"""
This is an implementation of the Linear Fascicle Evaluation (LiFE) algorithm
described in :footcite:p:`Pestilli2014`.
References
----------
.. footbibliography::
"""
import numpy as np
import scipy.linalg as la
import scipy.sparse as sps
import dipy.core.optimize as opt
import dipy.data as dpd
from dipy.reconst.base import ReconstFit, ReconstModel
from dipy.testing.decorators import warning_for_keywords
from dipy.tracking.streamline import transform_streamlines
from dipy.tracking.utils import unique_rows
from dipy.tracking.vox2track import _voxel2streamline
[docs]
def gradient(f):
"""
Return the gradient of an N-dimensional array.
The gradient is computed using central differences in the interior
and first differences at the boundaries. The returned gradient hence has
the same shape as the input array.
Parameters
----------
f : array_like
An N-dimensional array containing samples of a scalar function.
Returns
-------
gradient : ndarray
N arrays of the same shape as `f` giving the derivative of `f` with
respect to each dimension.
Examples
--------
>>> x = np.array([1, 2, 4, 7, 11, 16], dtype=float)
>>> gradient(x)
array([ 1. , 1.5, 2.5, 3.5, 4.5, 5. ])
>>> gradient(np.array([[1, 2, 6], [3, 4, 5]], dtype=float))
[array([[ 2., 2., -1.],
[ 2., 2., -1.]]), array([[ 1. , 2.5, 4. ],
[ 1. , 1. , 1. ]])]
Notes
-----
This is a simplified implementation of gradient that is part of numpy
1.8. In order to mitigate the effects of changes added to this
implementation in version 1.9 of numpy, we include this implementation
here.
"""
f = np.asanyarray(f)
N = len(f.shape) # number of dimensions
dx = [1.0] * N
# use central differences on interior and first differences on endpoints
outvals = []
# create slice objects --- initially all are [:, :, ..., :]
slice1 = [slice(None)] * N
slice2 = [slice(None)] * N
slice3 = [slice(None)] * N
for axis in range(N):
# select out appropriate parts for this dimension
out = np.empty_like(f)
slice1[axis] = slice(1, -1)
slice2[axis] = slice(2, None)
slice3[axis] = slice(None, -2)
# 1D equivalent -- out[1:-1] = (f[2:] - f[:-2])/2.0
out[tuple(slice1)] = (f[tuple(slice2)] - f[tuple(slice3)]) / 2.0
slice1[axis] = 0
slice2[axis] = 1
slice3[axis] = 0
# 1D equivalent -- out[0] = (f[1] - f[0])
out[tuple(slice1)] = f[tuple(slice2)] - f[tuple(slice3)]
slice1[axis] = -1
slice2[axis] = -1
slice3[axis] = -2
# 1D equivalent -- out[-1] = (f[-1] - f[-2])
out[tuple(slice1)] = f[tuple(slice2)] - f[tuple(slice3)]
# divide by step size
outvals.append(out / dx[axis])
# reset the slice object in this dimension to ":"
slice1[axis] = slice(None)
slice2[axis] = slice(None)
slice3[axis] = slice(None)
if N == 1:
return outvals[0]
else:
return outvals
[docs]
def streamline_gradients(streamline):
"""
Calculate the gradients of the streamline along the spatial dimension
Parameters
----------
streamline : array-like of shape (n, 3)
The 3d coordinates of a single streamline
Returns
-------
Array of shape (3, n): Spatial gradients along the length of the
streamline.
"""
return np.array(gradient(np.asarray(streamline))[0])
[docs]
def grad_tensor(grad, evals):
"""
Calculate the 3 by 3 tensor for a given spatial gradient, given a canonical
tensor shape (also as a 3 by 3), pointing at [1,0,0]
Parameters
----------
grad : 1d array of shape (3,)
The spatial gradient (e.g between two nodes of a streamline).
evals: 1d array of shape (3,)
The eigenvalues of a canonical tensor to be used as a response
function.
"""
# This is the rotation matrix from [1, 0, 0] to this gradient of the sl:
R = la.svd([grad], overwrite_a=True)[2]
# This is the 3 by 3 tensor after rotation:
T = np.linalg.multi_dot([R, np.diag(evals), R.T])
return T
[docs]
@warning_for_keywords()
def streamline_tensors(streamline, *, evals=(0.001, 0, 0)):
"""
The tensors generated by this fiber.
Parameters
----------
streamline : array-like of shape (n, 3)
The 3d coordinates of a single streamline
evals : iterable with three entries, optional
The estimated eigenvalues of a single fiber tensor.
Returns
-------
An n_nodes by 3 by 3 array with the tensor for each node in the fiber.
Notes
-----
Estimates of the radial/axial diffusivities may rely on
empirical measurements (for example, the AD in the Corpus Callosum), or
may be based on a biophysical model of some kind.
"""
grad = streamline_gradients(streamline)
# Preallocate:
tensors = np.empty((grad.shape[0], 3, 3))
for grad_idx, this_grad in enumerate(grad):
tensors[grad_idx] = grad_tensor(this_grad, evals)
return tensors
[docs]
@warning_for_keywords()
def streamline_signal(streamline, gtab, *, evals=(0.001, 0, 0)):
"""
The signal from a single streamline estimate along each of its nodes.
Parameters
----------
streamline : a single streamline
Streamline data.
gtab : GradientTable class instance
Gradient table.
evals : array-like of length 3, optional
The eigenvalues of the canonical tensor used as an estimate of the
signal generated by each node of the streamline.
"""
# Gotta have those tensors:
tensors = streamline_tensors(streamline, evals=evals)
sig = np.empty((len(streamline), np.sum(~gtab.b0s_mask)))
# Extract them once:
bvecs = gtab.bvecs[~gtab.b0s_mask]
bvals = gtab.bvals[~gtab.b0s_mask]
for ii, tensor in enumerate(tensors):
ADC = np.diag(np.linalg.multi_dot([bvecs, tensor, bvecs.T]))
# Use the Stejskal-Tanner equation with the ADC as input, and S0 = 1:
sig[ii] = np.exp(-bvals * ADC)
return sig - np.mean(sig)
[docs]
class LifeSignalMaker:
"""
A class for generating signals from streamlines in an efficient and speedy
manner.
"""
@warning_for_keywords()
def __init__(self, gtab, *, evals=(0.001, 0, 0), sphere=None):
"""
Initialize a signal maker
Parameters
----------
gtab : GradientTable class instance
The gradient table on which the signal is calculated.
evals : array-like of 3 items, optional
The eigenvalues of the canonical tensor to use in calculating the
signal.
sphere : `dipy.core.Sphere` class instance, optional
The discrete sphere to use as an approximation for the continuous
sphere on which the signal is represented. If integer - we will use
an instance of one of the symmetric spheres cached in
`dps.get_sphere`. If a 'dipy.core.Sphere' class instance is
provided, we will use this object. Default: the :mod:`dipy.data`
symmetric sphere with 724 vertices
"""
self.sphere = sphere or dpd.default_sphere
self.gtab = gtab
self.evals = evals
# Initialize an empty dict to fill with signals for each of the sphere
# vertices:
self.signal = np.empty((self.sphere.vertices.shape[0], np.sum(~gtab.b0s_mask)))
# We'll need to keep track of what we've already calculated:
self._calculated = []
[docs]
def calc_signal(self, xyz):
idx = self.sphere.find_closest(xyz)
if idx not in self._calculated:
bvecs = self.gtab.bvecs[~self.gtab.b0s_mask]
bvals = self.gtab.bvals[~self.gtab.b0s_mask]
tensor = grad_tensor(self.sphere.vertices[idx], self.evals)
ADC = np.diag(np.linalg.multi_dot([bvecs, tensor, bvecs.T]))
sig = np.exp(-bvals * ADC)
sig = sig - np.mean(sig)
self.signal[idx] = sig
self._calculated.append(idx)
return self.signal[idx]
[docs]
def streamline_signal(self, streamline):
"""
Approximate the signal for a given streamline
"""
grad = streamline_gradients(streamline)
sig_out = np.zeros((grad.shape[0], self.signal.shape[-1]))
for ii, g in enumerate(grad):
sig_out[ii] = self.calc_signal(g)
return sig_out
[docs]
@warning_for_keywords()
def voxel2streamline(streamline, affine, *, unique_idx=None):
"""
Maps voxels to streamlines and streamlines to voxels, for setting up
the LiFE equations matrix
Parameters
----------
streamline : list
A collection of streamlines, each n by 3, with n being the number of
nodes in the fiber.
affine : array_like (4, 4)
The mapping from voxel coordinates to streamline points.
The voxel_to_rasmm matrix, typically from a NIFTI file.
unique_idx : array, optional.
The unique indices in the streamlines
Returns
-------
v2f, v2fn : tuple of dicts
The first dict in the tuple answers the question: Given a voxel (from
the unique indices in this model), which fibers pass through it?
The second answers the question: Given a streamline, for each voxel that
this streamline passes through, which nodes of that streamline are in that
voxel?
"""
transformed_streamline = transform_streamlines(streamline, affine)
if unique_idx is None:
all_coords = np.concatenate(transformed_streamline)
unique_idx = unique_rows(np.round(all_coords))
return _voxel2streamline(transformed_streamline, unique_idx.astype(np.intp))
[docs]
class FiberModel(ReconstModel):
"""
A class for representing and solving predictive models based on
tractography solutions.
Notes
-----
This is an implementation of the LiFE model described in
:footcite:p:`Pestilli2014`.
References
----------
.. footbibliography::
"""
def __init__(self, gtab):
"""
Parameters
----------
gtab : a GradientTable class instance
"""
# Initialize the super-class:
ReconstModel.__init__(self, gtab)
[docs]
@warning_for_keywords()
def setup(self, streamline, affine, *, evals=(0.001, 0, 0), sphere=None):
"""
Set up the necessary components for the LiFE model: the matrix of
fiber-contributions to the DWI signal, and the coordinates of voxels
for which the equations will be solved
Parameters
----------
streamline : list
Streamlines, each is an array of shape (n, 3)
affine : array_like (4, 4)
The mapping from voxel coordinates to streamline points.
The voxel_to_rasmm matrix, typically from a NIFTI file.
evals : array-like (3 items, optional)
The eigenvalues of the canonical tensor used as a response
function. Default:[0.001, 0, 0].
sphere: `dipy.core.Sphere` instance, optional
Whether to approximate (and cache) the signal on a discrete
sphere. This may confer a significant speed-up in setting up the
problem, but is not as accurate. If `False`, we use the exact
gradients along the streamlines to calculate the matrix, instead of
an approximation. Defaults to use the 724-vertex symmetric sphere
from :mod:`dipy.data`
"""
if sphere is not False:
SignalMaker = LifeSignalMaker(self.gtab, evals=evals, sphere=sphere)
streamline = transform_streamlines(streamline, affine)
# Assign some local variables, for shorthand:
all_coords = np.concatenate(streamline)
vox_coords = unique_rows(np.round(all_coords).astype(np.intp))
del all_coords
# We only consider the diffusion-weighted signals:
n_bvecs = self.gtab.bvals[~self.gtab.b0s_mask].shape[0]
v2f, v2fn = voxel2streamline(streamline, np.eye(4), unique_idx=vox_coords)
# How many fibers in each voxel (this will determine how many
# components are in the matrix):
n_unique_f = len(np.hstack(list(v2f.values())))
# Preallocate these, which will be used to generate the sparse
# matrix:
f_matrix_sig = np.zeros(n_unique_f * n_bvecs, dtype=float)
f_matrix_row = np.zeros(n_unique_f * n_bvecs, dtype=np.intp)
f_matrix_col = np.zeros(n_unique_f * n_bvecs, dtype=np.intp)
fiber_signal = []
for _, s in enumerate(streamline):
if sphere is not False:
fiber_signal.append(SignalMaker.streamline_signal(s))
else:
fiber_signal.append(streamline_signal(s, self.gtab, evals=evals))
del streamline
if sphere is not False:
del SignalMaker
keep_ct = 0
range_bvecs = np.arange(n_bvecs).astype(int)
# In each voxel:
for v_idx in range(vox_coords.shape[0]):
mat_row_idx = (range_bvecs + v_idx * n_bvecs).astype(np.intp)
# For each fiber in that voxel:
for f_idx in v2f[v_idx]:
# For each fiber-voxel combination, store the row/column
# indices in the pre-allocated linear arrays
f_matrix_row[keep_ct : keep_ct + n_bvecs] = mat_row_idx
f_matrix_col[keep_ct : keep_ct + n_bvecs] = f_idx
vox_fiber_sig = np.zeros(n_bvecs)
for node_idx in v2fn[f_idx][v_idx]:
# Sum the signal from each node of the fiber in that voxel:
vox_fiber_sig += fiber_signal[f_idx][node_idx]
# And add the summed thing into the corresponding rows:
f_matrix_sig[keep_ct : keep_ct + n_bvecs] += vox_fiber_sig
keep_ct = keep_ct + n_bvecs
del v2f, v2fn
# Allocate the sparse matrix, using the more memory-efficient 'csr'
# format:
life_matrix = sps.csr_array((f_matrix_sig, [f_matrix_row, f_matrix_col]))
return life_matrix, vox_coords
def _signals(self, data, vox_coords):
"""
Helper function to extract and separate all the signals we need to fit
and evaluate a fit of this model
Parameters
----------
data : 4D array
vox_coords: n by 3 array
The coordinates into the data array of the fiber nodes.
"""
# Fitting is done on the S0-normalized-and-demeaned diffusion-weighted
# signal:
idx_tuple = (vox_coords[:, 0], vox_coords[:, 1], vox_coords[:, 2])
# We'll look at a 2D array, extracting the data from the voxels:
vox_data = data[idx_tuple]
weighted_signal = vox_data[:, ~self.gtab.b0s_mask]
b0_signal = np.mean(vox_data[:, self.gtab.b0s_mask], -1)
relative_signal = weighted_signal / b0_signal[:, None]
# The mean of the relative signal across directions in each voxel:
mean_sig = np.mean(relative_signal, -1)
to_fit = (relative_signal - mean_sig[:, None]).ravel()
to_fit[np.isnan(to_fit)] = 0
return (to_fit, weighted_signal, b0_signal, relative_signal, mean_sig, vox_data)
[docs]
@warning_for_keywords()
def fit(self, data, streamline, affine, *, evals=(0.001, 0, 0), sphere=None):
"""
Fit the LiFE FiberModel for data and a set of streamlines associated
with this data
Parameters
----------
data : 4D array
Diffusion-weighted data
streamline : list
A bunch of streamlines
affine : array_like (4, 4)
The mapping from voxel coordinates to streamline points.
The voxel_to_rasmm matrix, typically from a NIFTI file.
evals : array-like, optional
The eigenvalues of the tensor response function used in
constructing the model signal. Default: [0.001, 0, 0]
sphere: `dipy.core.Sphere` instance or False, optional
Whether to approximate (and cache) the signal on a discrete
sphere. This may confer a significant speed-up in setting up the
problem, but is not as accurate. If `False`, we use the exact
gradients along the streamlines to calculate the matrix, instead of
an approximation.
Returns
-------
FiberFit class instance
"""
if affine is None:
affine = np.eye(4)
sl_len = np.array([len(s) for s in streamline])
if np.any(sl_len < 2):
raise ValueError(
"Input contains streamlines with only one node."
" The LiFE model cannot be fit with these streamlines included."
)
life_matrix, vox_coords = self.setup(
streamline, affine, evals=evals, sphere=sphere
)
(to_fit, weighted_signal, b0_signal, relative_signal, mean_sig, vox_data) = (
self._signals(data, vox_coords)
)
beta = opt.sparse_nnls(to_fit, life_matrix)
return FiberFit(
self,
life_matrix,
vox_coords,
to_fit,
beta,
weighted_signal,
b0_signal,
relative_signal,
mean_sig,
vox_data,
streamline,
affine,
evals,
)
[docs]
class FiberFit(ReconstFit):
"""
A fit of the LiFE model to diffusion data
"""
def __init__(
self,
fiber_model,
life_matrix,
vox_coords,
to_fit,
beta,
weighted_signal,
b0_signal,
relative_signal,
mean_sig,
vox_data,
streamline,
affine,
evals,
):
"""
Parameters
----------
fiber_model : A FiberModel class instance
params : the parameters derived from a fit of the model to the data.
"""
ReconstFit.__init__(self, fiber_model, vox_data)
self.life_matrix = life_matrix
self.vox_coords = vox_coords
self.fit_data = to_fit
self.beta = beta
self.weighted_signal = weighted_signal
self.b0_signal = b0_signal
self.relative_signal = relative_signal
self.mean_signal = mean_sig
self.streamline = streamline
self.affine = affine
self.evals = evals
[docs]
@warning_for_keywords()
def predict(self, *, gtab=None, S0=None):
"""
Predict the signal
Parameters
----------
gtab : GradientTable
Default: use self.gtab
S0 : float or array
The non-diffusion-weighted signal in the voxels for which a
prediction is made. Default: use self.b0_signal
Returns
-------
prediction : ndarray of shape (voxels, bvecs)
An array with a prediction of the signal in each voxel/direction
"""
# We generate the prediction and in each voxel, we add the
# offset, according to the isotropic part of the signal, which was
# removed prior to fitting:
if gtab is None:
_matrix = self.life_matrix
gtab = self.model.gtab
else:
_model = FiberModel(gtab)
_matrix, _ = _model.setup(self.streamline, self.affine, evals=self.evals)
pred_weighted = np.reshape(
opt.spdot(_matrix, self.beta),
(self.vox_coords.shape[0], np.sum(~gtab.b0s_mask)),
)
pred = np.empty((self.vox_coords.shape[0], gtab.bvals.shape[0]))
if S0 is None:
S0 = self.b0_signal
pred[..., gtab.b0s_mask] = S0[:, None]
pred[..., ~gtab.b0s_mask] = (pred_weighted + self.mean_signal[:, None]) * S0[
:, None
]
return pred