def sim_response(sh_order, bvals, evals=evals_d, csf_md=csf_md, gm_md=gm_md):
bvals = np.array(bvals, copy=True)
evecs = np.zeros((3, 3))
z = np.array([0, 0, 1.])
evecs[:, 0] = z
evecs[:2, 1:] = np.eye(2)
n = np.arange(0, sh_order + 1, 2)
m = np.zeros_like(n)
big_sphere = default_sphere.subdivide()
theta, phi = big_sphere.theta, big_sphere.phi
B = shm.real_sph_harm(m, n, theta[:, None], phi[:, None])
A = shm.real_sph_harm(0, 0, 0, 0)
response = np.empty([len(bvals), len(n) + 2])
for i, bvalue in enumerate(bvals):
gtab = GradientTable(big_sphere.vertices * bvalue)
wm_response = single_tensor(gtab, 1., evals, evecs, snr=None)
response[i, 2:] = np.linalg.lstsq(B, wm_response)[0]
response[i, 0] = np.exp(-bvalue * csf_md) / A
response[i, 1] = np.exp(-bvalue * gm_md) / A
return MultiShellResponse(response, sh_order, bvals)
def multi_shell_fiber_response(sh_order,
bvals,
evals,
csf_md,
gm_md,
sphere=None):
"""Fiber response function estimation for multi-shell data.
Parameters
----------
sh_order : int
Maximum spherical harmonics order.
bvals : ndarray
Array containing the b-values.
evals : (3,) ndarray
Eigenvalues of the diffusion tensor.
csf_md : float
CSF tissue mean diffusivity value.
gm_md : float
GM tissue mean diffusivity value.
sphere : `dipy.core.Sphere` instance, optional
Sphere where the signal will be evaluated.
Returns
-------
MultiShellResponse
MultiShellResponse object.
"""
bvals = np.array(bvals, copy=True)
evecs = np.zeros((3, 3))
z = np.array([0, 0, 1.])
evecs[:, 0] = z
evecs[:2, 1:] = np.eye(2)
n = np.arange(0, sh_order + 1, 2)
m = np.zeros_like(n)
if sphere is None:
sphere = default_sphere
big_sphere = sphere.subdivide()
theta, phi = big_sphere.theta, big_sphere.phi
B = shm.real_sph_harm(m, n, theta[:, None], phi[:, None])
A = shm.real_sph_harm(0, 0, 0, 0)
response = np.empty([len(bvals), len(n) + 2])
for i, bvalue in enumerate(bvals):
gtab = GradientTable(big_sphere.vertices * bvalue)
wm_response = single_tensor(gtab, 1., evals, evecs, snr=None)
response[i, 2:] = np.linalg.lstsq(B, wm_response)[0]
response[i, 0] = np.exp(-bvalue * csf_md) / A
response[i, 1] = np.exp(-bvalue * gm_md) / A
return MultiShellResponse(response, sh_order, bvals)
def shore_psi_matrix(radial_order, mu, rgrad):
"""Computes the K matrix without optimization.
"""
r, theta, phi = cart2sphere(rgrad[:, 0], rgrad[:, 1], rgrad[:, 2])
theta[np.isnan(theta)] = 0
ind_mat = shore_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 = n + 2 - 2 * j
const = (-1) ** (j - 1) * \
(np.sqrt(2) * np.pi * mu ** 3) ** (-1) *\
(r ** 2 / (2 * mu ** 2)) ** (l / 2) *\
np.exp(- r ** 2 / (2 * mu ** 2)) *\
genlaguerre(j - 1, l + 0.5)(r ** 2 / mu ** 2)
for m in range(-l, l + 1):
K[:, counter] = const * real_sph_harm(m, l, theta, phi)
counter += 1
return K
def test_mcsd_model_delta():
sh_order = 8
gtab = get_3shell_gtab()
shells = np.unique(gtab.bvals // 100.) * 100.
response = sim_response(sh_order, shells, evals_d)
model = MultiShellDeconvModel(gtab, response)
iso = response.iso
theta, phi = default_sphere.theta, default_sphere.phi
B = shm.real_sph_harm(response.m, response.n, theta[:, None], phi[:, None])
wm_delta = model.delta.copy()
# set isotropic components to zero
wm_delta[:iso] = 0.
wm_delta = _expand(model.m, iso, wm_delta)
for i, s in enumerate(shells):
g = GradientTable(default_sphere.vertices * s)
signal = model.predict(wm_delta, g)
expected = np.dot(response.response[i, iso:], B.T)
npt.assert_array_almost_equal(signal, expected)
signal = model.predict(wm_delta, gtab)
fit = model.fit(signal)
m = model.m
npt.assert_array_almost_equal(fit.shm_coeff[m != 0], 0., 2)
def shore_phi_matrix(radial_order, mu, q):
'''Computed the Q matrix completely without separation into
mu-depenent / -independent. See shore_Q_mu_independent for help.
'''
qval, theta, phi = cart2sphere(q[:, 0], q[:, 1], q[:, 2])
theta[np.isnan(theta)] = 0
ind_mat = shore_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 = n + 2 - 2 * j
const = (-1) ** (l / 2) * np.sqrt(4.0 * np.pi) *\
(2 * np.pi ** 2 * mu ** 2 * qval ** 2) ** (l / 2) *\
np.exp(-2 * np.pi ** 2 * mu ** 2 * qval ** 2) *\
genlaguerre(j - 1, l + 0.5)(4 * np.pi ** 2 * mu ** 2 *
qval ** 2)
for m in range(-l, l + 1):
M[:, counter] = const * real_sph_harm(m, l, theta, phi)
counter += 1
return M
def shore_matrix_odf(radial_order, zeta, sphere_vertices):
r"""Compute the SHORE ODF matrix [1]_"
Parameters
----------
radial_order : unsigned int,
an even integer that represent the order of the basis
zeta : unsigned int,
scale factor
sphere_vertices : array, shape (N,3)
vertices of the odf sphere
References
----------
.. [1] Merlet S. et. al, "Continuous diffusion signal, EAP and
ODF estimation via Compressive Sensing in diffusion MRI", Medical
Image Analysis, 2013.
"""
r, theta, phi = cart2sphere(sphere_vertices[:, 0], sphere_vertices[:, 1],
sphere_vertices[:, 2])
theta[np.isnan(theta)] = 0
F = radial_order / 2
n_c = int(np.round(1 / 6.0 * (F + 1) * (F + 2) * (4 * F + 3)))
upsilon = np.zeros((len(sphere_vertices), n_c))
counter = 0
for l in range(0, radial_order + 1, 2):
for n in range(l, int((radial_order + l) / 2) + 1):
for m in range(-l, l + 1):
upsilon[:, counter] = (-1) ** (n - l / 2.0) * _kappa_odf(zeta, n, l) * \
hyp2f1(l - n, l / 2.0 + 1.5, l + 1.5, 2.0) * \
real_sph_harm(m, l, theta, phi)
counter += 1
return upsilon
def rtap(self, direction=None):
""" Recovers the directional Return To Axis Probability (RTAP) [1,2].
Its value is only valid along the direction of the fiber. If no
direction is given the biggest eigenvector of the tensor is used.
Its value is defined as:
..math::
:nowrap:
\begin{equation}
RTAP=\int_{\mathbb{R}}P(\textbf{r}_{\parallel})
d\textbf{r}_{\parallel}=\frac{1}{(2\pi) u_0^2}
\sum_{N=0}^{N_{max}}\sum_{\{j,l,m\}}\textbf{c}_{\{j,l,m\}}
(-1)^{j-1}2^{-l/2}\kappa(j,l)Y_l^m(\textbf{u}_{fiber})
The vector is precomputed to speed up the computation.
"""
if direction is None:
if self.R is None:
warn('Tensor linearity too low to use main tensor eigenvalue '
'as fiber direction. Returning 0.')
return 0.
else:
direction = np.array(self.R[:, 0], ndmin=2)
r, theta, phi = cart2sphere(direction[:, 0], direction[:, 1],
direction[:, 2])
else:
r, theta, phi = cart2sphere(direction[0], direction[1],
direction[2])
inx = self.model.ind_mat
rtap_vec = self.model.rtap_vec
rtap = self._shore_coef * (1 / self.mu ** 2) *\
rtap_vec * real_sph_harm(inx[:, 2], inx[:, 1], theta, phi)
return rtap.sum()
def multi_tissue_basis(gtab, sh_order, iso_comp):
"""
Builds a basis for multi-shell multi-tissue CSD model.
Parameters
----------
gtab : GradientTable
sh_order : int
iso_comp: int
Number of tissue compartments for running the MSMT-CSD. Minimum
number of compartments required is 2.
Returns
-------
B : ndarray
Matrix of the spherical harmonics model used to fit the data
m : int ``|m| <= n``
The order of the harmonic.
n : int ``>= 0``
The degree of the harmonic.
"""
if iso_comp < 2:
msg = ("Multi-tissue CSD requires at least 2 tissue compartments")
raise ValueError(msg)
r, theta, phi = geo.cart2sphere(*gtab.gradients.T)
m, n = shm.sph_harm_ind_list(sh_order)
B = shm.real_sph_harm(m, n, theta[:, None], phi[:, None])
B[np.ix_(gtab.b0s_mask, n > 0)] = 0.
iso = np.empty([B.shape[0], iso_comp])
iso[:] = SH_CONST
B = np.concatenate([iso, B], axis=1)
return B, m, n
def __init__(self, gtab, response, sh_order, lambda_=1, tau=0.1):
super(NumberSmallfODF, self).__init__()
m, n = sph_harm_ind_list(sh_order)
# x, y, z = gtab.gradients[~gtab.b0s_mask].T
# r, theta, phi = cart2sphere(x, y, z)
# self.B_dwi = real_sph_harm(m, n, theta[:, None], phi[:, None])
self.B_dwi = shm.get_B_matrix(gtab, sh_order)
self.sphere = get_sphere('symmetric362')
r, theta, phi = cart2sphere(self.sphere.x, self.sphere.y,
self.sphere.z)
self.B_reg = real_sph_harm(m, n, theta[:, None], phi[:, None])
S_r = shm.estimate_response(gtab, response[0:3], response[3])
r_sh = np.linalg.lstsq(self.B_dwi, S_r[~gtab.b0s_mask], rcond=-1)[0]
n_response = n
m_response = m
r_rh = sh_to_rh(r_sh, m_response, n_response)
R = forward_sdeconv_mat(r_rh, n)
# scale lambda_ to account for differences in the number of
# SH coefficients and number of mapped directions
# This is exactly what is done in [4]_
lambda_ = (lambda_ * R.shape[0] * r_rh[0] /
(np.sqrt(self.B_reg.shape[0]) * np.sqrt(362.)))
self.B_reg = self.B_reg * lambda_
self.B_reg = nn.Parameter(torch.FloatTensor(self.B_reg),
requires_grad=False)
self.tau = tau
def shore_matrix_odf(radial_order, zeta, sphere_vertices):
r"""Compute the SHORE ODF matrix [1]_"
Parameters
----------
radial_order : unsigned int,
an even integer that represent the order of the basis
zeta : unsigned int,
scale factor
sphere_vertices : array, shape (N,3)
vertices of the odf sphere
References
----------
.. [1] Merlet S. et. al, "Continuous diffusion signal, EAP and
ODF estimation via Compressive Sensing in diffusion MRI", Medical
Image Analysis, 2013.
"""
r, theta, phi = cart2sphere(sphere_vertices[:, 0], sphere_vertices[:, 1],
sphere_vertices[:, 2])
theta[np.isnan(theta)] = 0
F = radial_order / 2
n_c = np.round(1 / 6.0 * (F + 1) * (F + 2) * (4 * F + 3))
upsilon = np.zeros((len(sphere_vertices), n_c))
counter = 0
for l in range(0, radial_order + 1, 2):
for n in range(l, int((radial_order + l) / 2) + 1):
for m in range(-l, l + 1):
upsilon[:, counter] = (-1) ** (n - l / 2.0) * _kappa_odf(zeta, n, l) * \
hyp2f1(l - n, l / 2.0 + 1.5, l + 1.5, 2.0) * \
real_sph_harm(m, l, theta, phi)
counter += 1
return upsilon
def gen_dirac(pol, azi, sh_order):
""" Generate Dirac delta function orientated in (theta, phi) = (azi, pol)
on the sphere. The spherical harmonics (SH) representation of this Dirac is
returned.
Parameters
----------
pol : float [0, pi]
The polar (colatitudinal) coordinate (phi)
az : float [0, 2*pi]
The azimuthal (longitudinal) coordinate (theta)
sh_order : int
maximal SH order of the SH representation
Returns
-------
dirac : ndarray (``(sh_order + 1)(sh_order + 2)/2``,)
SH coefficients representing the Dirac function
"""
m, n = sph_harm_ind_list(sh_order)
dirac = np.zeros(m.shape)
i = 0
for l in np.arange(0, sh_order + 1, 2):
for m in np.arange(-l, l + 1):
if m == 0:
dirac[i] = real_sph_harm(0, l, azi, pol)
i = i + 1
return dirac
def gen_dirac(m, n, theta, phi):
""" Generate Dirac delta function orientated in (theta, phi) on the sphere
The spherical harmonics (SH) representation of this Dirac is returned as
coefficients to spherical harmonic functions produced by
`shm.real_sph_harm`.
Parameters
----------
m : ndarray (N,)
The order of the spherical harmonic function associated with each
coefficient.
n : ndarray (N,)
The degree of the spherical harmonic function associated with each
coefficient.
theta : float [0, 2*pi]
The azimuthal (longitudinal) coordinate.
phi : float [0, pi]
The polar (colatitudinal) coordinate.
See Also
--------
shm.real_sph_harm, shm.real_sym_sh_basis
Returns
-------
dirac : ndarray
SH coefficients representing the Dirac function
"""
return real_sph_harm(m, n, theta, phi)
def SHOREmatrix_pdf(radial_order, zeta, rtab):
"""Compute the SHORE matrix"
Parameters
----------
radial_order : unsigned int,
Radial Order
zeta : unsigned int,
scale factor
rtab : array, shape (N,3)
r-space points in which calculates the pdf
"""
r, theta, phi = cart2sphere(rtab[:, 0], rtab[:, 1], rtab[:, 2])
theta[np.isnan(theta)] = 0
psi = np.zeros(
(r.shape[0], (radial_order + 1) * ((radial_order + 1) / 2) * (2 * radial_order + 1)))
counter = 0
for n in range(radial_order + 1):
for l in range(0, n + 1, 2):
for m in range(-l, l + 1):
psi[:, counter] = real_sph_harm(m, l, theta, phi) * \
genlaguerre(n - l, l + 0.5)(4 * np.pi ** 2 * zeta * r ** 2 ) *\
np.exp(-2 * np.pi ** 2 * zeta * r ** 2) *\
__kappa_pdf(zeta, n, l) *\
(4 * np.pi ** 2 * zeta * r ** 2) ** (l / 2) * \
(-1) ** (n - l / 2)
counter += 1
return psi[:, 0:counter]
def shore_Q_mu_independent(radial_order, q):
r'''Computed the u0 independent part of the design matrix.
..math::
:nowrap:
\begin{align}
\Xi_{jlm(i)}(u_0,\textbf{q}_k)=&\overbrace{u_0^{l(i)}
e^{-2\pi^2u_0^2q_k^2}L_{j(i)-1}^{l(i)+1/2}
(4\pi^2u_0^2q_k^2)}^{u_0\,dependent}
\overbrace{\sqrt{4\pi}i^{-l(i)}(2\pi^2q_k^2)^{l(i)/2}
Y_{l(i)}^{m(i)}(\textbf{u}_{q_k})}^{u_0\,independent}
=&A_{jl(i)}(q_k)B_{lm(i)}(\textbf{q}_k)
\end{align}
'''
ind_mat = shore_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 = n + 2 - 2 * j
const = np.sqrt(4 * np.pi) * (-1) ** (-l / 2) * \
(2 * np.pi ** 2 * qval ** 2) ** (l / 2)
for m in range(-1 * (n + 2 - 2 * j), (n + 3 - 2 * j)):
Q_mu_independent[:, counter] = const * \
real_sph_harm(m, l, theta, phi)
counter += 1
return Q_mu_independent
def real_sym_rh_basis(sh_order, theta, phi):
r"""Samples a real symmetric rotational 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 fibernavigator, where the real harmonic $Y^m_n$ is defined to
be:
$Y^0_n$ if m = 0
Parameters
-----------
sh_order : int
even int > 0, max spherical harmonic degree
theta : float [0, 2*pi]
The azimuthal (longitudinal) coordinate.
phi : float [0, pi]
The polar (colatitudinal) coordinate.
Returns
--------
real_rh_matrix : array of shape ()
The real harmonic $Y^0_n$ sampled at `theta` and `phi`
"""
n = np.arange(0, sh_order + 1, 2)
m = np.zeros(sh_order // 2 + 1)
phi = np.reshape(phi, [-1, 1])
theta = np.reshape(theta, [-1, 1])
real_rh_matrix = real_sph_harm(m, n, theta, phi)
return real_rh_matrix
def sh_convolution_matrix(self, kernel="rank1"):
"""
Parameters
----------
kernel
Returns
-------
"""
if self.kernel_type != kernel:
self.set_kernel(kernel)
# Build matrix that maps ODF to signal
M = np.zeros((self.gtab.bvals.shape[0], esh.LENGTH[self.order]))
r, theta, phi = cart2sphere(self.gtab.bvecs[:, 0],
self.gtab.bvecs[:, 1], self.gtab.bvecs[:,
2])
theta[np.isnan(theta)] = 0
counter = 0
for l in range(0, self.order + 1, 2):
for m in range(-l, l + 1):
M[:, counter] = (real_sph_harm(m, l, theta, phi) *
self.kernel_wm[l // 2])
counter += 1
return M
def SHOREmatrix_odf(radial_order, zeta, sphere_vertices):
"""Compute the SHORE matrix"
Parameters
----------
radial_order : unsigned int,
Radial Order
zeta : unsigned int,
scale factor
sphere_vertices : array, shape (N,3)
vertices of the odf sphere
"""
r, theta, phi = cart2sphere(sphere_vertices[:, 0], sphere_vertices[:, 1], sphere_vertices[:, 2])
theta[np.isnan(theta)] = 0
counter = 0
upsilon = np.zeros(
(len(sphere_vertices), (radial_order + 1) * ((radial_order + 1) / 2) * (2 * radial_order + 1)))
for n in range(radial_order + 1):
for l in range(0, n + 1, 2):
for m in range(-l, l + 1):
upsilon[:, counter] = (-1) ** (n - l / 2.0) * __kappa_odf(zeta, n, l) * \
hyp2f1(l - n, l / 2.0 + 1.5, l + 1.5, 2.0) * \
real_sph_harm(m, l, theta, phi)
counter += 1
return upsilon[:, 0:counter]
def gen_dirac(pol, azi, sh_order):
""" Generate Dirac delta function orientated in (theta, phi) = (azi, pol)
on the sphere. The spherical harmonics (SH) representation of this Dirac is
returned.
Parameters
----------
pol : float [0, pi]
The polar (colatitudinal) coordinate (phi)
az : float [0, 2*pi]
The azimuthal (longitudinal) coordinate (theta)
sh_order : int
maximal SH order of the SH representation
Returns
-------
dirac : ndarray (``(sh_order + 1)(sh_order + 2)/2``,)
SH coefficients representing the Dirac function
"""
m, n = sph_harm_ind_list(sh_order)
dirac = np.zeros(m.shape)
i = 0
for l in np.arange(0, sh_order + 1, 2):
for m in np.arange(-l, l + 1):
if m == 0:
dirac[i] = real_sph_harm(0, l, azi, pol)
i = i + 1
return dirac
def sh_smooth(data, bvals, bvecs, sh_order=4, similarity_threshold=50, regul=0.006):
"""Smooth the raw diffusion signal with spherical harmonics.
data : ndarray
The diffusion data to smooth.
gtab : gradient table object
Corresponding gradients table object to data.
sh_order : int, default 8
Order of the spherical harmonics to fit.
similarity_threshold : int, default 50
All b-values such that |b_1 - b_2| < similarity_threshold
will be considered as identical for smoothing purpose.
Must be lower than 200.
regul : float, default 0.006
Amount of regularization to apply to sh coefficients computation.
Return
---------
pred_sig : ndarray
The smoothed diffusion data, fitted through spherical harmonics.
"""
if similarity_threshold > 200:
raise ValueError("similarity_threshold = {}, which is higher than 200,"
" please use a lower value".format(similarity_threshold))
m, n = sph_harm_ind_list(sh_order)
L = -n * (n + 1)
where_b0s = bvals == 0
pred_sig = np.zeros_like(data, dtype=np.float32)
# Round similar bvals together for identifying similar shells
rounded_bvals = np.zeros_like(bvals)
for unique_bval in np.unique(bvals):
idx = np.abs(unique_bval - bvals) < similarity_threshold
rounded_bvals[idx] = unique_bval
# process each b-value separately
for unique_bval in np.unique(rounded_bvals):
idx = rounded_bvals == unique_bval
# Just give back the signal for the b0s since we can't really do anything about it
if np.all(idx == where_b0s):
if np.sum(where_b0s) > 1:
pred_sig[..., idx] = np.mean(data[..., idx], axis=-1, keepdims=True)
else:
pred_sig[..., idx] = data[..., idx]
continue
x, y, z = bvecs[:, idx]
r, theta, phi = cart2sphere(x, y, z)
# Find the sh coefficients to smooth the signal
B_dwi = real_sph_harm(m, n, theta[:, None], phi[:, None])
invB = smooth_pinv(B_dwi, np.sqrt(regul) * L)
sh_coeff = np.dot(data[..., idx], invB.T)
# Find the smoothed signal from the sh fit for the given gtab
pred_sig[..., idx] = np.dot(sh_coeff, B_dwi.T)
return pred_sig
def odf(self, sphere):
sampling_matrix = self.model.cache_get("sampling_matrix", sphere)
if sampling_matrix is None:
phi = sphere.phi[:, np.newaxis] #sphere.phi.reshape((-1, 1))
theta = sphere.theta.reshape((-1, 1))
sampling_matrix = real_sph_harm(self.model.m, self.model.n, theta, phi)
self.model.cache_set("sampling_matrix", sphere, sampling_matrix)
return np.dot(self.shm_coeff, sampling_matrix.T)
def shore_odf_matrix(radial_order, mu, s, vertices):
r"""The ODF in terms of SHORE coefficients for arbitrary radial moment
can be given as [2]:
..math::
:nowrap:
\begin{equation}
ODF_s(u_0,\textbf{v})=\sum_{N=0}^{N_{max}}
\sum_{\{j,l,m\}}\textbf{c}_{\{j,l,m\}}
\Omega_s^{jlm}(u_0,\textbf{v})
\end{equation}
with $\textbf{v}$ an orientation on the unit sphere and the ODF
basis function:
..math::
:nowrap:
\begin{equation}
\Omega_s^{jlm}(u_0,\textbf{v})=\frac{u_0^s}{\pi}(-1)^{j-1}
2^{-l/2}\kappa(j,l,s)Y^l_m(\textbf{v})
\end{equation}
with
..math::
:nowrap:
\begin{equation}
\kappa(j,l,s)=\sum_{k=0}^{j-1}\frac{(-1)^k}{k!}
\binom{j+l-1/2}{j-k-1}
\frac{\Gamma((l+s+3)/2+k)}{2^{-((l+s)/2+k)}}.
\end{equation}
"""
r, theta, phi = cart2sphere(vertices[:, 0], vertices[:, 1], vertices[:, 2])
theta[np.isnan(theta)] = 0
ind_mat = shore_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):
l = n + 2 - 2 * j
kappa = ((-1)**(j - 1) * 2**(-(l + 3) / 2.0) * mu**s) / np.pi
matsum = 0
for k in range(0, j):
matsum += ((-1) ** k * binomialfloat(j + l - 0.5, j - k - 1) *
gamma((l + s + 3) / 2.0 + k)) /\
(factorial(k) * 0.5 ** ((l + s + 3) / 2.0 + k))
for m in range(-l, l + 1):
odf_mat[:, counter] = kappa * matsum *\
real_sph_harm(m, l, theta, phi)
counter += 1
return odf_mat
def rho_matrix(sh_order, vecs):
r"""Compute the SH matrix $\rho$
"""
r, theta, phi = cart2sphere(vecs[:, 0], vecs[:, 1], vecs[:, 2])
theta[np.isnan(theta)] = 0
n_c = int((sh_order + 1) * (sh_order + 2) / 2)
rho = np.zeros((vecs.shape[0], n_c))
counter = 0
for l in range(0, sh_order + 1, 2):
for m in range(-l, l + 1):
rho[:, counter] = real_sph_harm(m, l, theta, phi)
counter += 1
return rho
def rho_matrix(sh_order, vecs):
r"""Compute the SH matrix $\rho$
"""
r, theta, phi = cart2sphere(vecs[:, 0], vecs[:, 1], vecs[:, 2])
theta[np.isnan(theta)] = 0
n_c = int((sh_order + 1) * (sh_order + 2) / 2)
rho = np.zeros((vecs.shape[0], n_c))
counter = 0
for l in range(0, sh_order + 1, 2):
for m in range(-l, l + 1):
rho[:, counter] = real_sph_harm(m, l, theta, phi)
counter += 1
return rho
def sh_smooth(data, gtab, sh_order=4):
"""Smooth the raw diffusion signal with spherical harmonics
data : ndarray
The diffusion data to smooth.
gtab : gradient table object
Corresponding gradients table object to data.
sh_order : int, default 4
Order of the spherical harmonics to fit.
Return
---------
pred_sig : ndarray
The smoothed diffusion data, fitted through spherical harmonics.
"""
m, n = sph_harm_ind_list(sh_order)
where_b0s = lazy_index(gtab.b0s_mask)
where_dwi = lazy_index(~gtab.b0s_mask)
x, y, z = gtab.gradients[where_dwi].T
r, theta, phi = cart2sphere(x, y, z)
# Find the sh coefficients to smooth the signal
B_dwi = real_sph_harm(m, n, theta[:, None], phi[:, None])
sh_shape = (np.prod(data.shape[:-1]), -1)
sh_coeff = np.linalg.lstsq(B_dwi, data[...,
where_dwi].reshape(sh_shape).T)[0]
# Find the smoothed signal from the sh fit for the given gtab
smoothed_signal = np.dot(B_dwi,
sh_coeff).T.reshape(data.shape[:-1] + (-1, ))
pred_sig = np.zeros(smoothed_signal.shape[:-1] + (gtab.bvals.shape[0], ))
pred_sig[..., ~gtab.b0s_mask] = smoothed_signal
# Just give back the signal for the b0s since we can't really do anything about it
if np.sum(gtab.b0s_mask) > 1:
pred_sig[..., where_b0s] = np.mean(data[..., where_b0s], axis=-1)
else:
pred_sig[..., where_b0s] = data[..., where_b0s]
return pred_sig
def test_hat_and_lcr():
hemi = hemi_icosahedron.subdivide(3)
m, n = sph_harm_ind_list(8)
B = real_sph_harm(m, n, hemi.theta[:, None], hemi.phi[:, None])
H = hat(B)
B_hat = np.dot(H, B)
assert_array_almost_equal(B, B_hat)
R = lcr_matrix(H)
d = np.arange(len(hemi.theta))
r = d - np.dot(H, d)
lev = np.sqrt(1 - H.diagonal())
r /= lev
r -= r.mean()
r2 = np.dot(R, d)
assert_array_almost_equal(r, r2)
r3 = np.dot(d, R.T)
assert_array_almost_equal(r, r3)
def test_hat_and_lcr():
hemi = hemi_icosahedron.subdivide(3)
m, n = sph_harm_ind_list(8)
B = real_sph_harm(m, n, hemi.theta[:, None], hemi.phi[:, None])
H = hat(B)
B_hat = np.dot(H, B)
assert_array_almost_equal(B, B_hat)
R = lcr_matrix(H)
d = np.arange(len(hemi.theta))
r = d - np.dot(H, d)
lev = np.sqrt(1 - H.diagonal())
r /= lev
r -= r.mean()
r2 = np.dot(R, d)
assert_array_almost_equal(r, r2)
r3 = np.dot(d, R.T)
assert_array_almost_equal(r, r3)
def test_hat_and_lcr():
v, e, f = create_half_unit_sphere(6)
m, n = sph_harm_ind_list(8)
r, pol, azi = cart2sphere(*v.T)
B = real_sph_harm(m, n, azi[:, None], pol[:, None])
H = hat(B)
B_hat = np.dot(H, B)
assert_array_almost_equal(B, B_hat)
R = lcr_matrix(H)
d = np.arange(len(azi))
r = d - np.dot(H, d)
lev = np.sqrt(1 - H.diagonal())
r /= lev
r -= r.mean()
r2 = np.dot(R, d)
assert_array_almost_equal(r, r2)
r3 = np.dot(d, R.T)
assert_array_almost_equal(r, r3)
def test_hat_and_lcr():
v, e, f = create_half_unit_sphere(6)
m, n = sph_harm_ind_list(8)
r, pol, azi = cart2sphere(*v.T)
B = real_sph_harm(m, n, azi[:, None], pol[:, None])
H = hat(B)
B_hat = np.dot(H, B)
assert_array_almost_equal(B, B_hat)
R = lcr_matrix(H)
d = np.arange(len(azi))
r = d - np.dot(H, d)
lev = np.sqrt(1 - H.diagonal())
r /= lev
r -= r.mean()
r2 = np.dot(R, d)
assert_array_almost_equal(r, r2)
r3 = np.dot(d, R.T)
assert_array_almost_equal(r, r3)
def esh_matrix(order, gtab):
""" Matrix that evaluates SH coeffs in the given directions
Parameters
----------
order
gtab
Returns
-------
"""
bvecs = gtab.bvecs
r, theta, phi = cart2sphere(bvecs[:, 0], bvecs[:, 1], bvecs[:, 2])
theta[np.isnan(theta)] = 0
M = np.zeros((bvecs.shape[0], esh.LENGTH[order]))
counter = 0
for l in range(0, order + 1, 2):
for m in range(-l, l + 1):
M[:, counter] = real_sph_harm(m, l, theta, phi)
counter += 1
return M
def SHOREmatrix(radial_order, zeta, gtab, tau=1 / (4 * np.pi ** 2)):
"""Compute the SHORE matrix"
Parameters
----------
radial_order : unsigned int,
Radial Order
zeta : unsigned int,
scale factor
gtab : GradientTable,
Gradient directions and bvalues container class
tau : float,
diffusion time. By default the value that makes q=sqrt(b).
"""
qvals = np.sqrt(gtab.bvals / (4 * np.pi ** 2 * tau))
bvecs = gtab.bvecs
qgradients = qvals[:, None] * bvecs
r, theta, phi = cart2sphere(
qgradients[:, 0], qgradients[:, 1], qgradients[:, 2])
theta[np.isnan(theta)] = 0
M = np.zeros(
(r.shape[0], (radial_order + 1) * ((radial_order + 1) / 2) * (2 * radial_order + 1)))
counter = 0
for n in range(radial_order + 1):
for l in range(0, n + 1, 2):
for m in range(-l, l + 1):
M[:, counter] = real_sph_harm(m, l, theta, phi) * \
genlaguerre(n - l, l + 0.5)(r ** 2 / float(zeta)) * \
np.exp(- r ** 2 / (2.0 * zeta)) * \
__kappa(zeta, n, l) * \
(r ** 2 / float(zeta)) ** (l / 2)
counter += 1
return M[:, 0:counter]
def test_smooth_pinv():
hemi = hemi_icosahedron.subdivide(2)
m, n = sph_harm_ind_list(4)
B = real_sph_harm(m, n, hemi.theta[:, None], hemi.phi[:, None])
L = np.zeros(len(m))
C = smooth_pinv(B, L)
D = np.dot(npl.inv(np.dot(B.T, B)), B.T)
assert_array_almost_equal(C, D)
L = n * (n + 1) * .05
C = smooth_pinv(B, L)
L = np.diag(L)
D = np.dot(npl.inv(np.dot(B.T, B) + L * L), B.T)
assert_array_almost_equal(C, D)
L = np.arange(len(n)) * .05
C = smooth_pinv(B, L)
L = np.diag(L)
D = np.dot(npl.inv(np.dot(B.T, B) + L * L), B.T)
assert_array_almost_equal(C, D)
def test_smooth_pinv():
hemi = hemi_icosahedron.subdivide(2)
m, n = sph_harm_ind_list(4)
B = real_sph_harm(m, n, hemi.theta[:, None], hemi.phi[:, None])
L = np.zeros(len(m))
C = smooth_pinv(B, L)
D = np.dot(npl.inv(np.dot(B.T, B)), B.T)
assert_array_almost_equal(C, D)
L = n * (n + 1) * .05
C = smooth_pinv(B, L)
L = np.diag(L)
D = np.dot(npl.inv(np.dot(B.T, B) + L * L), B.T)
assert_array_almost_equal(C, D)
L = np.arange(len(n)) * .05
C = smooth_pinv(B, L)
L = np.diag(L)
D = np.dot(npl.inv(np.dot(B.T, B) + L * L), B.T)
assert_array_almost_equal(C, D)
def test_smooth_pinv():
v, e, f = create_half_unit_sphere(3)
m, n = sph_harm_ind_list(4)
r, pol, azi = cart2sphere(*v.T)
B = real_sph_harm(m, n, azi[:, None], pol[:, None])
L = np.zeros(len(m))
C = smooth_pinv(B, L)
D = np.dot(npl.inv(np.dot(B.T, B)), B.T)
assert_array_almost_equal(C, D)
L = n * (n + 1) * 0.05
C = smooth_pinv(B, L)
L = np.diag(L)
D = np.dot(npl.inv(np.dot(B.T, B) + L * L), B.T)
assert_array_almost_equal(C, D)
L = np.arange(len(n)) * 0.05
C = smooth_pinv(B, L)
L = np.diag(L)
D = np.dot(npl.inv(np.dot(B.T, B) + L * L), B.T)
assert_array_almost_equal(C, D)
def test_smooth_pinv():
v, e, f = create_half_unit_sphere(3)
m, n = sph_harm_ind_list(4)
r, pol, azi = cart2sphere(*v.T)
B = real_sph_harm(m, n, azi[:, None], pol[:, None])
L = np.zeros(len(m))
C = smooth_pinv(B, L)
D = np.dot(npl.inv(np.dot(B.T, B)), B.T)
assert_array_almost_equal(C, D)
L = n * (n + 1) * .05
C = smooth_pinv(B, L)
L = np.diag(L)
D = np.dot(npl.inv(np.dot(B.T, B) + L * L), B.T)
assert_array_almost_equal(C, D)
L = np.arange(len(n)) * .05
C = smooth_pinv(B, L)
L = np.diag(L)
D = np.dot(npl.inv(np.dot(B.T, B) + L * L), B.T)
assert_array_almost_equal(C, D)
def shore_K_mu_independent(radial_order, rgrad):
'''Computes mu independent part of K [2]. Same trick as with Q.
'''
r, theta, phi = cart2sphere(rgrad[:, 0], rgrad[:, 1], rgrad[:, 2])
theta[np.isnan(theta)] = 0
ind_mat = shore_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 = n + 2 - 2 * j
const = (-1) ** (j - 1) *\
(np.sqrt(2) * np.pi) ** (-1) *\
(r ** 2 / 2) ** (l / 2)
for m in range(-l, l + 1):
K[:, counter] = const * real_sph_harm(m, l, theta, phi)
counter += 1
return K
def shore_matrix_pdf(radial_order, zeta, rtab):
r"""Compute the SHORE propagator matrix [1]_"
Parameters
----------
radial_order : unsigned int,
an even integer that represent the order of the basis
zeta : unsigned int,
scale factor
rtab : array, shape (N,3)
real space points in which calculates the pdf
References
----------
.. [1] Merlet S. et. al, "Continuous diffusion signal, EAP and
ODF estimation via Compressive Sensing in diffusion MRI", Medical
Image Analysis, 2013.
"""
r, theta, phi = cart2sphere(rtab[:, 0], rtab[:, 1], rtab[:, 2])
theta[np.isnan(theta)] = 0
F = radial_order / 2
n_c = int(np.round(1 / 6.0 * (F + 1) * (F + 2) * (4 * F + 3)))
psi = np.zeros((r.shape[0], n_c))
counter = 0
for l in range(0, radial_order + 1, 2):
for n in range(l, int((radial_order + l) / 2) + 1):
for m in range(-l, l + 1):
psi[:, counter] = real_sph_harm(m, l, theta, phi) * \
genlaguerre(n - l, l + 0.5)(4 * np.pi ** 2 *
zeta * r ** 2) *\
np.exp(-2 * np.pi ** 2 * zeta * r ** 2) *\
_kappa_pdf(zeta, n, l) *\
(4 * np.pi ** 2 * zeta * r ** 2) ** (l / 2) * \
(-1) ** (n - l / 2)
counter += 1
return psi
def shore_matrix_pdf(radial_order, zeta, rtab):
r"""Compute the SHORE propagator matrix [1]_"
Parameters
----------
radial_order : unsigned int,
an even integer that represent the order of the basis
zeta : unsigned int,
scale factor
rtab : array, shape (N,3)
real space points in which calculates the pdf
References
----------
.. [1] Merlet S. et al., "Continuous diffusion signal, EAP and
ODF estimation via Compressive Sensing in diffusion MRI", Medical
Image Analysis, 2013.
"""
r, theta, phi = cart2sphere(rtab[:, 0], rtab[:, 1], rtab[:, 2])
theta[np.isnan(theta)] = 0
F = radial_order / 2
n_c = int(np.round(1 / 6.0 * (F + 1) * (F + 2) * (4 * F + 3)))
psi = np.zeros((r.shape[0], n_c))
counter = 0
for l in range(0, radial_order + 1, 2):
for n in range(l, int((radial_order + l) / 2) + 1):
for m in range(-l, l + 1):
psi[:, counter] = real_sph_harm(m, l, theta, phi) * \
genlaguerre(n - l, l + 0.5)(4 * np.pi ** 2 *
zeta * r ** 2) *\
np.exp(-2 * np.pi ** 2 * zeta * r ** 2) *\
_kappa_pdf(zeta, n, l) *\
(4 * np.pi ** 2 * zeta * r ** 2) ** (l / 2) * \
(-1) ** (n - l / 2)
counter += 1
return psi
def 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=default_sphere):
""" Recursive calibration of response function using peak threshold
Parameters
----------
gtab : GradientTable
data : ndarray
diffusion data
mask : ndarray, optional
mask for recursive calibration, for example a white matter mask. It has
shape `data.shape[0:3]` and dtype=bool. Default: use the entire data
array.
sh_order : int, optional
maximal spherical harmonics order. Default: 8
peak_thr : float, 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_fa : float, optional
FA of the initial 'fat' response function (tensor). Default: 0.08
init_trace : float, optional
trace of the initial 'fat' response function (tensor). Default: 0.0021
iter : int, optional
maximum number of iterations for calibration. Default: 8.
convergence : float, optional
convergence criterion, maximum relative change of SH
coefficients. Default: 0.001.
parallel : bool, 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()).
sphere : Sphere, optional.
The sphere used for peak finding. Default: default_sphere.
Returns
-------
response : ndarray
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.
"""
S0 = 1.
evals = fa_trace_to_lambdas(init_fa, init_trace)
res_obj = (evals, S0)
if mask is None:
data = data.reshape(-1, data.shape[-1])
else:
data = data[mask]
n = np.arange(0, sh_order + 1, 2)
where_dwi = lazy_index(~gtab.b0s_mask)
response_p = np.ones(len(n))
for num_it in range(iter):
r_sh_all = np.zeros(len(n))
csd_model = ConstrainedSphericalDeconvModel(gtab, res_obj,
sh_order=sh_order)
csd_peaks = peaks_from_model(model=csd_model,
data=data,
sphere=sphere,
relative_peak_threshold=peak_thr,
min_separation_angle=25,
parallel=parallel,
nbr_processes=nbr_processes)
dirs = csd_peaks.peak_dirs
vals = csd_peaks.peak_values
single_peak_mask = (vals[:, 1] / vals[:, 0]) < peak_thr
data = data[single_peak_mask]
dirs = dirs[single_peak_mask]
for num_vox in range(data.shape[0]):
rotmat = vec2vec_rotmat(dirs[num_vox, 0], np.array([0, 0, 1]))
rot_gradients = np.dot(rotmat, gtab.gradients.T).T
x, y, z = rot_gradients[where_dwi].T
r, theta, phi = cart2sphere(x, y, z)
# for the gradient sphere
B_dwi = real_sph_harm(0, n, theta[:, None], phi[:, None])
r_sh_all += np.linalg.lstsq(B_dwi, data[num_vox, where_dwi])[0]
response = r_sh_all / data.shape[0]
res_obj = AxSymShResponse(data[:, gtab.b0s_mask].mean(), response)
change = abs((response_p - response) / response_p)
if all(change < convergence):
break
response_p = response
return res_obj
def 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=default_sphere):
""" Recursive calibration of response function using peak threshold
Parameters
----------
gtab : GradientTable
data : ndarray
diffusion data
mask : ndarray, optional
mask for recursive calibration, for example a white matter mask. It has
shape `data.shape[0:3]` and dtype=bool. Default: use the entire data
array.
sh_order : int, optional
maximal spherical harmonics order. Default: 8
peak_thr : float, 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_fa : float, optional
FA of the initial 'fat' response function (tensor). Default: 0.08
init_trace : float, optional
trace of the initial 'fat' response function (tensor). Default: 0.0021
iter : int, optional
maximum number of iterations for calibration. Default: 8.
convergence : float, optional
convergence criterion, maximum relative change of SH
coefficients. Default: 0.001.
parallel : bool, 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()).
sphere : Sphere, optional.
The sphere used for peak finding. Default: default_sphere.
Returns
-------
response : ndarray
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.
"""
S0 = 1.
evals = fa_trace_to_lambdas(init_fa, init_trace)
res_obj = (evals, S0)
if mask is None:
data = data.reshape(-1, data.shape[-1])
else:
data = data[mask]
n = np.arange(0, sh_order + 1, 2)
where_dwi = lazy_index(~gtab.b0s_mask)
response_p = np.ones(len(n))
for _ in range(iter):
r_sh_all = np.zeros(len(n))
csd_model = ConstrainedSphericalDeconvModel(gtab,
res_obj,
sh_order=sh_order)
csd_peaks = peaks_from_model(model=csd_model,
data=data,
sphere=sphere,
relative_peak_threshold=peak_thr,
min_separation_angle=25,
parallel=parallel,
nbr_processes=nbr_processes)
dirs = csd_peaks.peak_dirs
vals = csd_peaks.peak_values
single_peak_mask = (vals[:, 1] / vals[:, 0]) < peak_thr
data = data[single_peak_mask]
dirs = dirs[single_peak_mask]
for num_vox in range(data.shape[0]):
rotmat = vec2vec_rotmat(dirs[num_vox, 0], np.array([0, 0, 1]))
rot_gradients = np.dot(rotmat, gtab.gradients.T).T
x, y, z = rot_gradients[where_dwi].T
r, theta, phi = cart2sphere(x, y, z)
# for the gradient sphere
B_dwi = real_sph_harm(0, n, theta[:, None], phi[:, None])
r_sh_all += np.linalg.lstsq(B_dwi,
data[num_vox, where_dwi],
rcond=-1)[0]
response = r_sh_all / data.shape[0]
res_obj = AxSymShResponse(data[:, gtab.b0s_mask].mean(), response)
change = abs((response_p - response) / response_p)
if all(change < convergence):
break
response_p = response
return res_obj
def __init__(self,
gtab,
response,
reg_sphere=None,
sh_order=8,
lambda_=1,
tau=0.1,
convergence=50):
r""" 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
----------
gtab : GradientTable
response : tuple 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. This is to be able to
generate a single fiber synthetic signal. The response function
will be used as deconvolution kernel ([1]_)
reg_sphere : Sphere (optional)
sphere used to build the regularization B matrix.
Default: 'symmetric362'.
sh_order : int (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
tau : float (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
convergence : int
Maximum number of iterations to allow the deconvolution to converge.
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
.. [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
"""
# Initialize the parent class:
SphHarmModel.__init__(self, gtab)
m, n = sph_harm_ind_list(sh_order)
self.m, self.n = m, n
self._where_b0s = lazy_index(gtab.b0s_mask)
self._where_dwi = lazy_index(~gtab.b0s_mask)
no_params = ((sh_order + 1) * (sh_order + 2)) / 2
if no_params > np.sum(~gtab.b0s_mask):
msg = "Number of parameters required for the fit are more "
msg += "than the actual data points"
warnings.warn(msg, UserWarning)
x, y, z = gtab.gradients[self._where_dwi].T
r, theta, phi = cart2sphere(x, y, z)
# for the gradient sphere
self.B_dwi = real_sph_harm(m, n, theta[:, None], phi[:, None])
# for the sphere used in the regularization positivity constraint
if reg_sphere is None:
self.sphere = small_sphere
else:
self.sphere = reg_sphere
r, theta, phi = cart2sphere(self.sphere.x, self.sphere.y,
self.sphere.z)
self.B_reg = real_sph_harm(m, n, theta[:, None], phi[:, None])
if response is None:
response = (np.array([0.0015, 0.0003, 0.0003]), 1)
self.response = response
if isinstance(response, AxSymShResponse):
r_sh = response.dwi_response
self.response_scaling = response.S0
n_response = response.n
m_response = response.m
else:
self.S_r = estimate_response(gtab, self.response[0],
self.response[1])
r_sh = np.linalg.lstsq(self.B_dwi,
self.S_r[self._where_dwi],
rcond=-1)[0]
n_response = n
m_response = m
self.response_scaling = response[1]
r_rh = sh_to_rh(r_sh, m_response, n_response)
self.R = forward_sdeconv_mat(r_rh, n)
# scale lambda_ to account for differences in the number of
# SH coefficients and number of mapped directions
# This is exactly what is done in [4]_
lambda_ = (lambda_ * self.R.shape[0] * r_rh[0] /
(np.sqrt(self.B_reg.shape[0]) * np.sqrt(362.)))
self.B_reg *= lambda_
self.sh_order = sh_order
self.tau = tau
self.convergence = convergence
self._X = X = self.R.diagonal() * self.B_dwi
self._P = np.dot(X.T, X)
def basis(self, sphere):
"""A basis that maps the response coefficients onto a sphere."""
theta = sphere.theta[:, None]
phi = sphere.phi[:, None]
return real_sph_harm(self.m, self.n, theta, phi)
def __init__(self,
gtab,
ratio,
reg_sphere=None,
sh_order=8,
lambda_=1.,
tau=0.1):
r""" 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
----------
gtab : GradientTable
ratio : float
ratio of the smallest vs the largest eigenvalue of the single
prolate tensor response function
reg_sphere : Sphere
sphere used to build the regularization B matrix
sh_order : int
maximal spherical harmonics order
lambda_ : float
weight given to the constrained-positivity regularization part of
the deconvolution equation
tau : float
threshold (tau *mean(fODF)) controlling the amplitude below
which the corresponding fODF is assumed to be zero.
References
----------
.. [1] Descoteaux, M., et al. IEEE TMI 2009. Deterministic and
Probabilistic Tractography Based on Complex Fibre Orientation
Distributions.
"""
SphHarmModel.__init__(self, gtab)
m, n = sph_harm_ind_list(sh_order)
self.m, self.n = m, n
self._where_b0s = lazy_index(gtab.b0s_mask)
self._where_dwi = lazy_index(~gtab.b0s_mask)
no_params = ((sh_order + 1) * (sh_order + 2)) / 2
if no_params > np.sum(~gtab.b0s_mask):
msg = "Number of parameters required for the fit are more "
msg += "than the actual data points"
warnings.warn(msg, UserWarning)
x, y, z = gtab.gradients[self._where_dwi].T
r, theta, phi = cart2sphere(x, y, z)
# for the gradient sphere
self.B_dwi = real_sph_harm(m, n, theta[:, None], phi[:, None])
# for the odf sphere
if reg_sphere is None:
self.sphere = get_sphere('symmetric362')
else:
self.sphere = reg_sphere
r, theta, phi = cart2sphere(self.sphere.x, self.sphere.y,
self.sphere.z)
self.B_reg = real_sph_harm(m, n, theta[:, None], phi[:, None])
self.R, self.P = forward_sdt_deconv_mat(ratio, n)
# scale lambda_ to account for differences in the number of
# SH coefficients and number of mapped directions
self.lambda_ = (lambda_ * self.R.shape[0] * self.R[0, 0] /
self.B_reg.shape[0])
self.tau = tau
self.sh_order = sh_order
def shore_matrix(radial_order, zeta, gtab, tau=1 / (4 * np.pi**2)):
r"""Compute the SHORE matrix for modified Merlet's 3D-SHORE [1]_
..math::
:nowrap:
\begin{equation}
\textbf{E}(q\textbf{u})=\sum_{l=0, even}^{N_{max}}
\sum_{n=l}^{(N_{max}+l)/2}
\sum_{m=-l}^l c_{nlm}
\phi_{nlm}(q\textbf{u})
\end{equation}
where $\phi_{nlm}$ is
..math::
:nowrap:
\begin{equation}
\phi_{nlm}^{SHORE}(q\textbf{u})=\Biggl[\dfrac{2(n-l)!}
{\zeta^{3/2} \Gamma(n+3/2)} \Biggr]^{1/2}
\Biggl(\dfrac{q^2}{\zeta}\Biggr)^{l/2}
exp\Biggl(\dfrac{-q^2}{2\zeta}\Biggr)
L^{l+1/2}_{n-l} \Biggl(\dfrac{q^2}{\zeta}\Biggr)
Y_l^m(\textbf{u}).
\end{equation}
Parameters
----------
radial_order : unsigned int,
an even integer that represent the order of the basis
zeta : unsigned int,
scale factor
gtab : GradientTable,
gradient directions and bvalues container class
tau : float,
diffusion time. By default the value that makes q=sqrt(b).
References
----------
.. [1] Merlet S. et al., "Continuous diffusion signal, EAP and
ODF estimation via Compressive Sensing in diffusion MRI", Medical
Image Analysis, 2013.
"""
qvals = np.sqrt(gtab.bvals / (4 * np.pi**2 * tau))
qvals[gtab.b0s_mask] = 0
bvecs = gtab.bvecs
qgradients = qvals[:, None] * bvecs
r, theta, phi = cart2sphere(qgradients[:, 0], qgradients[:, 1],
qgradients[:, 2])
theta[np.isnan(theta)] = 0
F = radial_order / 2
n_c = int(np.round(1 / 6.0 * (F + 1) * (F + 2) * (4 * F + 3)))
M = np.zeros((r.shape[0], n_c))
counter = 0
for l in range(0, radial_order + 1, 2):
for n in range(l, int((radial_order + l) / 2) + 1):
for m in range(-l, l + 1):
M[:, counter] = real_sph_harm(m, l, theta, phi) * \
genlaguerre(n - l, l + 0.5)(r ** 2 / zeta) * \
np.exp(- r ** 2 / (2.0 * zeta)) * \
_kappa(zeta, n, l) * \
(r ** 2 / zeta) ** (l / 2)
counter += 1
return M
def multi_shell_fiber_response(sh_order,
bvals,
wm_rf,
gm_rf,
csf_rf,
sphere=None,
tol=20):
"""Fiber response function estimation for multi-shell data.
Parameters
----------
sh_order : int
Maximum spherical harmonics order.
bvals : ndarray
Array containing the b-values. Must be unique b-values, like outputed
by `dipy.core.gradients.unique_bvals_tolerance`.
wm_rf : (4, len(bvals)) ndarray
Response function of the WM tissue, for each bvals.
gm_rf : (4, len(bvals)) ndarray
Response function of the GM tissue, for each bvals.
csf_rf : (4, len(bvals)) ndarray
Response function of the CSF tissue, for each bvals.
sphere : `dipy.core.Sphere` instance, optional
Sphere where the signal will be evaluated.
Returns
-------
MultiShellResponse
MultiShellResponse object.
"""
NUMPY_1_14_PLUS = LooseVersion(np.__version__) >= LooseVersion('1.14.0')
rcond_value = None if NUMPY_1_14_PLUS else -1
bvals = np.array(bvals, copy=True)
evecs = np.zeros((3, 3))
z = np.array([0, 0, 1.])
evecs[:, 0] = z
evecs[:2, 1:] = np.eye(2)
n = np.arange(0, sh_order + 1, 2)
m = np.zeros_like(n)
if sphere is None:
sphere = default_sphere
big_sphere = sphere.subdivide()
theta, phi = big_sphere.theta, big_sphere.phi
B = shm.real_sph_harm(m, n, theta[:, None], phi[:, None])
A = shm.real_sph_harm(0, 0, 0, 0)
response = np.empty([len(bvals), len(n) + 2])
if bvals[0] < tol:
gtab = GradientTable(big_sphere.vertices * 0)
wm_response = single_tensor(gtab,
wm_rf[0, 3],
wm_rf[0, :3],
evecs,
snr=None)
response[0, 2:] = np.linalg.lstsq(B, wm_response, rcond=rcond_value)[0]
response[0, 1] = gm_rf[0, 3] / A
response[0, 0] = csf_rf[0, 3] / A
for i, bvalue in enumerate(bvals[1:]):
gtab = GradientTable(big_sphere.vertices * bvalue)
wm_response = single_tensor(gtab,
wm_rf[i, 3],
wm_rf[i, :3],
evecs,
snr=None)
response[i + 1, 2:] = np.linalg.lstsq(B,
wm_response,
rcond=rcond_value)[0]
response[i + 1,
1] = gm_rf[i, 3] * np.exp(-bvalue * gm_rf[i, 0]) / A
response[i + 1,
0] = csf_rf[i, 3] * np.exp(-bvalue * csf_rf[i, 0]) / A
S0 = [csf_rf[0, 3], gm_rf[0, 3], wm_rf[0, 3]]
else:
warnings.warn("""No b0 given. Proceeding either way.""", UserWarning)
for i, bvalue in enumerate(bvals):
gtab = GradientTable(big_sphere.vertices * bvalue)
wm_response = single_tensor(gtab,
wm_rf[i, 3],
wm_rf[i, :3],
evecs,
snr=None)
response[i, 2:] = np.linalg.lstsq(B,
wm_response,
rcond=rcond_value)[0]
response[i, 1] = gm_rf[i, 3] * np.exp(-bvalue * gm_rf[i, 0]) / A
response[i, 0] = csf_rf[i, 3] * np.exp(-bvalue * csf_rf[i, 0]) / A
S0 = [csf_rf[0, 3], gm_rf[0, 3], wm_rf[0, 3]]
return MultiShellResponse(response, sh_order, bvals, S0=S0)
def csd_predict(sh_coeff, gtab, response=None, S0=1, R=None):
"""
Compute a signal prediction given spherical harmonic coefficients and
(optionally) a response function for the provided GradientTable class
instance
Parameters
----------
sh_coeff : ndarray
Spherical harmonic coefficients
gtab : GradientTable class instance
response : tuple
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.
Default: (np.array([0.0015, 0.0003, 0.0003]), 1)
S0 : ndarray or float
The non diffusion-weighted signal value.
R : ndarray
Optionally, provide an R matrix. If not provided, calculated from the
gtab, response function, etc.
Returns
-------
pred_sig : ndarray
The signal predicted from the provided SH coefficients for a
measurement with the provided GradientTable. The last dimension of the
resulting array is the same as the number of bvals/bvecs in the
GradientTable. The first dimensions have shape: `sh_coeff.shape[:-1]`.
"""
n_coeff = sh_coeff.shape[-1]
sh_order = order_from_ncoef(n_coeff)
x, y, z = gtab.gradients[~gtab.b0s_mask].T
r, theta, phi = cart2sphere(x, y, z)
SH_basis, m, n = real_sym_sh_basis(sh_order, theta, phi)
if R is None:
# for the gradient sphere
B_dwi = real_sph_harm(m, n, theta[:, None], phi[:, None])
if response is None:
response = (np.array([0.0015, 0.0003, 0.0003]), 1)
else:
response = response
S_r = estimate_response(gtab, response[0], response[1])
r_sh = np.linalg.lstsq(B_dwi, S_r[~gtab.b0s_mask])[0]
r_rh = sh_to_rh(r_sh, m, n)
R = forward_sdeconv_mat(r_rh, n)
predict_matrix = np.dot(SH_basis, R)
if np.iterable(S0):
# If it's an array, we need to give it one more dimension:
S0 = S0[..., None]
# This is the key operation: convolve and multiply by S0:
pre_pred_sig = S0 * np.dot(predict_matrix, sh_coeff)
# Now put everything in its right place:
pred_sig = np.zeros(pre_pred_sig.shape[:-1] + (gtab.bvals.shape[0],))
pred_sig[..., ~gtab.b0s_mask] = pre_pred_sig
pred_sig[..., gtab.b0s_mask] = S0
return pred_sig
def __init__(self, gtab, ratio, reg_sphere=None, sh_order=8, lambda_=1., tau=0.1):
r""" 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 Contrained 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
----------
gtab : GradientTable
ratio : float
ratio of the smallest vs the largest eigenvalue of the single prolate tensor response function
reg_sphere : Sphere
sphere used to build the regularization B matrix
sh_order : int
maximal spherical harmonics order
lambda_ : float
weight given to the constrained-positivity regularization part of the
deconvolution equation
tau : float
threshold (tau *mean(fODF)) controlling the amplitude below
which the corresponding fODF is assumed to be zero.
References
----------
.. [1] Descoteaux, M., et al. IEEE TMI 2009. Deterministic and Probabilistic Tractography Based
on Complex Fibre Orientation Distributions.
"""
m, n = sph_harm_ind_list(sh_order)
self.m, self.n = m, n
self._where_b0s = lazy_index(gtab.b0s_mask)
self._where_dwi = lazy_index(~gtab.b0s_mask)
no_params = ((sh_order + 1) * (sh_order + 2)) / 2
if no_params > np.sum(gtab.b0s_mask == False):
msg = "Number of parameters required for the fit are more "
msg += "than the actual data points"
warnings.warn(msg, UserWarning)
x, y, z = gtab.gradients[self._where_dwi].T
r, theta, phi = cart2sphere(x, y, z)
# for the gradient sphere
self.B_dwi = real_sph_harm(m, n, theta[:, None], phi[:, None])
# for the odf sphere
if reg_sphere is None:
self.sphere = get_sphere('symmetric362')
else:
self.sphere = reg_sphere
r, theta, phi = cart2sphere(self.sphere.x, self.sphere.y, self.sphere.z)
self.B_reg = real_sph_harm(m, n, theta[:, None], phi[:, None])
self.R, self.P = forward_sdt_deconv_mat(ratio, sh_order)
# scale lambda_ to account for differences in the number of
# SH coefficients and number of mapped directions
self.lambda_ = lambda_ * self.R.shape[0] * self.R[0, 0] / self.B_reg.shape[0]
self.tau = tau
self.sh_order = sh_order
def basis(self, sphere):
"""A basis that maps the response coefficients onto a sphere."""
theta = sphere.theta[:, None]
phi = sphere.phi[:, None]
return real_sph_harm(self.m, self.n, theta, phi)
def shore_matrix(radial_order, zeta, gtab, tau=1 / (4 * np.pi ** 2)):
r"""Compute the SHORE matrix for modified Merlet's 3D-SHORE [1]_
..math::
:nowrap:
\begin{equation}
\textbf{E}(q\textbf{u})=\sum_{l=0, even}^{N_{max}}
\sum_{n=l}^{(N_{max}+l)/2}
\sum_{m=-l}^l c_{nlm}
\phi_{nlm}(q\textbf{u})
\end{equation}
where $\phi_{nlm}$ is
..math::
:nowrap:
\begin{equation}
\phi_{nlm}^{SHORE}(q\textbf{u})=\Biggl[\dfrac{2(n-l)!}
{\zeta^{3/2} \Gamma(n+3/2)} \Biggr]^{1/2}
\Biggl(\dfrac{q^2}{\zeta}\Biggr)^{l/2}
exp\Biggl(\dfrac{-q^2}{2\zeta}\Biggr)
L^{l+1/2}_{n-l} \Biggl(\dfrac{q^2}{\zeta}\Biggr)
Y_l^m(\textbf{u}).
\end{equation}
Parameters
----------
radial_order : unsigned int,
an even integer that represent the order of the basis
zeta : unsigned int,
scale factor
gtab : GradientTable,
gradient directions and bvalues container class
tau : float,
diffusion time. By default the value that makes q=sqrt(b).
References
----------
.. [1] Merlet S. et. al, "Continuous diffusion signal, EAP and
ODF estimation via Compressive Sensing in diffusion MRI", Medical
Image Analysis, 2013.
"""
qvals = np.sqrt(gtab.bvals / (4 * np.pi ** 2 * tau))
qvals[gtab.b0s_mask] = 0
bvecs = gtab.bvecs
qgradients = qvals[:, None] * bvecs
r, theta, phi = cart2sphere(qgradients[:, 0], qgradients[:, 1],
qgradients[:, 2])
theta[np.isnan(theta)] = 0
F = radial_order / 2
n_c = int(np.round(1 / 6.0 * (F + 1) * (F + 2) * (4 * F + 3)))
M = np.zeros((r.shape[0], n_c))
counter = 0
for l in range(0, radial_order + 1, 2):
for n in range(l, int((radial_order + l) / 2) + 1):
for m in range(-l, l + 1):
M[:, counter] = real_sph_harm(m, l, theta, phi) * \
genlaguerre(n - l, l + 0.5)(r ** 2 / zeta) * \
np.exp(- r ** 2 / (2.0 * zeta)) * \
_kappa(zeta, n, l) * \
(r ** 2 / zeta) ** (l / 2)
counter += 1
return M
def __init__(self, gtab, response, reg_sphere=None, sh_order=8, lambda_=1, tau=0.1):
r""" 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 thus,
comes as input to the ConstrainedSphericalDeconvModel. It will be used as deconvolution
kernel, as described in [1]_.
Parameters
----------
gtab : GradientTable
response : tuple or callable
If tuple, then it should have 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.
This is to be able to generate a single fiber synthetic signal. If callable then the function
should return an ndarray with the all the signal values for the response function. The response
function will be used as deconvolution kernel ([1]_)
reg_sphere : Sphere
sphere used to build the regularization B matrix
sh_order : int
maximal spherical harmonics order
lambda_ : float
weight given to the constrained-positivity regularization part of the
deconvolution equation (see [1]_)
tau : float
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]_)
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
.. [2] Descoteaux, M., et al. IEEE TMI 2009. Deterministic and Probabilistic Tractography Based
on Complex Fibre Orientation Distributions
.. [3] C\^ot\'e, 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
"""
m, n = sph_harm_ind_list(sh_order)
self.m, self.n = m, n
self._where_b0s = lazy_index(gtab.b0s_mask)
self._where_dwi = lazy_index(~gtab.b0s_mask)
no_params = ((sh_order + 1) * (sh_order + 2)) / 2
if no_params > np.sum(gtab.b0s_mask == False):
msg = "Number of parameters required for the fit are more "
msg += "than the actual data points"
warnings.warn(msg, UserWarning)
x, y, z = gtab.gradients[self._where_dwi].T
r, theta, phi = cart2sphere(x, y, z)
# for the gradient sphere
self.B_dwi = real_sph_harm(m, n, theta[:, None], phi[:, None])
# for the sphere used in the regularization positivity constraint
if reg_sphere is None:
self.sphere = get_sphere('symmetric362')
else:
self.sphere = reg_sphere
r, theta, phi = cart2sphere(self.sphere.x, self.sphere.y, self.sphere.z)
self.B_reg = real_sph_harm(m, n, theta[:, None], phi[:, None])
if callable(response):
S_r = response
else:
if response is None:
S_r = estimate_response(gtab, np.array([0.0015, 0.0003, 0.0003]), 1)
else:
S_r = estimate_response(gtab, response[0], response[1])
r_sh = np.linalg.lstsq(self.B_dwi, S_r[self._where_dwi])[0]
r_rh = sh_to_rh(r_sh, sh_order)
self.R = forward_sdeconv_mat(r_rh, sh_order)
# scale lambda_ to account for differences in the number of
# SH coefficients and number of mapped directions
# This is exactly what is done in [4]_
self.lambda_ = lambda_ * self.R.shape[0] * r_rh[0] / self.B_reg.shape[0]
self.sh_order = sh_order
self.tau = tau
def sh_smooth(data,
bvals,
bvecs,
sh_order=4,
similarity_threshold=50,
regul=0.006):
"""Smooth the raw diffusion signal with spherical harmonics.
data : ndarray
The diffusion data to smooth.
gtab : gradient table object
Corresponding gradients table object to data.
sh_order : int, default 8
Order of the spherical harmonics to fit.
similarity_threshold : int, default 50
All b-values such that |b_1 - b_2| < similarity_threshold
will be considered as identical for smoothing purpose.
Must be lower than 200.
regul : float, default 0.006
Amount of regularization to apply to sh coefficients computation.
Return
---------
pred_sig : ndarray
The smoothed diffusion data, fitted through spherical harmonics.
"""
if similarity_threshold > 200:
raise ValueError(
"similarity_threshold = {}, which is higher than 200,"
" please use a lower value".format(similarity_threshold))
m, n = sph_harm_ind_list(sh_order)
L = -n * (n + 1)
where_b0s = bvals == 0
pred_sig = np.zeros_like(data, dtype=np.float32)
# Round similar bvals together for identifying similar shells
rounded_bvals = np.zeros_like(bvals)
for unique_bval in np.unique(bvals):
idx = np.abs(unique_bval - bvals) < similarity_threshold
rounded_bvals[idx] = unique_bval
# process each b-value separately
for unique_bval in np.unique(rounded_bvals):
idx = rounded_bvals == unique_bval
# Just give back the signal for the b0s since we can't really do anything about it
if np.all(idx == where_b0s):
if np.sum(where_b0s) > 1:
pred_sig[..., idx] = np.mean(data[..., idx],
axis=-1,
keepdims=True)
else:
pred_sig[..., idx] = data[..., idx]
continue
x, y, z = bvecs[:, idx]
r, theta, phi = cart2sphere(x, y, z)
# Find the sh coefficients to smooth the signal
B_dwi = real_sph_harm(m, n, theta[:, None], phi[:, None])
invB = smooth_pinv(B_dwi, np.sqrt(regul) * L)
sh_coeff = np.dot(data[..., idx], invB.T)
# Find the smoothed signal from the sh fit for the given gtab
pred_sig[..., idx] = np.dot(sh_coeff, B_dwi.T)
return pred_sig
