdot hat{mathbf{r}}^{2} dr_x dr_y dr_z

end{equation}

where $$\hat{\mathbf{r}}$$ is a point in the 3D Propagator space (see Wu et al. [1]).

Parameters
normalizedboolean, optional

Whether to normalize the propagator by its sum in order to obtain a pdf. Default: True

Returns
msdfloat

the mean square displacement

References

1

Wu Y. et al., “Hybrid diffusion imaging”, NeuroImage, vol 36,

1. 617-629, 2007.

odf(self, sphere)

Calculates the real discrete odf for a given discrete sphere

..math::
nowrap
begin{equation}

psi_{DSI}(hat{mathbf{u}})=int_{0}^{infty}P(rhat{mathbf{u}})r^{2}dr

end{equation}

where $$\hat{\mathbf{u}}$$ is the unit vector which corresponds to a sphere point.

pdf(self, normalized=True)

Applies the 3D FFT in the q-space grid to generate the diffusion propagator

rtop_pdf(self, normalized=True)

Calculates the return to origin probability from the propagator, which is the propagator evaluated at zero (see Descoteaux et Al. [1], Tuch [2], Wu et al. [3]) rtop = P(0)

Parameters
normalizedboolean, optional

Whether to normalize the propagator by its sum in order to obtain a pdf. Default: True.

Returns
rtopfloat

References

1

Descoteaux M. et al., “Multiple q-shell diffusion propagator

imaging”, Medical Image Analysis, vol 15, No. 4, p. 603-621, 2011.

2

Tuch D.S., “Diffusion MRI of Complex Tissue Structure”, PhD Thesis, 2002.

3

Wu Y. et al., “Computation of Diffusion Function Measures

in q -Space Using Magnetic Resonance Hybrid Diffusion Imaging”, IEEE TRANSACTIONS ON MEDICAL IMAGING, vol. 27, No. 6, p. 858-865, 2008

rtop_signal(self, filtering=True)

Calculates the return to origin probability (rtop) from the signal rtop equals to the sum of all signal values

Parameters
filteringboolean, optional

Whether to perform Hanning filtering. Default: True

Returns
rtopfloat

### DiffusionSpectrumModel

class dipy.reconst.dsi.DiffusionSpectrumModel(gtab, qgrid_size=17, r_start=2.1, r_end=6.0, r_step=0.2, filter_width=32, normalize_peaks=False)

Methods

 cache_clear(self) Clear the cache. cache_get(self, tag, key[, default]) Retrieve a value from the cache. cache_set(self, tag, key, value) Store a value in the cache. fit(self, data[, mask]) Fit method for every voxel in data
__init__(self, gtab, qgrid_size=17, r_start=2.1, r_end=6.0, r_step=0.2, filter_width=32, normalize_peaks=False)

Diffusion Spectrum Imaging

The theoretical idea underlying this method is that the diffusion propagator $$P(\mathbf{r})$$ (probability density function of the average spin displacements) can be estimated by applying 3D FFT to the signal values $$S(\mathbf{q})$$

..math::
nowrap
begin{eqnarray}

P(mathbf{r}) & = & S_{0}^{-1}int S(mathbf{q})exp(-i2pimathbf{q}cdotmathbf{r})dmathbf{r}

end{eqnarray}

where $$\mathbf{r}$$ is the displacement vector and $$\mathbf{q}$$ is the wave vector which corresponds to different gradient directions. Method used to calculate the ODFs. Here we implement the method proposed by Wedeen et al. [1].

The main assumption for this model is fast gradient switching and that the acquisition gradients will sit on a keyhole Cartesian grid in q_space [3].

Parameters

Gradient directions and bvalues container class

qgrid_sizeint,

has to be an odd number. Sets the size of the q_space grid. For example if qgrid_size is 17 then the shape of the grid will be (17, 17, 17).

r_startfloat,

ODF is sampled radially in the PDF. This parameters shows where the sampling should start.

r_endfloat,

r_stepfloat,

Step size of the ODf sampling from r_start to r_end

filter_widthfloat,

Strength of the hanning filter

dipy.reconst.gqi.GeneralizedQSampling

Notes

A. Have in mind that DSI expects gradients on both hemispheres. If your gradients span only one hemisphere you need to duplicate the data and project them to the other hemisphere before calling this class. The function dipy.reconst.dsi.half_to_full_qspace can be used for this purpose.

B. If you increase the size of the grid (parameter qgrid_size) you will most likely also need to update the r_* parameters. This is because the added zero padding from the increase of gqrid_size also introduces a scaling of the PDF.

1. We assume that data only one b0 volume is provided.

References

1

Wedeen V.J et al., “Mapping Complex Tissue Architecture With

Diffusion Spectrum Magnetic Resonance Imaging”, MRM 2005.

2

Canales-Rodriguez E.J et al., “Deconvolution in Diffusion

Spectrum Imaging”, Neuroimage, 2010.

3

Garyfallidis E, “Towards an accurate brain tractography”, PhD

thesis, University of Cambridge, 2012.

Examples

In this example where we provide the data, a gradient table and a reconstruction sphere, we calculate generalized FA for the first voxel in the data with the reconstruction performed using DSI.

>>> import warnings
>>> from dipy.data import dsi_voxels, default_sphere
>>> data, gtab = dsi_voxels()
>>> from dipy.reconst.dsi import DiffusionSpectrumModel
>>> ds = DiffusionSpectrumModel(gtab)
>>> dsfit = ds.fit(data)
>>> from dipy.reconst.odf import gfa
>>> np.round(gfa(dsfit.odf(default_sphere))[0, 0, 0], 2)
0.11

fit(self, data, mask=None)

Fit method for every voxel in data

### OdfFit

class dipy.reconst.dsi.OdfFit(model, data)

Methods

 odf(self, sphere) To be implemented but specific odf models
__init__(self, model, data)

Initialize self. See help(type(self)) for accurate signature.

odf(self, sphere)

To be implemented but specific odf models

### OdfModel

class dipy.reconst.dsi.OdfModel(gtab)

An abstract class to be sub-classed by specific odf models

All odf models should provide a fit method which may take data as it’s first and only argument.

Methods

 fit(self, data) To be implemented by specific odf models
__init__(self, gtab)

Initialization of the abstract class for signal reconstruction models

Parameters
fit(self, data)

To be implemented by specific odf models

### LR_deconv

dipy.reconst.dsi.LR_deconv(prop, psf, numit=5, acc_factor=1)

Perform Lucy-Richardson deconvolution algorithm on a 3D array.

Parameters
prop3-D ndarray of dtype float

The 3D volume to be deconvolve

psf3-D ndarray of dtype float

The filter that will be used for the deconvolution.

numitint

Number of Lucy-Richardson iteration to perform.

acc_factorfloat

Exponential acceleration factor as in [1].

References

1

Biggs David S.C. et al., “Acceleration of Iterative Image Restoration Algorithms”, Applied Optics, vol. 36, No. 8, p. 1766-1775, 1997.

### create_qspace

dipy.reconst.dsi.create_qspace(gtab, origin)

create the 3D grid which holds the signal values (q-space)

Parameters
origin(3,) ndarray

center of qspace

Returns
qgridndarray

qspace coordinates

### create_qtable

dipy.reconst.dsi.create_qtable(gtab, origin)

create a normalized version of gradients

Parameters
origin(3,) ndarray

center of qspace

Returns
qtablendarray

### fftn

dipy.reconst.dsi.fftn(x, shape=None, axes=None, overwrite_x=False)

Return multidimensional discrete Fourier transform.

The returned array contains:

y[j_1,..,j_d] = sum[k_1=0..n_1-1, ..., k_d=0..n_d-1]
x[k_1,..,k_d] * prod[i=1..d] exp(-sqrt(-1)*2*pi/n_i * j_i * k_i)


where d = len(x.shape) and n = x.shape.

Parameters
xarray_like

The (n-dimensional) array to transform.

shapeint or array_like of ints or None, optional

The shape of the result. If both shape and axes (see below) are None, shape is x.shape; if shape is None but axes is not None, then shape is scipy.take(x.shape, axes, axis=0). If shape[i] > x.shape[i], the i-th dimension is padded with zeros. If shape[i] < x.shape[i], the i-th dimension is truncated to length shape[i]. If any element of shape is -1, the size of the corresponding dimension of x is used.

axesint or array_like of ints or None, optional

The axes of x (y if shape is not None) along which the transform is applied. The default is over all axes.

overwrite_xbool, optional

If True, the contents of x can be destroyed. Default is False.

Returns
ycomplex-valued n-dimensional numpy array

The (n-dimensional) DFT of the input array.

ifftn

Notes

If x is real-valued, then y[..., j_i, ...] == y[..., n_i-j_i, ...].conjugate().

Both single and double precision routines are implemented. Half precision inputs will be converted to single precision. Non floating-point inputs will be converted to double precision. Long-double precision inputs are not supported.

Examples

>>> from scipy.fftpack import fftn, ifftn
>>> y = (-np.arange(16), 8 - np.arange(16), np.arange(16))
>>> np.allclose(y, fftn(ifftn(y)))
True


### fftshift

dipy.reconst.dsi.fftshift(x, axes=None)

Shift the zero-frequency component to the center of the spectrum.

This function swaps half-spaces for all axes listed (defaults to all). Note that y[0] is the Nyquist component only if len(x) is even.

Parameters
xarray_like

Input array.

axesint or shape tuple, optional

Axes over which to shift. Default is None, which shifts all axes.

Returns
yndarray

The shifted array.

ifftshift

The inverse of fftshift.

Examples

>>> freqs = np.fft.fftfreq(10, 0.1)
>>> freqs
array([ 0.,  1.,  2., ..., -3., -2., -1.])
>>> np.fft.fftshift(freqs)
array([-5., -4., -3., -2., -1.,  0.,  1.,  2.,  3.,  4.])


Shift the zero-frequency component only along the second axis:

>>> freqs = np.fft.fftfreq(9, d=1./9).reshape(3, 3)
>>> freqs
array([[ 0.,  1.,  2.],
[ 3.,  4., -4.],
[-3., -2., -1.]])
>>> np.fft.fftshift(freqs, axes=(1,))
array([[ 2.,  0.,  1.],
[-4.,  3.,  4.],
[-1., -3., -2.]])


### gen_PSF

dipy.reconst.dsi.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

dipy.reconst.dsi.half_to_full_qspace(data, gtab)

Half to full Cartesian grid mapping

Useful when dMRI data are provided in one qspace hemisphere as DiffusionSpectrum expects data to be in full qspace.

Parameters
dataarray, shape (X, Y, Z, W)

where (X, Y, Z) volume size and W number of gradient directions

container for b-values and b-vectors (gradient directions)

Returns
new_dataarray, shape (X, Y, Z, 2 * W -1)

Notes

We assume here that only on b0 is provided with the initial data. If that is not the case then you will need to write your own preparation function before providing the gradients and the data to the DiffusionSpectrumModel class.

### hanning_filter

dipy.reconst.dsi.hanning_filter(gtab, filter_width, origin)

create a hanning window

The signal is premultiplied by a Hanning window before Fourier transform in order to ensure a smooth attenuation of the signal at high q values.

Parameters
filter_widthint
origin(3,) ndarray

center of qspace

Returns
filter(N,) ndarray

where N is the number of non-b0 gradient directions

### ifftshift

dipy.reconst.dsi.ifftshift(x, axes=None)

The inverse of fftshift. Although identical for even-length x, the functions differ by one sample for odd-length x.

Parameters
xarray_like

Input array.

axesint or shape tuple, optional

Axes over which to calculate. Defaults to None, which shifts all axes.

Returns
yndarray

The shifted array.

fftshift

Shift zero-frequency component to the center of the spectrum.

Examples

>>> freqs = np.fft.fftfreq(9, d=1./9).reshape(3, 3)
>>> freqs
array([[ 0.,  1.,  2.],
[ 3.,  4., -4.],
[-3., -2., -1.]])
>>> np.fft.ifftshift(np.fft.fftshift(freqs))
array([[ 0.,  1.,  2.],
[ 3.,  4., -4.],
[-3., -2., -1.]])


### map_coordinates

dipy.reconst.dsi.map_coordinates(input, coordinates, output=None, order=3, mode='constant', cval=0.0, prefilter=True)

Map the input array to new coordinates by interpolation.

The array of coordinates is used to find, for each point in the output, the corresponding coordinates in the input. The value of the input at those coordinates is determined by spline interpolation of the requested order.

The shape of the output is derived from that of the coordinate array by dropping the first axis. The values of the array along the first axis are the coordinates in the input array at which the output value is found.

Parameters
inputarray_like

The input array.

coordinatesarray_like

The coordinates at which input is evaluated.

outputarray or dtype, optional

The array in which to place the output, or the dtype of the returned array. By default an array of the same dtype as input will be created.

orderint, optional

The order of the spline interpolation, default is 3. The order has to be in the range 0-5.

mode{‘reflect’, ‘constant’, ‘nearest’, ‘mirror’, ‘wrap’}, optional

The mode parameter determines how the input array is extended beyond its boundaries. Default is ‘constant’. Behavior for each valid value is as follows:

‘reflect’ (d c b a | a b c d | d c b a)

The input is extended by reflecting about the edge of the last pixel.

‘constant’ (k k k k | a b c d | k k k k)

The input is extended by filling all values beyond the edge with the same constant value, defined by the cval parameter.

‘nearest’ (a a a a | a b c d | d d d d)

The input is extended by replicating the last pixel.

‘mirror’ (d c b | a b c d | c b a)

The input is extended by reflecting about the center of the last pixel.

‘wrap’ (a b c d | a b c d | a b c d)

The input is extended by wrapping around to the opposite edge.

cvalscalar, optional

Value to fill past edges of input if mode is ‘constant’. Default is 0.0.

prefilterbool, optional

Determines if the input array is prefiltered with spline_filter before interpolation. The default is True, which will create a temporary float64 array of filtered values if order > 1. If setting this to False, the output will be slightly blurred if order > 1, unless the input is prefiltered, i.e. it is the result of calling spline_filter on the original input.

Returns
map_coordinatesndarray

The result of transforming the input. The shape of the output is derived from that of coordinates by dropping the first axis.

spline_filter, geometric_transform, scipy.interpolate

Examples

>>> from scipy import ndimage
>>> a = np.arange(12.).reshape((4, 3))
>>> a
array([[  0.,   1.,   2.],
[  3.,   4.,   5.],
[  6.,   7.,   8.],
[  9.,  10.,  11.]])
>>> ndimage.map_coordinates(a, [[0.5, 2], [0.5, 1]], order=1)
array([ 2.,  7.])


Above, the interpolated value of a[0.5, 0.5] gives output[0], while a[2, 1] is output[1].

>>> inds = np.array([[0.5, 2], [0.5, 4]])
>>> ndimage.map_coordinates(a, inds, order=1, cval=-33.3)
array([  2. , -33.3])
>>> ndimage.map_coordinates(a, inds, order=1, mode='nearest')
array([ 2.,  8.])
>>> ndimage.map_coordinates(a, inds, order=1, cval=0, output=bool)
array([ True, False], dtype=bool)


### multi_voxel_fit

dipy.reconst.dsi.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

dipy.reconst.dsi.pdf_interp_coords(sphere, rradius, origin)

Precompute coordinates for ODF calculation from the PDF

Parameters
sphereobject,

Sphere

line interpolation points

originarray, shape (3,)

center of the grid

### pdf_odf

dipy.reconst.dsi.pdf_odf(Pr, rradius, interp_coords)

Calculates the real ODF from the diffusion propagator(PDF) Pr

Parameters
Prarray, shape (X, X, X)

probability density function

interp_coordsarray, shape (3, M, N)

coordinates in the pdf for interpolating the odf

### project_hemisph_bvecs

dipy.reconst.dsi.project_hemisph_bvecs(gtab)

Project any near identical bvecs to the other hemisphere

Parameters
gtabobject,

Notes

Useful only when working with some types of dsi data.

### threshold_propagator

dipy.reconst.dsi.threshold_propagator(P, estimated_snr=15.0)

Applies hard threshold on the propagator to remove background noise for the deconvolution.

### ReconstModel

class dipy.reconst.dti.ReconstModel(gtab)

Bases: object

Abstract class for signal reconstruction models

Methods

 fit
__init__(self, gtab)

Initialization of the abstract class for signal reconstruction models

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

### TensorFit

class dipy.reconst.dti.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(self) Axial diffusivity (AD) calculated from cached eigenvalues. adc(self, sphere) Calculate the apparent diffusion coefficient (ADC) in each direction on color_fa(self) Color fractional anisotropy of diffusion tensor fa(self) Fractional anisotropy (FA) calculated from cached eigenvalues. ga(self) Geodesic anisotropy (GA) calculated from cached eigenvalues. linearity(self) Returns md(self) Mean diffusivity (MD) calculated from cached eigenvalues. mode(self) Tensor mode calculated from cached eigenvalues. odf(self, sphere) The diffusion orientation distribution function (dODF). planarity(self) Returns predict(self, gtab[, S0, step]) Given a model fit, predict the signal on the vertices of a sphere rd(self) Radial diffusivity (RD) calculated from cached eigenvalues. sphericity(self) Returns trace(self) Trace of the tensor calculated from cached eigenvalues.
 lower_triangular
__init__(self, model, model_params, model_S0=None)

Initialize a TensorFit class instance.

property S0_hat
ad(self)

Axial diffusivity (AD) calculated from cached eigenvalues.

Returns

Notes

RD is calculated with the following equation:

$AD = \lambda_1$
adc(self, sphere)

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

Parameters
sphereSphere class instance
Returns

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(self)

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(self)

Fractional anisotropy (FA) calculated from cached eigenvalues.

ga(self)

Geodesic anisotropy (GA) calculated from cached eigenvalues.

linearity(self)
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(self, b0=None)
md(self)

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(self)

Tensor mode calculated from cached eigenvalues.

odf(self, 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 [Aganj2010]. To re-derive it from scratch, follow steps in [Descoteaux2008], Section 7.9 Equation 7.24 but with an $$r^2$$ term in the integral.

References

Aganj2010

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

Descoteaux2008

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

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

Parameters

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(self)

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(self)
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(self)

Trace of the tensor calculated from cached eigenvalues.

Returns
tracearray (V, 1)

Calculated trace.

Notes

The trace is calculated with the following equation:

$trace = \lambda_1 + \lambda_2 + \lambda_3$

### TensorModel

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

Diffusion Tensor

Methods

 fit(self, data[, mask]) Fit method of the DTI model class predict(self, dti_params[, S0]) Predict a signal for this TensorModel class instance given parameters.
__init__(self, gtab, fit_method='WLS', return_S0_hat=False, *args, **kwargs)

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

Parameters
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(self, data, mask=None)

Fit method of the DTI model class

Parameters
dataarray

The measured signal from one voxel.

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

predict(self, 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

### apparent_diffusion_coef

dipy.reconst.dti.apparent_diffusion_coef(q_form, sphere)

Calculate the apparent diffusion coefficient (ADC) in each direction of a sphere.

Parameters
q_formndarray

The quadratic form of a tensor, or an array with quadratic forms of tensors. Should be of shape (…, 3, 3)

spherea Sphere class instance

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

ec{b} Q ec{b}^T

Where Q is the quadratic form of the tensor.

### auto_attr

dipy.reconst.dti.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


### axial_diffusivity

dipy.reconst.dti.axial_diffusivity(evals, axis=-1)

Axial Diffusivity (AD) of a diffusion tensor. Also called parallel diffusivity.

Parameters
evalsarray-like

Eigenvalues of a diffusion tensor, must be sorted in descending order along axis.

axisint

Axis of evals which contains 3 eigenvalues.

Returns

Notes

AD is calculated with the following equation:

$AD = \lambda_1$

### color_fa

dipy.reconst.dti.color_fa(fa, evecs)

Color fractional anisotropy of diffusion tensor

Parameters
faarray-like

Array of the fractional anisotropy (can be 1D, 2D or 3D)

evecsarray-like

eigen vectors from the tensor model

Returns
rgbArray with 3 channels for each color as the last dimension.

Colormap of the FA with red for the x value, y for the green value and z for the blue value.

ec{e})) imes fa

### decompose_tensor

dipy.reconst.dti.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.dti.design_matrix(gtab, dtype=None)

Constructs design matrix for DTI weighted least squares or least squares fitting. (Basser et al., 1994a)

Parameters
dtypestring

Parameter to control the dtype of returned designed matrix

Returns
design_matrixarray (g,7)

Design matrix or B matrix assuming Gaussian distributed tensor model design_matrix[j, :] = (Bxx, Byy, Bzz, Bxy, Bxz, Byz, dummy)

### determinant

dipy.reconst.dti.determinant(q_form)

The determinant of a tensor, given in quadratic form

Parameters
q_formndarray

The quadratic form of a tensor, or an array with quadratic forms of tensors. Should be of shape (x, y, z, 3, 3) or (n, 3, 3) or (3, 3).

Returns
detarray

The determinant of the tensor in each spatial coordinate

### deviatoric

dipy.reconst.dti.deviatoric(q_form)

Calculate the deviatoric (anisotropic) part of the tensor [1].

Parameters
q_formndarray

The quadratic form of a tensor, or an array with quadratic forms of tensors. Should be of shape (x,y,z,3,3) or (n, 3, 3) or (3,3).

Returns
A_squigglendarray

The deviatoric part of the tensor in each spatial coordinate.

Notes

The deviatoric part of the tensor is defined as (equations 3-5 in [1]):

$\widetilde{A} = A - \bar{A}$

Where $$A$$ is the tensor quadratic form and $$\bar{A}$$ is the anisotropic part of the tensor.

References

1(1,2,3)

Daniel B. Ennis and G. Kindlmann, “Orthogonal Tensor Invariants and the Analysis of Diffusion Tensor Magnetic Resonance Images”, Magnetic Resonance in Medicine, vol. 55, no. 1, pp. 136-146, 2006.

### eig_from_lo_tri

dipy.reconst.dti.eig_from_lo_tri(data, min_diffusivity=0)

Calculates tensor eigenvalues/eigenvectors from an array containing the lower diagonal form of the six unique tensor elements.

Parameters
dataarray_like (…, 6)

diffusion tensors elements stored in lower triangular order

min_diffusivityfloat

See decompose_tensor()

Returns
dti_paramsarray (…, 12)

Eigen-values and eigen-vectors of the same array.

### fractional_anisotropy

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

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:

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

### from_lower_triangular

dipy.reconst.dti.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

### geodesic_anisotropy

dipy.reconst.dti.geodesic_anisotropy(evals, axis=-1)

Geodesic anisotropy (GA) of a diffusion tensor.

Parameters
evalsarray-like

Eigenvalues of a diffusion tensor.

axisint

Axis of evals which contains 3 eigenvalues.

Returns
gaarray

Calculated GA. In the range 0 to +infinity

Notes

GA is calculated using the following equation given in [1]:

$GA = \sqrt{\sum_{i=1}^3 \log^2{\left ( \lambda_i/<\mathbf{D}> \right )}}, \quad \textrm{where} \quad <\mathbf{D}> = (\lambda_1\lambda_2\lambda_3)^{1/3}$

Note that the notation, $$<D>$$, is often used as the mean diffusivity (MD) of the diffusion tensor and can lead to confusions in the literature (see [1] versus [2] versus [3] for example). Reference [2] defines geodesic anisotropy (GA) with $$<D>$$ as the MD in the denominator of the sum. This is wrong. The original paper [1] defines GA with $$<D> = det(D)^{1/3}$$, as the isotropic part of the distance. This might be an explanation for the confusion. The isotropic part of the diffusion tensor in Euclidean space is the MD whereas the isotropic part of the tensor in log-Euclidean space is $$det(D)^{1/3}$$. The Appendix of [1] and log-Euclidean derivations from [3] are clear on this. Hence, all that to say that $$<D> = det(D)^{1/3}$$ here for the GA definition and not MD.

References

1(1,2,3,4)

P. G. Batchelor, M. Moakher, D. Atkinson, F. Calamante, A. Connelly, “A rigorous framework for diffusion tensor calculus”, Magnetic Resonance in Medicine, vol. 53, pp. 221-225, 2005.

2(1,2)

M. M. Correia, V. F. Newcombe, G.B. Williams. “Contrast-to-noise ratios for indices of anisotropy obtained from diffusion MRI: a study with standard clinical b-values at 3T”. NeuroImage, vol. 57, pp. 1103-1115, 2011.

3(1,2)

A. D. Lee, etal, P. M. Thompson. “Comparison of fractional and geodesic anisotropy in diffusion tensor images of 90 monozygotic and dizygotic twins”. 5th IEEE International Symposium on Biomedical Imaging (ISBI), pp. 943-946, May 2008.

4

V. Arsigny, P. Fillard, X. Pennec, N. Ayache. “Log-Euclidean metrics for fast and simple calculus on diffusion tensors.” Magnetic Resonance in Medecine, vol 56, pp. 411-421, 2006.

### get_sphere

dipy.reconst.dti.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"


dipy.reconst.dti.gradient_table(bvals, bvecs=None, big_delta=None, small_delta=None, b0_threshold=50, atol=0.01)

A general function for creating diffusion MR gradients.

It reads, loads and prepares scanner parameters like the b-values and b-vectors so that they can be useful during the reconstruction process.

Parameters
bvalscan be any of the four options
1. an array of shape (N,) or (1, N) or (N, 1) with the b-values.

2. a path for the file which contains an array like the above (1).

3. an array of shape (N, 4) or (4, N). Then this parameter is considered to be a b-table which contains both bvals and bvecs. In this case the next parameter is skipped.

4. a path for the file which contains an array like the one at (3).

bvecscan be any of two options
1. an array of shape (N, 3) or (3, N) with the b-vectors.

2. a path for the file which contains an array like the previous.

big_deltafloat

acquisition pulse separation time in seconds (default None)

small_deltafloat

acquisition pulse duration time in seconds (default None)

b0_thresholdfloat

All b-values with values less than or equal to bo_threshold are considered as b0s i.e. without diffusion weighting.

atolfloat

All b-vectors need to be unit vectors up to a tolerance.

Returns

Notes

1. Often b0s (b-values which correspond to images without diffusion weighting) have 0 values however in some cases the scanner cannot provide b0s of an exact 0 value and it gives a bit higher values e.g. 6 or 12. This is the purpose of the b0_threshold in the __init__.

2. We assume that the minimum number of b-values is 7.

3. B-vectors should be unit vectors.

Examples

>>> from dipy.core.gradients import gradient_table
>>> bvals = 1500 * np.ones(7)
>>> bvals[0] = 0
>>> sq2 = np.sqrt(2) / 2
>>> bvecs = np.array([[0, 0, 0],
...                   [1, 0, 0],
...                   [0, 1, 0],
...                   [0, 0, 1],
...                   [sq2, sq2, 0],
...                   [sq2, 0, sq2],
...                   [0, sq2, sq2]])
>>> gt.bvecs.shape == bvecs.shape
True
>>> gt.bvecs.shape == bvecs.T.shape
False


### isotropic

dipy.reconst.dti.isotropic(q_form)

Calculate the isotropic part of the tensor [Rd0568a744381-1].

Parameters
q_formndarray

The quadratic form of a tensor, or an array with quadratic forms of tensors. Should be of shape (x,y,z,3,3) or (n, 3, 3) or (3,3).

Returns
A_hat: ndarray

The isotropic part of the tensor in each spatial coordinate

rac{1}{2} tr(A) I

### iter_fit_tensor

dipy.reconst.dti.iter_fit_tensor(step=10000.0)

Wrap a fit_tensor func and iterate over chunks of data with given length

Splits data into a number of chunks of specified size and iterates the decorated fit_tensor function over them. This is useful to counteract the temporary but significant memory usage increase in fit_tensor functions that use vectorized operations and need to store large temporary arrays for their vectorized operations.

Parameters
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.

### linearity

dipy.reconst.dti.linearity(evals, axis=-1)

The linearity of the tensor 1

Parameters
evalsarray-like

Eigenvalues of a diffusion tensor.

axisint

Axis of evals which contains 3 eigenvalues.

Returns
linearityarray

Calculated linearity of the diffusion tensor.

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

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

Returns the six lower triangular values of the tensor 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.dti.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:

$MD = \frac{\lambda_1 + \lambda_2 + \lambda_3}{3}$

### mode

dipy.reconst.dti.mode(q_form)

Mode (MO) of a diffusion tensor [1].

Parameters
q_formndarray

The quadratic form of a tensor, or an array with quadratic forms of tensors. Should be of shape (x, y, z, 3, 3) or (n, 3, 3) or (3, 3).

Returns
modearray

Calculated tensor mode in each spatial coordinate.

Notes

Mode ranges between -1 (planar anisotropy) and +1 (linear anisotropy) with 0 representing orthotropy. Mode is calculated with the following equation (equation 9 in [1]):

$Mode = 3*\sqrt{6}*det(\widetilde{A}/norm(\widetilde{A}))$

Where $$\widetilde{A}$$ is the deviatoric part of the tensor quadratic form.

References

1(1,2,3)

Daniel B. Ennis and G. Kindlmann, “Orthogonal Tensor Invariants and the Analysis of Diffusion Tensor Magnetic Resonance Images”, Magnetic Resonance in Medicine, vol. 55, no. 1, pp. 136-146, 2006.

### nlls_fit_tensor

dipy.reconst.dti.nlls_fit_tensor(design_matrix, data, weighting=None, sigma=None, jac=True, return_S0_hat=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.

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

voxel.

### norm

dipy.reconst.dti.norm(q_form)

Calculate the Frobenius norm of a tensor quadratic form

Parameters
q_form: ndarray

The quadratic form of a tensor, or an array with quadratic forms of tensors. Should be of shape (x,y,z,3,3) or (n, 3, 3) or (3,3).

Returns
normndarray

The Frobenius norm of the 3,3 tensor q_form in each spatial coordinate.

np.linalg.norm

Notes

The Frobenius norm is defined as:

Math

||A||_F = [sum_{i,j} abs(a_{i,j})^2]^{1/2}

### ols_fit_tensor

dipy.reconst.dti.ols_fit_tensor(design_matrix, data, return_S0_hat=False)

Computes ordinary least squares (OLS) fit to calculate self-diffusion tensor using a linear regression model [Rd310240b4eed-1].

Parameters
design_matrixarray (g, 7)

Design matrix holding the covariants used to solve for the regression coefficients.

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.

return_S0_hatbool

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

Returns
eigvalsarray (…, 3)

Eigenvalues from eigen decomposition of the tensor.

eigvecsarray (…, 3, 3)

Associated eigenvectors from eigen decomposition of the tensor. Eigenvectors are columnar (e.g. eigvecs[:,j] is associated with eigvals[j])

WLS_fit_tensor, decompose_tensor, design_matrix

Notes

\begin{align}\begin{aligned}\begin{split}y = \mathrm{data} \\ X = \mathrm{design matrix} \\\end{split}\\\hat{\beta}_\mathrm{OLS} = (X^T X)^{-1} X^T y\end{aligned}\end{align}

References

1(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27)

Chung, SW., Lu, Y., Henry, R.G., 2006. Comparison of bootstrap approaches for estimation of uncertainties of DTI parameters. NeuroImage 33, 531-541.

### pinv

dipy.reconst.dti.pinv(a, rcond=1e-15)

Vectorized version of numpy.linalg.pinv

If numpy version is less than 1.8, it falls back to iterating over np.linalg.pinv since there isn’t a vectorized version of np.linalg.svd available.

Parameters
aarray_like (…, M, N)

Matrix to be pseudo-inverted.

rcondfloat

Cutoff for small singular values.

Returns
Bndarray (…, N, M)

The pseudo-inverse of a.

Raises
LinAlgError

If the SVD computation does not converge.

np.linalg.pinv

### planarity

dipy.reconst.dti.planarity(evals, axis=-1)

The planarity of the tensor 1

Parameters
evalsarray-like

Eigenvalues of a diffusion tensor.

axisint

Axis of evals which contains 3 eigenvalues.

Returns
linearityarray

Calculated linearity of the diffusion tensor.

Notes

Planarity is calculated with the following equation:

$Planarity = \frac{2 (\lambda_2-\lambda_3)}{\lambda_1+\lambda_2+\lambda_3}$

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.

### quantize_evecs

dipy.reconst.dti.quantize_evecs(evecs, odf_vertices=None)

Find the closest orientation of an evenly distributed sphere

Parameters
evecsndarray
odf_verticesNone or ndarray

If None, then set vertices from symmetric362 sphere. Otherwise use passed ndarray as vertices

Returns
INndarray

dipy.reconst.dti.radial_diffusivity(evals, axis=-1)

Radial Diffusivity (RD) of a diffusion tensor. Also called perpendicular diffusivity.

Parameters
evalsarray-like

Eigenvalues of a diffusion tensor, must be sorted in descending order along axis.

axisint

Axis of evals which contains 3 eigenvalues.

Returns
rdarray

Calculated RD.

Notes

RD is calculated with the following equation:

$RD = \frac{\lambda_2 + \lambda_3}{2}$

### restore_fit_tensor

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

Use the RESTORE algorithm [Chang2005] 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

An estimate of the variance. [Chang2005] 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).

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.

Returns
restore_paramsan estimate of the tensor parameters in each voxel.

References

Chang, L-C, Jones, DK and Pierpaoli, C (2005). RESTORE: robust estimation of tensors by outlier rejection. MRM, 53: 1088-95.

### sphericity

dipy.reconst.dti.sphericity(evals, axis=-1)

The sphericity of the tensor 1

Parameters
evalsarray-like

Eigenvalues of a diffusion tensor.

axisint

Axis of evals which contains 3 eigenvalues.

Returns
sphericityarray

Calculated sphericity of the diffusion tensor.

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.

### tensor_prediction

dipy.reconst.dti.tensor_prediction(dti_params, gtab, S0)

Predict a signal given tensor parameters.

Parameters
dti_paramsndarray

Tensor parameters. The last dimension should have 12 tensor parameters: 3 eigenvalues, followed by the 3 corresponding eigenvectors.

The gradient table for this prediction

S0float or ndarray

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

Notes

The predicted signal is given by: $$S( heta, b) = S_0 * e^{-b ADC}$$, where $$ADC = heta Q heta^T$$, :math: heta is a unit vector pointing at any direction on the sphere for which a signal is to be predicted, $$b$$ is the b value provided in the GradientTable input for that direction, $$Q$$ is the quadratic form of the tensor determined by the input parameters.

### trace

dipy.reconst.dti.trace(evals, axis=-1)

Trace of a diffusion tensor.

Parameters
evalsarray-like

Eigenvalues of a diffusion tensor.

axisint

Axis of evals which contains 3 eigenvalues.

Returns
tracearray

Calculated trace of the diffusion tensor.

Notes

Trace is calculated with the following equation:

$Trace = \lambda_1 + \lambda_2 + \lambda_3$

### vec_val_vect

dipy.reconst.dti.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]])


### vector_norm

dipy.reconst.dti.vector_norm(vec, axis=-1, keepdims=False)

Return vector Euclidean (L2) norm

Parameters
vecarray_like

Vectors to norm.

axisint

Axis over which to norm. By default norm over last axis. If axis is None, vec is flattened then normed.

keepdimsbool

If True, the output will have the same number of dimensions as vec, with shape 1 on axis.

Returns
normarray

Euclidean norms of vectors.

Examples

>>> import numpy as np
>>> vec = [[8, 15, 0], [0, 36, 77]]
>>> vector_norm(vec)
array([ 17.,  85.])
>>> vector_norm(vec, keepdims=True)
array([[ 17.],
[ 85.]])
>>> vector_norm(vec, axis=0)
array([  8.,  39.,  77.])


### wls_fit_tensor

dipy.reconst.dti.wls_fit_tensor(design_matrix, data, return_S0_hat=False)

Computes weighted least squares (WLS) fit to calculate self-diffusion tensor using a linear regression model [1].

Parameters
design_matrixarray (g, 7)

Design matrix holding the covariants used to solve for the regression coefficients.

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.

return_S0_hatbool

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

Returns
eigvalsarray (…, 3)

Eigenvalues from eigen decomposition of the tensor.

eigvecsarray (…, 3, 3)

Associated eigenvectors from eigen decomposition of the tensor. Eigenvectors are columnar (e.g. eigvecs[:,j] is associated with eigvals[j])

Notes

In Chung, et al. 2006, the regression of the WLS fit needed an unbiased preliminary estimate of the weights and therefore the ordinary least squares (OLS) estimates were used. A “two pass” method was implemented:

1. calculate OLS estimates of the data

2. apply the OLS estimates as weights to the WLS fit of the data

This ensured heteroscedasticity could be properly modeled for various types of bootstrap resampling (namely residual bootstrap).

$\begin{split}y = \mathrm{data} \\ X = \mathrm{design matrix} \\ \hat{\beta}_\mathrm{WLS} = \mathrm{desired regression coefficients (e.g. tensor)}\\ \\ \hat{\beta}_\mathrm{WLS} = (X^T W X)^{-1} X^T W y \\ \\ W = \mathrm{diag}((X \hat{\beta}_\mathrm{OLS})^2), \mathrm{where} \hat{\beta}_\mathrm{OLS} = (X^T X)^{-1} X^T y\end{split}$

References

1(1,2)

Chung, SW., Lu, Y., Henry, R.G., 2006. Comparison of bootstrap approaches for estimation of uncertainties of DTI parameters. NeuroImage 33, 531-541.

### Cache

class dipy.reconst.forecast.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(self) Clear the cache. cache_get(self, tag, key[, default]) Retrieve a value from the cache. cache_set(self, tag, key, value) Store a value in the cache.
__init__(self, /, *args, **kwargs)

Initialize self. See help(type(self)) for accurate signature.

cache_clear(self)

Clear the cache.

cache_get(self, 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(self, 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


### ForecastFit

class dipy.reconst.forecast.ForecastFit(model, data, sh_coef, d_par, d_perp)
Attributes
dpar

The parallel diffusivity

dperp

The perpendicular diffusivity

sh_coeff

The FORECAST SH coefficients

Methods

 Calculates the fractional anisotropy. Calculates the mean diffusivity. odf(self, sphere[, clip_negative]) Calculates the fODF for a given discrete sphere. predict(self[, gtab, S0]) Calculates the fODF for a given discrete sphere.
__init__(self, model, data, sh_coef, d_par, d_perp)

Calculates diffusion properties for a single voxel

Parameters
modelobject,

AnalyticalModel

data1d ndarray,

fitted data

sh_coef1d ndarray,

forecast sh coefficients

d_parfloat,

parallel diffusivity

d_perpfloat,

perpendicular diffusivity

property dpar

The parallel diffusivity

property dperp

The perpendicular diffusivity

fractional_anisotropy(self)

Calculates the fractional anisotropy.

mean_diffusivity(self)

Calculates the mean diffusivity.

odf(self, sphere, clip_negative=True)

Calculates the fODF for a given discrete sphere.

Parameters
sphereSphere,

the odf sphere

clip_negativeboolean, optional

if True clip the negative odf values to 0, default True

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

Calculates the fODF for a given discrete sphere.

Parameters

gradient directions and bvalues container class.

S0float, optional

the signal at b-value=0

property sh_coeff

The FORECAST SH coefficients

### ForecastModel

class dipy.reconst.forecast.ForecastModel(gtab, sh_order=8, lambda_lb=0.001, dec_alg='CSD', sphere=None, lambda_csd=1.0)

Fiber ORientation Estimated using Continuous Axially Symmetric Tensors (FORECAST) [1,2,3]_. FORECAST is a Spherical Deconvolution reconstruction model for multi-shell diffusion data which enables the calculation of a voxel adaptive response function using the Spherical Mean Tecnique (SMT) [2,3]_.

With FORECAST it is possible to calculate crossing invariant parallel diffusivity, perpendicular diffusivity, mean diffusivity, and fractional anisotropy [R1340df73dba1-2]

References

R1340df73dba1-1

Anderson A. W., “Measurement of Fiber Orientation Distributions Using High Angular Resolution Diffusion Imaging”, Magnetic Resonance in Medicine, 2005.

R1340df73dba1-2

Kaden E. et al., “Quantitative Mapping of the Per-Axon Diffusion Coefficients in Brain White Matter”, Magnetic Resonance in Medicine, 2016.

R1340df73dba1-3

Zucchelli E. et al., “A generalized SMT-based framework for Diffusion MRI microstructural model estimation”, MICCAI Workshop on Computational DIFFUSION MRI (CDMRI), 2017.

The implementation of FORECAST may require CVXPY (http://www.cvxpy.org/).

Methods

 cache_clear(self) Clear the cache. cache_get(self, tag, key[, default]) Retrieve a value from the cache. cache_set(self, tag, key, value) Store a value in the cache. fit(self, data[, mask]) Fit method for every voxel in data
__init__(self, gtab, sh_order=8, lambda_lb=0.001, dec_alg='CSD', sphere=None, lambda_csd=1.0)

Analytical and continuous modeling of the diffusion signal with respect to the FORECAST basis [1,2,3]_. This implementation is a modification of the original FORECAST model presented in [1] adapted for multi-shell data as in [2,3]_ .

The main idea is to model the diffusion signal as the combination of a single fiber response function $$F(\mathbf{b})$$ times the fODF $$\rho(\mathbf{v})$$

..math::
nowrap
begin{equation}

E(mathbf{b}) = int_{mathbf{v} in mathcal{S}^2} rho(mathbf{v}) F({mathbf{b}} | mathbf{v}) d mathbf{v}

end{equation}

where $$\mathbf{b}$$ is the b-vector (b-value times gradient direction) and $$\mathbf{v}$$ is an unit vector representing a fiber direction.

In FORECAST $$\rho$$ is modeled using real symmetric Spherical Harmonics (SH) and $$F(\mathbf(b))$$ is an axially symmetric tensor.

Parameters

gradient directions and bvalues container class.

sh_orderunsigned int,

an even integer that represent the SH order of the basis (max 12)

lambda_lb: float,

Laplace-Beltrami regularization weight.

dec_algstr,

Spherical deconvolution algorithm. The possible values are Weighted Least Squares (‘WLS’), Positivity Constraints using CVXPY (‘POS’) and the Constraint Spherical Deconvolution algorithm (‘CSD’). Default is ‘CSD’.

spherearray, shape (N,3),

sphere points where to enforce positivity when ‘POS’ or ‘CSD’ dec_alg are selected.

lambda_csdfloat,

CSD regularization weight.

References

1

Anderson A. W., “Measurement of Fiber Orientation Distributions Using High Angular Resolution Diffusion Imaging”, Magnetic Resonance in Medicine, 2005.

2

Kaden E. et al., “Quantitative Mapping of the Per-Axon Diffusion Coefficients in Brain White Matter”, Magnetic Resonance in Medicine, 2016.

3

Zucchelli M. et al., “A generalized SMT-based framework for Diffusion MRI microstructural model estimation”, MICCAI Workshop on Computational DIFFUSION MRI (CDMRI), 2017.

Examples

In this example, where the data, gradient table and sphere tessellation used for reconstruction are provided, we model the diffusion signal with respect to the FORECAST and compute the fODF, parallel and perpendicular diffusivity.

>>> from dipy.data import default_sphere, get_3shell_gtab
>>> gtab = get_3shell_gtab()
>>> from dipy.sims.voxel import multi_tensor
>>> mevals = np.array(([0.0017, 0.0003, 0.0003],
...                    [0.0017, 0.0003, 0.0003]))
>>> angl = [(0, 0), (60, 0)]
>>> data, sticks = multi_tensor(gtab,
...                             mevals,
...                             S0=100.0,
...                             angles=angl,
...                             fractions=[50, 50],
...                             snr=None)
>>> from dipy.reconst.forecast import ForecastModel
>>> fm = ForecastModel(gtab, sh_order=6)
>>> f_fit = fm.fit(data)
>>> d_par = f_fit.dpar
>>> d_perp = f_fit.dperp
>>> fodf = f_fit.odf(default_sphere)

fit(self, data, mask=None)

Fit method for every voxel in data

### OdfFit

class dipy.reconst.forecast.OdfFit(model, data)

Methods

 odf(self, sphere) To be implemented but specific odf models
__init__(self, model, data)

Initialize self. See help(type(self)) for accurate signature.

odf(self, sphere)

To be implemented but specific odf models

### OdfModel

class dipy.reconst.forecast.OdfModel(gtab)

An abstract class to be sub-classed by specific odf models

All odf models should provide a fit method which may take data as it’s first and only argument.

Methods

 fit(self, data) To be implemented by specific odf models
__init__(self, gtab)

Initialization of the abstract class for signal reconstruction models

Parameters
fit(self, data)

To be implemented by specific odf models

### cart2sphere

dipy.reconst.forecast.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

thetaarray

inclination (polar) angle

phiarray

azimuth angle

### csdeconv

dipy.reconst.forecast.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.

### find_signal_means

dipy.reconst.forecast.find_signal_means(b_unique, data_norm, bvals, rho, lb_matrix, w=0.001)

Calculate the mean signal for each shell.

Parameters
b_unique1d ndarray,

unique b-values in a vector excluding zero

data_norm1d ndarray,

normalized diffusion signal

bvals1d ndarray,

the b-values

rho2d ndarray,

SH basis matrix for fitting the signal on each shell

lb_matrix2d ndarray,

Laplace-Beltrami regularization matrix

wfloat,

weight for the Laplace-Beltrami regularization

Returns
means1d ndarray

the average of the signal for each b-values

### forecast_error_func

dipy.reconst.forecast.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

dipy.reconst.forecast.forecast_matrix(sh_order, d_par, d_perp, bvals)

### lb_forecast

dipy.reconst.forecast.lb_forecast(sh_order)

Returns the Laplace-Beltrami regularization matrix for FORECAST

### leastsq

dipy.reconst.forecast.leastsq(func, x0, args=(), Dfun=None, full_output=0, col_deriv=0, ftol=1.49012e-08, xtol=1.49012e-08, gtol=0.0, maxfev=0, epsfcn=None, factor=100, diag=None)

Minimize the sum of squares of a set of equations.

x = arg min(sum(func(y)**2,axis=0))
y

Parameters
funccallable

should take at least one (possibly length N vector) argument and returns M floating point numbers. It must not return NaNs or fitting might fail.

x0ndarray

The starting estimate for the minimization.

argstuple, optional

Any extra arguments to func are placed in this tuple.

Dfuncallable, optional

A function or method to compute the Jacobian of func with derivatives across the rows. If this is None, the Jacobian will be estimated.

full_outputbool, optional

non-zero to return all optional outputs.

col_derivbool, optional

non-zero to specify that the Jacobian function computes derivatives down the columns (faster, because there is no transpose operation).

ftolfloat, optional

Relative error desired in the sum of squares.

xtolfloat, optional

Relative error desired in the approximate solution.

gtolfloat, optional

Orthogonality desired between the function vector and the columns of the Jacobian.

maxfevint, optional

The maximum number of calls to the function. If Dfun is provided then the default maxfev is 100*(N+1) where N is the number of elements in x0, otherwise the default maxfev is 200*(N+1).

epsfcnfloat, optional

A variable used in determining a suitable step length for the forward- difference approximation of the Jacobian (for Dfun=None). Normally the actual step length will be sqrt(epsfcn)*x If epsfcn is less than the machine precision, it is assumed that the relative errors are of the order of the machine precision.

factorfloat, optional

A parameter determining the initial step bound (factor * || diag * x||). Should be in interval (0.1, 100).

diagsequence, optional

N positive entries that serve as a scale factors for the variables.

Returns
xndarray

The solution (or the result of the last iteration for an unsuccessful call).

cov_xndarray

The inverse of the Hessian. fjac and ipvt are used to construct an estimate of the Hessian. A value of None indicates a singular matrix, which means the curvature in parameters x is numerically flat. To obtain the covariance matrix of the parameters x, cov_x must be multiplied by the variance of the residuals – see curve_fit.

infodictdict

a dictionary of optional outputs with the keys:

nfev

The number of function calls

fvec

The function evaluated at the output

fjac

A permutation of the R matrix of a QR factorization of the final approximate Jacobian matrix, stored column wise. Together with ipvt, the covariance of the estimate can be approximated.

ipvt

An integer array of length N which defines a permutation matrix, p, such that fjac*p = q*r, where r is upper triangular with diagonal elements of nonincreasing magnitude. Column j of p is column ipvt(j) of the identity matrix.

qtf

The vector (transpose(q) * fvec).

mesgstr

A string message giving information about the cause of failure.

ierint

An integer flag. If it is equal to 1, 2, 3 or 4, the solution was found. Otherwise, the solution was not found. In either case, the optional output variable ‘mesg’ gives more information.

Notes

“leastsq” is a wrapper around MINPACK’s lmdif and lmder algorithms.

cov_x is a Jacobian approximation to the Hessian of the least squares objective function. This approximation assumes that the objective function is based on the difference between some observed target data (ydata) and a (non-linear) function of the parameters f(xdata, params)

func(params) = ydata - f(xdata, params)


so that the objective function is

  min   sum((ydata - f(xdata, params))**2, axis=0)
params


The solution, x, is always a 1D array, regardless of the shape of x0, or whether x0 is a scalar.

### multi_voxel_fit

dipy.reconst.forecast.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

dipy.reconst.forecast.optional_package(name, trip_msg=None)

Return package-like thing and module setup for package name

Parameters
namestr

package name

trip_msgNone or str

message to give when someone tries to use the return package, but we could not import it, and have returned a TripWire object instead. Default message if None.

Returns
pkg_likemodule or TripWire instance

If we can import the package, return it. Otherwise return an object raising an error when accessed

have_pkgbool

True if import for package was successful, false otherwise

module_setupfunction

callable usually set as setup_module in calling namespace, to allow skipping tests.

Examples

Typical use would be something like this at the top of a module using an optional package:

>>> from dipy.utils.optpkg import optional_package
>>> pkg, have_pkg, setup_module = optional_package('not_a_package')


Of course in this case the package doesn’t exist, and so, in the module:

>>> have_pkg
False


and

>>> pkg.some_function()
Traceback (most recent call last):
...
TripWireError: We need package not_a_package for these functions, but
import not_a_package raised an ImportError


If the module does exist - we get the module

>>> pkg, _, _ = optional_package('os')
>>> hasattr(pkg, 'path')
True


Or a submodule if that’s what we asked for

>>> subpkg, _, _ = optional_package('os.path')
>>> hasattr(subpkg, 'dirname')
True


### psi_l

dipy.reconst.forecast.psi_l(l, b)

### real_sph_harm

dipy.reconst.forecast.real_sph_harm(m, n, theta, phi)

Compute real spherical harmonics.

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 broadcasted against each other.

Parameters
mint |m| <= n

The order of the harmonic.

nint >= 0

The degree of the harmonic.

thetafloat [0, 2*pi]

The azimuthal (longitudinal) coordinate.

phifloat [0, pi]

The polar (colatitudinal) coordinate.

Returns
y_mnreal float

The real harmonic $$Y^m_n$$ sampled at theta and phi.

scipy.special.sph_harm

### rho_matrix

dipy.reconst.forecast.rho_matrix(sh_order, vecs)

Compute the SH matrix $$\rho$$

### warn

dipy.reconst.forecast.warn(message, category=None, stacklevel=1, source=None)

Issue a warning, or maybe ignore it or raise an exception.

### FreeWaterTensorFit

class dipy.reconst.fwdti.FreeWaterTensorFit(model, model_params)

Class for fitting the Free Water Tensor 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

f

Returns the free water diffusion volume fraction f

quadratic_form

Calculates the 3x3 diffusion tensor for each voxel

shape

Methods

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