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 test_sph_harm_ind_list():
m_list, n_list = sph_harm_ind_list(8)
assert_equal(m_list.shape, n_list.shape)
assert_equal(m_list.shape, (45,))
assert_true(np.all(np.abs(m_list) <= n_list))
assert_array_equal(n_list % 2, 0)
assert_raises(ValueError, sph_harm_ind_list, 1)
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 test_sph_harm_ind_list():
m_list, n_list = sph_harm_ind_list(8)
assert_equal(m_list.shape, n_list.shape)
assert_equal(m_list.shape, (45, ))
assert_true(np.all(np.abs(m_list) <= n_list))
assert_array_equal(n_list % 2, 0)
assert_raises(ValueError, sph_harm_ind_list, 1)
def main():
parser = _build_arg_parser()
args = parser.parse_args()
assert_inputs_exist(parser, args.in_sh, optional=args.mask)
# Load data
sh_img = nib.load(args.in_sh)
sh = sh_img.get_fdata(dtype=np.float32)
mask = None
if args.mask:
mask = get_data_as_mask(nib.load(args.mask), dtype=bool)
# Precompute output filenames to check if they exist
sh_order = order_from_ncoef(sh.shape[-1], full_basis=args.full_basis)
_, order_ids = sph_harm_ind_list(sh_order, full_basis=args.full_basis)
orders = sorted(np.unique(order_ids))
output_fnames = ["{}{}.nii.gz".format(args.out_prefix, i) for i in orders]
assert_outputs_exist(parser, args, output_fnames)
# Compute RISH features
rish, final_orders = compute_rish(sh, mask, full_basis=args.full_basis)
# Make sure the precomputed orders match the orders returned
assert np.all(orders == np.array(final_orders))
# Save each RISH feature as a separate file
for i, fname in enumerate(output_fnames):
nib.save(nib.Nifti1Image(rish[..., i], sh_img.affine), fname)
def forward_sdeconv_mat(r_rh, sh_order):
""" Build forward spherical deconvolution matrix
Parameters
----------
r_rh : ndarray (``(sh_order + 1)*(sh_order + 2)/2``,)
ndarray of rotational harmonics coefficients for the single
fiber response function
sh_order : int
maximal SH order
Returns
-------
R : ndarray (``(sh_order + 1)*(sh_order + 2)/2``, ``(sh_order + 1)*(sh_order + 2)/2``)
"""
m, n = sph_harm_ind_list(sh_order)
b = np.zeros(m.shape)
i = 0
for l in np.arange(0, sh_order + 1, 2):
for m in np.arange(-l, l + 1):
b[i] = r_rh[l / 2]
i = i + 1
return np.diag(b)
def test_hat_and_lcr():
hemi = hemi_icosahedron.subdivide(3)
m, n = sph_harm_ind_list(8)
with warnings.catch_warnings():
warnings.filterwarnings("ignore",
message=descoteaux07_legacy_msg,
category=PendingDeprecationWarning)
B = real_sh_descoteaux_from_index(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 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 get_deconv_matrix(gtab, response, sh_order):
""" Compute the deconvolution of the response of a single fiber
Attributes:
gtab (dipy GradientTable): the gradient of the dwi in wich the response
is represented.
response (float[4]): first 3 elements: eigenvalues, 4th: mean b0 value
sh_order (int): the sh_order of the DWI used for the deconvolution
Returns:
R:
r_rh:
B_dwi (np.array(nb_sh_coeff, nb_dwi_directions)): matrix to fit DWi->SH
"""
m, n = sph_harm_ind_list(sh_order)
# x, y, z = gtab.gradients[~gtab.b0s_mask].T
# r, theta, phi = cart2sphere(x, y, z)
# # for the gradient sphere
# B_dwi = real_sph_harm(m, n, theta[:, None], phi[:, None])
B_dwi, _ = get_B_matrix(gtab, sh_order)
S_r = estimate_response(gtab, response[0:3], response[3])
r_sh = np.linalg.lstsq(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)
# X = R.diagonal() * B_dwi
return R, r_rh, B_dwi
def test_smooth_pinv():
hemi = hemi_icosahedron.subdivide(2)
m, n = sph_harm_ind_list(4)
with warnings.catch_warnings():
warnings.filterwarnings("ignore",
message=descoteaux07_legacy_msg,
category=PendingDeprecationWarning)
B = real_sh_descoteaux_from_index(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 mats_odfdeconv(sphere, basis=None, ratio=3 / 15., sh_order=8, lambda_=1., tau=0.1, r2=True):
m, n = sph_harm_ind_list(sh_order)
r, theta, phi = cart2sphere(sphere.x, sphere.y, sphere.z)
real_sym_sh = sph_harm_lookup[basis]
B_reg, m, n = real_sym_sh(sh_order, theta[:, None], phi[:, None])
R, P = forward_sdt_deconv_mat(ratio, sh_order, r2_term=r2)
lambda_ = lambda_ * R.shape[0] * R[0, 0] / B_reg.shape[0]
return R, B_reg
def test_forward_sdeconv_mat():
m, n = sph_harm_ind_list(4)
mat = forward_sdeconv_mat(np.array([0, 2, 4]), n)
expected = np.diag([0, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4])
npt.assert_array_equal(mat, expected)
sh_order = 8
expected_size = (sh_order + 1) * (sh_order + 2) / 2
r_rh = np.arange(0, sh_order + 1, 2)
m, n = sph_harm_ind_list(sh_order)
mat = forward_sdeconv_mat(r_rh, n)
npt.assert_equal(mat.shape, (expected_size, expected_size))
npt.assert_array_equal(mat.diagonal(), n)
# Odd spherical harmonic degrees should raise a ValueError
n[2] = 3
npt.assert_raises(ValueError, forward_sdeconv_mat, r_rh, n)
def test_forward_sdeconv_mat():
m, n = sph_harm_ind_list(4)
mat = forward_sdeconv_mat(np.array([0, 2, 4]), n)
expected = np.diag([0, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4])
npt.assert_array_equal(mat, expected)
sh_order = 8
expected_size = (sh_order + 1) * (sh_order + 2) / 2
r_rh = np.arange(0, sh_order + 1, 2)
m, n = sph_harm_ind_list(sh_order)
mat = forward_sdeconv_mat(r_rh, n)
npt.assert_equal(mat.shape, (expected_size, expected_size))
npt.assert_array_equal(mat.diagonal(), n)
# Odd spherical harmonic degrees should raise a ValueError
n[2] = 3
npt.assert_raises(ValueError, forward_sdeconv_mat, r_rh, n)
def test_order_from_ncoeff():
"""
"""
# Just try some out:
for sh_order in [2, 4, 6, 8, 12, 24]:
m, n = sph_harm_ind_list(sh_order)
n_coef = m.shape[0]
npt.assert_equal(order_from_ncoef(n_coef), sh_order)
def test_order_from_ncoeff():
"""
"""
# Just try some out:
for sh_order in [2, 4, 6, 8, 12, 24]:
m, n = sph_harm_ind_list(sh_order)
n_coef = m.shape[0]
assert_equal(order_from_ncoef(n_coef), sh_order)
def __init__(self,
acquisition_scheme,
model,
x0_vector=None,
sh_order=8,
unity_constraint=True,
lambda_lb=0.):
self.model = model
self.acquisition_scheme = acquisition_scheme
self.sh_order = sh_order
self.Ncoef = int((sh_order + 2) * (sh_order + 1) // 2)
self.Nmodels = len(self.model.models)
self.lambda_lb = lambda_lb
self.unity_constraint = unity_constraint
self.sphere_jacobian = 2 * np.sqrt(np.pi)
sphere = get_sphere('symmetric724')
hemisphere = HemiSphere(phi=sphere.phi, theta=sphere.theta)
self.L_positivity = real_sym_sh_mrtrix(self.sh_order, hemisphere.theta,
hemisphere.phi)[0]
x0_single_voxel = np.reshape(x0_vector, (-1, x0_vector.shape[-1]))[0]
if np.all(np.isnan(x0_single_voxel)):
self.single_convolution_kernel = True
self.A = self._construct_convolution_kernel(x0_single_voxel)
else:
self.single_convolution_kernel = False
self.Ncoef_total = 0
vf_array = []
if self.model.volume_fractions_fixed:
self.sh_start = 0
self.Ncoef_total = self.Ncoef
self.vf_indices = np.array([0])
else:
for model in self.model.models:
if 'orientation' in model.parameter_types.values():
self.sh_start = self.Ncoef_total
sh_model = np.zeros(self.Ncoef)
sh_model[0] = 1
vf_array.append(sh_model)
self.Ncoef_total += self.Ncoef
else:
vf_array.append(1)
self.Ncoef_total += 1
self.vf_indices = np.where(np.hstack(vf_array))[0]
sh_l = sph_harm_ind_list(sh_order)[1]
lb_weights = sh_l**2 * (sh_l + 1)**2 # laplace-beltrami
if self.model.volume_fractions_fixed:
self.R_smoothness = np.diag(lb_weights)
else:
diagonal = np.zeros(self.Ncoef_total)
diagonal[self.sh_start:self.sh_start + self.Ncoef] = lb_weights
self.R_smoothness = np.diag(diagonal)
def odf_sh_to_sharp(odfs_sh, sphere, basis=None, ratio=3 / 15., sh_order=8, lambda_=1., tau=0.1):
r""" Sharpen odfs using the spherical deconvolution transform [1]_
This function can be used to sharpen any smooth ODF spherical function. In theory, this should
only be used to sharpen QballModel ODFs, but in practice, one can play with the deconvolution
ratio and sharpen almost any ODF-like spherical function. The constrained-regularization is stable
and will not only sharp the ODF peaks but also regularize the noisy peaks.
Parameters
----------
odfs_sh : ndarray (``(sh_order + 1)*(sh_order + 2)/2``, )
array of odfs expressed as spherical harmonics coefficients
sphere : Sphere
sphere used to build the regularization matrix
basis : {None, 'mrtrix', 'fibernav'}
different spherical harmonic basis. None is the fibernav basis as well.
ratio : float,
ratio of the smallest vs the largest eigenvalue of the single prolate tensor response function
(:math:`\frac{\lambda_2}{\lambda_1}`)
sh_order : int
maximal SH order of the SH representation
lambda_ : float
lambda parameter (see odfdeconv) (default 1.0)
tau : float
tau parameter in the L matrix construction (see odfdeconv) (default 0.1)
Returns
-------
fodf_sh : ndarray
sharpened odf expressed as spherical harmonics coefficients
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)
r, theta, phi = cart2sphere(sphere.x, sphere.y, sphere.z)
real_sym_sh = sph_harm_lookup[basis]
B_reg, m, n = real_sym_sh(sh_order, theta[:, None], phi[:, None])
R, 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
lambda_ = lambda_ * R.shape[0] * R[0, 0] / B_reg.shape[0]
fodf_sh = np.zeros(odfs_sh.shape)
for index in ndindex(odfs_sh.shape[:-1]):
fodf_sh[index], num_it = odf_deconv(odfs_sh[index], sh_order, R, B_reg, lambda_=lambda_, tau=tau)
return fodf_sh
def compute_odd_power_map(sh_coeffs, order, mask):
_, l_list = sph_harm_ind_list(order, full_basis=True)
odd_l_list = (l_list % 2 == 1).reshape((1, 1, 1, -1))
odd_order_norm = np.linalg.norm(sh_coeffs * odd_l_list, ord=2, axis=-1)
full_order_norm = np.linalg.norm(sh_coeffs, ord=2, axis=-1)
asym_map = np.zeros(sh_coeffs.shape[:-1])
mask = np.logical_and(full_order_norm > 0, mask)
asym_map[mask] = odd_order_norm[mask] / full_order_norm[mask]
return asym_map
def test_order_from_ncoeff():
"""
"""
# Just try some out:
for sh_order in [2, 4, 6, 8, 12, 24]:
m, n = sph_harm_ind_list(sh_order)
n_coef = m.shape[0]
assert_equal(order_from_ncoef(n_coef), sh_order)
# Try out full basis
m_full, n_full = sph_harm_full_ind_list(sh_order)
n_coef_full = m_full.shape[0]
assert_equal(order_from_ncoef(n_coef_full, True), sh_order)
def __init__(self,
acquisition_scheme,
model,
x0_vector=None,
sh_order=8,
lambda_pos=1.,
lambda_lb=5e-4,
tau=0.1,
max_iter=50,
unity_constraint=True,
init_sh_order=4):
self.model = model
self.acquisition_scheme = acquisition_scheme
self.sh_order = sh_order
self.Ncoef = int((sh_order + 2) * (sh_order + 1) // 2)
self.Ncoef4 = int((init_sh_order + 2) * (init_sh_order + 1) // 2)
self.Nmodels = len(self.model.models)
self.lambda_pos = lambda_pos
self.lambda_lb = lambda_lb
self.tau = tau
self.max_iter = max_iter
self.unity_constraint = unity_constraint
self.sphere_jacobian = 1 / (2 * np.sqrt(np.pi))
# step 1: prepare positivity grid on sphere
sphere = get_sphere('symmetric724')
hemisphere = HemiSphere(phi=sphere.phi, theta=sphere.theta)
self.L_positivity = real_sym_sh_mrtrix(self.sh_order, hemisphere.theta,
hemisphere.phi)[0]
sh_l = sph_harm_ind_list(sh_order)[1]
self.R_smoothness = np.diag(sh_l**2 * (sh_l + 1)**2)
# check if there is only one model. If so, precompute rh array.
if self.model.volume_fractions_fixed:
x0_single_voxel = np.reshape(x0_vector,
(-1, x0_vector.shape[-1]))[0]
if np.all(np.isnan(x0_single_voxel)):
self.single_convolution_kernel = True
parameters_dict = self.model.parameter_vector_to_parameters(
x0_single_voxel)
self.A = self.model._construct_convolution_kernel(
**parameters_dict)
self.AT_A = np.dot(self.A.T, self.A)
else:
self.single_convolution_kernel = False
else:
msg = "This CSD optimizer cannot estimate volume fractions."
raise ValueError(msg)
def compute_cos_asym_map(sh_coeffs, order, mask):
_, l_list = sph_harm_ind_list(order, full_basis=True)
sign = np.power(-1.0, l_list)
sign = np.reshape(sign, (1, 1, 1, len(l_list)))
sh_squared = sh_coeffs**2
mask = np.logical_and(sh_squared.sum(axis=-1) > 0., mask)
cos_asym_map = np.zeros(sh_coeffs.shape[:-1])
cos_asym_map[mask] = np.sum(sh_squared * sign, axis=-1)[mask] / \
np.sum(sh_squared, axis=-1)[mask]
cos_asym_map = np.sqrt(1 - cos_asym_map**2) * mask
return cos_asym_map
def compute_asymmetry_map(sh_coeffs):
order = order_from_ncoef(sh_coeffs.shape[-1], full_basis=True)
_, l_list = sph_harm_ind_list(order, full_basis=True)
sign = np.power(-1.0, l_list)
sign = np.reshape(sign, (1, 1, 1, len(l_list)))
sh_squared = sh_coeffs**2
mask = sh_squared.sum(axis=-1) > 0.
asym_map = np.zeros(sh_coeffs.shape[:-1])
asym_map[mask] = np.sum(sh_squared * sign, axis=-1)[mask] / \
np.sum(sh_squared, axis=-1)[mask]
asym_map = np.sqrt(1 - asym_map**2) * mask
return asym_map
def test_faster_sph_harm():
sh_order = 8
m, n = sph_harm_ind_list(sh_order)
theta = np.array([1.61491146, 0.76661665, 0.11976141, 1.20198246,
1.74066314, 1.5925956, 2.13022055, 0.50332859,
1.19868988, 0.78440679, 0.50686938, 0.51739718,
1.80342999, 0.73778957, 2.28559395, 1.29569064,
1.86877091, 0.39239191, 0.54043037, 1.61263047,
0.72695314, 1.90527318, 1.58186125, 0.23130073,
2.51695237, 0.99835604, 1.2883426, 0.48114057,
1.50079318, 1.07978624, 1.9798903, 2.36616966,
2.49233299, 2.13116602, 1.36801518, 1.32932608,
0.95926683, 1.070349, 0.76355762, 2.07148422,
1.50113501, 1.49823314, 0.89248164, 0.22187079,
1.53805373, 1.9765295, 1.13361568, 1.04908355,
1.68737368, 1.91732452, 1.01937457, 1.45839,
0.49641525, 0.29087155, 0.52824641, 1.29875871,
1.81023541, 1.17030475, 2.24953206, 1.20280498,
0.76399964, 2.16109722, 0.79780421, 0.87154509])
phi = np.array([-1.5889514, -3.11092733, -0.61328674, -2.4485381,
2.88058822, 2.02165946, -1.99783366, 2.71235211,
1.41577992, -2.29413676, -2.24565773, -1.55548635,
2.59318232, -1.84672472, -2.33710739, 2.12111948,
1.87523722, -1.05206575, -2.85381987,
-2.22808984, 2.3202034, -2.19004474, -1.90358372,
2.14818373, 3.1030696, -2.86620183, -2.19860123,
-0.45468447, -3.0034923, 1.73345011, -2.51716288,
2.49961525, -2.68782986, 2.69699056, 1.78566133,
-1.59119705, -2.53378963, -2.02476738, 1.36924987,
2.17600517, 2.38117241, 2.99021511, -1.4218007,
-2.44016802, -2.52868164, 3.01531658, 2.50093627,
-1.70745826, -2.7863931, -2.97359741, 2.17039906,
2.68424643, 1.77896086, 0.45476215, 0.99734418,
-2.73107896, 2.28815009, 2.86276506, 3.09450274,
-3.09857384, -1.06955885, -2.83826831, 1.81932195,
2.81296654])
sh = spherical_harmonics(m, n, theta[:, None], phi[:, None])
sh2 = sph_harm_sp(m, n, theta[:, None], phi[:, None])
assert_array_almost_equal(sh, sh2, 8)
sh = spherical_harmonics(m, n, theta[:, None], phi[:, None],
use_scipy=False)
assert_array_almost_equal(sh, sh2, 8)
def __init__(self,
gtab,
response,
sh_order,
lambda_=1,
tau=0.1,
size=3,
method='center'):
"""
Attributes:
gtab (dipy GradientTable): the gradient of the dwi in wich the
response is represented.
response (float[4]): first 3 elements: eigenvalues
4th one: mean b0 value
sh_order (int): the sh_order of the DWI used for the deconvolution
lambda_ (float): the coefficient to rescale the loss
better to keep to 1. and change the 'coeff' param
when defining the loss
tau (float): the coefficient to rescale the threshold used
size (int): the size of the subsample taken to compute the loss
take size**3 voxels
method (str): the method to take the subsample (random or center)
"""
super(NegativefODFLoss, self).__init__()
m, n = sph_harm_ind_list(sh_order)
self.sphere = get_sphere('symmetric362')
r, theta, phi = cart2sphere(self.sphere.x, self.sphere.y,
self.sphere.z)
# B_reg = real_sph_harm(m, n, theta[:, None], phi[:, None])
B_reg = shm.get_B_matrix(theta=theta, phi=phi, sh_order=sh_order)
R, r_rh, B_dwi = shm.get_deconv_matrix(gtab, response, 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]_
lambda_ = (lambda_ * R.shape[0] * r_rh[0] /
(np.sqrt(B_reg.shape[0]) * np.sqrt(362.)))
B_reg = torch.FloatTensor(B_reg * lambda_)
self.B_reg = nn.Parameter(B_reg, requires_grad=False)
self.tau = tau
self.size = size
self.method = method
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 main():
parser = _build_arg_parser()
args = parser.parse_args()
if args.verbose:
logging.basicConfig(level=logging.INFO)
# Checking args
outputs = [args.out_sh]
if args.out_sym:
outputs.append(args.out_sym)
assert_outputs_exist(parser, args, outputs)
assert_inputs_exist(parser, args.in_sh)
nbr_processes = validate_nbr_processes(parser, args)
# Prepare data
sh_img = nib.load(args.in_sh)
data = sh_img.get_fdata(dtype=np.float32)
sh_order, full_basis = get_sh_order_and_fullness(data.shape[-1])
t0 = time.perf_counter()
logging.info('Executing angle-aware bilateral filtering.')
asym_sh = angle_aware_bilateral_filtering(
data, sh_order=sh_order,
sh_basis=args.sh_basis,
in_full_basis=full_basis,
sphere_str=args.sphere,
sigma_spatial=args.sigma_spatial,
sigma_angular=args.sigma_angular,
sigma_range=args.sigma_range,
use_gpu=args.use_gpu,
nbr_processes=nbr_processes)
t1 = time.perf_counter()
logging.info('Elapsed time (s): {0}'.format(t1 - t0))
logging.info('Saving filtered SH to file {0}.'.format(args.out_sh))
nib.save(nib.Nifti1Image(asym_sh, sh_img.affine), args.out_sh)
if args.out_sym:
_, orders = sph_harm_ind_list(sh_order, full_basis=True)
logging.info('Saving symmetric SH to file {0}.'.format(args.out_sym))
nib.save(nib.Nifti1Image(asym_sh[..., orders % 2 == 0], sh_img.affine),
args.out_sym)
def test_faster_sph_harm():
sh_order = 8
m, n = sph_harm_ind_list(sh_order)
theta = np.array([1.61491146, 0.76661665, 0.11976141, 1.20198246,
1.74066314, 1.5925956, 2.13022055, 0.50332859,
1.19868988, 0.78440679, 0.50686938, 0.51739718,
1.80342999, 0.73778957, 2.28559395, 1.29569064,
1.86877091, 0.39239191, 0.54043037, 1.61263047,
0.72695314, 1.90527318, 1.58186125, 0.23130073,
2.51695237, 0.99835604, 1.2883426, 0.48114057,
1.50079318, 1.07978624, 1.9798903, 2.36616966,
2.49233299, 2.13116602, 1.36801518, 1.32932608,
0.95926683, 1.070349, 0.76355762, 2.07148422,
1.50113501, 1.49823314, 0.89248164, 0.22187079,
1.53805373, 1.9765295, 1.13361568, 1.04908355,
1.68737368, 1.91732452, 1.01937457, 1.45839,
0.49641525, 0.29087155, 0.52824641, 1.29875871,
1.81023541, 1.17030475, 2.24953206, 1.20280498,
0.76399964, 2.16109722, 0.79780421, 0.87154509])
phi = np.array([-1.5889514, -3.11092733, -0.61328674, -2.4485381,
2.88058822, 2.02165946, -1.99783366, 2.71235211,
1.41577992, -2.29413676, -2.24565773, -1.55548635,
2.59318232, -1.84672472, -2.33710739, 2.12111948,
1.87523722, -1.05206575, -2.85381987,
-2.22808984, 2.3202034, -2.19004474, -1.90358372,
2.14818373, 3.1030696, -2.86620183, -2.19860123,
-0.45468447, -3.0034923, 1.73345011, -2.51716288,
2.49961525, -2.68782986, 2.69699056, 1.78566133,
-1.59119705, -2.53378963, -2.02476738, 1.36924987,
2.17600517, 2.38117241, 2.99021511, -1.4218007,
-2.44016802, -2.52868164, 3.01531658, 2.50093627,
-1.70745826, -2.7863931, -2.97359741, 2.17039906,
2.68424643, 1.77896086, 0.45476215, 0.99734418,
-2.73107896, 2.28815009, 2.86276506, 3.09450274,
-3.09857384, -1.06955885, -2.83826831, 1.81932195,
2.81296654])
sh = spherical_harmonics(m, n, theta[:, None], phi[:, None])
sh2 = sph_harm_sp(m, n, theta[:, None], phi[:, None])
assert_array_almost_equal(sh, sh2, 8)
def forward_sdt_deconv_mat(ratio, sh_order):
""" Build forward sharpening deconvolution transform (SDT) matrix
Parameters
----------
ratio : float
ratio = $\frac{\lambda_2}{\lambda_1}$ of the single fiber response function
sh_order : int
spherical harmonic order
Returns
-------
R : ndarray (``(sh_order + 1)*(sh_order + 2)/2``, ``(sh_order + 1)*(sh_order + 2)/2``)
SDT deconvolution matrix
P : ndarray (``(sh_order + 1)*(sh_order + 2)/2``, ``(sh_order + 1)*(sh_order + 2)/2``)
Funk-Radon Transform (FRT) matrix
"""
m, n = sph_harm_ind_list(sh_order)
sdt = np.zeros(m.shape) # SDT matrix
frt = np.zeros(m.shape) # FRT (Funk-Radon transform) q-ball matrix
b = np.zeros(m.shape)
bb = np.zeros(m.shape)
for l in np.arange(0, sh_order + 1, 2):
from scipy.integrate import quad
sharp = quad(
lambda z: lpn(l, z)[0][-1] * np.sqrt(1 / (1 -
(1 - ratio) * z * z)),
-1., 1.)
sdt[l / 2] = sharp[0]
frt[l / 2] = 2 * np.pi * lpn(l, 0)[0][-1]
i = 0
for l in np.arange(0, sh_order + 1, 2):
for m in np.arange(-l, l + 1):
b[i] = sdt[l / 2]
bb[i] = frt[l / 2]
i = i + 1
return np.diag(b), np.diag(bb)
def norm_of_laplacian_fod(self):
"""
Estimates the squared norm of the Laplacian of the FOD. Similar to
the anisotropy index, a higher norm means there are larger higher-order
coefficients in the FOD spherical harmonics expansion. This indicates
the FOD is more anisotropic overall. This kind of measure was explored
in e.g. [1]_.
References
----------
.. [1] Descoteaux, Maxime, et al. "Regularized, fast, and robust
analytical Q-ball imaging." Magnetic Resonance in Medicine: An
Official Journal of the International Society for Magnetic
Resonance in Medicine 58.3 (2007): 497-510.
"""
sh_coef = self.fitted_parameters['sh_coeff']
sh_l = sph_harm_ind_list(self.model.sh_order)[1]
lb_weights = sh_l**2 * (sh_l + 1)**2 # laplace-beltrami
norm_laplacian = np.sum(sh_coef**2 * lb_weights, axis=-1)
return norm_laplacian
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 forward_sdt_deconv_mat(ratio, sh_order):
""" Build forward sharpening deconvolution transform (SDT) matrix
Parameters
----------
ratio : float
ratio = $\frac{\lambda_2}{\lambda_1}$ of the single fiber response function
sh_order : int
spherical harmonic order
Returns
-------
R : ndarray (``(sh_order + 1)*(sh_order + 2)/2``, ``(sh_order + 1)*(sh_order + 2)/2``)
SDT deconvolution matrix
P : ndarray (``(sh_order + 1)*(sh_order + 2)/2``, ``(sh_order + 1)*(sh_order + 2)/2``)
Funk-Radon Transform (FRT) matrix
"""
m, n = sph_harm_ind_list(sh_order)
sdt = np.zeros(m.shape) # SDT matrix
frt = np.zeros(m.shape) # FRT (Funk-Radon transform) q-ball matrix
b = np.zeros(m.shape)
bb = np.zeros(m.shape)
for l in np.arange(0, sh_order + 1, 2):
from scipy.integrate import quad
sharp = quad(lambda z: lpn(l, z)[0][-1] * np.sqrt(1 / (1 - (1 - ratio) * z * z)), -1., 1.)
sdt[l / 2] = sharp[0]
frt[l / 2] = 2 * np.pi * lpn(l, 0)[0][-1]
i = 0
for l in np.arange(0, sh_order + 1, 2):
for m in np.arange(-l, l + 1):
b[i] = sdt[l / 2]
bb[i] = frt[l / 2]
i = i + 1
return np.diag(b), np.diag(bb)
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 reconst_flow_core(flow):
with TemporaryDirectory() as out_dir:
data_path, bval_path, bvec_path = get_fnames('small_64D')
volume, affine = load_nifti(data_path)
mask = np.ones_like(volume[:, :, :, 0])
mask_path = pjoin(out_dir, 'tmp_mask.nii.gz')
save_nifti(mask_path, mask.astype(np.uint8), affine)
reconst_flow = flow()
for sh_order in [4, 6, 8]:
if flow.get_short_name() == 'csd':
reconst_flow.run(data_path,
bval_path,
bvec_path,
mask_path,
sh_order=sh_order,
out_dir=out_dir,
extract_pam_values=True)
elif flow.get_short_name() == 'csa':
reconst_flow.run(data_path,
bval_path,
bvec_path,
mask_path,
sh_order=sh_order,
odf_to_sh_order=sh_order,
out_dir=out_dir,
extract_pam_values=True)
gfa_path = reconst_flow.last_generated_outputs['out_gfa']
gfa_data = load_nifti_data(gfa_path)
npt.assert_equal(gfa_data.shape, volume.shape[:-1])
peaks_dir_path =\
reconst_flow.last_generated_outputs['out_peaks_dir']
peaks_dir_data = load_nifti_data(peaks_dir_path)
npt.assert_equal(peaks_dir_data.shape[-1], 15)
npt.assert_equal(peaks_dir_data.shape[:-1], volume.shape[:-1])
peaks_idx_path = \
reconst_flow.last_generated_outputs['out_peaks_indices']
peaks_idx_data = load_nifti_data(peaks_idx_path)
npt.assert_equal(peaks_idx_data.shape[-1], 5)
npt.assert_equal(peaks_idx_data.shape[:-1], volume.shape[:-1])
peaks_vals_path = \
reconst_flow.last_generated_outputs['out_peaks_values']
peaks_vals_data = load_nifti_data(peaks_vals_path)
npt.assert_equal(peaks_vals_data.shape[-1], 5)
npt.assert_equal(peaks_vals_data.shape[:-1], volume.shape[:-1])
shm_path = reconst_flow.last_generated_outputs['out_shm']
shm_data = load_nifti_data(shm_path)
# Test that the number of coefficients is what you would expect
# given the order of the sh basis:
npt.assert_equal(shm_data.shape[-1],
sph_harm_ind_list(sh_order)[0].shape[0])
npt.assert_equal(shm_data.shape[:-1], volume.shape[:-1])
pam = load_peaks(reconst_flow.last_generated_outputs['out_pam'])
npt.assert_allclose(pam.peak_dirs.reshape(peaks_dir_data.shape),
peaks_dir_data)
npt.assert_allclose(pam.peak_values, peaks_vals_data)
npt.assert_allclose(pam.peak_indices, peaks_idx_data)
npt.assert_allclose(pam.shm_coeff, shm_data)
npt.assert_allclose(pam.gfa, gfa_data)
bvals, bvecs = read_bvals_bvecs(bval_path, bvec_path)
bvals[0] = 5.
bvecs = generate_bvecs(len(bvals))
tmp_bval_path = pjoin(out_dir, "tmp.bval")
tmp_bvec_path = pjoin(out_dir, "tmp.bvec")
np.savetxt(tmp_bval_path, bvals)
np.savetxt(tmp_bvec_path, bvecs.T)
reconst_flow._force_overwrite = True
if flow.get_short_name() == 'csd':
reconst_flow = flow()
reconst_flow._force_overwrite = True
reconst_flow.run(data_path,
bval_path,
bvec_path,
mask_path,
out_dir=out_dir,
frf=[15, 5, 5])
reconst_flow = flow()
reconst_flow._force_overwrite = True
reconst_flow.run(data_path,
bval_path,
bvec_path,
mask_path,
out_dir=out_dir,
frf='15, 5, 5')
reconst_flow = flow()
reconst_flow._force_overwrite = True
reconst_flow.run(data_path,
bval_path,
bvec_path,
mask_path,
out_dir=out_dir,
frf=None)
reconst_flow2 = flow()
reconst_flow2._force_overwrite = True
reconst_flow2.run(data_path,
bval_path,
bvec_path,
mask_path,
out_dir=out_dir,
frf=None,
roi_center=[5, 5, 5])
else:
with npt.assert_raises(BaseException):
npt.assert_warns(UserWarning,
reconst_flow.run,
data_path,
tmp_bval_path,
tmp_bvec_path,
mask_path,
out_dir=out_dir,
extract_pam_values=True)
# test parallel implementation
reconst_flow = flow()
reconst_flow._force_overwrite = True
reconst_flow.run(data_path,
bval_path,
bvec_path,
mask_path,
out_dir=out_dir,
parallel=True,
nbr_processes=None)
reconst_flow = flow()
reconst_flow._force_overwrite = True
reconst_flow.run(data_path,
bval_path,
bvec_path,
mask_path,
out_dir=out_dir,
parallel=True,
nbr_processes=2)
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 __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 reconst_flow_core(flow):
with TemporaryDirectory() as out_dir:
data_path, bval_path, bvec_path = get_fnames('small_64D')
vol_img = nib.load(data_path)
volume = vol_img.get_data()
mask = np.ones_like(volume[:, :, :, 0])
mask_img = nib.Nifti1Image(mask.astype(np.uint8), vol_img.affine)
mask_path = pjoin(out_dir, 'tmp_mask.nii.gz')
nib.save(mask_img, mask_path)
reconst_flow = flow()
for sh_order in [4, 6, 8]:
if flow.get_short_name() == 'csd':
reconst_flow.run(data_path, bval_path, bvec_path, mask_path,
sh_order=sh_order,
out_dir=out_dir, extract_pam_values=True)
elif flow.get_short_name() == 'csa':
reconst_flow.run(data_path, bval_path, bvec_path, mask_path,
sh_order=sh_order,
odf_to_sh_order=sh_order,
out_dir=out_dir, extract_pam_values=True)
gfa_path = reconst_flow.last_generated_outputs['out_gfa']
gfa_data = nib.load(gfa_path).get_data()
npt.assert_equal(gfa_data.shape, volume.shape[:-1])
peaks_dir_path =\
reconst_flow.last_generated_outputs['out_peaks_dir']
peaks_dir_data = nib.load(peaks_dir_path).get_data()
npt.assert_equal(peaks_dir_data.shape[-1], 15)
npt.assert_equal(peaks_dir_data.shape[:-1], volume.shape[:-1])
peaks_idx_path = \
reconst_flow.last_generated_outputs['out_peaks_indices']
peaks_idx_data = nib.load(peaks_idx_path).get_data()
npt.assert_equal(peaks_idx_data.shape[-1], 5)
npt.assert_equal(peaks_idx_data.shape[:-1], volume.shape[:-1])
peaks_vals_path = \
reconst_flow.last_generated_outputs['out_peaks_values']
peaks_vals_data = nib.load(peaks_vals_path).get_data()
npt.assert_equal(peaks_vals_data.shape[-1], 5)
npt.assert_equal(peaks_vals_data.shape[:-1], volume.shape[:-1])
shm_path = reconst_flow.last_generated_outputs['out_shm']
shm_data = nib.load(shm_path).get_data()
# Test that the number of coefficients is what you would expect
# given the order of the sh basis:
npt.assert_equal(shm_data.shape[-1],
sph_harm_ind_list(sh_order)[0].shape[0])
npt.assert_equal(shm_data.shape[:-1], volume.shape[:-1])
pam = load_peaks(reconst_flow.last_generated_outputs['out_pam'])
npt.assert_allclose(pam.peak_dirs.reshape(peaks_dir_data.shape),
peaks_dir_data)
npt.assert_allclose(pam.peak_values, peaks_vals_data)
npt.assert_allclose(pam.peak_indices, peaks_idx_data)
npt.assert_allclose(pam.shm_coeff, shm_data)
npt.assert_allclose(pam.gfa, gfa_data)
bvals, bvecs = read_bvals_bvecs(bval_path, bvec_path)
bvals[0] = 5.
bvecs = generate_bvecs(len(bvals))
tmp_bval_path = pjoin(out_dir, "tmp.bval")
tmp_bvec_path = pjoin(out_dir, "tmp.bvec")
np.savetxt(tmp_bval_path, bvals)
np.savetxt(tmp_bvec_path, bvecs.T)
reconst_flow._force_overwrite = True
with npt.assert_raises(BaseException):
npt.assert_warns(UserWarning, reconst_flow.run, data_path,
tmp_bval_path, tmp_bvec_path, mask_path,
out_dir=out_dir, extract_pam_values=True)
if flow.get_short_name() == 'csd':
reconst_flow = flow()
reconst_flow._force_overwrite = True
reconst_flow.run(data_path, bval_path, bvec_path, mask_path,
out_dir=out_dir, frf=[15, 5, 5])
reconst_flow = flow()
reconst_flow._force_overwrite = True
reconst_flow.run(data_path, bval_path, bvec_path, mask_path,
out_dir=out_dir, frf='15, 5, 5')
reconst_flow = flow()
reconst_flow._force_overwrite = True
reconst_flow.run(data_path, bval_path, bvec_path, mask_path,
out_dir=out_dir, frf=None)
reconst_flow2 = flow()
reconst_flow2._force_overwrite = True
reconst_flow2.run(data_path, bval_path, bvec_path, mask_path,
out_dir=out_dir, frf=None,
roi_center=[10, 10, 10])
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 __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 odf_deconv(odf_sh, sh_order, R, B_reg, lambda_=1., tau=0.1):
r""" ODF constrained-regularized sherical deconvolution using
the Sharpening Deconvolution Transform (SDT) [1]_, [2]_.
Parameters
----------
odf_sh : ndarray (``(sh_order + 1)*(sh_order + 2)/2``,)
ndarray of SH coefficients for the ODF spherical function to be deconvolved
sh_order : int
maximal SH order of the SH representation
R : ndarray (``(sh_order + 1)(sh_order + 2)/2``, ``(sh_order + 1)(sh_order + 2)/2``)
SDT matrix in SH basis
B_reg : ndarray (``(sh_order + 1)(sh_order + 2)/2``, ``(sh_order + 1)(sh_order + 2)/2``)
SH basis matrix used for deconvolution
lambda_ : float
lambda parameter in minimization equation (default 1.0)
tau : float
threshold (tau *max(fODF)) controlling the amplitude below
which the corresponding fODF is assumed to be zero.
Returns
-------
fodf_sh : ndarray (``(sh_order + 1)(sh_order + 2)/2``,)
Spherical harmonics coefficients of the constrained-regularized fiber ODF
num_it : int
Number of iterations in the constrained-regularization used for convergence
References
----------
.. [1] Descoteaux, M., et al. IEEE TMI 2009. Deterministic and Probabilistic Tractography Based
on Complex Fibre Orientation Distributions
.. [2] Descoteaux, M, PhD thesis, INRIA Sophia-Antipolis, 2008.
"""
m, n = sph_harm_ind_list(sh_order)
# Generate initial fODF estimate, which is the ODF truncated at SH order 4
fodf_sh = np.linalg.lstsq(R, odf_sh)[0]
fodf_sh[15:] = 0
fodf = np.dot(B_reg, fodf_sh)
Z = np.linalg.norm(fodf)
fodf_sh /= Z
fodf = np.dot(B_reg, fodf_sh)
threshold = tau * np.max(np.dot(B_reg, fodf_sh))
#print(np.min(fodf), np.max(fodf), np.mean(fodf), threshold, tau)
k = []
convergence = 50
for num_it in range(1, convergence + 1):
A = np.dot(B_reg, fodf_sh)
k2 = np.nonzero(A < threshold)[0]
if (k2.shape[0] + R.shape[0]) < B_reg.shape[1]:
warnings.warn(
'too few negative directions identified - failed to converge')
return fodf_sh, num_it
if num_it > 1 and k.shape[0] == k2.shape[0]:
if (k == k2).all():
return fodf_sh, num_it
k = k2
M = np.concatenate((R, lambda_ * B_reg[k, :]))
ODF = np.concatenate((odf_sh, np.zeros(k.shape)))
fodf_sh = np.linalg.lstsq(M, ODF)[0]
warnings.warn('maximum number of iterations exceeded - failed to converge')
return fodf_sh, num_it
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 odf_deconv(odf_sh, sh_order, R, B_reg, lambda_=1., tau=0.1):
r""" ODF constrained-regularized sherical deconvolution using
the Sharpening Deconvolution Transform (SDT) [1]_, [2]_.
Parameters
----------
odf_sh : ndarray (``(sh_order + 1)*(sh_order + 2)/2``,)
ndarray of SH coefficients for the ODF spherical function to be deconvolved
sh_order : int
maximal SH order of the SH representation
R : ndarray (``(sh_order + 1)(sh_order + 2)/2``, ``(sh_order + 1)(sh_order + 2)/2``)
SDT matrix in SH basis
B_reg : ndarray (``(sh_order + 1)(sh_order + 2)/2``, ``(sh_order + 1)(sh_order + 2)/2``)
SH basis matrix used for deconvolution
lambda_ : float
lambda parameter in minimization equation (default 1.0)
tau : float
threshold (tau *max(fODF)) controlling the amplitude below
which the corresponding fODF is assumed to be zero.
Returns
-------
fodf_sh : ndarray (``(sh_order + 1)(sh_order + 2)/2``,)
Spherical harmonics coefficients of the constrained-regularized fiber ODF
num_it : int
Number of iterations in the constrained-regularization used for convergence
References
----------
.. [1] Descoteaux, M., et al. IEEE TMI 2009. Deterministic and Probabilistic Tractography Based
on Complex Fibre Orientation Distributions
.. [2] Descoteaux, M, PhD thesis, INRIA Sophia-Antipolis, 2008.
"""
m, n = sph_harm_ind_list(sh_order)
# Generate initial fODF estimate, which is the ODF truncated at SH order 4
fodf_sh = np.linalg.lstsq(R, odf_sh)[0]
fodf_sh[15:] = 0
fodf = np.dot(B_reg, fodf_sh)
Z = np.linalg.norm(fodf)
fodf_sh /= Z
fodf = np.dot(B_reg, fodf_sh)
threshold = tau * np.max(np.dot(B_reg, fodf_sh))
#print(np.min(fodf), np.max(fodf), np.mean(fodf), threshold, tau)
k = []
convergence = 50
for num_it in range(1, convergence + 1):
A = np.dot(B_reg, fodf_sh)
k2 = np.nonzero(A < threshold)[0]
if (k2.shape[0] + R.shape[0]) < B_reg.shape[1]:
warnings.warn('too few negative directions identified - failed to converge')
return fodf_sh, num_it
if num_it > 1 and k.shape[0] == k2.shape[0]:
if (k == k2).all():
return fodf_sh, num_it
k = k2
M = np.concatenate((R, lambda_ * B_reg[k, :]))
ODF = np.concatenate((odf_sh, np.zeros(k.shape)))
fodf_sh = np.linalg.lstsq(M, ODF)[0]
warnings.warn('maximum number of iterations exceeded - failed to converge')
return fodf_sh, num_it
def odf_sh_to_sharp(odfs_sh,
sphere,
basis=None,
ratio=3 / 15.,
sh_order=8,
lambda_=1.,
tau=0.1):
r""" Sharpen odfs using the spherical deconvolution transform [1]_
This function can be used to sharpen any smooth ODF spherical function. In theory, this should
only be used to sharpen QballModel ODFs, but in practice, one can play with the deconvolution
ratio and sharpen almost any ODF-like spherical function. The constrained-regularization is stable
and will not only sharp the ODF peaks but also regularize the noisy peaks.
Parameters
----------
odfs_sh : ndarray (``(sh_order + 1)*(sh_order + 2)/2``, )
array of odfs expressed as spherical harmonics coefficients
sphere : Sphere
sphere used to build the regularization matrix
basis : {None, 'mrtrix', 'fibernav'}
different spherical harmonic basis. None is the fibernav basis as well.
ratio : float,
ratio of the smallest vs the largest eigenvalue of the single prolate tensor response function
(:math:`\frac{\lambda_2}{\lambda_1}`)
sh_order : int
maximal SH order of the SH representation
lambda_ : float
lambda parameter (see odfdeconv) (default 1.0)
tau : float
tau parameter in the L matrix construction (see odfdeconv) (default 0.1)
Returns
-------
fodf_sh : ndarray
sharpened odf expressed as spherical harmonics coefficients
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)
r, theta, phi = cart2sphere(sphere.x, sphere.y, sphere.z)
real_sym_sh = sph_harm_lookup[basis]
B_reg, m, n = real_sym_sh(sh_order, theta[:, None], phi[:, None])
R, 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
lambda_ = lambda_ * R.shape[0] * R[0, 0] / B_reg.shape[0]
fodf_sh = np.zeros(odfs_sh.shape)
for index in ndindex(odfs_sh.shape[:-1]):
fodf_sh[index], num_it = odf_deconv(odfs_sh[index],
sh_order,
R,
B_reg,
lambda_=lambda_,
tau=tau)
return fodf_sh
