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csdeconv.py
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csdeconv.py
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import warnings
import numpy as np
import numbers
from scipy.integrate import quad
from scipy.special import lpn, gamma
import scipy.linalg as la
import scipy.linalg.lapack as ll
from dipy.data import small_sphere, get_sphere, default_sphere
from dipy.core.geometry import cart2sphere
from dipy.core.ndindex import ndindex
from dipy.sims.voxel import single_tensor
from dipy.reconst.multi_voxel import multi_voxel_fit
from dipy.reconst.dti import TensorModel, fractional_anisotropy
from dipy.reconst.shm import (sph_harm_ind_list, real_sh_descoteaux_from_index,
sph_harm_lookup, lazy_index, SphHarmFit,
real_sh_descoteaux, sh_to_rh, forward_sdeconv_mat,
SphHarmModel)
from dipy.reconst.utils import _roi_in_volume, _mask_from_roi
from dipy.direction.peaks import peaks_from_model
from dipy.core.geometry import vec2vec_rotmat
from dipy.utils.deprecator import deprecate_with_version, deprecated_params
@deprecate_with_version("dipy.reconst.csdeconv.auto_response is deprecated, "
"Please use "
"dipy.reconst.csdeconv.auto_response_ssst instead",
since='1.2', until='1.4')
def auto_response(gtab, data, roi_center=None, roi_radius=10, fa_thr=0.7,
fa_callable=None, return_number_of_voxels=None):
""" Automatic estimation of ssst response function using FA.
Parameters
----------
gtab : GradientTable
data : ndarray
diffusion data
roi_center : array-like, (3,)
Center of ROI in data. If center is None, it is assumed that it is
the center of the volume with shape `data.shape[:3]`.
roi_radius : int
radius of cubic ROI
fa_thr : float
FA threshold
fa_callable : callable
A callable that defines an operation that compares FA with the fa_thr.
The operator should have two positional arguments
(e.g., `fa_operator(FA, fa_thr)`) and it should return a bool array.
return_number_of_voxels : bool
If True, returns the number of voxels used for estimating the response
function
Returns
-------
response : tuple, (2,)
(`evals`, `S0`)
ratio : float
The ratio between smallest versus largest eigenvalue of the response.
Notes
-----
In CSD there is an important pre-processing step: the estimation of the
fiber response function. In order to do this, we look for voxels with very
anisotropic configurations. We get this information from
csdeconv.mask_for_response_ssst(), which returns a mask of selected voxels
(more details are available in the description of the function).
With the mask, we compute the response function by using
csdeconv.response_from_mask_ssst(), which returns the `response` and the
`ratio` (more details are available in the description of the function).
"""
msg = """Parameter `roi_radius` is now called `roi_radii` and can be an
array-like (3,). Parameters `fa_callable` and `return_number_of_voxels`
are not used anymore."""
warnings.warn(msg, UserWarning)
return auto_response_ssst(gtab, data, roi_center, roi_radius, fa_thr)
@deprecate_with_version("dipy.reconst.csdeconv.response_from_mask is "
"deprecated, Please use "
"dipy.reconst.csdeconv.response_from_mask_ssst instead",
since='1.2', until='1.4')
def response_from_mask(gtab, data, mask):
""" Computation of single-shell single-tissue (ssst) response
function from a given mask.
Parameters
----------
gtab : GradientTable
data : ndarray
diffusion data
mask : ndarray
mask from where to compute the response function
Returns
-------
response : tuple, (2,)
(`evals`, `S0`)
ratio : float
The ratio between smallest versus largest eigenvalue of the response.
Notes
-----
In CSD there is an important pre-processing step: the estimation of the
fiber response function. In order to do this, we look for voxels with very
anisotropic configurations. This information can be obtained by using
csdeconv.mask_for_response_ssst() through a mask of selected voxels
(see[1]_). The present function uses such a mask to compute the ssst
response function.
For the response we also need to find the average S0 in the ROI. This is
possible using `gtab.b0s_mask()` we can find all the S0 volumes (which
correspond to b-values equal 0) in the dataset.
The `response` consists always of a prolate tensor created by averaging
the highest and second highest eigenvalues in the ROI with FA higher than
threshold. We also include the average S0s.
We also return the `ratio` which is used for the SDT models.
References
----------
.. [1] Tournier, J.D., et al. NeuroImage 2004. Direct estimation of the
fiber orientation density function from diffusion-weighted MRI
data using spherical deconvolution
"""
return response_from_mask_ssst(gtab, data, mask)
class AxSymShResponse:
"""A simple wrapper for response functions represented using only axially
symmetric, even spherical harmonic functions (ie, m == 0 and n even).
Parameters
----------
S0 : float
Signal with no diffusion weighting.
dwi_response : array
Response function signal as coefficients to axially symmetric, even
spherical harmonic.
"""
def __init__(self, S0, dwi_response, bvalue=None):
self.S0 = S0
self.dwi_response = dwi_response
self.bvalue = bvalue
self.m = np.zeros(len(dwi_response))
self.sh_order = 2 * (len(dwi_response) - 1)
self.n = np.arange(0, self.sh_order + 1, 2)
def basis(self, sphere):
"""A basis that maps the response coefficients onto a sphere."""
theta = sphere.theta[:, None]
phi = sphere.phi[:, None]
return real_sh_descoteaux_from_index(self.m, self.n, theta, phi)
def on_sphere(self, sphere):
"""Evaluates the response function on sphere."""
B = self.basis(sphere)
return np.dot(self.dwi_response, B.T)
class ConstrainedSphericalDeconvModel(SphHarmModel):
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 (i.e. S0). This is to be able
to generate a single fiber synthetic signal. The response function
will be used as deconvolution kernel ([1]_).
reg_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_sh_descoteaux_from_index(
m, n, theta[:, None], phi[:, None])
# for the sphere used in the regularization positivity constraint
self.sphere = reg_sphere or small_sphere
r, theta, phi = cart2sphere(
self.sphere.x,
self.sphere.y,
self.sphere.z
)
self.B_reg = real_sh_descoteaux_from_index(
m, n, theta[:, None], phi[:, None])
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)
@multi_voxel_fit
def fit(self, data):
dwi_data = data[self._where_dwi]
shm_coeff, _ = csdeconv(dwi_data, self._X, self.B_reg, self.tau,
convergence=self.convergence, P=self._P)
return SphHarmFit(self, shm_coeff, None)
def predict(self, sh_coeff, gtab=None, S0=1.):
"""Compute a signal prediction given spherical harmonic coefficients
for the provided GradientTable class instance.
Parameters
----------
sh_coeff : ndarray
The spherical harmonic representation of the FOD from which to make
the signal prediction.
gtab : GradientTable
The gradients for which the signal will be predicted. Uses the
model's gradient table by default.
S0 : ndarray or float
The non diffusion-weighted signal value.
Returns
-------
pred_sig : ndarray
The predicted signal.
"""
if gtab is None or gtab is self.gtab:
SH_basis = self.B_dwi
gtab = self.gtab
else:
x, y, z = gtab.gradients[~gtab.b0s_mask].T
r, theta, phi = cart2sphere(x, y, z)
SH_basis, m, n = real_sh_descoteaux(self.sh_order, theta, phi)
# Because R is diagonal, the matrix multiply is written as a multiply
predict_matrix = SH_basis * self.R.diagonal()
S0 = np.asarray(S0)[..., None]
scaling = S0 / self.response_scaling
# This is the key operation: convolve and multiply by S0:
pre_pred_sig = scaling * np.dot(sh_coeff, predict_matrix.T)
# 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
class ConstrainedSDTModel(SphHarmModel):
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_sh_descoteaux_from_index(
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_sh_descoteaux_from_index(
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
@multi_voxel_fit
def fit(self, data):
s_sh = np.linalg.lstsq(self.B_dwi, data[self._where_dwi],
rcond=-1)[0]
# initial ODF estimation
odf_sh = np.dot(self.P, s_sh)
qball_odf = np.dot(self.B_reg, odf_sh)
Z = np.linalg.norm(qball_odf)
shm_coeff, num_it = np.zeros_like(odf_sh), 0
if Z:
# normalize ODF
odf_sh /= Z
shm_coeff, num_it = odf_deconv(odf_sh, self.R, self.B_reg,
self.lambda_, self.tau)
# print 'SDT CSD converged after %d iterations' % num_it
return SphHarmFit(self, shm_coeff, None)
def estimate_response(gtab, evals, S0):
""" Estimate single fiber response function
Parameters
----------
gtab : GradientTable
evals : ndarray
S0 : float
non diffusion weighted
Returns
-------
S : estimated signal
"""
evecs = np.array([[0, 0, 1],
[0, 1, 0],
[1, 0, 0]])
return single_tensor(gtab, S0, evals, evecs, snr=None)
def forward_sdt_deconv_mat(ratio, n, r2_term=False):
r""" Build forward sharpening deconvolution transform (SDT) matrix
Parameters
----------
ratio : float
ratio = $\frac{\lambda_2}{\lambda_1}$ of the single fiber response
function
n : ndarray (N,)
The degree of spherical harmonic function associated with each row of
the deconvolution matrix. Only even degrees are allowed.
r2_term : bool
True if ODF comes from an ODF computed from a model using the $r^2$
term in the integral. For example, DSI, GQI, SHORE, CSA, Tensor,
Multi-tensor ODFs. This results in using the proper analytical response
function solution solving from the single-fiber ODF with the r^2 term.
This derivation is not published anywhere but is very similar to [1]_.
Returns
-------
R : ndarray (N, N)
SDT deconvolution matrix
P : ndarray (N, N)
Funk-Radon Transform (FRT) matrix
References
----------
.. [1] Descoteaux, M. PhD Thesis. INRIA Sophia-Antipolis. 2008.
"""
if np.any(n % 2):
raise ValueError("n has odd degrees, expecting only even degrees")
n_degrees = n.max() // 2 + 1
sdt = np.zeros(n_degrees) # SDT matrix
frt = np.zeros(n_degrees) # FRT (Funk-Radon transform) q-ball matrix
for l in np.arange(0, n_degrees * 2, 2):
if r2_term:
sharp = quad(lambda z: lpn(l, z)[0][-1] * gamma(1.5) *
np.sqrt(ratio / (4 * np.pi ** 3)) /
np.power((1 - (1 - ratio) * z ** 2), 1.5), -1., 1.)
else:
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]
idx = n // 2
b = sdt[idx]
bb = frt[idx]
return np.diag(b), np.diag(bb)
potrf, potrs = ll.get_lapack_funcs(('potrf', 'potrs'))
def _solve_cholesky(Q, z):
L, info = potrf(Q, lower=False, overwrite_a=False, clean=False)
if info > 0:
msg = "%d-th leading minor not positive definite" % info
raise la.LinAlgError(msg)
if info < 0:
msg = 'illegal value in %d-th argument of internal potrf' % -info
raise ValueError(msg)
f, info = potrs(L, z, lower=False, overwrite_b=False)
if info != 0:
msg = 'illegal value in %d-th argument of internal potrs' % -info
raise ValueError(msg)
return f
def csdeconv(dwsignal, X, B_reg, tau=0.1, convergence=50, P=None):
r""" Constrained-regularized spherical deconvolution (CSD) [1]_
Deconvolves the axially symmetric single fiber response function `r_rh` in
rotational harmonics coefficients from the diffusion weighted signal in
`dwsignal`.
Parameters
----------
dwsignal : array
Diffusion weighted signals to be deconvolved.
X : array
Prediction matrix which estimates diffusion weighted signals from FOD
coefficients.
B_reg : array (N, B)
SH basis matrix which maps FOD coefficients to FOD values on the
surface of the sphere. B_reg should be scaled to account for lambda.
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
max fODF amplitude (here, 10% by default). This is similar to peak
detection where peaks below 0.1 amplitude are usually considered noise
peaks. Because SDT is based on a q-ball ODF deconvolution, and not
signal deconvolution, using the max instead of mean (as in CSD), is
more stable.
convergence : int
Maximum number of iterations to allow the deconvolution to converge.
P : ndarray
This is an optimization to avoid computing ``dot(X.T, X)`` many times.
If the same ``X`` is used many times, ``P`` can be precomputed and
passed to this function.
Returns
-------
fodf_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.
Notes
-----
This section describes how the fitting of the SH coefficients is done.
Problem is to minimise per iteration:
$F(f_n) = ||Xf_n - S||^2 + \lambda^2 ||H_{n-1} f_n||^2$
Where $X$ maps current FOD SH coefficients $f_n$ to DW signals $s$ and
$H_{n-1}$ maps FOD SH coefficients $f_n$ to amplitudes along set of
negative directions identified in previous iteration, i.e. the matrix
formed by the rows of $B_{reg}$ for which $Hf_{n-1}<0$ where $B_{reg}$
maps $f_n$ to FOD amplitude on a sphere.
Solve by differentiating and setting to zero:
$\Rightarrow \frac{\delta F}{\delta f_n} = 2X^T(Xf_n - S) + 2 \lambda^2
H_{n-1}^TH_{n-1}f_n=0$
Or:
$(X^TX + \lambda^2 H_{n-1}^TH_{n-1})f_n = X^Ts$
Define $Q = X^TX + \lambda^2 H_{n-1}^TH_{n-1}$ , which by construction is a
square positive definite symmetric matrix of size $n_{SH} by n_{SH}$. If
needed, positive definiteness can be enforced with a small minimum norm
regulariser (helps a lot with poorly conditioned direction sets and/or
superresolution):
$Q = X^TX + (\lambda H_{n-1}^T) (\lambda H_{n-1}) + \mu I$
Solve $Qf_n = X^Ts$ using Cholesky decomposition:
$Q = LL^T$
where $L$ is lower triangular. Then problem can be solved by
back-substitution:
$L_y = X^Ts$
$L^Tf_n = y$
To speeds things up further, form $P = X^TX + \mu I$, and update to form
$Q$ by rankn update with $H_{n-1}$. The dipy implementation looks like:
form initially $P = X^T X + \mu I$ and $\lambda B_{reg}$
for each voxel: form $z = X^Ts$
estimate $f_0$ by solving $Pf_0=z$. We use a simplified $l_{max}=4$
solution here, but it might not make a big difference.
Then iterate until no change in rows of $H$ used in $H_n$
form $H_{n}$ given $f_{n-1}$
form $Q = P + (\lambda H_{n-1}^T) (\lambda H_{n-1}$) (this can
be done by rankn update, but we currently do not use rankn
update).
solve $Qf_n = z$ using Cholesky decomposition
We'd like to thanks Donald Tournier for his help with describing and
implementing this algorithm.
References
----------
.. [1] Tournier, J.D., et al. NeuroImage 2007. Robust determination of the
fibre orientation distribution in diffusion MRI: Non-negativity
constrained super-resolved spherical deconvolution.
"""
mu = 1e-5
if P is None:
P = np.dot(X.T, X)
z = np.dot(X.T, dwsignal)
try:
fodf_sh = _solve_cholesky(P, z)
except la.LinAlgError:
P = P + mu * np.eye(P.shape[0])
fodf_sh = _solve_cholesky(P, z)
# For the first iteration we use a smooth FOD that only uses SH orders up
# to 4 (the first 15 coefficients).
fodf = np.dot(B_reg[:, :15], fodf_sh[:15])
# The mean of an fodf can be computed by taking $Y_{0,0} * coeff_{0,0}$
threshold = B_reg[0, 0] * fodf_sh[0] * tau
where_fodf_small = (fodf < threshold).nonzero()[0]
# If the low-order fodf does not have any values less than threshold, the
# full-order fodf is used.
if len(where_fodf_small) == 0:
fodf = np.dot(B_reg, fodf_sh)
where_fodf_small = (fodf < threshold).nonzero()[0]
# If the fodf still has no values less than threshold, return the fodf.
if len(where_fodf_small) == 0:
return fodf_sh, 0
for num_it in range(1, convergence + 1):
# This is the super-resolved trick. Wherever there is a negative
# amplitude value on the fODF, it concatenates a value to the S vector
# so that the estimation can focus on trying to eliminate it. In a
# sense, this "adds" a measurement, which can help to better estimate
# the fodf_sh, even if you have more SH coefficients to estimate than
# actual S measurements.
H = B_reg.take(where_fodf_small, axis=0)
# We use the Cholesky decomposition to solve for the SH coefficients.
Q = P + np.dot(H.T, H)
fodf_sh = _solve_cholesky(Q, z)
# Sample the FOD using the regularization sphere and compute k.
fodf = np.dot(B_reg, fodf_sh)
where_fodf_small_last = where_fodf_small
where_fodf_small = (fodf < threshold).nonzero()[0]
if (len(where_fodf_small) == len(where_fodf_small_last) and
(where_fodf_small == where_fodf_small_last).all()):
break
else:
msg = 'maximum number of iterations exceeded - failed to converge'
warnings.warn(msg)
return fodf_sh, num_it
def odf_deconv(odf_sh, R, B_reg, lambda_=1., tau=0.1, r2_term=False):
r""" ODF constrained-regularized spherical 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
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.
r2_term : bool
True if ODF is computed from model that uses the $r^2$ term in the
integral. Recall that Tuch's ODF (used in Q-ball Imaging [1]_) and
the true normalized ODF definition differ from a $r^2$ term in the ODF
integral. The original Sharpening Deconvolution Transform (SDT)
technique [2]_ is expecting Tuch's ODF without the $r^2$ (see [3]_ for
the mathematical details). Now, this function supports ODF that have
been computed using the $r^2$ term because the proper analytical
response function has be derived. For example, models such as DSI,
GQI, SHORE, CSA, Tensor, Multi-tensor ODFs, should now be deconvolved
with the r2_term=True.
Returns
-------
fodf_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] Tuch, D. MRM 2004. Q-Ball Imaging.
.. [2] Descoteaux, M., et al. IEEE TMI 2009. Deterministic and
Probabilistic Tractography Based on Complex Fibre Orientation
Distributions
.. [3] Descoteaux, M, PhD thesis, INRIA Sophia-Antipolis, 2008.
"""
# In ConstrainedSDTModel.fit, odf_sh is divided by its norm (Z) and
# sometimes the norm is 0 which creates NaNs.
if np.any(np.isnan(odf_sh)):
return np.zeros_like(odf_sh), 0
# Generate initial fODF estimate, which is the ODF truncated at SH order 4
fodf_sh = np.linalg.lstsq(R, odf_sh, rcond=-1)[0]
fodf_sh[15:] = 0
fodf = np.dot(B_reg, fodf_sh)
# if sharpening a q-ball odf (it is NOT properly normalized), we need to
# force normalization otherwise, for DSI, CSA, SHORE, Tensor odfs, they are
# normalized by construction
if ~r2_term:
Z = np.linalg.norm(fodf)
fodf_sh /= Z
threshold = tau * np.max(np.dot(B_reg, fodf_sh))
k = np.empty([])
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)))
try:
fodf_sh = np.linalg.lstsq(M, ODF, rcond=-1)[0]
except np.linalg.LinAlgError:
# SVD did not converge in Linear Least Squares in current
# voxel. Proceeding with initial SH estimate for this voxel.
pass
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, r2_term=False):
r""" Sharpen odfs using the sharpening deconvolution transform [2]_
This function can be used to sharpen any smooth ODF spherical function. In
theory, this should only be used to sharpen QballModel ODFs, but in
practice, one can play with the deconvolution ratio and sharpen almost any
ODF-like spherical function. The constrained-regularization is stable and
will not only sharpen the ODF peaks but also regularize the noisy peaks.
Parameters
----------
odfs_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, 'tournier07', 'descoteaux07'}
different spherical harmonic basis:
``None`` for the default DIPY basis,
``tournier07`` for the Tournier 2007 [4]_ basis, and
``descoteaux07`` for the Descoteaux 2007 [3]_ basis
(``None`` defaults to ``descoteaux07``).
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)
r2_term : bool
True if ODF is computed from model that uses the $r^2$ term in the
integral. Recall that Tuch's ODF (used in Q-ball Imaging [1]_) and
the true normalized ODF definition differ from a $r^2$ term in the ODF
integral. The original Sharpening Deconvolution Transform (SDT)
technique [2]_ is expecting Tuch's ODF without the $r^2$ (see [3]_ for
the mathematical details). Now, this function supports ODF that have
been computed using the $r^2$ term because the proper analytical
response function has be derived. For example, models such as DSI,
GQI, SHORE, CSA, Tensor, Multi-tensor ODFs, should now be deconvolved
with the r2_term=True.
Returns
-------
fodf_sh : ndarray
sharpened odf expressed as spherical harmonics coefficients
References
----------
.. [1] Tuch, D. MRM 2004. Q-Ball Imaging.
.. [2] Descoteaux, M., et al. IEEE TMI 2009. Deterministic and
Probabilistic Tractography Based on Complex Fibre Orientation
Distributions
.. [3] Descoteaux, M., Angelino, E., Fitzgibbons, S. and Deriche, R.
Regularized, Fast, and Robust Analytical Q-ball Imaging.
Magn. Reson. Med. 2007;58:497-510.
.. [4] Tournier J.D., Calamante F. and Connelly A. Robust determination
of the fibre orientation distribution in diffusion MRI:
Non-negativity constrained super-resolved spherical deconvolution.
NeuroImage. 2007;35(4):1459-1472.
"""
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, phi)
R, P = forward_sdt_deconv_mat(ratio, n, r2_term=r2_term)
# 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], R, B_reg,
lambda_=lambda_, tau=tau,
r2_term=r2_term)
return fodf_sh
def mask_for_response_ssst(gtab, data, roi_center=None, roi_radii=10,
fa_thr=0.7):
""" Computation of mask for single-shell single-tissue (ssst) response
function using FA.
Parameters
----------
gtab : GradientTable
data : ndarray
diffusion data (4D)
roi_center : array-like, (3,)
Center of ROI in data. If center is None, it is assumed that it is
the center of the volume with shape `data.shape[:3]`.
roi_radii : int or array-like, (3,)
radii of cuboid ROI
fa_thr : float
FA threshold
Returns
-------
mask : ndarray
Mask of voxels within the ROI and with FA above the FA threshold.
Notes
-----
In CSD there is an important pre-processing step: the estimation of the
fiber response function. In order to do this, we look for voxels with very
anisotropic configurations. This function aims to accomplish that by
returning a mask of voxels within a ROI, that have a FA value above a
given threshold. For example we can use a ROI (20x20x20) at
the center of the volume and store the signal values for the voxels with
FA values higher than 0.7 (see [1]_).
References
----------
.. [1] Tournier, J.D., et al. NeuroImage 2004. Direct estimation of the
fiber orientation density function from diffusion-weighted MRI
data using spherical deconvolution
"""
if len(data.shape) < 4:
msg = """Data must be 4D (3D image + directions). To use a 2D image,
please reshape it into a (N, N, 1, ndirs) array."""
raise ValueError(msg)
if isinstance(roi_radii, numbers.Number):
roi_radii = (roi_radii, roi_radii, roi_radii)
if roi_center is None:
roi_center = np.array(data.shape[:3]) // 2
roi_radii = _roi_in_volume(data.shape, np.asarray(roi_center),
np.asarray(roi_radii))
roi_mask = _mask_from_roi(data.shape[:3], roi_center, roi_radii)
ten = TensorModel(gtab)
tenfit = ten.fit(data, mask=roi_mask)
fa = fractional_anisotropy(tenfit.evals)
fa[np.isnan(fa)] = 0
mask = np.zeros(fa.shape, dtype=np.int64)
mask[fa > fa_thr] = 1
if np.sum(mask) == 0:
msg = """No voxel with a FA higher than {} were found.
Try a larger roi or a lower threshold.""".format(str(fa_thr))
warnings.warn(msg, UserWarning)
return mask
def response_from_mask_ssst(gtab, data, mask):
""" Computation of single-shell single-tissue (ssst) response
function from a given mask.
Parameters
----------
gtab : GradientTable
data : ndarray
diffusion data
mask : ndarray
mask from where to compute the response function
Returns
-------
response : tuple, (2,)
(`evals`, `S0`)
ratio : float
The ratio between smallest versus largest eigenvalue of the response.
Notes
-----
In CSD there is an important pre-processing step: the estimation of the
fiber response function. In order to do this, we look for voxels with very
anisotropic configurations. This information can be obtained by using
csdeconv.mask_for_response_ssst() through a mask of selected voxels
(see[1]_). The present function uses such a mask to compute the ssst
response function.
For the response we also need to find the average S0 in the ROI. This is
possible using `gtab.b0s_mask()` we can find all the S0 volumes (which
correspond to b-values equal 0) in the dataset.
The `response` consists always of a prolate tensor created by averaging
the highest and second highest eigenvalues in the ROI with FA higher than
threshold. We also include the average S0s.
We also return the `ratio` which is used for the SDT models.
References
----------
.. [1] Tournier, J.D., et al. NeuroImage 2004. Direct estimation of the
fiber orientation density function from diffusion-weighted MRI