import numpy as np
from scipy.special import gamma, genlaguerre, hermite
from dipy.reconst.base import ReconstFit, ReconstModel
from dipy.reconst.cache import Cache
from dipy.reconst.multi_voxel import multi_voxel_fit
try: # preferred scipy >= 0.14, required scipy >= 1.0
from scipy.special import factorial as sfactorial, factorial2
except ImportError:
from scipy.misc import factorial as sfactorial, factorial2
from math import factorial as mfactorial
from warnings import warn
from dipy.core.geometry import cart2sphere
from dipy.core.gradients import gradient_table
from dipy.core.optimize import Optimizer, PositiveDefiniteLeastSquares
from dipy.data import load_sdp_constraints
import dipy.reconst.dti as dti
from dipy.reconst.shm import real_sh_descoteaux_from_index, sph_harm_ind_list
from dipy.testing.decorators import warning_for_keywords
from dipy.utils.optpkg import optional_package
cvxpy, have_cvxpy, _ = optional_package("cvxpy", min_version="1.4.1")
[docs]
class MapmriModel(ReconstModel, Cache):
r"""Mean Apparent Propagator MRI (MAPMRI) of the diffusion signal.
The main idea in MAPMRI footcite:p:`Ozarslan2013` is to model the diffusion
signal as a linear combination of the continuous functions presented in
footcite:p:`Ozarslan2008` but extending it in three dimensions.
The main difference with the SHORE proposed in footcite:p:`Merlet2013` is
that MAPMRI 3D extension is provided using a set of three basis functions
for the radial part, one for the signal along x, one for y and one for z,
while footcite:p:`Merlet2013` uses one basis function to model the radial
part and real Spherical Harmonics to model the angular part.
From the MAPMRI coefficients is possible to use the analytical formulae
to estimate the ODF.
See :footcite:p:`Avram2015` for additional tissue microstructure insights
provided by MAPMRI.
See also footcite:p:`Fick2016b`, footcite:p:`Cheng2012`,
footcite:p:`Hosseinbor2013`, footcite:p:`Craven1979`, and
footcite:p:`DelaHaije2020` for additional insight into to the model.
References
----------
.. footbibliography::
"""
@warning_for_keywords()
def __init__(
self,
gtab,
*,
radial_order=6,
laplacian_regularization=True,
laplacian_weighting=0.2,
positivity_constraint=False,
global_constraints=False,
pos_grid=15,
pos_radius="adaptive",
anisotropic_scaling=True,
eigenvalue_threshold=1e-04,
bval_threshold=np.inf,
dti_scale_estimation=True,
static_diffusivity=0.7e-3,
cvxpy_solver=None,
):
r"""Analytical and continuous modeling of the diffusion signal with
respect to the MAPMRI basis.
The main idea of the MAPMRI :footcite:p:`Ozarslan2013` is to model the
diffusion signal as a linear combination of the continuous functions
presented in :footcite:p:`Ozarslan2008` but extending it in three
dimensions.
The main difference with the SHORE proposed in
:footcite:p:`Ozarslan2009` is that MAPMRI 3D extension is provided using
a set of three basis functions for the radial part, one for the signal
along x, one for y and one for z, while :footcite:p:`Ozarslan2009` uses
one basis function to model the radial part and real Spherical Harmonics
to model the angular part.
From the MAPMRI coefficients it is possible to estimate various
q-space indices, the PDF and the ODF.
The fitting procedure can be constrained using the positivity
constraint proposed in :footcite:p:`Ozarslan2013` or
:footcite:p:`DelaHaije2020` and/or the laplacian regularization proposed
in :footcite:p:`Fick2016b`.
For the estimation of q-space indices we recommend using the 'regular'
anisotropic implementation of MAPMRI. However, it has been shown that
the ODF estimation in this implementation has a bias which
'squeezes together' the ODF peaks when there is a crossing at an angle
smaller than 90 degrees :footcite:p:`Fick2016b`. When you want to
estimate ODFs for tractography we therefore recommend using the
isotropic implementation (which is equivalent to
:footcite:p:`Ozarslan2009`).
The switch between isotropic and anisotropic can be easily made through
the anisotropic_scaling option.
Parameters
----------
gtab : GradientTable,
gradient directions and bvalues container class.
the gradient table has to include b0-images.
radial_order : unsigned int,
an even integer that represent the order of the basis
laplacian_regularization: bool,
Regularize using the Laplacian of the MAP-MRI basis.
laplacian_weighting: string or scalar,
The string 'GCV' makes it use generalized cross-validation
:footcite:p:`Craven1979` to find the regularization weight
:footcite:p:`DelaHaije2020`. A scalar sets the regularization
weight to that value and an array will make it selected the optimal
weight from the values in the array.
positivity_constraint : bool,
Constrain the propagator to be positive.
global_constraints : bool, optional
If set to False, positivity is enforced on a grid determined by
pos_grid and pos_radius. If set to True, positivity is enforced
everywhere using the constraints of :footcite:p:`Merlet2013`. Global
constraints are currently supported for anisotropic_scaling=True and
for radial_order <= 10.
pos_grid : int, optional
The number of points in the grid that is used in the local
positivity constraint.
pos_radius : float or string, optional
If set to a float, the maximum distance the local positivity
constraint constrains to posivity is that value. If set to
'adaptive', the maximum distance is dependent on the estimated
tissue diffusivity. If 'infinity', semidefinite programming
constraints are used :footcite:p:`DelaHaije2020`.
anisotropic_scaling : bool, optional
If True, uses the standard anisotropic MAP-MRI basis. If False,
uses the isotropic MAP-MRI basis (equal to 3D-SHORE).
eigenvalue_threshold : float, optional
Sets the minimum of the tensor eigenvalues in order to avoid
stability problem.
bval_threshold : float, optional
Sets the b-value threshold to be used in the scale factor
estimation. In order for the estimated non-Gaussianity to have
meaning this value should set to a lower value (b<2000 s/mm^2)
such that the scale factors are estimated on signal points that
reasonably represent the spins at Gaussian diffusion.
dti_scale_estimation : bool, optional
Whether or not DTI fitting is used to estimate the isotropic scale
factor for isotropic MAP-MRI.
When set to False the algorithm presets the isotropic tissue
diffusivity to static_diffusivity. This vastly increases fitting
speed but at the cost of slightly reduced fitting quality. Can
still be used in combination with regularization and constraints.
static_diffusivity : float, optional
the tissue diffusivity that is used when dti_scale_estimation is
set to False. The default is that of typical white matter
D=0.7e-3 :footcite:p:`Fick2016b`.
cvxpy_solver : str, optional
cvxpy solver name. Optionally optimize the positivity constraint
with a particular cvxpy solver. See https://www.cvxpy.org/ for
details.
Default: None (cvxpy chooses its own solver)
References
----------
.. footbibliography::
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 SHORE basis and compute the real and analytical
ODF.
>>> from dipy.data import dsi_voxels, default_sphere
>>> from dipy.core.gradients import gradient_table
>>> _, gtab_ = dsi_voxels()
>>> gtab = gradient_table(gtab_.bvals, bvecs=gtab_.bvecs,
... b0_threshold=gtab_.bvals.min())
>>> from dipy.sims.voxel import sticks_and_ball
>>> data, golden_directions = sticks_and_ball(gtab, d=0.0015, S0=1,
... angles=[(0, 0),
... (90, 0)],
... fractions=[50, 50],
... snr=None)
>>> from dipy.reconst.mapmri import MapmriModel
>>> radial_order = 4
>>> map_model = MapmriModel(gtab, radial_order=radial_order)
>>> mapfit = map_model.fit(data)
>>> odf = mapfit.odf(default_sphere)
"""
if np.sum(gtab.b0s_mask) == 0:
raise ValueError(
"gtab does not have any b0s, check in the "
"gradient_table if b0_threshold needs to be "
"increased."
)
self.gtab = gtab
if radial_order < 0 or radial_order % 2:
raise ValueError("radial_order must be a positive, even number.")
self.radial_order = radial_order
self.bval_threshold = bval_threshold
self.dti_scale_estimation = dti_scale_estimation
if laplacian_regularization:
msg = (
"Laplacian Regularization weighting must be 'GCV',"
" a positive float or an array of positive floats."
)
if isinstance(laplacian_weighting, str):
if not laplacian_weighting == "GCV":
raise ValueError(msg)
elif isinstance(laplacian_weighting, (float, np.ndarray)):
if np.sum(laplacian_weighting < 0) > 0:
raise ValueError(msg)
self.laplacian_weighting = laplacian_weighting
self.laplacian_regularization = laplacian_regularization
if positivity_constraint:
if not have_cvxpy:
raise ImportError("CVXPY package needed to enforce constraints.")
if cvxpy_solver is not None:
if cvxpy_solver not in cvxpy.installed_solvers():
installed_solvers = ", ".join(cvxpy.installed_solvers())
raise ValueError(
f"Input `cvxpy_solver` was set to"
f" {cvxpy_solver}. One of"
f" {installed_solvers} was expected."
)
self.cvxpy_solver = cvxpy_solver
if global_constraints:
if not anisotropic_scaling:
raise ValueError(
"Global constraints only available for"
" anistropic_scaling=True."
)
if radial_order > 10:
self.sdp_constraints = load_sdp_constraints("hermite", order=10)
warn(
"Global constraints are currently supported for"
" radial_order <= 10.",
stacklevel=2,
)
else:
self.sdp_constraints = load_sdp_constraints(
"hermite", order=radial_order
)
m = (2 + radial_order) * (4 + radial_order) * (3 + 2 * radial_order)
m = m // 24
self.sdp = PositiveDefiniteLeastSquares(m, A=self.sdp_constraints)
else:
msg = "pos_radius must be 'adaptive' or a positive float."
if isinstance(pos_radius, str):
if pos_radius != "adaptive":
raise ValueError(msg)
elif isinstance(pos_radius, (float, int)):
if pos_radius <= 0:
raise ValueError(msg)
self.constraint_grid = create_rspace(pos_grid, pos_radius)
if not anisotropic_scaling:
self.pos_K_independent = mapmri_isotropic_K_mu_independent(
radial_order, self.constraint_grid
)
else:
raise ValueError(msg)
self.pos_grid = pos_grid
self.pos_radius = pos_radius
self.global_constraints = global_constraints
self.positivity_constraint = positivity_constraint
self.anisotropic_scaling = anisotropic_scaling
if (gtab.big_delta is None) or (gtab.small_delta is None):
self.tau = 1 / (4 * np.pi**2)
else:
self.tau = gtab.big_delta - gtab.small_delta / 3.0
self.eigenvalue_threshold = eigenvalue_threshold
self.cutoff = gtab.bvals < self.bval_threshold
gtab_cutoff = gradient_table(
bvals=self.gtab.bvals[self.cutoff], bvecs=self.gtab.bvecs[self.cutoff]
)
self.tenmodel = dti.TensorModel(gtab_cutoff)
if self.anisotropic_scaling:
self.ind_mat = mapmri_index_matrix(self.radial_order)
self.Bm = b_mat(self.ind_mat)
self.S_mat, self.T_mat, self.U_mat = mapmri_STU_reg_matrices(radial_order)
else:
self.ind_mat = mapmri_isotropic_index_matrix(self.radial_order)
self.Bm = b_mat_isotropic(self.ind_mat)
self.laplacian_matrix = mapmri_isotropic_laplacian_reg_matrix(
radial_order, 1.0
)
qvals = np.sqrt(self.gtab.bvals / self.tau) / (2 * np.pi)
q = gtab.bvecs * qvals[:, None]
if self.dti_scale_estimation:
self.M_mu_independent = mapmri_isotropic_M_mu_independent(
self.radial_order, q
)
else:
D = static_diffusivity
mumean = np.sqrt(2 * D * self.tau)
self.mu = np.array([mumean, mumean, mumean])
self.M = mapmri_isotropic_phi_matrix(radial_order, mumean, q)
if (
self.laplacian_regularization
and isinstance(laplacian_weighting, float)
and not positivity_constraint
):
MMt = (
np.dot(self.M.T, self.M)
+ laplacian_weighting * mumean * self.laplacian_matrix
)
self.MMt_inv_Mt = np.dot(np.linalg.pinv(MMt), self.M.T)
@multi_voxel_fit
def fit(self, data, **kwargs):
errorcode = 0
tenfit = self.tenmodel.fit(data[self.cutoff])
evals = tenfit.evals
R = tenfit.evecs
evals = np.clip(evals, self.eigenvalue_threshold, evals.max())
qvals = np.sqrt(self.gtab.bvals / self.tau) / (2 * np.pi)
mu_max = max(np.sqrt(evals * 2 * self.tau)) # used for constraint
if self.anisotropic_scaling:
mu = np.sqrt(evals * 2 * self.tau)
qvecs = np.dot(self.gtab.bvecs, R)
q = qvecs * qvals[:, None]
M = mapmri_phi_matrix(self.radial_order, mu, q)
else:
try:
# self.MMt_inv_Mt
lopt = self.laplacian_weighting
coef = np.dot(self.MMt_inv_Mt, data)
coef = coef / sum(coef * self.Bm)
return MapmriFit(self, coef, self.mu, R, lopt, errorcode=errorcode)
except AttributeError:
try:
M = self.M
mu = self.mu
except AttributeError:
u0 = isotropic_scale_factor(evals * 2 * self.tau)
mu = np.array([u0, u0, u0])
M_mu_dependent = mapmri_isotropic_M_mu_dependent(
self.radial_order, mu[0], qvals
)
M = M_mu_dependent * self.M_mu_independent
if self.laplacian_regularization:
if self.anisotropic_scaling:
laplacian_matrix = mapmri_laplacian_reg_matrix(
self.ind_mat, mu, self.S_mat, self.T_mat, self.U_mat
)
else:
laplacian_matrix = self.laplacian_matrix * mu[0]
if (
isinstance(self.laplacian_weighting, str)
and self.laplacian_weighting.upper() == "GCV"
):
try:
lopt = generalized_crossvalidation(data, M, laplacian_matrix)
except np.linalg.linalg.LinAlgError:
# 1/0.
lopt = 0.05
errorcode = 1
elif np.isscalar(self.laplacian_weighting):
lopt = self.laplacian_weighting
else:
lopt = generalized_crossvalidation_array(
data, M, laplacian_matrix, weights_array=self.laplacian_weighting
)
else:
lopt = 0.0
laplacian_matrix = np.ones((self.ind_mat.shape[0], self.ind_mat.shape[0]))
if self.positivity_constraint:
data_norm = np.asarray(data / data[self.gtab.b0s_mask].mean())
if self.global_constraints:
coef = self.sdp.solve(M, data_norm, solver=self.cvxpy_solver)
else:
c = cvxpy.Variable(M.shape[1])
design_matrix = cvxpy.Constant(M) @ c
# workaround for the bug on cvxpy 1.0.15 when lopt = 0
# See https://github.com/cvxgrp/cvxpy/issues/672
if not lopt:
objective = cvxpy.Minimize(
cvxpy.sum_squares(design_matrix - data_norm)
)
else:
objective = cvxpy.Minimize(
cvxpy.sum_squares(design_matrix - data_norm)
+ lopt * cvxpy.quad_form(c, laplacian_matrix)
)
if self.pos_radius == "adaptive":
# custom constraint grid based on scale factor [Avram2015]
constraint_grid = create_rspace(self.pos_grid, np.sqrt(5) * mu_max)
else:
constraint_grid = self.constraint_grid
if self.anisotropic_scaling:
K = mapmri_psi_matrix(self.radial_order, mu, constraint_grid)
else:
if self.pos_radius == "adaptive":
# grid changes per voxel. Recompute entire K matrix.
K = mapmri_isotropic_psi_matrix(
self.radial_order, mu[0], constraint_grid
)
else:
# grid is static. Only compute mu-dependent part of K.
K_dependent = mapmri_isotropic_K_mu_dependent(
self.radial_order, mu[0], constraint_grid
)
K = K_dependent * self.pos_K_independent
M0 = M[self.gtab.b0s_mask, :]
constraints = [(M0[0] @ c) == 1, (K @ c) >= -0.1]
prob = cvxpy.Problem(objective, constraints)
try:
prob.solve(solver=self.cvxpy_solver)
coef = np.asarray(c.value).squeeze()
except Exception:
errorcode = 2
warn("Optimization did not find a solution", stacklevel=2)
try:
coef = np.dot(np.linalg.pinv(M), data) # least squares
except np.linalg.linalg.LinAlgError:
errorcode = 3
coef = np.zeros(M.shape[1])
return MapmriFit(self, coef, mu, R, lopt, errorcode=errorcode)
else:
try:
pseudoInv = np.dot(
np.linalg.inv(np.dot(M.T, M) + lopt * laplacian_matrix), M.T
)
coef = np.dot(pseudoInv, data)
except np.linalg.linalg.LinAlgError:
errorcode = 1
coef = np.zeros(M.shape[1])
return MapmriFit(self, coef, mu, R, lopt, errorcode=errorcode)
coef = coef / sum(coef * self.Bm)
return MapmriFit(self, coef, mu, R, lopt, errorcode=errorcode)
[docs]
class MapmriFit(ReconstFit):
@warning_for_keywords()
def __init__(self, model, mapmri_coef, mu, R, lopt, *, errorcode=0):
"""Calculates diffusion properties for a single voxel
Parameters
----------
model : object,
AnalyticalModel
mapmri_coef : 1d ndarray,
mapmri coefficients
mu : array, shape (3,)
scale parameters vector for x, y and z
R : array, shape (3,3)
rotation matrix
lopt : float,
regularization weight used for laplacian regularization
errorcode : int
provides information on whether errors occurred in the fitting
of each voxel. 0 means no problem, 1 means a LinAlgError
occurred when trying to invert the design matrix. 2 means the
positivity constraint was unable to solve the problem. 3 means
that after positivity constraint failed, also matrix inversion
failed.
"""
self.model = model
self._mapmri_coef = mapmri_coef
self.gtab = model.gtab
self.radial_order = model.radial_order
self.mu = mu
self.R = R
self.lopt = lopt
self.errorcode = errorcode
@property
def mapmri_mu(self):
"""The MAPMRI scale factors"""
return self.mu
@property
def mapmri_R(self):
"""The MAPMRI rotation matrix"""
return self.R
@property
def mapmri_coeff(self):
"""The MAPMRI coefficients"""
return self._mapmri_coef
[docs]
@warning_for_keywords()
def odf(self, sphere, *, s=2):
"""Calculates the analytical Orientation Distribution Function (ODF)
from the signal.
See :footcite:p:`Ozarslan2013` Eq. (32).
Parameters
----------
sphere : Sphere
A Sphere instance with vertices, edges and faces attributes.
s : unsigned int
radial moment of the ODF
References
----------
.. footbibliography::
"""
if self.model.anisotropic_scaling:
v_ = sphere.vertices
v = np.dot(v_, self.R)
I_s = mapmri_odf_matrix(self.radial_order, self.mu, s, v)
odf = np.dot(I_s, self._mapmri_coef)
else:
_I = self.model.cache_get("ODF_matrix", key=(sphere, s))
if _I is None:
_I = mapmri_isotropic_odf_matrix(
self.radial_order, 1, s, sphere.vertices
)
self.model.cache_set("ODF_matrix", (sphere, s), _I)
odf = self.mu[0] ** s * np.dot(_I, self._mapmri_coef)
return odf
[docs]
@warning_for_keywords()
def odf_sh(self, *, s=2):
"""Calculates the real analytical odf for a given discrete sphere.
Computes the design matrix of the ODF for the given sphere vertices
and radial moment :footcite:p:`Ozarslan2013` eq. (32). The radial moment
s acts as a sharpening method. The analytical equation for the spherical
ODF basis is given in :footcite:p:`Fick2016b` eq. (C8).
References
----------
.. footbibliography::
"""
if self.model.anisotropic_scaling:
raise ValueError(
"odf in spherical harmonics not yet implemented "
"for anisotropic implementation"
)
_I = self.model.cache_get("ODF_sh_matrix", key=(self.radial_order, s))
if _I is None:
_I = mapmri_isotropic_odf_sh_matrix(self.radial_order, 1, s)
self.model.cache_set("ODF_sh_matrix", (self.radial_order, s), _I)
odf = self.mu[0] ** s * np.dot(_I, self._mapmri_coef)
return odf
[docs]
def rtpp(self):
"""Calculates the analytical return to the plane probability (RTPP).
RTPP is defined in :footcite:p:`Ozarslan2013` eq. (42). The analytical
formula for the isotropic MAP-MRI basis was derived in
:footcite:p:`Fick2016b` eq. (C11).
References
----------
.. footbibliography::
"""
Bm = self.model.Bm
ind_mat = self.model.ind_mat
if self.model.anisotropic_scaling:
sel = Bm > 0.0 # select only relevant coefficients
const = 1 / (np.sqrt(2 * np.pi) * self.mu[0])
ind_sum = (-1.0) ** (ind_mat[sel, 0] / 2.0)
rtpp_vec = const * Bm[sel] * ind_sum * self._mapmri_coef[sel]
rtpp = rtpp_vec.sum()
return rtpp
else:
rtpp_vec = np.zeros((ind_mat.shape[0]))
count = 0
for n in range(0, self.model.radial_order + 1, 2):
for j in range(1, 2 + n // 2):
ell = n + 2 - 2 * j
const = (-1 / 2.0) ** (ell / 2) / np.sqrt(np.pi)
matsum = 0
for k in range(0, j):
matsum += (
(-1) ** k
* binomialfloat(j + ell - 0.5, j - k - 1)
* gamma(ell / 2 + k + 1 / 2.0)
/ (sfactorial(k) * 0.5 ** (ell / 2 + 1 / 2.0 + k))
)
for _ in range(-ell, ell + 1):
rtpp_vec[count] = const * matsum
count += 1
direction = np.array(self.R[:, 0], ndmin=2)
r, theta, phi = cart2sphere(
direction[:, 0], direction[:, 1], direction[:, 2]
)
rtpp = (
self._mapmri_coef
* (1 / self.mu[0])
* rtpp_vec
* real_sh_descoteaux_from_index(
ind_mat[:, 2], ind_mat[:, 1], theta, phi
)
)
return rtpp.sum()
[docs]
def rtap(self):
"""Calculates the analytical return to the axis probability (RTAP).
RTAP is defined in :footcite:p:`Ozarslan2013` eq. (40, 44a). The
analytical formula for the isotropic MAP-MRI basis was derived in
:footcite:p:`Fick2016b` eq. (C11).
References
----------
.. footbibliography::
"""
Bm = self.model.Bm
ind_mat = self.model.ind_mat
if self.model.anisotropic_scaling:
sel = Bm > 0.0 # select only relevant coefficients
const = 1 / (2 * np.pi * np.prod(self.mu[1:]))
ind_sum = (-1.0) ** (np.sum(ind_mat[sel, 1:], axis=1) / 2.0)
rtap_vec = const * Bm[sel] * ind_sum * self._mapmri_coef[sel]
rtap = np.sum(rtap_vec)
else:
rtap_vec = np.zeros((ind_mat.shape[0]))
count = 0
for n in range(0, self.model.radial_order + 1, 2):
for j in range(1, 2 + n // 2):
ell = n + 2 - 2 * j
kappa = ((-1) ** (j - 1) * 2 ** (-(ell + 3) / 2.0)) / np.pi
matsum = 0
for k in range(0, j):
matsum += (
(-1) ** k
* binomialfloat(j + ell - 0.5, j - k - 1)
* gamma((ell + 1) / 2.0 + k)
) / (sfactorial(k) * 0.5 ** ((ell + 1) / 2.0 + k))
for _ in range(-ell, ell + 1):
rtap_vec[count] = kappa * matsum
count += 1
rtap_vec *= 2
direction = np.array(self.R[:, 0], ndmin=2)
r, theta, phi = cart2sphere(
direction[:, 0], direction[:, 1], direction[:, 2]
)
rtap_vec = (
self._mapmri_coef
* (1 / self.mu[0] ** 2)
* rtap_vec
* real_sh_descoteaux_from_index(
ind_mat[:, 2], ind_mat[:, 1], theta, phi
)
)
rtap = rtap_vec.sum()
return rtap
[docs]
def rtop(self):
"""Calculates the analytical return to the origin probability (RTOP).
RTOP is defined in :footcite:p:`Ozarslan2013` eq. (36, 43). The
analytical formula for the isotropic MAP-MRI basis was derived in
:footcite:p:`Fick2016b` eq. (C11).
References
----------
.. footbibliography::
"""
Bm = self.model.Bm
if self.model.anisotropic_scaling:
const = 1 / (np.sqrt(8 * np.pi**3) * np.prod(self.mu))
ind_sum = (-1.0) ** (np.sum(self.model.ind_mat, axis=1) / 2)
rtop_vec = const * ind_sum * Bm * self._mapmri_coef
rtop = rtop_vec.sum()
else:
const = 1 / (2 * np.sqrt(2.0) * np.pi ** (3 / 2.0))
rtop_vec = const * (-1.0) ** (self.model.ind_mat[:, 0] - 1) * Bm
rtop = (1 / self.mu[0] ** 3) * rtop_vec * self._mapmri_coef
rtop = rtop.sum()
return rtop
[docs]
def msd(self):
"""Calculates the analytical Mean Squared Displacement (MSD).
It is defined as the Laplacian of the origin of the estimated signal
:footcite:p:`Cheng2012`. The analytical formula for the MAP-MRI basis
was derived in :footcite:p:`Fick2016b` eq. (C13, D1).
References
----------
.. footbibliography::
"""
mu = self.mu
ind_mat = self.model.ind_mat
Bm = self.model.Bm
sel = self.model.Bm > 0.0 # select only relevant coefficients
mapmri_coef = self._mapmri_coef[sel]
if self.model.anisotropic_scaling:
ind_sum = np.sum(ind_mat[sel], axis=1)
nx, ny, nz = ind_mat[sel].T
numerator = (
(-1) ** (0.5 * (-ind_sum))
* np.pi ** (3 / 2.0)
* (
(1 + 2 * nx) * mu[0] ** 2
+ (1 + 2 * ny) * mu[1] ** 2
+ (1 + 2 * nz) * mu[2] ** 2
)
)
denominator = (
np.sqrt(
2.0 ** (-ind_sum) * sfactorial(nx) * sfactorial(ny) * sfactorial(nz)
)
* gamma(0.5 - 0.5 * nx)
* gamma(0.5 - 0.5 * ny)
* gamma(0.5 - 0.5 * nz)
)
msd_vec = self._mapmri_coef[sel] * (numerator / denominator)
msd = msd_vec.sum()
else:
msd_vec = (4 * ind_mat[sel, 0] - 1) * Bm[sel]
msd = self.mu[0] ** 2 * msd_vec * mapmri_coef
msd = msd.sum()
return msd
[docs]
def qiv(self):
"""Calculates the analytical Q-space Inverse Variance (QIV).
It is defined as the inverse of the Laplacian of the origin of the
estimated propagator :footcite:p:`Hosseinbor2013` eq. (22). The
analytical formula for the MAP-MRI basis was derived in
:footcite:p:`Fick2016b` eq. (C14, D2).
References
----------
.. footbibliography::
"""
ux, uy, uz = self.mu
ind_mat = self.model.ind_mat
if self.model.anisotropic_scaling:
sel = self.model.Bm > 0 # select only relevant coefficients
nx, ny, nz = ind_mat[sel].T
numerator = (
8
* np.pi**2
* (ux * uy * uz) ** 3
* np.sqrt(sfactorial(nx) * sfactorial(ny) * sfactorial(nz))
* gamma(0.5 - 0.5 * nx)
* gamma(0.5 - 0.5 * ny)
* gamma(0.5 - 0.5 * nz)
)
denominator = np.sqrt(2.0 ** (-1 + nx + ny + nz)) * (
(1 + 2 * nx) * uy**2 * uz**2
+ ux**2 * ((1 + 2 * nz) * uy**2 + (1 + 2 * ny) * uz**2)
)
qiv_vec = self._mapmri_coef[sel] * (numerator / denominator)
qiv = qiv_vec.sum()
else:
sel = self.model.Bm > 0.0 # select only relevant coefficients
j = ind_mat[sel, 0]
qiv_vec = (8 * (-1.0) ** (1 - j) * np.sqrt(2) * np.pi ** (7 / 2.0)) / (
(4.0 * j - 1) * self.model.Bm[sel]
)
qiv = ux**5 * qiv_vec * self._mapmri_coef[sel]
qiv = qiv.sum()
return qiv
[docs]
def ng(self):
r"""Calculates the analytical non-Gaussiannity (NG).
For the NG to be meaningful the mapmri scale factors must be estimated
only on data representing Gaussian diffusion of spins, i.e., bvals
smaller than about 2000 s/mm^2 :footcite:p:`Avram2015`.
See :footcite:p:`Ozarslan2013` for a definition of the metric.
References
----------
.. footbibliography::
"""
if self.model.bval_threshold > 2000.0:
warn(
"model bval_threshold must be lower than 2000 for the "
"non_Gaussianity to be physically meaningful [2].",
stacklevel=2,
)
if not self.model.anisotropic_scaling:
raise ValueError(
"Parallel non-Gaussianity is not defined using isotropic scaling."
)
coef = self._mapmri_coef
return np.sqrt(1 - coef[0] ** 2 / np.sum(coef**2))
[docs]
def ng_parallel(self):
r"""Calculates the analytical parallel non-Gaussiannity (NG).
For the NG to be meaningful the mapmri scale factors must be estimated
only on data representing Gaussian diffusion of spins, i.e., bvals
smaller than about 2000 s/mm^2 :footcite:p:`Avram2015`.
See :footcite:p:`Ozarslan2013` for a definition of the metric.
References
----------
.. footbibliography::
"""
if self.model.bval_threshold > 2000.0:
warn(
"Model bval_threshold must be lower than 2000 for the "
"non_Gaussianity to be physically meaningful [2].",
stacklevel=2,
)
if not self.model.anisotropic_scaling:
raise ValueError(
"Parallel non-Gaussianity is not defined using isotropic scaling."
)
ind_mat = self.model.ind_mat
coef = self._mapmri_coef
a_par = np.zeros_like(coef)
a0 = np.zeros_like(coef)
for i in range(coef.shape[0]):
n1, n2, n3 = ind_mat[i]
if (n2 % 2 + n3 % 2) == 0:
a_par[i] = (
coef[i]
* (-1) ** ((n2 + n3) / 2)
* np.sqrt(sfactorial(n2) * sfactorial(n3))
/ (factorial2(n2) * factorial2(n3))
)
if n1 == 0:
a0[i] = a_par[i]
return np.sqrt(1 - np.sum(a0**2) / np.sum(a_par**2))
[docs]
def ng_perpendicular(self):
r"""Calculates the analytical perpendicular non-Gaussiannity (NG)
For the NG to be meaningful the mapmri scale factors must be estimated
only on data representing Gaussian diffusion of spins, i.e., bvals
smaller than about 2000 s/mm^2 :footcite:p:`Avram2015`.
See :footcite:p:`Ozarslan2013` for a definition of the metric.
References
----------
.. footbibliography::
"""
if self.model.bval_threshold > 2000.0:
warn(
"model bval_threshold must be lower than 2000 for the "
"non_Gaussianity to be physically meaningful [2].",
stacklevel=2,
)
if not self.model.anisotropic_scaling:
raise ValueError(
"Parallel non-Gaussianity is not defined using isotropic scaling."
)
ind_mat = self.model.ind_mat
coef = self._mapmri_coef
a_perp = np.zeros_like(coef)
a00 = np.zeros_like(coef)
for i in range(coef.shape[0]):
n1, n2, n3 = ind_mat[i]
if n1 % 2 == 0:
if n2 % 2 == 0 and n3 % 2 == 0:
a_perp[i] = (
coef[i]
* (-1) ** (n1 / 2)
* np.sqrt(sfactorial(n1))
/ factorial2(n1)
)
if n2 == 0 and n3 == 0:
a00[i] = a_perp[i]
return np.sqrt(1 - np.sum(a00**2) / np.sum(a_perp**2))
[docs]
def norm_of_laplacian_signal(self):
"""Calculates the norm of the laplacian of the fitted signal.
This information could be useful to assess if the extrapolation of the
fitted signal contains spurious oscillations. A high laplacian may
indicate that these are present, and any q-space indices that use
integrals of the signal may be corrupted (e.g. RTOP, RTAP, RTPP, QIV).
See :footcite:p:`Fick2016b` for a definition of the metric.
References
----------
.. footbibliography::
"""
if self.model.anisotropic_scaling:
laplacian_matrix = mapmri_laplacian_reg_matrix(
self.model.ind_mat,
self.mu,
self.model.S_mat,
self.model.T_mat,
self.model.U_mat,
)
else:
laplacian_matrix = self.mu[0] * self.model.laplacian_matrix
norm_of_laplacian = np.linalg.multi_dot(
[self._mapmri_coef, laplacian_matrix, self._mapmri_coef]
)
return norm_of_laplacian
[docs]
@warning_for_keywords()
def fitted_signal(self, *, gtab=None):
"""Recovers the fitted signal for the given gradient table. If no
gradient table is given it recovers the signal for the gtab of the model
object.
"""
if gtab is None:
E = self.predict(self.model.gtab, S0=1.0)
else:
E = self.predict(gtab, S0=1.0)
return E
[docs]
@warning_for_keywords()
def predict(self, qvals_or_gtab, *, S0=100.0):
"""Recovers the reconstructed signal for any qvalue array or gradient
table.
"""
if isinstance(qvals_or_gtab, np.ndarray):
q = qvals_or_gtab
# qvals = np.linalg.norm(q, axis=1)
else:
gtab = qvals_or_gtab
qvals = np.sqrt(gtab.bvals / self.model.tau) / (2 * np.pi)
q = qvals[:, None] * gtab.bvecs
if self.model.anisotropic_scaling:
q_rot = np.dot(q, self.R)
M = mapmri_phi_matrix(self.radial_order, self.mu, q_rot)
else:
M = mapmri_isotropic_phi_matrix(self.radial_order, self.mu[0], q)
E = S0 * np.dot(M, self._mapmri_coef)
return E
[docs]
def pdf(self, r_points):
"""Diffusion propagator on a given set of real points.
if the array r_points is non writeable, then intermediate
results are cached for faster recalculation
"""
if self.model.anisotropic_scaling:
r_point_rotated = np.dot(r_points, self.R)
K = mapmri_psi_matrix(self.radial_order, self.mu, r_point_rotated)
EAP = np.dot(K, self._mapmri_coef)
else:
if not r_points.flags.writeable:
K_independent = self.model.cache_get(
"mapmri_matrix_pdf_independent", key=hash(r_points.data)
)
if K_independent is None:
K_independent = mapmri_isotropic_K_mu_independent(
self.radial_order, r_points
)
self.model.cache_set(
"mapmri_matrix_pdf_independent",
hash(r_points.data),
K_independent,
)
K_dependent = mapmri_isotropic_K_mu_dependent(
self.radial_order, self.mu[0], r_points
)
K = K_dependent * K_independent
else:
K = mapmri_isotropic_psi_matrix(self.radial_order, self.mu[0], r_points)
EAP = np.dot(K, self._mapmri_coef)
return EAP
[docs]
def isotropic_scale_factor(mu_squared):
"""Estimated isotropic scaling factor.
See :footcite:p:`Ozarslan2013` Eq. (49).
Parameters
----------
mu_squared : array, shape (N,3)
squared scale factors of mapmri basis in x, y, z
Returns
-------
u0 : float
closest isotropic scale factor for the isotropic basis
References
----------
.. footbibliography::
"""
X, Y, Z = mu_squared
coef_array = np.array([-3, -(X + Y + Z), (X * Y + X * Z + Y * Z), 3 * X * Y * Z])
# take the real, positive root of the problem.
u0 = np.sqrt(np.real(np.roots(coef_array).max()))
return u0
[docs]
def mapmri_index_matrix(radial_order):
"""Calculates the indices for the MAPMRI basis in x, y and z.
See :footcite:p:`Ozarslan2013` for a definition of MAPMRI.
Parameters
----------
radial_order : unsigned int
radial order of MAPMRI basis
Returns
-------
index_matrix : array, shape (N,3)
ordering of the basis in x, y, z
References
----------
.. footbibliography::
"""
index_matrix = []
for n in range(0, radial_order + 1, 2):
for i in range(0, n + 1):
for j in range(0, n - i + 1):
index_matrix.append([n - i - j, j, i])
return np.array(index_matrix)
[docs]
def b_mat(index_matrix):
"""Calculates the B coefficients from
See :footcite:p:`Ozarslan2013` Eq. (27).
Parameters
----------
index_matrix : array, shape (N,3)
ordering of the basis in x, y, z
Returns
-------
B : array, shape (N,)
B coefficients for the basis
References
----------
.. footbibliography::
"""
B = np.zeros(index_matrix.shape[0])
for i in range(index_matrix.shape[0]):
n1, n2, n3 = index_matrix[i]
K = int(not (n1 % 2) and not (n2 % 2) and not (n3 % 2))
B[i] = (
K
* np.sqrt(sfactorial(n1) * sfactorial(n2) * sfactorial(n3))
/ (factorial2(n1) * factorial2(n2) * factorial2(n3))
)
return B
[docs]
def b_mat_isotropic(index_matrix):
"""Calculates the isotropic B coefficients.
See :footcite:p:`Ozarslan2013` Fig 8.
Parameters
----------
index_matrix : array, shape (N,3)
ordering of the isotropic basis in j, l, m
Returns
-------
B : array, shape (N,)
B coefficients for the isotropic basis
References
----------
.. footbibliography::
"""
B = np.zeros((index_matrix.shape[0]))
for i in range(index_matrix.shape[0]):
if index_matrix[i, 1] == 0:
B[i] = genlaguerre(index_matrix[i, 0] - 1, 0.5)(0)
return B
[docs]
def mapmri_phi_1d(n, q, mu):
"""One dimensional MAPMRI basis function.
See :footcite:p:`Ozarslan2013` Eq. (4).
Parameters
----------
n : unsigned int
order of the basis
q : array, shape (N,)
points in the q-space in which evaluate the basis
mu : float
scale factor of the basis
References
----------
.. footbibliography::
"""
qn = 2 * np.pi * mu * q
H = hermite(n)(qn)
i = complex(0, 1)
f = mfactorial(n)
k = i ** (-n) / np.sqrt(2**n * f)
phi = k * np.exp(-(qn**2) / 2) * H
return phi
[docs]
def mapmri_phi_matrix(radial_order, mu, q_gradients):
"""Compute the MAPMRI phi matrix for the signal.
See :footcite:p:`Ozarslan2013` eq. (23).
Parameters
----------
radial_order : unsigned int,
an even integer that represent the order of the basis
mu : array, shape (3,)
scale factors of the basis for x, y, z
q_gradients : array, shape (N,3)
points in the q-space in which evaluate the basis
References
----------
.. footbibliography::
"""
ind_mat = mapmri_index_matrix(radial_order)
n_elem = ind_mat.shape[0]
n_qgrad = q_gradients.shape[0]
qx, qy, qz = q_gradients.T
mux, muy, muz = mu
Mx_storage = np.array(np.zeros((n_qgrad, radial_order + 1)), dtype=complex)
My_storage = np.array(np.zeros((n_qgrad, radial_order + 1)), dtype=complex)
Mz_storage = np.array(np.zeros((n_qgrad, radial_order + 1)), dtype=complex)
M = np.zeros((n_qgrad, n_elem))
for n in range(radial_order + 1):
Mx_storage[:, n] = mapmri_phi_1d(n, qx, mux)
My_storage[:, n] = mapmri_phi_1d(n, qy, muy)
Mz_storage[:, n] = mapmri_phi_1d(n, qz, muz)
counter = 0
for nx, ny, nz in ind_mat:
M[:, counter] = np.real(
Mx_storage[:, nx] * My_storage[:, ny] * Mz_storage[:, nz]
)
counter += 1
return M
[docs]
def mapmri_psi_1d(n, x, mu):
"""One dimensional MAPMRI propagator basis function.
See :footcite:p:`Ozarslan2013` Eq. (10).
Parameters
----------
n : unsigned int
order of the basis
x : array, shape (N,)
points in the r-space in which evaluate the basis
mu : float
scale factor of the basis
References
----------
.. footbibliography::
"""
H = hermite(n)(x / mu)
f = mfactorial(n)
k = 1 / (np.sqrt(2 ** (n + 1) * np.pi * f) * mu)
psi = k * np.exp(-(x**2) / (2 * mu**2)) * H
return psi
[docs]
def mapmri_psi_matrix(radial_order, mu, rgrad):
"""Compute the MAPMRI psi matrix for the propagator.
See :footcite:p:`Ozarslan2013` eq. (22).
Parameters
----------
radial_order : unsigned int,
an even integer that represent the order of the basis
mu : array, shape (3,)
scale factors of the basis for x, y, z
rgrad : array, shape (N,3)
points in the r-space in which evaluate the EAP
References
----------
.. footbibliography::
"""
ind_mat = mapmri_index_matrix(radial_order)
n_elem = ind_mat.shape[0]
n_qgrad = rgrad.shape[0]
rx, ry, rz = rgrad.T
mux, muy, muz = mu
Kx_storage = np.zeros((n_qgrad, radial_order + 1))
Ky_storage = np.zeros((n_qgrad, radial_order + 1))
Kz_storage = np.zeros((n_qgrad, radial_order + 1))
K = np.zeros((n_qgrad, n_elem))
for n in range(radial_order + 1):
Kx_storage[:, n] = mapmri_psi_1d(n, rx, mux)
Ky_storage[:, n] = mapmri_psi_1d(n, ry, muy)
Kz_storage[:, n] = mapmri_psi_1d(n, rz, muz)
counter = 0
for nx, ny, nz in ind_mat:
K[:, counter] = Kx_storage[:, nx] * Ky_storage[:, ny] * Kz_storage[:, nz]
counter += 1
return K
[docs]
def mapmri_odf_matrix(radial_order, mu, s, vertices):
"""Compute the MAPMRI ODF matrix.
See :footcite:p:`Ozarslan2013` Eq. (33).
Parameters
----------
radial_order : unsigned int,
an even integer that represent the order of the basis
mu : array, shape (3,)
scale factors of the basis for x, y, z
s : unsigned int
radial moment of the ODF
vertices : array, shape (N,3)
points of the sphere shell in the r-space in which evaluate the ODF
References
----------
.. footbibliography::
"""
ind_mat = mapmri_index_matrix(radial_order)
n_vert = vertices.shape[0]
n_elem = ind_mat.shape[0]
odf_mat = np.zeros((n_vert, n_elem))
mux, muy, muz = mu
# Eq, 35a
rho = 1.0 / np.sqrt(
(vertices[:, 0] / mux) ** 2
+ (vertices[:, 1] / muy) ** 2
+ (vertices[:, 2] / muz) ** 2
)
# Eq, 35b
alpha = 2 * rho * (vertices[:, 0] / mux)
# Eq, 35c
beta = 2 * rho * (vertices[:, 1] / muy)
# Eq, 35d
gamma = 2 * rho * (vertices[:, 2] / muz)
const = rho ** (3 + s) / np.sqrt(
2 ** (2 - s) * np.pi**3 * (mux**2 * muy**2 * muz**2)
)
for j in range(n_elem):
n1, n2, n3 = ind_mat[j]
f = np.sqrt(sfactorial(n1) * sfactorial(n2) * sfactorial(n3))
odf_mat[:, j] = const * f * _odf_cfunc(n1, n2, n3, alpha, beta, gamma, s)
return odf_mat
def _odf_cfunc(n1, n2, n3, a, b, g, s):
"""Compute the MAPMRI ODF function.
See :footcite:p:`Ozarslan2013` Eq. (34).
References
----------
.. footbibliography::
"""
f = mfactorial
f2 = factorial2
sumc = 0
for i in range(0, n1 + 1, 2):
for j in range(0, n2 + 1, 2):
for k in range(0, n3 + 1, 2):
nn = n1 + n2 + n3 - i - j - k
gam = (-1) ** ((i + j + k) / 2.0) * gamma((3 + s + nn) / 2.0)
num1 = a ** (n1 - i)
num2 = b ** (n2 - j)
num3 = g ** (n3 - k)
num = gam * num1 * num2 * num3
denom = f(n1 - i) * f(n2 - j) * f(n3 - k) * f2(i) * f2(j) * f2(k)
sumc += num / denom
return sumc
[docs]
def mapmri_isotropic_phi_matrix(radial_order, mu, q):
"""Three dimensional isotropic MAPMRI signal basis function
See :footcite:p:`Ozarslan2013` Eq. (61).
Parameters
----------
radial_order : unsigned int,
radial order of the mapmri basis.
mu : float,
positive isotropic scale factor of the basis
q : array, shape (N,3)
points in the q-space in which evaluate the basis
References
----------
.. footbibliography::
"""
qval, theta, phi = cart2sphere(q[:, 0], q[:, 1], q[:, 2])
theta[np.isnan(theta)] = 0
ind_mat = mapmri_isotropic_index_matrix(radial_order)
n_elem = ind_mat.shape[0]
n_qgrad = q.shape[0]
M = np.zeros((n_qgrad, n_elem))
counter = 0
for n in range(0, radial_order + 1, 2):
for j in range(1, 2 + n // 2):
l_value = n + 2 - 2 * j
const = mapmri_isotropic_radial_signal_basis(j, l_value, mu, qval)
for m_value in range(-l_value, l_value + 1):
M[:, counter] = const * real_sh_descoteaux_from_index(
m_value, l_value, theta, phi
)
counter += 1
return M
[docs]
def mapmri_isotropic_radial_signal_basis(j, l_value, mu, qval):
"""Radial part of the isotropic 1D-SHORE signal basis.
See :footcite:p:`Ozarslan2013` eq. (61).
Parameters
----------
j : unsigned int,
a positive integer related to the radial order
l_value : unsigned int,
the spherical harmonic order (l)
mu : float,
isotropic scale factor of the basis
qval : float,
points in the q-space in which evaluate the basis
References
----------
.. footbibliography::
"""
pi2_mu2_q2 = 2 * np.pi**2 * mu**2 * qval**2
const = (
(-1) ** (l_value / 2)
* np.sqrt(4.0 * np.pi)
* pi2_mu2_q2 ** (l_value / 2)
* np.exp(-pi2_mu2_q2)
* genlaguerre(j - 1, l_value + 0.5)(2 * pi2_mu2_q2)
)
return const
[docs]
def mapmri_isotropic_M_mu_independent(radial_order, q):
r"""Computed the mu independent part of the signal design matrix."""
ind_mat = mapmri_isotropic_index_matrix(radial_order)
qval, theta, phi = cart2sphere(q[:, 0], q[:, 1], q[:, 2])
theta[np.isnan(theta)] = 0
n_elem = ind_mat.shape[0]
n_rgrad = theta.shape[0]
Q_mu_independent = np.zeros((n_rgrad, n_elem))
counter = 0
for n in range(0, radial_order + 1, 2):
for j in range(1, 2 + n // 2):
l_value = n + 2 - 2 * j
const = (
np.sqrt(4 * np.pi)
* (-1) ** (-l_value / 2)
* (2 * np.pi**2 * qval**2) ** (l_value / 2)
)
for m_value in range(-1 * (n + 2 - 2 * j), (n + 3 - 2 * j)):
Q_mu_independent[:, counter] = const * real_sh_descoteaux_from_index(
m_value, l_value, theta, phi
)
counter += 1
return Q_mu_independent
[docs]
def mapmri_isotropic_M_mu_dependent(radial_order, mu, qval):
"""Computed the mu dependent part of the signal design matrix."""
ind_mat = mapmri_isotropic_index_matrix(radial_order)
n_elem = ind_mat.shape[0]
n_qgrad = qval.shape[0]
Q_u0_dependent = np.zeros((n_qgrad, n_elem))
pi2q2mu2 = 2 * np.pi**2 * mu**2 * qval**2
counter = 0
for n in range(0, radial_order + 1, 2):
for j in range(1, 2 + n // 2):
l_value = n + 2 - 2 * j
const = (
mu**l_value
* np.exp(-pi2q2mu2)
* genlaguerre(j - 1, l_value + 0.5)(2 * pi2q2mu2)
)
for _ in range(-l_value, l_value + 1):
Q_u0_dependent[:, counter] = const
counter += 1
return Q_u0_dependent
[docs]
def mapmri_isotropic_psi_matrix(radial_order, mu, rgrad):
"""Three dimensional isotropic MAPMRI propagator basis function.
See :footcite:p:`Ozarslan2013` Eq. (61).
Parameters
----------
radial_order : unsigned int,
radial order of the mapmri basis.
mu : float,
positive isotropic scale factor of the basis
rgrad : array, shape (N,3)
points in the r-space in which evaluate the basis
References
----------
.. footbibliography::
"""
r, theta, phi = cart2sphere(rgrad[:, 0], rgrad[:, 1], rgrad[:, 2])
theta[np.isnan(theta)] = 0
ind_mat = mapmri_isotropic_index_matrix(radial_order)
n_elem = ind_mat.shape[0]
n_rgrad = rgrad.shape[0]
K = np.zeros((n_rgrad, n_elem))
counter = 0
for n in range(0, radial_order + 1, 2):
for j in range(1, 2 + n // 2):
l_value = n + 2 - 2 * j
const = mapmri_isotropic_radial_pdf_basis(j, l_value, mu, r)
for m_value in range(-l_value, l_value + 1):
K[:, counter] = const * real_sh_descoteaux_from_index(
m_value, l_value, theta, phi
)
counter += 1
return K
[docs]
def mapmri_isotropic_radial_pdf_basis(j, l_value, mu, r):
"""Radial part of the isotropic 1D-SHORE propagator basis.
See :footcite:p:`Ozarslan2013` eq. (61).
Parameters
----------
j : unsigned int,
a positive integer related to the radial order
l_value : unsigned int,
the spherical harmonic order (l)
mu : float,
isotropic scale factor of the basis
r : float,
points in the r-space in which evaluate the basis
References
----------
.. footbibliography::
"""
r2u2 = r**2 / (2 * mu**2)
const = (
(-1) ** (j - 1)
/ (np.sqrt(2) * np.pi * mu**3)
* r2u2 ** (l_value / 2)
* np.exp(-r2u2)
* genlaguerre(j - 1, l_value + 0.5)(2 * r2u2)
)
return const
[docs]
def mapmri_isotropic_K_mu_independent(radial_order, rgrad):
"""Computes mu independent part of K. Same trick as with M."""
r, theta, phi = cart2sphere(rgrad[:, 0], rgrad[:, 1], rgrad[:, 2])
theta[np.isnan(theta)] = 0
ind_mat = mapmri_isotropic_index_matrix(radial_order)
n_elem = ind_mat.shape[0]
n_rgrad = rgrad.shape[0]
K = np.zeros((n_rgrad, n_elem))
counter = 0
for n in range(0, radial_order + 1, 2):
for j in range(1, 2 + n // 2):
ell = n + 2 - 2 * j
const = (
(-1) ** (j - 1) * (np.sqrt(2) * np.pi) ** (-1) * (r**2 / 2) ** (ell / 2)
)
for m in range(-ell, ell + 1):
K[:, counter] = const * real_sh_descoteaux_from_index(
m, ell, theta, phi
)
counter += 1
return K
[docs]
def mapmri_isotropic_K_mu_dependent(radial_order, mu, rgrad):
"""Computes mu dependent part of M. Same trick as with M."""
r, theta, phi = cart2sphere(rgrad[:, 0], rgrad[:, 1], rgrad[:, 2])
theta[np.isnan(theta)] = 0
ind_mat = mapmri_isotropic_index_matrix(radial_order)
n_elem = ind_mat.shape[0]
n_rgrad = rgrad.shape[0]
K = np.zeros((n_rgrad, n_elem))
r2mu2 = r**2 / (2 * mu**2)
counter = 0
for n in range(0, radial_order + 1, 2):
for j in range(1, 2 + n // 2):
ell = n + 2 - 2 * j
const = (
(mu**3) ** (-1)
* mu ** (-ell)
* np.exp(-r2mu2)
* genlaguerre(j - 1, ell + 0.5)(2 * r2mu2)
)
for _ in range(-ell, ell + 1):
K[:, counter] = const
counter += 1
return K
[docs]
def binomialfloat(n, k):
"""Custom Binomial function"""
return sfactorial(n) / (sfactorial(n - k) * sfactorial(k))
[docs]
def mapmri_isotropic_odf_matrix(radial_order, mu, s, vertices):
"""Compute the isotropic MAPMRI ODF matrix.
The computation follows :footcite:p:`Ozarslan2013` Eq. 32, but it is done
for the isotropic propagator in footcite:p:`Ozarslan2013` eq. (60).
Analytical derivation in :footcite:p:`Fick2016b` eq. (C8).
Parameters
----------
radial_order : unsigned int,
an even integer that represent the order of the basis
mu : float,
isotropic scale factor of the isotropic MAP-MRI basis
s : unsigned int
radial moment of the ODF
vertices : array, shape (N,3)
points of the sphere shell in the r-space in which evaluate the ODF
Returns
-------
odf_mat : Matrix, shape (N_vertices, N_mapmri_coef)
ODF design matrix to discrete sphere function
References
----------
.. footbibliography::
"""
r, theta, phi = cart2sphere(vertices[:, 0], vertices[:, 1], vertices[:, 2])
theta[np.isnan(theta)] = 0
ind_mat = mapmri_isotropic_index_matrix(radial_order)
n_vert = vertices.shape[0]
n_elem = ind_mat.shape[0]
odf_mat = np.zeros((n_vert, n_elem))
counter = 0
for n in range(0, radial_order + 1, 2):
for j in range(1, 2 + n // 2):
ell = n + 2 - 2 * j
kappa = ((-1) ** (j - 1) * 2 ** (-(ell + 3) / 2.0) * mu**s) / np.pi
matsum = 0
for k in range(0, j):
matsum += (
(-1) ** k
* binomialfloat(j + ell - 0.5, j - k - 1)
* gamma((ell + s + 3) / 2.0 + k)
) / (mfactorial(k) * 0.5 ** ((ell + s + 3) / 2.0 + k))
for m in range(-ell, ell + 1):
odf_mat[:, counter] = (
kappa * matsum * real_sh_descoteaux_from_index(m, ell, theta, phi)
)
counter += 1
return odf_mat
[docs]
def mapmri_isotropic_odf_sh_matrix(radial_order, mu, s):
"""Compute the isotropic MAPMRI ODF matrix.
The computation follows :footcite:p:`Ozarslan2013` Eq. 32, but it is done
for the isotropic propagator in :footcite:p:`Ozarslan2013` eq. (60). Here
we do not compute the sphere function but the spherical harmonics by only
integrating the radial part of the propagator. We use the same derivation of
the ODF in the isotropic implementation as in :footcite:p:`Fick2016b` eq.
(C8).
Parameters
----------
radial_order : unsigned int,
an even integer that represent the order of the basis
mu : float,
isotropic scale factor of the isotropic MAP-MRI basis
s : unsigned int
radial moment of the ODF
Returns
-------
odf_sh_mat : Matrix, shape (N_sh_coef, N_mapmri_coef)
ODF design matrix to spherical harmonics
References
----------
.. footbibliography::
"""
sh_mat = sph_harm_ind_list(radial_order)
ind_mat = mapmri_isotropic_index_matrix(radial_order)
n_elem_shore = ind_mat.shape[0]
n_elem_sh = sh_mat[0].shape[0]
odf_sh_mat = np.zeros((n_elem_sh, n_elem_shore))
counter = 0
for n in range(0, radial_order + 1, 2):
for j in range(1, 2 + n // 2):
ell = n + 2 - 2 * j
kappa = ((-1) ** (j - 1) * 2 ** (-(ell + 3) / 2.0) * mu**s) / np.pi
matsum = 0
for k in range(0, j):
matsum += (
(-1) ** k
* binomialfloat(j + ell - 0.5, j - k - 1)
* gamma((ell + s + 3) / 2.0 + k)
) / (mfactorial(k) * 0.5 ** ((ell + s + 3) / 2.0 + k))
for m in range(-ell, ell + 1):
index_overlap = np.all([ell == sh_mat[1], m == sh_mat[0]], 0)
odf_sh_mat[:, counter] = kappa * matsum * index_overlap
counter += 1
return odf_sh_mat
[docs]
def mapmri_isotropic_laplacian_reg_matrix(radial_order, mu):
"""Computes the Laplacian regularization matrix for MAP-MRI's isotropic
implementation.
See :footcite:p:`Fick2016b` eq. (C7).
Parameters
----------
radial_order : unsigned int,
an even integer that represent the order of the basis
mu : float,
isotropic scale factor of the isotropic MAP-MRI basis
Returns
-------
LR : Matrix, shape (N_coef, N_coef)
Laplacian regularization matrix
References
----------
.. footbibliography::
"""
ind_mat = mapmri_isotropic_index_matrix(radial_order)
return mapmri_isotropic_laplacian_reg_matrix_from_index_matrix(ind_mat, mu)
[docs]
def mapmri_isotropic_laplacian_reg_matrix_from_index_matrix(ind_mat, mu):
"""Computes the Laplacian regularization matrix for MAP-MRI's isotropic
implementation.
See :footcite:p:`Fick2016b` eq. (C7).
Parameters
----------
ind_mat : matrix (N_coef, 3),
Basis order matrix
mu : float,
isotropic scale factor of the isotropic MAP-MRI basis
Returns
-------
LR : Matrix, shape (N_coef, N_coef)
Laplacian regularization matrix
References
----------
.. footbibliography::
"""
n_elem = ind_mat.shape[0]
LR = np.zeros((n_elem, n_elem))
for i in range(n_elem):
for k in range(i, n_elem):
if ind_mat[i, 1] == ind_mat[k, 1] and ind_mat[i, 2] == ind_mat[k, 2]:
ji = ind_mat[i, 0]
jk = ind_mat[k, 0]
ell = ind_mat[i, 1]
if ji == (jk + 2):
LR[i, k] = LR[k, i] = (
2.0 ** (2 - ell)
* np.pi**2
* mu
* gamma(5 / 2.0 + jk + ell)
/ gamma(jk)
)
elif ji == (jk + 1):
LR[i, k] = LR[k, i] = (
2.0 ** (2 - ell)
* np.pi**2
* mu
* (-3 + 4 * ji + 2 * ell)
* gamma(3 / 2.0 + jk + ell)
/ gamma(jk)
)
elif ji == jk:
LR[i, k] = (
2.0 ** (-ell)
* np.pi**2
* mu
* (
3
+ 24 * ji**2
+ 4 * (-2 + ell) * ell
+ 12 * ji * (-1 + 2 * ell)
)
* gamma(1 / 2.0 + ji + ell)
/ gamma(ji)
)
elif ji == (jk - 1):
LR[i, k] = LR[k, i] = (
2.0 ** (2 - ell)
* np.pi**2
* mu
* (-3 + 4 * jk + 2 * ell)
* gamma(3 / 2.0 + ji + ell)
/ gamma(ji)
)
elif ji == (jk - 2):
LR[i, k] = LR[k, i] = (
2.0 ** (2 - ell)
* np.pi**2
* mu
* gamma(5 / 2.0 + ji + ell)
/ gamma(ji)
)
return LR
[docs]
def mapmri_isotropic_index_matrix(radial_order):
"""Calculates the indices for the isotropic MAPMRI basis.
See :footcite:p:`Ozarslan2013` Fig 8.
Parameters
----------
radial_order : unsigned int
radial order of isotropic MAPMRI basis
Returns
-------
index_matrix : array, shape (N,3)
ordering of the basis in x, y, z
References
----------
.. footbibliography::
"""
index_matrix = []
for n in range(0, radial_order + 1, 2):
for j in range(1, 2 + n // 2):
for m in range(-1 * (n + 2 - 2 * j), (n + 3 - 2 * j)):
index_matrix.append([j, n + 2 - 2 * j, m])
return np.array(index_matrix)
[docs]
def create_rspace(gridsize, radius_max):
"""Create the real space table, that contains the points in which
to compute the pdf.
Parameters
----------
gridsize : unsigned int
dimension of the propagator grid
radius_max : float
maximal radius in which compute the propagator
Returns
-------
tab : array, shape (N,3)
real space points in which calculates the pdf
"""
radius = gridsize // 2
vecs = []
for i in range(-radius, radius + 1):
for j in range(-radius, radius + 1):
for k in range(0, radius + 1):
vecs.append([i, j, k])
vecs = np.array(vecs, dtype=np.float32)
# there are points in the corners farther than sphere radius
points_inside_sphere = np.sqrt(np.einsum("ij,ij->i", vecs, vecs)) <= radius
vecs_inside_sphere = vecs[points_inside_sphere]
tab = vecs_inside_sphere / radius
tab = tab * radius_max
return tab
[docs]
def delta(n, m):
if n == m:
return 1
return 0
[docs]
def map_laplace_u(n, m):
r"""S(n, m) static matrix for Laplacian regularization.
See :footcite:p:`Fick2016b` eq. (13).
Parameters
----------
n, m : unsigned int
basis order of the MAP-MRI basis in different directions
Returns
-------
U : float,
Analytical integral of :math:`\phi_n(q) * \phi_m(q)`
References
----------
.. footbibliography::
"""
return (-1) ** n * delta(n, m) / (2 * np.sqrt(np.pi))
[docs]
def map_laplace_t(n, m):
r"""L(m, n) static matrix for Laplacian regularization.
See :footcite:p:`Fick2016b` eq. (12).
Parameters
----------
n, m : unsigned int
basis order of the MAP-MRI basis in different directions
Returns
-------
T : float
Analytical integral of :math:`\phi_n(q) * \phi_m''(q)`
References
----------
.. footbibliography::
"""
a = np.sqrt((m - 1) * m) * delta(m - 2, n)
b = np.sqrt((n - 1) * n) * delta(n - 2, m)
c = (2 * n + 1) * delta(m, n)
return np.pi ** (3 / 2.0) * (-1) ** (n + 1) * (a + b + c)
[docs]
def map_laplace_s(n, m):
r"""R(m,n) static matrix for Laplacian regularization.
See :footcite:p:`Fick2016b` eq. (11).
Parameters
----------
n, m : unsigned int
basis order of the MAP-MRI basis in different directions
Returns
-------
S : float
Analytical integral of :math:`\phi_n''(q) * \phi_m''(q)`
References
----------
.. footbibliography::
"""
k = 2 * np.pi ** (7 / 2.0) * (-1) ** n
a0 = 3 * (2 * n**2 + 2 * n + 1) * delta(n, m)
sqmn = np.sqrt(gamma(m + 1) / gamma(n + 1))
sqnm = 1 / sqmn
an2 = 2 * (2 * n + 3) * sqmn * delta(m, n + 2)
an4 = sqmn * delta(m, n + 4)
am2 = 2 * (2 * m + 3) * sqnm * delta(m + 2, n)
am4 = sqnm * delta(m + 4, n)
return k * (a0 + an2 + an4 + am2 + am4)
[docs]
def mapmri_STU_reg_matrices(radial_order):
"""Generate the static portions of the Laplacian regularization matrix.
See :footcite:p:`Fick2016b` eq. (11, 12, 13).
Parameters
----------
radial_order : unsigned int,
an even integer that represent the order of the basis
Returns
-------
S, T, U : Matrices, shape (N_coef,N_coef)
Regularization submatrices
References
----------
.. footbibliography::
"""
S = np.zeros((radial_order + 1, radial_order + 1))
for i in range(radial_order + 1):
for j in range(radial_order + 1):
S[i, j] = map_laplace_s(i, j)
T = np.zeros((radial_order + 1, radial_order + 1))
for i in range(radial_order + 1):
for j in range(radial_order + 1):
T[i, j] = map_laplace_t(i, j)
U = np.zeros((radial_order + 1, radial_order + 1))
for i in range(radial_order + 1):
for j in range(radial_order + 1):
U[i, j] = map_laplace_u(i, j)
return S, T, U
[docs]
def mapmri_laplacian_reg_matrix(ind_mat, mu, S_mat, T_mat, U_mat):
"""Put the Laplacian regularization matrix together.
See :footcite:p:`Fick2016b` eq. (10).
The static parts in S, T and U are multiplied and divided by the
voxel-specific scale factors.
Parameters
----------
ind_mat : matrix (N_coef, 3),
Basis order matrix
mu : array, shape (3,)
scale factors of the basis for x, y, z
S, T, U : matrices, shape (N_coef,N_coef)
Regularization submatrices
Returns
-------
LR : matrix (N_coef, N_coef),
Voxel-specific Laplacian regularization matrix
References
----------
.. footbibliography::
"""
ux, uy, uz = mu
x, y, z = ind_mat.T
n_elem = ind_mat.shape[0]
LR = np.zeros((n_elem, n_elem))
for i in range(n_elem):
for j in range(i, n_elem):
if (
(x[i] - x[j]) % 2 == 0
and (y[i] - y[j]) % 2 == 0
and (z[i] - z[j]) % 2 == 0
):
LR[i, j] = LR[j, i] = (
(ux**3 / (uy * uz))
* S_mat[x[i], x[j]]
* U_mat[y[i], y[j]]
* U_mat[z[i], z[j]]
+ (uy**3 / (ux * uz))
* S_mat[y[i], y[j]]
* U_mat[z[i], z[j]]
* U_mat[x[i], x[j]]
+ (uz**3 / (ux * uy))
* S_mat[z[i], z[j]]
* U_mat[x[i], x[j]]
* U_mat[y[i], y[j]]
+ 2
* ((ux * uy) / uz)
* T_mat[x[i], x[j]]
* T_mat[y[i], y[j]]
* U_mat[z[i], z[j]]
+ 2
* ((ux * uz) / uy)
* T_mat[x[i], x[j]]
* T_mat[z[i], z[j]]
* U_mat[y[i], y[j]]
+ 2
* ((uz * uy) / ux)
* T_mat[z[i], z[j]]
* T_mat[y[i], y[j]]
* U_mat[x[i], x[j]]
)
return LR
[docs]
@warning_for_keywords()
def generalized_crossvalidation_array(data, M, LR, *, weights_array=None):
"""Generalized Cross Validation Function.
See :footcite:p:`Fick2016b` eq. (15).
Here weights_array is a numpy array with all values that should be
considered in the GCV. It will run through the weights until the cost
function starts to increase, then stop and take the last value as the
optimum weight.
Parameters
----------
data : array (N),
Basis order matrix
M : matrix, shape (N, Ncoef)
mapmri observation matrix
LR : matrix, shape (N_coef, N_coef)
regularization matrix
weights_array : array (N_of_weights)
array of optional regularization weights
References
----------
.. footbibliography::
"""
if weights_array is None:
lrange = np.linspace(0.05, 1, 20) # reasonably fast standard range
else:
lrange = weights_array
samples = lrange.shape[0]
MMt = np.dot(M.T, M)
K = len(data)
gcvold = gcvnew = 10e10 # set initialization gcv threshold very high
i = -1
while gcvold >= gcvnew and i < samples - 2:
gcvold = gcvnew
i = i + 1
S = np.linalg.multi_dot([M, np.linalg.pinv(MMt + lrange[i] * LR), M.T])
trS = np.trace(S)
normyytilde = np.linalg.norm(data - np.dot(S, data), 2)
gcvnew = normyytilde / (K - trS)
lopt = lrange[i - 1]
return lopt
[docs]
@warning_for_keywords()
def generalized_crossvalidation(data, M, LR, *, gcv_startpoint=5e-2):
"""Generalized Cross Validation Function.
Finds optimal regularization weight based on generalized cross-validation.
See :footcite:p:`Craven1979` eq. (15).
Parameters
----------
data : array (N),
data array
M : matrix, shape (N, Ncoef)
mapmri observation matrix
LR : matrix, shape (N_coef, N_coef)
regularization matrix
gcv_startpoint : float
startpoint for the gcv optimization
Returns
-------
optimal_lambda : float,
optimal regularization weight
References
----------
.. footbibliography::
"""
MMt = np.dot(M.T, M)
K = len(data)
bounds = ((1e-5, 10),)
solver = Optimizer(
fun=gcv_cost_function,
x0=(gcv_startpoint,),
args=((data, M, MMt, K, LR),),
bounds=bounds,
)
optimal_lambda = solver.xopt
return optimal_lambda
[docs]
def gcv_cost_function(weight, args):
"""The GCV cost function that is iterated.
See :footcite:p:`Fick2016b` for further details about the method.
References
----------
.. footbibliography::
"""
data, M, MMt, K, LR = args
S = np.linalg.multi_dot([M, np.linalg.pinv(MMt + weight * LR), M.T])
trS = np.trace(S)
normyytilde = np.linalg.norm(data - np.dot(S, data), 2)
gcv_value = normyytilde / (K - trS)
return gcv_value