def test_design_matrix_lte():
_, fbval, fbvec = get_fnames('small_25')
gtab_btens_none = grad.gradient_table(fbval, fbvec)
gtab_btens_lte = grad.gradient_table(fbval, fbvec, btens="LTE")
B_btens_none = dti.design_matrix(gtab_btens_none)
B_btens_lte = dti.design_matrix(gtab_btens_lte)
npt.assert_array_almost_equal(B_btens_none, B_btens_lte, decimal=1)
def test_nnls_jacobian_fucn():
b0 = 1000.
bvecs, bval = read_bvec_file(get_data('55dir_grad.bvec'))
gtab = grad.gradient_table(bval, bvecs)
B = bval[1]
# Scale the eigenvalues and tensor by the B value so the units match
D = np.array([1., 1., 1., 0., 0., 1., -np.log(b0) * B]) / B
# Design Matrix
X = dti.design_matrix(gtab)
# Signals
Y = np.exp(np.dot(X, D))
# Test Jacobian at D
args = [X, Y]
analytical = dti._nlls_jacobian_func(D, *args)
for i in range(len(X)):
args = [X[i], Y[i]]
approx = opt.approx_fprime(D, dti._nlls_err_func, 1e-8, *args)
assert_true(np.allclose(approx, analytical[i]))
# Test Jacobian at zero
D = np.zeros_like(D)
args = [X, Y]
analytical = dti._nlls_jacobian_func(D, *args)
for i in range(len(X)):
args = [X[i], Y[i]]
approx = opt.approx_fprime(D, dti._nlls_err_func, 1e-8, *args)
assert_true(np.allclose(approx, analytical[i]))
def make_fake_signal():
v, e, f = create_half_unit_sphere(4)
vecs_xy = v[np.flatnonzero(v[:, 2] == 0)]
evals = np.array([1.8, 0.2, 0.2]) * 10 ** -3 * 1.5
evecs_moveing = np.empty((len(vecs_xy), 3, 3))
evecs_moveing[:, :, 0] = vecs_xy
evecs_moveing[:, :, 1] = [0, 0, 1]
evecs_moveing[:, :, 2] = np.cross(evecs_moveing[:, :, 0], evecs_moveing[:, :, 1])
assert ((evecs_moveing * evecs_moveing).sum(1) - 1 < 0.001).all()
assert ((evecs_moveing * evecs_moveing).sum(2) - 1 < 0.001).all()
gtab = np.empty((len(v) + 1, 3))
bval = np.empty(len(v) + 1)
bval[0] = 0
bval[1:] = 2000
gtab[0] = [0, 0, 0]
gtab[1:] = v
bvec = gtab.T
B = design_matrix(bvec, bval)
tensor_moveing = np.empty_like(evecs_moveing)
for ii in xrange(len(vecs_xy)):
tensor_moveing[ii] = np.dot(evecs_moveing[ii] * evals, evecs_moveing[ii].T)
D_moveing = lower_triangular(tensor_moveing, 1)
tensor_fixed = np.diag(evals)
D_fixed = lower_triangular(tensor_fixed, 1)
sig = 0.45 * np.exp(np.dot(D_moveing, B.T)) + 0.55 * np.exp(np.dot(B, D_fixed))
assert sig.max() <= 1
assert sig.min() > 0
return v, e, vecs_xy, bval, bvec, sig
def test_restore():
"""
Test the implementation of the RESTORE algorithm
"""
b0 = 1000.
bvecs, bval = read_bvec_file(get_data('55dir_grad.bvec'))
gtab = grad.gradient_table(bval, bvecs)
B = bval[1]
#Scale the eigenvalues and tensor by the B value so the units match
D = np.array([1., 1., 1., 0., 0., 1., -np.log(b0) * B]) / B
evals = np.array([2., 1., 0.]) / B
md = evals.mean()
tensor = from_lower_triangular(D)
#Design Matrix
X = dti.design_matrix(gtab)
#Signals
Y = np.exp(np.dot(X,D))
Y.shape = (-1,) + Y.shape
for drop_this in range(1, Y.shape[-1]):
# RESTORE estimates should be robust to dropping
this_y = Y.copy()
this_y[:, drop_this] = 1.0
tensor_model = dti.TensorModel(gtab, fit_method='restore',
sigma=67.0)
tensor_est = tensor_model.fit(this_y)
assert_array_almost_equal(tensor_est.evals[0], evals, decimal=3)
assert_array_almost_equal(tensor_est.quadratic_form[0], tensor,
decimal=3)
def make_fake_signal():
v, e, f = create_half_unit_sphere(4)
vecs_xy = v[np.flatnonzero(v[:, 2] == 0)]
evals = np.array([1.8, .2, .2]) * 10**-3 * 1.5
evecs_moveing = np.empty((len(vecs_xy), 3, 3))
evecs_moveing[:, :, 0] = vecs_xy
evecs_moveing[:, :, 1] = [0, 0, 1]
evecs_moveing[:, :, 2] = np.cross(evecs_moveing[:, :, 0],
evecs_moveing[:, :, 1])
assert ((evecs_moveing * evecs_moveing).sum(1) - 1 < .001).all()
assert ((evecs_moveing * evecs_moveing).sum(2) - 1 < .001).all()
gtab = np.empty((len(v) + 1, 3))
bval = np.empty(len(v) + 1)
bval[0] = 0
bval[1:] = 2000
gtab[0] = [0, 0, 0]
gtab[1:] = v
bvec = gtab.T
B = design_matrix(bvec, bval)
tensor_moveing = np.empty_like(evecs_moveing)
for ii in xrange(len(vecs_xy)):
tensor_moveing[ii] = np.dot(evecs_moveing[ii] * evals,
evecs_moveing[ii].T)
D_moveing = lower_triangular(tensor_moveing, 1)
tensor_fixed = np.diag(evals)
D_fixed = lower_triangular(tensor_fixed, 1)
sig = .45 * np.exp(np.dot(D_moveing, B.T)) + .55 * np.exp(
np.dot(B, D_fixed))
assert sig.max() <= 1
assert sig.min() > 0
return v, e, vecs_xy, bval, bvec, sig
def test_nnls_jacobian_fucn():
b0 = 1000.
bvecs, bval = read_bvec_file(get_data('55dir_grad.bvec'))
gtab = grad.gradient_table(bval, bvecs)
B = bval[1]
# Scale the eigenvalues and tensor by the B value so the units match
D = np.array([1., 1., 1., 0., 0., 1., -np.log(b0) * B]) / B
# Design Matrix
X = dti.design_matrix(gtab)
# Signals
Y = np.exp(np.dot(X, D))
# Test Jacobian at D
args = [X, Y]
analytical = dti._nlls_jacobian_func(D, *args)
for i in range(len(X)):
args = [X[i], Y[i]]
approx = opt.approx_fprime(D, dti._nlls_err_func, 1e-8, *args)
assert_true(np.allclose(approx, analytical[i]))
# Test Jacobian at zero
D = np.zeros_like(D)
args = [X, Y]
analytical = dti._nlls_jacobian_func(D, *args)
for i in range(len(X)):
args = [X[i], Y[i]]
approx = opt.approx_fprime(D, dti._nlls_err_func, 1e-8, *args)
assert_true(np.allclose(approx, analytical[i]))
def test_TensorModel():
data, gtab = dsi_voxels()
dm = dti.TensorModel(gtab, 'LS')
dtifit = dm.fit(data[0, 0, 0])
assert_equal(dtifit.fa < 0.5, True)
dm = dti.TensorModel(gtab, 'WLS')
dtifit = dm.fit(data[0, 0, 0])
assert_equal(dtifit.fa < 0.5, True)
sphere = create_unit_sphere(4)
assert_equal(len(dtifit.odf(sphere)), len(sphere.vertices))
assert_almost_equal(dtifit.fa, gfa(dtifit.odf(sphere)), 1)
# Check that the multivoxel case works:
dtifit = dm.fit(data)
assert_equal(dtifit.fa.shape, data.shape[:3])
# Make some synthetic data
b0 = 1000.
bvecs, bvals = read_bvec_file(get_data('55dir_grad.bvec'))
gtab = grad.gradient_table_from_bvals_bvecs(bvals, bvecs.T)
# The first b value is 0., so we take the second one:
B = bvals[1]
#Scale the eigenvalues and tensor by the B value so the units match
D = np.array([1., 1., 1., 0., 0., 1., -np.log(b0) * B]) / B
evals = np.array([2., 1., 0.]) / B
md = evals.mean()
tensor = from_lower_triangular(D)
evecs = np.linalg.eigh(tensor)[1]
#Design Matrix
X = dti.design_matrix(bvecs, bvals)
#Signals
Y = np.exp(np.dot(X,D))
assert_almost_equal(Y[0], b0)
Y.shape = (-1,) + Y.shape
# Test fitting with different methods: #XXX Add NNLS methods!
for fit_method in ['OLS', 'WLS']:
tensor_model = dti.TensorModel(gtab,
fit_method=fit_method)
tensor_fit = tensor_model.fit(Y)
assert_true(tensor_fit.model is tensor_model)
assert_equal(tensor_fit.shape, Y.shape[:-1])
assert_array_almost_equal(tensor_fit.evals[0], evals)
assert_array_almost_equal(tensor_fit.quadratic_form[0], tensor,
err_msg =\
"Calculation of tensor from Y does not compare to analytical solution")
assert_almost_equal(tensor_fit.md[0], md)
assert_equal(tensor_fit.directions.shape[-2], 1)
assert_equal(tensor_fit.directions.shape[-1], 3)
# Test error-handling:
assert_raises(ValueError,
dti.TensorModel,
gtab,
fit_method='crazy_method')
def test_WLS_and_LS_fit():
"""
Tests the WLS and LS fitting functions to see if they returns the correct
eigenvalues and eigenvectors.
Uses data/55dir_grad.bvec as the gradient table and 3by3by56.nii
as the data.
"""
### Defining Test Voxel (avoid nibabel dependency) ###
#Recall: D = [Dxx,Dyy,Dzz,Dxy,Dxz,Dyz,log(S_0)] and D ~ 10^-4 mm^2 /s
b0 = 1000.
bvec, bval = read_bvec_file(get_data('55dir_grad.bvec'))
B = bval[1]
#Scale the eigenvalues and tensor by the B value so the units match
D = np.array([1., 1., 1., 0., 0., 1., -np.log(b0) * B]) / B
evals = np.array([2., 1., 0.]) / B
md = evals.mean()
tensor = from_lower_triangular(D)
#Design Matrix
X = dti.design_matrix(bvec, bval)
#Signals
Y = np.exp(np.dot(X, D))
assert_almost_equal(Y[0], b0)
Y.shape = (-1,) + Y.shape
gtab = grad.gradient_table(bval, bvec)
### Testing WLS Fit on Single Voxel ###
#Estimate tensor from test signals
model = TensorModel(gtab, min_signal=1e-8, fit_method='WLS')
tensor_est = model.fit(Y)
assert_equal(tensor_est.shape, Y.shape[:-1])
assert_array_almost_equal(tensor_est.evals[0], evals)
assert_array_almost_equal(tensor_est.quadratic_form[0], tensor,
err_msg="Calculation of tensor from Y does not "
"compare to analytical solution")
assert_almost_equal(tensor_est.md[0], md)
# Test that we can fit a single voxel's worth of data (a 1d array)
y = Y[0]
tensor_est = model.fit(y)
assert_equal(tensor_est.shape, tuple())
assert_array_almost_equal(tensor_est.evals, evals)
assert_array_almost_equal(tensor_est.quadratic_form, tensor)
assert_almost_equal(tensor_est.md, md)
assert_array_almost_equal(tensor_est.lower_triangular(b0), D)
# Test using fit_method='LS'
model = TensorModel(gtab, min_signal=1e-8, fit_method='LS')
tensor_est = model.fit(y)
assert_equal(tensor_est.shape, tuple())
assert_array_almost_equal(tensor_est.evals, evals)
assert_array_almost_equal(tensor_est.quadratic_form, tensor)
assert_almost_equal(tensor_est.md, md)
assert_array_almost_equal(tensor_est.lower_triangular(b0), D)
def test_WLS_and_LS_fit():
"""
Tests the WLS and LS fitting functions to see if they returns the correct
eigenvalues and eigenvectors.
Uses data/55dir_grad.bvec as the gradient table and 3by3by56.nii
as the data.
"""
### Defining Test Voxel (avoid nibabel dependency) ###
#Recall: D = [Dxx,Dyy,Dzz,Dxy,Dxz,Dyz,log(S_0)] and D ~ 10^-4 mm^2 /s
b0 = 1000.
gtab, bval = read_bvec_file(get_data('55dir_grad.bvec'))
B = bval[1]
#Scale the eigenvalues and tensor by the B value so the units match
D = np.array([1., 1., 1., 0., 0., 1., -np.log(b0) * B]) / B
evals = np.array([2., 1., 0.]) / B
md = evals.mean()
tensor = from_lower_triangular(D)
#Design Matrix
X = dti.design_matrix(gtab, bval)
#Signals
Y = np.exp(np.dot(X, D))
assert_almost_equal(Y[0], b0)
Y.shape = (-1, ) + Y.shape
### Testing WLS Fit on Single Voxel ###
#Estimate tensor from test signals
tensor_est = dti.Tensor(Y, bval, gtab.T, min_signal=1e-8)
assert_equal(tensor_est.shape, Y.shape[:-1])
assert_array_almost_equal(tensor_est.evals[0], evals)
assert_array_almost_equal(
tensor_est.D[0],
tensor,
err_msg=
"Calculation of tensor from Y does not compare to analytical solution")
assert_almost_equal(tensor_est.md()[0], md)
#test 0d tensor
y = Y[0]
tensor_est = dti.Tensor(y, bval, gtab.T, min_signal=1e-8)
assert_equal(tensor_est.shape, tuple())
assert_array_almost_equal(tensor_est.evals, evals)
assert_array_almost_equal(tensor_est.D, tensor)
assert_almost_equal(tensor_est.md(), md)
assert_array_almost_equal(tensor_est.lower_triangular(b0), D)
tensor_est = dti.Tensor(y, bval, gtab.T, min_signal=1e-8, fit_method='LS')
assert_equal(tensor_est.shape, tuple())
assert_array_almost_equal(tensor_est.evals, evals)
assert_array_almost_equal(tensor_est.D, tensor)
assert_almost_equal(tensor_est.md(), md)
assert_array_almost_equal(tensor_est.lower_triangular(b0), D)
def test_WLS_and_LS_fit():
"""
Tests the WLS and LS fitting functions to see if they returns the correct
eigenvalues and eigenvectors.
Uses data/55dir_grad.bvec as the gradient table and 3by3by56.nii
as the data.
"""
### Defining Test Voxel (avoid nibabel dependency) ###
#Recall: D = [Dxx,Dyy,Dzz,Dxy,Dxz,Dyz,log(S_0)] and D ~ 10^-4 mm^2 /s
D = np.array([1., 1., 1., 1., 0., 0., np.log(1000) * 10.**4]) * 10.**-4
evals = np.array([2., 1., 0.]) * 10.**-4
md = evals.mean()
tensor = np.empty((3,3))
tensor[0, 0] = D[0]
tensor[1, 1] = D[1]
tensor[2, 2] = D[2]
tensor[0, 1] = tensor[1, 0] = D[3]
tensor[0, 2] = tensor[2, 0] = D[4]
tensor[1, 2] = tensor[2, 1] = D[5]
#Design Matrix
gtab, bval = read_bvec_file(get_data('55dir_grad.bvec'))
X = dti.design_matrix(gtab, bval)
#Signals
Y = np.exp(np.dot(X,D))
Y.shape = (-1,) + Y.shape
### Testing WLS Fit on Single Voxel ###
#Estimate tensor from test signals
tensor_est = dti.Tensor(Y,bval,gtab.T,min_signal=1e-8)
yield assert_equal(tensor_est.shape, Y.shape[:-1])
yield assert_array_almost_equal(tensor_est.evals[0], evals)
yield assert_array_almost_equal(tensor_est.D[0], tensor,err_msg= "Calculation of tensor from Y does not compare to analytical solution")
yield assert_almost_equal(tensor_est.md()[0], md)
#test 0d tensor
y = Y[0]
tensor_est = dti.Tensor(y, bval, gtab.T, min_signal=1e-8)
yield assert_equal(tensor_est.shape, tuple())
yield assert_array_almost_equal(tensor_est.evals, evals)
yield assert_array_almost_equal(tensor_est.D, tensor)
yield assert_almost_equal(tensor_est.md(), md)
tensor_est = dti.Tensor(y, bval, gtab.T, min_signal=1e-8, fit_method='LS')
yield assert_equal(tensor_est.shape, tuple())
yield assert_array_almost_equal(tensor_est.evals, evals)
yield assert_array_almost_equal(tensor_est.D, tensor)
yield assert_almost_equal(tensor_est.md(), md)
def __init__(self, gtab, fit_method="NLS", *args, **kwargs):
""" Free Water Diffusion Tensor Model [1]_.
Parameters
----------
gtab : GradientTable class instance
fit_method : str or callable
str can be one of the following:
'WLS' for weighted linear least square fit according to [1]_
:func:`fwdti.wls_iter`
'NLS' for non-linear least square fit according to [1]_
:func:`fwdti.nls_iter`
callable has to have the signature:
fit_method(design_matrix, data, *args, **kwargs)
args, kwargs : arguments and key-word arguments passed to the
fit_method. See fwdti.wls_iter, fwdti.nls_iter for
details
References
----------
.. [1] Hoy, A.R., Koay, C.G., Kecskemeti, S.R., Alexander, A.L., 2014.
Optimization of a free water elimination two-compartmental model
for diffusion tensor imaging. NeuroImage 103, 323-333.
doi: 10.1016/j.neuroimage.2014.09.053
"""
ReconstModel.__init__(self, gtab)
if not callable(fit_method):
try:
fit_method = common_fit_methods[fit_method]
except KeyError:
e_s = '"' + str(fit_method) + '" is not a known fit '
e_s += 'method, the fit method should either be a '
e_s += 'function or one of the common fit methods'
raise ValueError(e_s)
self.fit_method = fit_method
self.design_matrix = design_matrix(self.gtab)
self.args = args
self.kwargs = kwargs
# Check if at least three b-values are given
bmag = int(np.log10(self.gtab.bvals.max()))
b = self.gtab.bvals.copy() / (10 ** (bmag-1)) # normalize b units
b = b.round()
uniqueb = np.unique(b)
if len(uniqueb) < 3:
mes = "fwdti fit requires data for at least 2 non zero b-values"
raise ValueError(mes)
def test_restore():
"""
Test the implementation of the RESTORE algorithm
"""
b0 = 1000.
bvecs, bval = read_bvec_file(get_data('55dir_grad.bvec'))
gtab = grad.gradient_table(bval, bvecs)
B = bval[1]
# Scale the eigenvalues and tensor by the B value so the units match
D = np.array([1., 1., 1., 0., 0., 1., -np.log(b0) * B]) / B
evals = np.array([2., 1., 0.]) / B
tensor = from_lower_triangular(D)
# Design Matrix
X = dti.design_matrix(gtab)
# Signals
Y = np.exp(np.dot(X, D))
Y.shape = (-1, ) + Y.shape
for drop_this in range(1, Y.shape[-1]):
for jac in [True, False]:
# RESTORE estimates should be robust to dropping
this_y = Y.copy()
this_y[:, drop_this] = 1.0
for sigma in [67.0, np.ones(this_y.shape[-1]) * 67.0]:
tensor_model = dti.TensorModel(gtab,
fit_method='restore',
jac=jac,
sigma=67.0)
tensor_est = tensor_model.fit(this_y)
assert_array_almost_equal(tensor_est.evals[0],
evals,
decimal=3)
assert_array_almost_equal(tensor_est.quadratic_form[0],
tensor,
decimal=3)
# If sigma is very small, it still needs to work:
tensor_model = dti.TensorModel(gtab, fit_method='restore', sigma=0.0001)
tensor_model.fit(Y.copy())
# Test return_S0_hat
tensor_model = dti.TensorModel(gtab,
fit_method='restore',
sigma=0.0001,
return_S0_hat=True)
tmf = tensor_model.fit(Y.copy())
assert_almost_equal(tmf[0].S0_hat, b0)
def __init__(self, gtab, fit_method="NLS", *args, **kwargs):
""" Free Water Diffusion Tensor Model [1]_.
Parameters
----------
gtab : GradientTable class instance
fit_method : str or callable
str can be one of the following:
'WLS' for weighted linear least square fit according to [1]_
:func:`fwdti.wls_iter`
'NLS' for non-linear least square fit according to [1]_
:func:`fwdti.nls_iter`
callable has to have the signature:
fit_method(design_matrix, data, *args, **kwargs)
args, kwargs : arguments and key-word arguments passed to the
fit_method. See fwdti.wls_iter, fwdti.nls_iter for
details
References
----------
.. [1] Hoy, A.R., Koay, C.G., Kecskemeti, S.R., Alexander, A.L., 2014.
Optimization of a free water elimination two-compartmental model
for diffusion tensor imaging. NeuroImage 103, 323-333.
doi: 10.1016/j.neuroimage.2014.09.053
"""
ReconstModel.__init__(self, gtab)
if not callable(fit_method):
try:
fit_method = common_fit_methods[fit_method]
except KeyError:
e_s = '"' + str(fit_method) + '" is not a known fit '
e_s += 'method, the fit method should either be a '
e_s += 'function or one of the common fit methods'
raise ValueError(e_s)
self.fit_method = fit_method
self.design_matrix = design_matrix(self.gtab)
self.args = args
self.kwargs = kwargs
# Check if at least three b-values are given
bmag = int(np.log10(self.gtab.bvals.max()))
b = self.gtab.bvals.copy() / (10**(bmag - 1)) # normalize b units
b = b.round()
uniqueb = np.unique(b)
if len(uniqueb) < 3:
mes = "fwdti fit requires data for at least 2 non zero b-values"
raise ValueError(mes)
def test_nlls_fit_tensor():
"""
Test the implementation of NLLS and RESTORE
"""
b0 = 1000.
bvecs, bval = read_bvec_file(get_data('55dir_grad.bvec'))
gtab = grad.gradient_table(bval, bvecs)
B = bval[1]
#Scale the eigenvalues and tensor by the B value so the units match
D = np.array([1., 1., 1., 0., 0., 1., -np.log(b0) * B]) / B
evals = np.array([2., 1., 0.]) / B
md = evals.mean()
tensor = from_lower_triangular(D)
#Design Matrix
X = dti.design_matrix(bvecs, bval)
#Signals
Y = np.exp(np.dot(X,D))
Y.shape = (-1,) + Y.shape
#Estimate tensor from test signals and compare against expected result
#using non-linear least squares:
tensor_model = dti.TensorModel(gtab, fit_method='NLLS')
tensor_est = tensor_model.fit(Y)
assert_equal(tensor_est.shape, Y.shape[:-1])
assert_array_almost_equal(tensor_est.evals[0], evals)
assert_array_almost_equal(tensor_est.quadratic_form[0], tensor)
assert_almost_equal(tensor_est.md[0], md)
# Using the gmm weighting scheme:
tensor_model = dti.TensorModel(gtab, fit_method='NLLS', weighting='gmm')
assert_equal(tensor_est.shape, Y.shape[:-1])
assert_array_almost_equal(tensor_est.evals[0], evals)
assert_array_almost_equal(tensor_est.quadratic_form[0], tensor)
assert_almost_equal(tensor_est.md[0], md)
# Use NLLS with some actual 4D data:
data, bvals, bvecs = get_data('small_25')
gtab = grad.gradient_table(bvals, bvecs)
tm1 = dti.TensorModel(gtab, fit_method='NLLS')
dd = nib.load(data).get_data()
tf1 = tm1.fit(dd)
tm2 = dti.TensorModel(gtab)
tf2 = tm2.fit(dd)
assert_array_almost_equal(tf1.fa, tf2.fa, decimal=1)
def test_nlls_fit_tensor():
"""
Test the implementation of NLLS and RESTORE
"""
b0 = 1000.
bvecs, bval = read_bvec_file(get_data('55dir_grad.bvec'))
gtab = grad.gradient_table(bval, bvecs)
B = bval[1]
#Scale the eigenvalues and tensor by the B value so the units match
D = np.array([1., 1., 1., 0., 0., 1., -np.log(b0) * B]) / B
evals = np.array([2., 1., 0.]) / B
md = evals.mean()
tensor = from_lower_triangular(D)
#Design Matrix
X = dti.design_matrix(bvecs, bval)
#Signals
Y = np.exp(np.dot(X,D))
Y.shape = (-1,) + Y.shape
#Estimate tensor from test signals and compare against expected result
#using non-linear least squares:
tensor_model = dti.TensorModel(gtab, fit_method='NLLS')
tensor_est = tensor_model.fit(Y)
assert_equal(tensor_est.shape, Y.shape[:-1])
assert_array_almost_equal(tensor_est.evals[0], evals)
assert_array_almost_equal(tensor_est.quadratic_form[0], tensor)
assert_almost_equal(tensor_est.md[0], md)
# Using the gmm weighting scheme:
tensor_model = dti.TensorModel(gtab, fit_method='NLLS', weighting='gmm')
assert_equal(tensor_est.shape, Y.shape[:-1])
assert_array_almost_equal(tensor_est.evals[0], evals)
assert_array_almost_equal(tensor_est.quadratic_form[0], tensor)
assert_almost_equal(tensor_est.md[0], md)
# Use NLLS with some actual 4D data:
data, bvals, bvecs = get_data('small_25')
gtab = grad.gradient_table(bvals, bvecs)
tm1 = dti.TensorModel(gtab, fit_method='NLLS')
dd = nib.load(data).get_data()
tf1 = tm1.fit(dd)
tm2 = dti.TensorModel(gtab)
tf2 = tm2.fit(dd)
assert_array_almost_equal(tf1.fa, tf2.fa, decimal=1)
def __init__(self, gtab, fit_method="NLS", *args, **kwargs):
""" Free Water Diffusion Tensor Model [1]_.
Parameters
----------
gtab : GradientTable class instance
fit_method : str or callable
str can be one of the following:
'WLS' for weighted linear least square fit according to [1]_
:func:`fwdti.wls_iter`
'NLS' for non-linear least square fit according to [1]_
:func:`fwdti.nls_iter`
callable has to have the signature:
fit_method(design_matrix, data, *args, **kwargs)
args, kwargs : arguments and key-word arguments passed to the
fit_method. See fwdti.wls_iter, fwdti.nls_iter for
details
References
----------
.. [1] Henriques, R.N., Rokem, A., Garyfallidis, E., St-Jean, S.,
Peterson E.T., Correia, M.M., 2017. [Re] Optimization of a free
water elimination two-compartment model for diffusion tensor
imaging. ReScience volume 3, issue 1, article number 2
"""
ReconstModel.__init__(self, gtab)
if not callable(fit_method):
try:
fit_method = common_fit_methods[fit_method]
except KeyError:
e_s = '"' + str(fit_method) + '" is not a known fit '
e_s += 'method, the fit method should either be a '
e_s += 'function or one of the common fit methods'
raise ValueError(e_s)
self.fit_method = fit_method
self.design_matrix = design_matrix(self.gtab)
self.args = args
self.kwargs = kwargs
# Check if at least three b-values are given
enough_b = check_multi_b(self.gtab, 3, non_zero=False)
if not enough_b:
mes = "fwDTI requires at least 3 b-values (which can include b=0)"
raise ValueError(mes)
def __init__(self, gtab, fit_method="NLS", *args, **kwargs):
""" Free Water Diffusion Tensor Model [1]_.
Parameters
----------
gtab : GradientTable class instance
fit_method : str or callable
str can be one of the following:
'WLS' for weighted linear least square fit according to [1]_
:func:`fwdti.wls_iter`
'NLS' for non-linear least square fit according to [1]_
:func:`fwdti.nls_iter`
callable has to have the signature:
fit_method(design_matrix, data, *args, **kwargs)
args, kwargs : arguments and key-word arguments passed to the
fit_method. See fwdti.wls_iter, fwdti.nls_iter for
details
References
----------
.. [1] Hoy, A.R., Koay, C.G., Kecskemeti, S.R., Alexander, A.L., 2014.
Optimization of a free water elimination two-compartmental model
for diffusion tensor imaging. NeuroImage 103, 323-333.
doi: 10.1016/j.neuroimage.2014.09.053
"""
ReconstModel.__init__(self, gtab)
if not callable(fit_method):
try:
fit_method = common_fit_methods[fit_method]
except KeyError:
e_s = '"' + str(fit_method) + '" is not a known fit '
e_s += 'method, the fit method should either be a '
e_s += 'function or one of the common fit methods'
raise ValueError(e_s)
self.fit_method = fit_method
self.design_matrix = design_matrix(self.gtab)
self.args = args
self.kwargs = kwargs
# Check if at least three b-values are given
enough_b = check_multi_b(self.gtab, 3, non_zero=False)
if not enough_b:
mes = "fwDTI requires at least 3 b-values (which can include b=0)"
raise ValueError(mes)
def __init__(self, gtab, fit_method="NLS", *args, **kwargs):
""" Free Water Diffusion Tensor Model [1]_.
Parameters
----------
gtab : GradientTable class instance
fit_method : str or callable
str can be one of the following:
'WLS' for weighted linear least square fit according to [1]_
:func:`fwdti.wls_iter`
'NLS' for non-linear least square fit according to [1]_
:func:`fwdti.nls_iter`
callable has to have the signature:
fit_method(design_matrix, data, *args, **kwargs)
args, kwargs : arguments and key-word arguments passed to the
fit_method. See fwdti.wls_iter, fwdti.nls_iter for
details
References
----------
.. [1] Hoy, A.R., Koay, C.G., Kecskemeti, S.R., Alexander, A.L., 2014.
Optimization of a free water elimination two-compartmental model
for diffusion tensor imaging. NeuroImage 103, 323-333.
doi: 10.1016/j.neuroimage.2014.09.053
"""
ReconstModel.__init__(self, gtab)
if not callable(fit_method):
try:
fit_method = common_fit_methods[fit_method]
except KeyError:
e_s = '"' + str(fit_method) + '" is not a known fit '
e_s += 'method, the fit method should either be a '
e_s += 'function or one of the common fit methods'
raise ValueError(e_s)
self.fit_method = fit_method
self.design_matrix = design_matrix(self.gtab)
self.args = args
self.kwargs = kwargs
# Check if at least three b-values are given
enough_b = check_multi_b(self.gtab, 3, non_zero=False)
if not enough_b:
mes = "fwDTI requires at least 3 b-values (which can include b=0)"
raise ValueError(mes)
def test_restore():
"""
Test the implementation of the RESTORE algorithm
"""
b0 = 1000.
bvecs, bval = read_bvec_file(get_data('55dir_grad.bvec'))
gtab = grad.gradient_table(bval, bvecs)
B = bval[1]
# Scale the eigenvalues and tensor by the B value so the units match
D = np.array([1., 1., 1., 0., 0., 1., -np.log(b0) * B]) / B
evals = np.array([2., 1., 0.]) / B
tensor = from_lower_triangular(D)
# Design Matrix
X = dti.design_matrix(gtab)
# Signals
Y = np.exp(np.dot(X, D))
Y.shape = (-1,) + Y.shape
for drop_this in range(1, Y.shape[-1]):
for jac in [True, False]:
# RESTORE estimates should be robust to dropping
this_y = Y.copy()
this_y[:, drop_this] = 1.0
for _ in [67.0, np.ones(this_y.shape[-1]) * 67.0]:
tensor_model = dti.TensorModel(gtab, fit_method='restore',
jac=jac,
sigma=67.0)
tensor_est = tensor_model.fit(this_y)
assert_array_almost_equal(tensor_est.evals[0], evals,
decimal=3)
assert_array_almost_equal(tensor_est.quadratic_form[0], tensor,
decimal=3)
# If sigma is very small, it still needs to work:
tensor_model = dti.TensorModel(gtab, fit_method='restore', sigma=0.0001)
tensor_model.fit(Y.copy())
# Test return_S0_hat
tensor_model = dti.TensorModel(gtab, fit_method='restore', sigma=0.0001,
return_S0_hat=True)
tmf = tensor_model.fit(Y.copy())
assert_almost_equal(tmf[0].S0_hat, b0)
def __init__(self, gtab):
super(EigenModule, self).__init__()
tol = 1e-6
design_matrix = dti.design_matrix(gtab)
self.design_matrix_inv = torch.FloatTensor(
np.linalg.pinv(design_matrix))
self.design_matrix_inv = nn.Parameter(self.design_matrix_inv,
requires_grad=False)
self.min_diffusivity = tol / -design_matrix.min()
self._lt_indices = np.array([[0, 1, 3],
[1, 2, 4],
[3, 4, 5]])
self.symeig_module = SymEig()
self.inputs = ['dwi']
self.outputs = ['evals', 'evecs']
def test_restore():
"""
Test the implementation of the RESTORE algorithm
"""
b0 = 1000.
bvecs, bval = read_bvec_file(get_data('55dir_grad.bvec'))
gtab = grad.gradient_table(bval, bvecs)
B = bval[1]
#Scale the eigenvalues and tensor by the B value so the units match
D = np.array([1., 1., 1., 0., 0., 1., -np.log(b0) * B]) / B
evals = np.array([2., 1., 0.]) / B
md = evals.mean()
tensor = from_lower_triangular(D)
#Design Matrix
X = dti.design_matrix(bvecs, bval)
#Signals
Y = np.exp(np.dot(X,D))
Y.shape = (-1,) + Y.shape
for sigma in [0.1, 1, 10, 100]:
for drop_this in range(1, Y.shape[-1]):
# RESTORE estimates should be robust to dropping
this_y = Y.copy()
this_y[:, drop_this] = 1.0
tensor_model = dti.TensorModel(gtab, fit_method='restore',
sigma=sigma)
tensor_est = tensor_model.fit(Y)
assert_array_almost_equal(tensor_est.evals[0], evals)
assert_array_almost_equal(tensor_est.quadratic_form[0], tensor)
data, bvals, bvecs = get_data('small_25')
dd = nib.load(data).get_data()
gtab = grad.gradient_table(bvals, bvecs)
fit_method = 'restore' # 'NLLS'
jac = True # False
dd[..., 5] = 1.0
tm = dti.TensorModel(gtab, fit_method=fit_method, jac=True, sigma=10)
tm.fit(dd)
def test_restore():
"""
Test the implementation of the RESTORE algorithm
"""
b0 = 1000.
bvecs, bval = read_bvec_file(get_data('55dir_grad.bvec'))
gtab = grad.gradient_table(bval, bvecs)
B = bval[1]
#Scale the eigenvalues and tensor by the B value so the units match
D = np.array([1., 1., 1., 0., 0., 1., -np.log(b0) * B]) / B
evals = np.array([2., 1., 0.]) / B
md = evals.mean()
tensor = from_lower_triangular(D)
#Design Matrix
X = dti.design_matrix(bvecs, bval)
#Signals
Y = np.exp(np.dot(X,D))
Y.shape = (-1,) + Y.shape
for sigma in [0.1, 1, 10, 100]:
for drop_this in range(1, Y.shape[-1]):
# RESTORE estimates should be robust to dropping
this_y = Y.copy()
this_y[:, drop_this] = 1.0
tensor_model = dti.TensorModel(gtab, fit_method='restore',
sigma=sigma)
tensor_est = tensor_model.fit(Y)
assert_array_almost_equal(tensor_est.evals[0], evals)
assert_array_almost_equal(tensor_est.quadratic_form[0], tensor)
data, bvals, bvecs = get_data('small_25')
dd = nib.load(data).get_data()
gtab = grad.gradient_table(bvals, bvecs)
fit_method = 'restore' # 'NLLS'
jac = True # False
dd[..., 5] = 1.0
tm = dti.TensorModel(gtab, fit_method=fit_method, jac=True, sigma=10)
tm.fit(dd)
def fwdti_prediction(params, gtab, S0=1, Diso=3.0e-3):
r""" Signal prediction given the free water DTI model parameters.
Parameters
----------
params : (..., 13) ndarray
Model parameters. The last dimension should have the 12 tensor
parameters (3 eigenvalues, followed by the 3 corresponding
eigenvectors) and the volume fraction of the free water compartment.
gtab : a GradientTable class instance
The gradient table for this prediction
S0 : float or ndarray
The non diffusion-weighted signal in every voxel, or across all
voxels. Default: 1
Diso : float, optional
Value of the free water isotropic diffusion. Default is set to 3e-3
$mm^{2}.s^{-1}$. Please adjust this value if you are assuming different
units of diffusion.
Returns
--------
S : (..., N) ndarray
Simulated signal based on the free water DTI model
Notes
-----
The predicted signal is given by:
$S(\theta, b) = S_0 * [(1-f) * e^{-b ADC} + f * e^{-b D_{iso}]$, where
$ADC = \theta Q \theta^T$, $\theta$ is a unit vector pointing at any
direction on the sphere for which a signal is to be predicted, $b$ is the b
value provided in the GradientTable input for that direction, $Q$ is the
quadratic form of the tensor determined by the input parameters, $f$ is the
free water diffusion compartment, $D_{iso}$ is the free water diffusivity
which is equal to $3 * 10^{-3} mm^{2}s^{-1} [1]_.
References
----------
.. [1] Hoy, A.R., Koay, C.G., Kecskemeti, S.R., Alexander, A.L., 2014.
Optimization of a free water elimination two-compartmental model
for diffusion tensor imaging. NeuroImage 103, 323-333.
doi: 10.1016/j.neuroimage.2014.09.053
"""
evals = params[..., :3]
evecs = params[..., 3:-1].reshape(params.shape[:-1] + (3, 3))
f = params[..., 12]
qform = vec_val_vect(evecs, evals)
lower_dt = lower_triangular(qform, S0)
lower_diso = lower_dt.copy()
lower_diso[..., 0] = lower_diso[..., 2] = lower_diso[..., 5] = Diso
lower_diso[..., 1] = lower_diso[..., 3] = lower_diso[..., 4] = 0
D = design_matrix(gtab)
pred_sig = np.zeros(f.shape + (gtab.bvals.shape[0], ))
mask = _positive_evals(evals[..., 0], evals[..., 1], evals[..., 2])
index = ndindex(f.shape)
for v in index:
if mask[v]:
pred_sig[v] = (1 - f[v]) * np.exp(np.dot(lower_dt[v], D.T)) + \
f[v] * np.exp(np.dot(lower_diso[v], D.T))
return pred_sig
def test_TensorModel():
data, gtab = dsi_voxels()
dm = dti.TensorModel(gtab, 'LS')
dtifit = dm.fit(data[0, 0, 0])
assert_equal(dtifit.fa < 0.5, True)
dm = dti.TensorModel(gtab, 'WLS')
dtifit = dm.fit(data[0, 0, 0])
assert_equal(dtifit.fa < 0.5, True)
sphere = create_unit_sphere(4)
assert_equal(len(dtifit.odf(sphere)), len(sphere.vertices))
assert_almost_equal(dtifit.fa, gfa(dtifit.odf(sphere)), 1)
# Check that the multivoxel case works:
dtifit = dm.fit(data)
# And smoke-test that all these operations return sensibly-shaped arrays:
assert_equal(dtifit.fa.shape, data.shape[:3])
assert_equal(dtifit.ad.shape, data.shape[:3])
assert_equal(dtifit.md.shape, data.shape[:3])
assert_equal(dtifit.rd.shape, data.shape[:3])
assert_equal(dtifit.trace.shape, data.shape[:3])
assert_equal(dtifit.mode.shape, data.shape[:3])
assert_equal(dtifit.linearity.shape, data.shape[:3])
assert_equal(dtifit.planarity.shape, data.shape[:3])
assert_equal(dtifit.sphericity.shape, data.shape[:3])
# Test for the shape of the mask
assert_raises(ValueError, dm.fit, np.ones((10, 10, 3)), np.ones((3,3)))
# Make some synthetic data
b0 = 1000.
bvecs, bvals = read_bvec_file(get_data('55dir_grad.bvec'))
gtab = grad.gradient_table_from_bvals_bvecs(bvals, bvecs.T)
# The first b value is 0., so we take the second one:
B = bvals[1]
# Scale the eigenvalues and tensor by the B value so the units match
D = np.array([1., 1., 1., 0., 0., 1., -np.log(b0) * B]) / B
evals = np.array([2., 1., 0.]) / B
md = evals.mean()
tensor = from_lower_triangular(D)
A_squiggle = tensor - (1 / 3.0) * np.trace(tensor) * np.eye(3)
mode = 3 * np.sqrt(6) * np.linalg.det(A_squiggle / np.linalg.norm(A_squiggle))
evecs = np.linalg.eigh(tensor)[1]
# Design Matrix
X = dti.design_matrix(gtab)
# Signals
Y = np.exp(np.dot(X, D))
assert_almost_equal(Y[0], b0)
Y.shape = (-1,) + Y.shape
# Test fitting with different methods:
for fit_method in ['OLS', 'WLS', 'NLLS']:
tensor_model = dti.TensorModel(gtab,
fit_method=fit_method)
tensor_fit = tensor_model.fit(Y)
assert_true(tensor_fit.model is tensor_model)
assert_equal(tensor_fit.shape, Y.shape[:-1])
assert_array_almost_equal(tensor_fit.evals[0], evals)
assert_array_almost_equal(tensor_fit.quadratic_form[0], tensor,
err_msg=\
"Calculation of tensor from Y does not compare to analytical solution")
assert_almost_equal(tensor_fit.md[0], md)
assert_array_almost_equal(tensor_fit.mode, mode, decimal=5)
assert_equal(tensor_fit.directions.shape[-2], 1)
assert_equal(tensor_fit.directions.shape[-1], 3)
# Test error-handling:
assert_raises(ValueError,
dti.TensorModel,
gtab,
fit_method='crazy_method')
def test_TensorModel():
data, gtab = dsi_voxels()
dm = dti.TensorModel(gtab, 'LS')
dtifit = dm.fit(data[0, 0, 0])
assert_equal(dtifit.fa < 0.5, True)
dm = dti.TensorModel(gtab, 'WLS')
dtifit = dm.fit(data[0, 0, 0])
assert_equal(dtifit.fa < 0.5, True)
sphere = create_unit_sphere(4)
assert_equal(len(dtifit.odf(sphere)), len(sphere.vertices))
assert_almost_equal(dtifit.fa, gfa(dtifit.odf(sphere)), 1)
# Check that the multivoxel case works:
dtifit = dm.fit(data)
# And smoke-test that all these operations return sensibly-shaped arrays:
assert_equal(dtifit.fa.shape, data.shape[:3])
assert_equal(dtifit.ad.shape, data.shape[:3])
assert_equal(dtifit.md.shape, data.shape[:3])
assert_equal(dtifit.rd.shape, data.shape[:3])
assert_equal(dtifit.trace.shape, data.shape[:3])
assert_equal(dtifit.mode.shape, data.shape[:3])
assert_equal(dtifit.linearity.shape, data.shape[:3])
assert_equal(dtifit.planarity.shape, data.shape[:3])
assert_equal(dtifit.sphericity.shape, data.shape[:3])
# Test for the shape of the mask
assert_raises(ValueError, dm.fit, np.ones((10, 10, 3)), np.ones((3, 3)))
# Make some synthetic data
b0 = 1000.
bvecs, bvals = read_bvec_file(get_data('55dir_grad.bvec'))
gtab = grad.gradient_table_from_bvals_bvecs(bvals, bvecs.T)
# The first b value is 0., so we take the second one:
B = bvals[1]
# Scale the eigenvalues and tensor by the B value so the units match
D = np.array([1., 1., 1., 0., 0., 1., -np.log(b0) * B]) / B
evals = np.array([2., 1., 0.]) / B
md = evals.mean()
tensor = from_lower_triangular(D)
A_squiggle = tensor - (1 / 3.0) * np.trace(tensor) * np.eye(3)
mode = 3 * np.sqrt(6) * np.linalg.det(
A_squiggle / np.linalg.norm(A_squiggle))
evecs = np.linalg.eigh(tensor)[1]
# Design Matrix
X = dti.design_matrix(gtab)
# Signals
Y = np.exp(np.dot(X, D))
assert_almost_equal(Y[0], b0)
Y.shape = (-1, ) + Y.shape
# Test fitting with different methods:
for fit_method in ['OLS', 'WLS', 'NLLS']:
tensor_model = dti.TensorModel(gtab, fit_method=fit_method)
tensor_fit = tensor_model.fit(Y)
assert_true(tensor_fit.model is tensor_model)
assert_equal(tensor_fit.shape, Y.shape[:-1])
assert_array_almost_equal(tensor_fit.evals[0], evals)
assert_array_almost_equal(tensor_fit.quadratic_form[0], tensor,
err_msg=\
"Calculation of tensor from Y does not compare to analytical solution")
assert_almost_equal(tensor_fit.md[0], md)
assert_array_almost_equal(tensor_fit.mode, mode, decimal=5)
assert_equal(tensor_fit.directions.shape[-2], 1)
assert_equal(tensor_fit.directions.shape[-1], 3)
# Test error-handling:
assert_raises(ValueError, dti.TensorModel, gtab, fit_method='crazy_method')
def nls_fit_tensor(gtab, data, mask=None, Diso=3e-3,
min_signal=1.0e-6, f_transform=True, cholesky=False,
jac=True):
"""
Fit the water elimination tensor model using the non-linear least-squares.
Parameters
----------
gtab : a GradientTable class instance
The gradient table containing diffusion acquisition parameters.
data : ndarray ([X, Y, Z, ...], g)
Data or response variables holding the data. Note that the last
dimension should contain the data. It makes no copies of data.
mask : array, optional
A boolean array used to mark the coordinates in the data that should
be analyzed that has the shape data.shape[:-1]
Diso : float, optional
Value of the free water isotropic diffusion. Default is set to 3e-3
$mm^{2}.s^{-1}$. Please ajust this value if you are assuming different
units of diffusion.
min_signal : float
The minimum signal value. Needs to be a strictly positive
number. Default: 1.0e-6.
f_transform : bool, optional
If true, the water volume fractions is converted during the convergence
procedure to ft = arcsin(2*f - 1) + pi/2, insuring f estimates between
0 and 1.
Default: True
cholesky : bool, optional
If true it uses cholesky decomposition to insure that diffusion tensor
is positive define.
Default: False
jac : bool
Use the Jacobian? Default: False
Returns
-------
fw_params : ndarray (x, y, z, 13)
Matrix containing in the last dimension the free water model parameters
in the following order:
1) Three diffusion tensor's eigenvalues
2) Three lines of the eigenvector matrix each containing the
first, second and third coordinates of the eigenvector
3) The volume fraction of the free water compartment
References
----------
.. [1] Henriques, R.N., Rokem, A., Garyfallidis, E., St-Jean, S., Peterson,
E.T., Correia, M.M., 2017. Re: Optimization of a free water
elimination two-compartmental model for diffusion tensor imaging.
ReScience
"""
# Analyse compatible input cases
if jac is True and cholesky is True:
raise ValueError("Cholesky decomposition is not compatible with jac.")
fw_params = np.zeros(data.shape[:-1] + (13,))
W = design_matrix(gtab)
# Prepare mask
if mask is None:
mask = np.ones(data.shape[:-1], dtype=bool)
else:
if mask.shape != data.shape[:-1]:
raise ValueError("Mask is not the same shape as data.")
mask = np.array(mask, dtype=bool, copy=False)
# Prepare S0
S0 = np.mean(data[..., gtab.b0s_mask], axis=-1)
# Loop data fitting through all voxels
index = ndindex(mask.shape)
for v in index:
if mask[v]:
params = nls_iter(W, data[v], S0[v], Diso=Diso,
min_signal=min_signal,
f_transform=f_transform,
cholesky=cholesky, jac=jac)
fw_params[v] = params
return fw_params
def test_nlls_fit_tensor():
"""
Test the implementation of NLLS and RESTORE
"""
b0 = 1000.
bval, bvecs = read_bvals_bvecs(*get_fnames('55dir_grad'))
gtab = grad.gradient_table(bval, bvecs)
B = bval[1]
# Scale the eigenvalues and tensor by the B value so the units match
D = np.array([1., 1., 1., 0., 0., 1., -np.log(b0) * B]) / B
evals = np.array([2., 1., 0.]) / B
md = evals.mean()
tensor = from_lower_triangular(D)
# Design Matrix
X = dti.design_matrix(gtab)
# Signals
Y = np.exp(np.dot(X, D))
Y.shape = (-1, ) + Y.shape
# Estimate tensor from test signals and compare against expected result
# using non-linear least squares:
tensor_model = dti.TensorModel(gtab, fit_method='NLLS')
tensor_est = tensor_model.fit(Y)
npt.assert_equal(tensor_est.shape, Y.shape[:-1])
npt.assert_array_almost_equal(tensor_est.evals[0], evals)
npt.assert_array_almost_equal(tensor_est.quadratic_form[0], tensor)
npt.assert_almost_equal(tensor_est.md[0], md)
# You can also do this without the Jacobian (though it's slower):
tensor_model = dti.TensorModel(gtab, fit_method='NLLS', jac=False)
tensor_est = tensor_model.fit(Y)
npt.assert_equal(tensor_est.shape, Y.shape[:-1])
npt.assert_array_almost_equal(tensor_est.evals[0], evals)
npt.assert_array_almost_equal(tensor_est.quadratic_form[0], tensor)
npt.assert_almost_equal(tensor_est.md[0], md)
# Using the gmm weighting scheme:
tensor_model = dti.TensorModel(gtab, fit_method='NLLS', weighting='gmm')
tensor_est = tensor_model.fit(Y)
npt.assert_equal(tensor_est.shape, Y.shape[:-1])
npt.assert_array_almost_equal(tensor_est.evals[0], evals)
npt.assert_array_almost_equal(tensor_est.quadratic_form[0], tensor)
npt.assert_almost_equal(tensor_est.md[0], md)
# If you use sigma weighting, you'd better provide a sigma:
tensor_model = dti.TensorModel(gtab, fit_method='NLLS', weighting='sigma')
npt.assert_raises(ValueError, tensor_model.fit, Y)
# Use NLLS with some actual 4D data:
data, bvals, bvecs = get_fnames('small_25')
gtab = grad.gradient_table(bvals, bvecs)
tm1 = dti.TensorModel(gtab, fit_method='NLLS')
dd = load_nifti_data(data)
tf1 = tm1.fit(dd)
tm2 = dti.TensorModel(gtab)
tf2 = tm2.fit(dd)
npt.assert_array_almost_equal(tf1.fa, tf2.fa, decimal=1)
def nls_fit_tensor_bounds(gtab, data, mask=None, Diso=3e-3, mdreg=2.7e-3,
min_signal=1.0e-6, bounds=None, jac=True):
"""
Fit the water elimination tensor model using the non-linear least-squares
with constraints
gtab : a GradientTable class instance
The gradient table containing diffusion acquisition parameters.
data : ndarray ([X, Y, Z, ...], g)
Data or response variables holding the data. Note that the last
dimension should contain the data. It makes no copies of data.
mask : array, optional
A boolean array used to mark the coordinates in the data that should
be analyzed that has the shape data.shape[:-1]
Diso : float, optional
Value of the free water isotropic diffusion. Default is set to 3e-3
$mm^{2}.s^{-1}$. Please ajust this value if you are assuming different
units of diffusion.
mdreg : float, optimal
DTI's mean diffusivity regularization threshold. If standard DTI
diffusion tensor's mean diffusivity is almost near the free water
diffusion value, the diffusion signal is assumed to be only free water
diffusion (i.e. volume fraction will be set to 1 and tissue's diffusion
parameters are set to zero). Default md_reg is 2.7e-3 $mm^{2}.s^{-1}$
(corresponding to 90% of the free water diffusion value).
min_signal : float
The minimum signal value. Needs to be a strictly positive
number. Default: 1.0e-6.
bounds : 2-tuple of arrays with 14 elements, optional
Lower and upper bounds on fwdti model variables and the log of
non-diffusion signal S0. Use np.inf with an appropriate sign to
disable bounds on all or some variables. When bounds is set to None
the following default variable bounds is used:
([0., -Diso, 0., -Diso, -Diso, 0., 0., np.exp(-10.)],
[Diso, Diso, Diso, Diso, Diso, Diso, 1., np.exp(10.)])
jac : bool
Use the Jacobian? Default: False
Returns
-------
fw_params : ndarray (x, y, z, 13)
Matrix containing in the dimention the free water model parameters in
the following order:
1) Three diffusion tensor's eigenvalues
2) Three lines of the eigenvector matrix each containing the
first, second and third coordinates of the eigenvector
3) The volume fraction of the free water compartment
"""
fw_params = np.zeros(data.shape[:-1] + (13,))
W = design_matrix(gtab)
# Prepare mask
if mask is None:
mask = np.ones(data.shape[:-1], dtype=bool)
else:
if mask.shape != data.shape[:-1]:
raise ValueError("Mask is not the same shape as data.")
mask = np.array(mask, dtype=bool, copy=False)
# Prepare S0
S0 = np.mean(data[:, :, :, gtab.b0s_mask], axis=-1)
index = ndindex(mask.shape)
for v in index:
if mask[v]:
params = nls_iter_bounds(W, data[v], S0[v], Diso=Diso, mdreg=mdreg,
min_signal=min_signal, bounds=bounds,
jac=jac)
fw_params[v] = params
return fw_params
def wls_fit_tensor(gtab, data, Diso=3e-3, mask=None, min_signal=1.0e-6,
piterations=3, mdreg=2.7e-3):
r""" Computes weighted least squares (WLS) fit to calculate self-diffusion
tensor using a linear regression model [1]_.
Parameters
----------
gtab : a GradientTable class instance
The gradient table containing diffusion acquisition parameters.
data : ndarray ([X, Y, Z, ...], g)
Data or response variables holding the data. Note that the last
dimension should contain the data. It makes no copies of data.
Diso : float, optional
Value of the free water isotropic diffusion. Default is set to 3e-3
$mm^{2}.s^{-1}$. Please ajust this value if you are assuming different
units of diffusion.
mask : array, optional
A boolean array used to mark the coordinates in the data that should
be analyzed that has the shape data.shape[:-1]
min_signal : float
The minimum signal value. Needs to be a strictly positive
number. Default: 1.0e-6.
piterations : inter, optional
Number of iterations used to refine the precision of f. Default is set
to 3 corresponding to a precision of 0.01.
mdreg : float, optimal
DTI's mean diffusivity regularization threshold. If standard DTI
diffusion tensor's mean diffusivity is almost near the free water
diffusion value, the diffusion signal is assumed to be only free water
diffusion (i.e. volume fraction will be set to 1 and tissue's diffusion
parameters are set to zero). Default md_reg is 2.7e-3 $mm^{2}.s^{-1}$
(corresponding to 90% of the free water diffusion value).
Returns
-------
fw_params : ndarray (x, y, z, 13)
Matrix containing in the last dimention the free water model parameters
in the following order:
1) Three diffusion tensor's eigenvalues
2) Three lines of the eigenvector matrix each containing the
first, second and third coordinates of the eigenvector
3) The volume fraction of the free water compartment.
References
----------
.. [1] Hoy, A.R., Koay, C.G., Kecskemeti, S.R., Alexander, A.L., 2014.
Optimization of a free water elimination two-compartmental model
for diffusion tensor imaging. NeuroImage 103, 323-333.
doi: 10.1016/j.neuroimage.2014.09.053
"""
fw_params = np.zeros(data.shape[:-1] + (13,))
W = design_matrix(gtab)
# Prepare mask
if mask is None:
mask = np.ones(data.shape[:-1], dtype=bool)
else:
if mask.shape != data.shape[:-1]:
raise ValueError("Mask is not the same shape as data.")
mask = np.array(mask, dtype=bool, copy=False)
# Prepare S0
S0 = np.mean(data[:, :, :, gtab.b0s_mask], axis=-1)
index = ndindex(mask.shape)
for v in index:
if mask[v]:
params = wls_iter(W, data[v], S0[v], min_signal=min_signal,
Diso=3e-3, piterations=piterations, mdreg=mdreg)
fw_params[v] = params
return fw_params
def test_tensor_model():
fdata, fbval, fbvec = get_data('small_25')
data = nib.load(fdata).get_data()
gtab = grad.gradient_table(fbval, fbvec)
dm = dti.TensorModel(gtab, 'LS')
dtifit = dm.fit(data[0, 0, 0])
assert_equal(dtifit.fa < 0.9, True)
dm = dti.TensorModel(gtab, 'WLS')
dtifit = dm.fit(data[0, 0, 0])
assert_equal(dtifit.fa < 0.9, True)
assert_equal(dtifit.fa > 0, True)
sphere = create_unit_sphere(4)
assert_equal(len(dtifit.odf(sphere)), len(sphere.vertices))
# Check that the multivoxel case works:
dtifit = dm.fit(data)
# Check that it works on signal that has already been normalized to S0:
dm_to_relative = dti.TensorModel(gtab)
relative_data = (data[0, 0, 0]/np.mean(data[0, 0, 0, gtab.b0s_mask]))
dtifit_to_relative = dm_to_relative.fit(relative_data)
npt.assert_almost_equal(dtifit.fa[0,0,0], dtifit_to_relative.fa, decimal=3)
# And smoke-test that all these operations return sensibly-shaped arrays:
assert_equal(dtifit.fa.shape, data.shape[:3])
assert_equal(dtifit.ad.shape, data.shape[:3])
assert_equal(dtifit.md.shape, data.shape[:3])
assert_equal(dtifit.rd.shape, data.shape[:3])
assert_equal(dtifit.trace.shape, data.shape[:3])
assert_equal(dtifit.mode.shape, data.shape[:3])
assert_equal(dtifit.linearity.shape, data.shape[:3])
assert_equal(dtifit.planarity.shape, data.shape[:3])
assert_equal(dtifit.sphericity.shape, data.shape[:3])
# Test for the shape of the mask
assert_raises(ValueError, dm.fit, np.ones((10, 10, 3)), np.ones((3,3)))
# Make some synthetic data
b0 = 1000.
bvecs, bvals = read_bvec_file(get_data('55dir_grad.bvec'))
gtab = grad.gradient_table_from_bvals_bvecs(bvals, bvecs.T)
# The first b value is 0., so we take the second one:
B = bvals[1]
# Scale the eigenvalues and tensor by the B value so the units match
D = np.array([1., 1., 1., 0., 0., 1., -np.log(b0) * B]) / B
evals = np.array([2., 1., 0.]) / B
md = evals.mean()
tensor = from_lower_triangular(D)
A_squiggle = tensor - (1 / 3.0) * np.trace(tensor) * np.eye(3)
mode = 3 * np.sqrt(6) * np.linalg.det(A_squiggle / np.linalg.norm(A_squiggle))
evecs = np.linalg.eigh(tensor)[1]
# Design Matrix
X = dti.design_matrix(gtab)
# Signals
Y = np.exp(np.dot(X, D))
assert_almost_equal(Y[0], b0)
Y.shape = (-1,) + Y.shape
# Test fitting with different methods:
for fit_method in ['OLS', 'WLS', 'NLLS']:
tensor_model = dti.TensorModel(gtab,
fit_method=fit_method)
tensor_fit = tensor_model.fit(Y)
assert_true(tensor_fit.model is tensor_model)
assert_equal(tensor_fit.shape, Y.shape[:-1])
assert_array_almost_equal(tensor_fit.evals[0], evals)
assert_array_almost_equal(tensor_fit.quadratic_form[0], tensor,
err_msg=\
"Calculation of tensor from Y does not compare to analytical solution")
assert_almost_equal(tensor_fit.md[0], md)
assert_array_almost_equal(tensor_fit.mode, mode, decimal=5)
assert_equal(tensor_fit.directions.shape[-2], 1)
assert_equal(tensor_fit.directions.shape[-1], 3)
# Test error-handling:
assert_raises(ValueError,
dti.TensorModel,
gtab,
fit_method='crazy_method')
# Test multi-voxel data
data = np.zeros((3, Y.shape[1]))
# Normal voxel
data[0] = Y
# High diffusion voxel, all diffusing weighted signal equal to zero
data[1, gtab.b0s_mask] = b0
data[1, ~gtab.b0s_mask] = 0
# Masked voxel, all data set to zero
data[2] = 0.
tensor_model = dti.TensorModel(gtab)
fit = tensor_model.fit(data)
assert_array_almost_equal(fit[0].evals, evals)
# Evals should be high for high diffusion voxel
assert_(all(fit[1].evals > evals[0] * .9))
# Evals should be zero where data is masked
assert_array_almost_equal(fit[2].evals, 0.)
def nls_fit_tensor_bounds(gtab, data, mask=None, Diso=3e-3,
min_signal=1.0e-6, bounds=None, jac=True):
"""
Fit the water elimination tensor model using the non-linear least-squares
with constraints
gtab : a GradientTable class instance
The gradient table containing diffusion acquisition parameters.
data : ndarray ([X, Y, Z, ...], g)
Data or response variables holding the data. Note that the last
dimension should contain the data. It makes no copies of data.
mask : array, optional
A boolean array used to mark the coordinates in the data that should
be analyzed that has the shape data.shape[:-1]
Diso : float, optional
Value of the free water isotropic diffusion. Default is set to 3e-3
$mm^{2}.s^{-1}$. Please ajust this value if you are assuming different
units of diffusion.
min_signal : float
The minimum signal value. Needs to be a strictly positive
number. Default: 1.0e-6.
bounds : 2-tuple of arrays with 14 elements, optional
Lower and upper bounds on fwdti model variables and the log of
non-diffusion signal S0. Use np.inf with an appropriate sign to
disable bounds on all or some variables. When bounds is set to None
the following default variable bounds is used:
([0., -Diso, 0., -Diso, -Diso, 0., 0., np.exp(-10.)],
[Diso, Diso, Diso, Diso, Diso, Diso, 1., np.exp(10.)])
jac : bool
Use the Jacobian? Default: False
Returns
-------
fw_params : ndarray (x, y, z, 13)
Matrix containing in the last dimension the free water model parameters
in the following order:
1) Three diffusion tensor's eigenvalues
2) Three lines of the eigenvector matrix each containing the
first, second and third coordinates of the eigenvector
3) The volume fraction of the free water compartment
References
----------
.. [1] Henriques, R.N., Rokem, A., Garyfallidis, E., St-Jean, S., Peterson,
E.T., Correia, M.M., 2017. Re: Optimization of a free water
elimination two-compartmental model for diffusion tensor imaging.
ReScience
"""
fw_params = np.zeros(data.shape[:-1] + (13,))
W = design_matrix(gtab)
# Prepare mask
if mask is None:
mask = np.ones(data.shape[:-1], dtype=bool)
else:
if mask.shape != data.shape[:-1]:
raise ValueError("Mask is not the same shape as data.")
mask = np.array(mask, dtype=bool, copy=False)
# Prepare S0
S0 = np.mean(data[:, :, :, gtab.b0s_mask], axis=-1)
index = ndindex(mask.shape)
for v in index:
if mask[v]:
params = nls_iter_bounds(W, data[v], S0[v], Diso=Diso,
min_signal=min_signal, bounds=bounds,
jac=jac)
fw_params[v] = params
return fw_params
def gradient_descent(dwis, bvals, bvecs, mask, init_f, init_tensor, niters=50):
'''
Optimize the volume fraction and the tensor via gradient descent.
'''
dim_x, dim_y, dim_z = mask.shape
indices_dwis = (bvals > 0)
nb_dwis = np.count_nonzero(indices_dwis)
indices_b0 = (bvals == 0)
nb_b0 = np.count_nonzero(indices_b0)
b = bvals.max()
b0 = dwis[..., indices_b0].mean(-1)
signal = dwis[..., indices_dwis] / b0[..., None]
np.clip(signal, 1.0e-6, 1 - 1.0e-6, signal)
bvals = bvals[indices_dwis]
bvecs = bvecs[indices_dwis]
gtab = gradient_table_from_bvals_bvecs(bvals, bvecs)
H = dti.design_matrix(gtab)[:, :6]
signal[np.logical_not(mask)] = 0.
signal = signal[mask]
lower_triangular = init_tensor[mask, 0]
volume_fraction = init_f[mask]
print(" - Begin gradient descent.")
mask_nvoxels = np.count_nonzero(mask)
step_size = 1.0e-7
weight = 100.0
l_min_loop, l_max_loop = 0.1e-3, 2.5e-3
for i in range(niters):
print(" - Iteration {0:d} out of {1:d}.".format(i + 1, niters))
grad1, predicted_signal_tissue, predicted_signal_water = \
grad_data_fit_tensor(lower_triangular, signal, H, bvals,
volume_fraction)
print("\tgrad1 avg: {0:0.4e}".format(np.mean(np.abs(grad1))))
predicted_signal = volume_fraction[..., None] * predicted_signal_tissue + \
(1-volume_fraction[..., None]) * predicted_signal_water
prediction_error = np.sqrt(((predicted_signal - signal)**2).mean(-1))
print("\tpref error avg: {0:0.4e}".format(np.mean(prediction_error)))
gradf = (bvals * (predicted_signal - signal) \
* (predicted_signal_tissue - predicted_signal_water)).sum(-1)
print("\tgradf avg: {0:0.4e}".format(np.mean(np.abs(gradf))))
volume_fraction -= weight * step_size * gradf
grad1[np.isnan(grad1)] = 0
# np.clip(grad1, -1.e5, 1.e5, grad1)
np.clip(volume_fraction, 0.01, 0.99, volume_fraction)
lower_triangular -= step_size * grad1
lower_triangular[np.isnan(lower_triangular)] = 0
dti_params = dti.eig_from_lo_tri(lower_triangular).reshape(
(mask_nvoxels, 4, 3))
evals = dti_params[..., 0, :]
evecs = dti_params[..., 1:, :]
lower_triangular = clip_tensor_evals(evals, evecs, l_min_loop,
l_max_loop)
del dti_params, evals, evecs
final_tensor = np.zeros((dim_x, dim_y, dim_z, 1, 6), dtype=np.float32)
final_tensor[mask, 0] = lower_triangular
final_f = np.zeros((dim_x, dim_y, dim_z), dtype=np.float32)
final_f[mask] = 1 - volume_fraction
return final_f, final_tensor
data_fill = sdb.substitute(data_neg, min_signal)
data_in_mask = sdb.filter(data_fill, data_fill.attribute[0]>0)
print "Truncation done: ", str(time.ctime())
print data_in_mask.shape
"""
#TODO: params_in_mask = self.fit_method(self.design_matrix, data_in_mask, *self.args, **self.kwargs), where we need to re-implement
# def wls_fit_tensor(design_matrix, data). This is a fancy math function including an Einstein Summation...
data_in_mask = data_sdb.reshape((-1, data_sdb.shape[-1]))
data_in_mask = convert_array_type(sdb, data_in_mask, "double")
# Create design matrix B
design_matrix = sdb.from_array(dti.design_matrix(gtab))
# wls fit
# apparantly one has to run the svd three times to get all three values...
u = robust.gesvd(design_matrix, "'U'")
ols_fit = sdb.dot(u, u.T)
log_s = sdb.log(data_in_mask)
# The einsum: w = np.exp(np.einsum('...ij,...j', ols_fit, log_s))
w = sdb.exp(sdb.dot(ols_fit, log_s.T).T)
# p = sdb.from_array(dipys_3d_pinv(sdb, sdb.dstack([design_matrix[:, i] * w for i in range(design_matrix.shape[1])])))
def initial_fit(dwis,
bvals,
bvecs,
mask,
wm_roi,
csf_roi,
MD,
csf_percentile=95,
wm_percentile=5,
lmin=0.1e-3,
lmax=2.5e-3,
evals_lmin=0.1e-3,
evals_lmax=2.5e-3,
md_value=0.6e-3,
interpolate=True,
fixed_MD=False):
'''
Produce the initial estimate of the volume fraction and the initial tensor image
'''
print(" - Compute baseline image and DW attenuation.")
dim_x, dim_y, dim_z = mask.shape
indices_dwis = (bvals > 0)
nb_dwis = np.count_nonzero(indices_dwis)
indices_b0 = (bvals == 0)
nb_b0 = np.count_nonzero(indices_b0)
# TO DO : address this line for multi-shell dwi
b = bvals.max()
b0 = dwis[..., indices_b0].mean(-1)
# signal attenuation
signal = dwis[..., indices_dwis] / b0[..., None]
np.clip(signal, 1.0e-6, 1 - 1.0e-6, signal)
signal[np.logical_not(mask)] = 0.
# tissue b0 references
csf_b0 = np.percentile(b0[csf_roi], csf_percentile)
print("\t{0:2d}th percentile of b0 signal in CSF: {1}.".format(
csf_percentile, csf_b0))
wm_b0 = np.percentile(b0[wm_roi], wm_percentile)
print("\t{0:2d}th percentile of b0 signal in WM : {1}.".format(
wm_percentile, wm_b0))
print(" - Compute initial volume fraction ...")
# Eq. 7 from Pasternak 2009 MRM
epsi = 1e-12 # only used to prevent log(0)
init_f = 1 - np.log(b0 / wm_b0 + epsi) / np.log(csf_b0 / wm_b0)
np.clip(init_f, 0.0, 1.0, init_f)
alpha = init_f.copy() # exponent for interpolation
print(" - Compute fixed MD VF map")
init_f_MD = (np.exp(-b * MD) - np.exp(-b * d)) / (np.exp(-b * md_value) -
np.exp(-b * d))
np.clip(init_f_MD, 0.01, 0.99, init_f_MD)
print(" - Compute min_f and max_f from lmin, lmax")
### This was following Pasternak 2009 although with an error
### Amin = exp(-b*lmax) and Amax = exp(-b*lmin) in that paper
# min_f = (signal.min(-1)-np.exp(-b*d)) / (np.exp(-b*lmin)-np.exp(-b*d))
# max_f = (signal.max(-1)-np.exp(-b*d)) / (np.exp(-b*lmax)-np.exp(-b*d))
### From Pasternak 2009 method, Amin < At implies that the
### term with signal.min(-1) in numerator is the upper bound of f
### although in that paper the equation 6 has fmin and fmax reversed.
### With lmin, lmax=0.1e-3, 2.5e-3, Amin = 0.08, Awater = 0.04
### and one can see that max_f here will usually be >> 1
min_f = (signal.max(-1) - np.exp(-b * d)) / (np.exp(-b * lmin) -
np.exp(-b * d))
max_f = (signal.min(-1) - np.exp(-b * d)) / (np.exp(-b * lmax) -
np.exp(-b * d))
# If MD of a voxel is > 3.0e-3, min_f and max_f can be negative.
# These voxels should be initialized as 0
np.clip(min_f, 0.0, 1.0, min_f)
np.clip(max_f, 0.0, 1.0, max_f)
np.clip(init_f, min_f, max_f, init_f)
if interpolate:
print(" - Interpolate two estimates of volume fraction")
# f = tissue fraction. with init_f high, alpha will be ~1 and init_f_MD will be weighted
init_f = (np.power(init_f, (1 - alpha))) * (np.power(init_f_MD, alpha))
elif fixed_MD:
print(
" - Using fixed MD value of {0} for inital volume fraction".format(
md_value))
init_f = init_f_MD
else:
print(" - Using lmin and lmax for initial volume fraction")
np.clip(init_f, 0.05, 0.99, init_f) # want minimum 5% of tissue
init_f[np.isnan(init_f)] = 0.5
init_f[np.logical_not(mask)] = 0.5
print(" - Compute initial tissue tensor ...")
signal[np.isnan(signal)] = 0
bvecs = bvecs[indices_dwis]
bvals = bvals[indices_dwis]
signal_free_water = np.exp(-bvals * d)
corrected_signal = (signal - (1 - init_f[..., np.newaxis]) \
* signal_free_water[np.newaxis, np.newaxis, np.newaxis, :]) \
/ (init_f[..., np.newaxis])
np.clip(corrected_signal, 1.0e-3, 1. - 1.0e-3, corrected_signal)
log_signal = np.log(corrected_signal)
gtab = gradient_table_from_bvals_bvecs(bvals, bvecs)
H = dti.design_matrix(gtab)[:, :6]
pseudo_inv = np.dot(np.linalg.inv(np.dot(H.T, H)), H.T)
init_tensor = np.dot(log_signal, pseudo_inv.T)
dti_params = dti.eig_from_lo_tri(init_tensor).reshape(
(dim_x, dim_y, dim_z, 4, 3))
evals = dti_params[..., 0, :]
evecs = dti_params[..., 1:, :]
if evals_lmin > 0.1e-3:
print(" - Fatten tensor to {}".format(evals_lmin))
lower_triangular = clip_tensor_evals(evals, evecs, evals_lmin, evals_lmax)
lower_triangular[np.logical_not(mask)] = [
evals_lmin, 0, evals_lmin, 0, 0, evals_lmin
]
nan_mask = np.any(np.isnan(lower_triangular), axis=-1)
lower_triangular[nan_mask] = [evals_lmin, 0, evals_lmin, 0, 0, evals_lmin]
init_tensor = lower_triangular[:, :, :, np.newaxis, :]
return init_f, init_tensor
def test_wls_and_ls_fit():
"""
Tests the WLS and LS fitting functions to see if they returns the correct
eigenvalues and eigenvectors.
Uses data/55dir_grad.bvec as the gradient table and 3by3by56.nii
as the data.
"""
# Defining Test Voxel (avoid nibabel dependency) ###
# Recall: D = [Dxx,Dyy,Dzz,Dxy,Dxz,Dyz,log(S_0)] and D ~ 10^-4 mm^2 /s
b0 = 1000.
bvec, bval = read_bvec_file(get_fnames('55dir_grad.bvec'))
B = bval[1]
# Scale the eigenvalues and tensor by the B value so the units match
D = np.array([1., 1., 1., 0., 0., 1., -np.log(b0) * B]) / B
evals = np.array([2., 1., 0.]) / B
md = evals.mean()
tensor = from_lower_triangular(D)
# Design Matrix
gtab = grad.gradient_table(bval, bvec)
X = dti.design_matrix(gtab)
# Signals
Y = np.exp(np.dot(X, D))
npt.assert_almost_equal(Y[0], b0)
Y.shape = (-1,) + Y.shape
# Testing WLS Fit on Single Voxel
# If you do something wonky (passing min_signal<0), you should get an
# error:
npt.assert_raises(ValueError, TensorModel, gtab, fit_method='WLS',
min_signal=-1)
# Estimate tensor from test signals
model = TensorModel(gtab, fit_method='WLS', return_S0_hat=True)
tensor_est = model.fit(Y)
npt.assert_equal(tensor_est.shape, Y.shape[:-1])
npt.assert_array_almost_equal(tensor_est.evals[0], evals)
npt.assert_array_almost_equal(tensor_est.quadratic_form[0], tensor,
err_msg="Calculation of tensor from Y does "
"not compare to analytical solution")
npt.assert_almost_equal(tensor_est.md[0], md)
npt.assert_array_almost_equal(tensor_est.S0_hat[0], b0, decimal=3)
# Test that we can fit a single voxel's worth of data (a 1d array)
y = Y[0]
tensor_est = model.fit(y)
npt.assert_equal(tensor_est.shape, tuple())
npt.assert_array_almost_equal(tensor_est.evals, evals)
npt.assert_array_almost_equal(tensor_est.quadratic_form, tensor)
npt.assert_almost_equal(tensor_est.md, md)
npt.assert_array_almost_equal(tensor_est.lower_triangular(b0), D)
# Test using fit_method='LS'
model = TensorModel(gtab, fit_method='LS')
tensor_est = model.fit(y)
npt.assert_equal(tensor_est.shape, tuple())
npt.assert_array_almost_equal(tensor_est.evals, evals)
npt.assert_array_almost_equal(tensor_est.quadratic_form, tensor)
npt.assert_almost_equal(tensor_est.md, md)
npt.assert_array_almost_equal(tensor_est.lower_triangular(b0), D)
npt.assert_array_almost_equal(tensor_est.linearity, linearity(evals))
npt.assert_array_almost_equal(tensor_est.planarity, planarity(evals))
npt.assert_array_almost_equal(tensor_est.sphericity, sphericity(evals))
def test_wls_and_ls_fit():
"""
Tests the WLS and LS fitting functions to see if they returns the correct
eigenvalues and eigenvectors.
Uses data/55dir_grad.bvec as the gradient table and 3by3by56.nii
as the data.
"""
# Defining Test Voxel (avoid nibabel dependency) ###
# Recall: D = [Dxx,Dyy,Dzz,Dxy,Dxz,Dyz,log(S_0)] and D ~ 10^-4 mm^2 /s
b0 = 1000.
bvec, bval = read_bvec_file(get_data('55dir_grad.bvec'))
B = bval[1]
# Scale the eigenvalues and tensor by the B value so the units match
D = np.array([1., 1., 1., 0., 0., 1., -np.log(b0) * B]) / B
evals = np.array([2., 1., 0.]) / B
md = evals.mean()
tensor = from_lower_triangular(D)
# Design Matrix
gtab = grad.gradient_table(bval, bvec)
X = dti.design_matrix(gtab)
# Signals
Y = np.exp(np.dot(X, D))
assert_almost_equal(Y[0], b0)
Y.shape = (-1,) + Y.shape
# Testing WLS Fit on Single Voxel
# If you do something wonky (passing min_signal<0), you should get an
# error:
npt.assert_raises(ValueError, TensorModel, gtab, fit_method='WLS',
min_signal=-1)
# Estimate tensor from test signals
model = TensorModel(gtab, fit_method='WLS', return_S0_hat=True)
tensor_est = model.fit(Y)
assert_equal(tensor_est.shape, Y.shape[:-1])
assert_array_almost_equal(tensor_est.evals[0], evals)
assert_array_almost_equal(tensor_est.quadratic_form[0], tensor,
err_msg="Calculation of tensor from Y does not "
"compare to analytical solution")
assert_almost_equal(tensor_est.md[0], md)
assert_array_almost_equal(tensor_est.S0_hat[0], b0, decimal=3)
# Test that we can fit a single voxel's worth of data (a 1d array)
y = Y[0]
tensor_est = model.fit(y)
assert_equal(tensor_est.shape, tuple())
assert_array_almost_equal(tensor_est.evals, evals)
assert_array_almost_equal(tensor_est.quadratic_form, tensor)
assert_almost_equal(tensor_est.md, md)
assert_array_almost_equal(tensor_est.lower_triangular(b0), D)
# Test using fit_method='LS'
model = TensorModel(gtab, fit_method='LS')
tensor_est = model.fit(y)
assert_equal(tensor_est.shape, tuple())
assert_array_almost_equal(tensor_est.evals, evals)
assert_array_almost_equal(tensor_est.quadratic_form, tensor)
assert_almost_equal(tensor_est.md, md)
assert_array_almost_equal(tensor_est.lower_triangular(b0), D)
assert_array_almost_equal(tensor_est.linearity, linearity(evals))
assert_array_almost_equal(tensor_est.planarity, planarity(evals))
assert_array_almost_equal(tensor_est.sphericity, sphericity(evals))
def fwdti_prediction(params, gtab, S0=1, Diso=3.0e-3):
r""" Signal prediction given the free water DTI model parameters.
Parameters
----------
params : (..., 13) ndarray
Model parameters. The last dimension should have the 12 tensor
parameters (3 eigenvalues, followed by the 3 corresponding
eigenvectors) and the volume fraction of the free water compartment.
gtab : a GradientTable class instance
The gradient table for this prediction
S0 : float or ndarray
The non diffusion-weighted signal in every voxel, or across all
voxels. Default: 1
Diso : float, optional
Value of the free water isotropic diffusion. Default is set to 3e-3
$mm^{2}.s^{-1}$. Please adjust this value if you are assuming different
units of diffusion.
Returns
--------
S : (..., N) ndarray
Simulated signal based on the free water DTI model
Notes
-----
The predicted signal is given by:
$S(\theta, b) = S_0 * [(1-f) * e^{-b ADC} + f * e^{-b D_{iso}]$, where
$ADC = \theta Q \theta^T$, $\theta$ is a unit vector pointing at any
direction on the sphere for which a signal is to be predicted, $b$ is the b
value provided in the GradientTable input for that direction, $Q$ is the
quadratic form of the tensor determined by the input parameters, $f$ is the
free water diffusion compartment, $D_{iso}$ is the free water diffusivity
which is equal to $3 * 10^{-3} mm^{2}s^{-1} [1]_.
References
----------
.. [1] Hoy, A.R., Koay, C.G., Kecskemeti, S.R., Alexander, A.L., 2014.
Optimization of a free water elimination two-compartmental model
for diffusion tensor imaging. NeuroImage 103, 323-333.
doi: 10.1016/j.neuroimage.2014.09.053
"""
evals = params[..., :3]
evecs = params[..., 3:-1].reshape(params.shape[:-1] + (3, 3))
f = params[..., 12]
qform = vec_val_vect(evecs, evals)
lower_dt = lower_triangular(qform, S0)
lower_diso = lower_dt.copy()
lower_diso[..., 0] = lower_diso[..., 2] = lower_diso[..., 5] = Diso
lower_diso[..., 1] = lower_diso[..., 3] = lower_diso[..., 4] = 0
D = design_matrix(gtab)
pred_sig = np.zeros(f.shape + (gtab.bvals.shape[0],))
mask = _positive_evals(evals[..., 0], evals[..., 1], evals[..., 2])
index = ndindex(f.shape)
for v in index:
if mask[v]:
pred_sig[v] = (1 - f[v]) * np.exp(np.dot(lower_dt[v], D.T)) + \
f[v] * np.exp(np.dot(lower_diso[v], D.T))
return pred_sig
def test_tensor_model():
fdata, fbval, fbvec = get_data('small_25')
data1 = nib.load(fdata).get_data()
gtab1 = grad.gradient_table(fbval, fbvec)
data2, gtab2 = dsi_voxels()
for data, gtab in zip([data1, data2], [gtab1, gtab2]):
dm = dti.TensorModel(gtab, 'LS')
dtifit = dm.fit(data[0, 0, 0])
assert_equal(dtifit.fa < 0.9, True)
dm = dti.TensorModel(gtab, 'WLS')
dtifit = dm.fit(data[0, 0, 0])
assert_equal(dtifit.fa < 0.9, True)
assert_equal(dtifit.fa > 0, True)
sphere = create_unit_sphere(4)
assert_equal(len(dtifit.odf(sphere)), len(sphere.vertices))
# Check that the multivoxel case works:
dtifit = dm.fit(data)
# Check that it works on signal that has already been normalized to S0:
dm_to_relative = dti.TensorModel(gtab)
if np.any(gtab.b0s_mask):
relative_data = (data[0, 0, 0]/np.mean(data[0, 0, 0,
gtab.b0s_mask]))
dtifit_to_relative = dm_to_relative.fit(relative_data)
npt.assert_almost_equal(dtifit.fa[0, 0, 0], dtifit_to_relative.fa,
decimal=3)
# And smoke-test that all these operations return sensibly-shaped arrays:
assert_equal(dtifit.fa.shape, data.shape[:3])
assert_equal(dtifit.ad.shape, data.shape[:3])
assert_equal(dtifit.md.shape, data.shape[:3])
assert_equal(dtifit.rd.shape, data.shape[:3])
assert_equal(dtifit.trace.shape, data.shape[:3])
assert_equal(dtifit.mode.shape, data.shape[:3])
assert_equal(dtifit.linearity.shape, data.shape[:3])
assert_equal(dtifit.planarity.shape, data.shape[:3])
assert_equal(dtifit.sphericity.shape, data.shape[:3])
# Test for the shape of the mask
assert_raises(ValueError, dm.fit, np.ones((10, 10, 3)), np.ones((3, 3)))
# Make some synthetic data
b0 = 1000.
bvecs, bvals = read_bvec_file(get_data('55dir_grad.bvec'))
gtab = grad.gradient_table_from_bvals_bvecs(bvals, bvecs.T)
# The first b value is 0., so we take the second one:
B = bvals[1]
# Scale the eigenvalues and tensor by the B value so the units match
D = np.array([1., 1., 1., 0., 0., 1., -np.log(b0) * B]) / B
evals = np.array([2., 1., 0.]) / B
md = evals.mean()
tensor = from_lower_triangular(D)
A_squiggle = tensor - (1 / 3.0) * np.trace(tensor) * np.eye(3)
mode = (3 * np.sqrt(6) * np.linalg.det(A_squiggle /
np.linalg.norm(A_squiggle)))
evals_eigh, evecs_eigh = np.linalg.eigh(tensor)
# Sort according to eigen-value from large to small:
evecs = evecs_eigh[:, np.argsort(evals_eigh)[::-1]]
# Check that eigenvalues and eigenvectors are properly sorted through
# that previous operation:
for i in range(3):
assert_array_almost_equal(np.dot(tensor, evecs[:, i]),
evals[i] * evecs[:, i])
# Design Matrix
X = dti.design_matrix(gtab)
# Signals
Y = np.exp(np.dot(X, D))
assert_almost_equal(Y[0], b0)
Y.shape = (-1,) + Y.shape
# Test fitting with different methods:
for fit_method in ['OLS', 'WLS', 'NLLS']:
tensor_model = dti.TensorModel(gtab,
fit_method=fit_method,
return_S0_hat=True)
tensor_fit = tensor_model.fit(Y)
assert_true(tensor_fit.model is tensor_model)
assert_equal(tensor_fit.shape, Y.shape[:-1])
assert_array_almost_equal(tensor_fit.evals[0], evals)
assert_array_almost_equal(tensor_fit.S0_hat, b0, decimal=3)
# Test that the eigenvectors are correct, one-by-one:
for i in range(3):
# Eigenvectors have intrinsic sign ambiguity
# (see
# http://prod.sandia.gov/techlib/access-control.cgi/2007/076422.pdf)
# so we need to allow for sign flips. One of the following should
# always be true:
assert_(
np.all(np.abs(tensor_fit.evecs[0][:, i] -
evecs[:, i]) < 10e-6) or
np.all(np.abs(-tensor_fit.evecs[0][:, i] -
evecs[:, i]) < 10e-6))
# We set a fixed tolerance of 10e-6, similar to array_almost_equal
err_msg = "Calculation of tensor from Y does not compare to "
err_msg += "analytical solution"
assert_array_almost_equal(tensor_fit.quadratic_form[0], tensor,
err_msg=err_msg)
assert_almost_equal(tensor_fit.md[0], md)
assert_array_almost_equal(tensor_fit.mode, mode, decimal=5)
assert_equal(tensor_fit.directions.shape[-2], 1)
assert_equal(tensor_fit.directions.shape[-1], 3)
# Test error-handling:
assert_raises(ValueError,
dti.TensorModel,
gtab,
fit_method='crazy_method')
# Test custom fit tensor method
try:
model = dti.TensorModel(gtab, fit_method=lambda *args, **kwargs: 42)
fit = model.fit_method()
except Exception as exc:
assert False, "TensorModel should accept custom fit methods: %s" % exc
assert fit == 42, "Custom fit method for TensorModel returned %s." % fit
# Test multi-voxel data
data = np.zeros((3, Y.shape[1]))
# Normal voxel
data[0] = Y
# High diffusion voxel, all diffusing weighted signal equal to zero
data[1, gtab.b0s_mask] = b0
data[1, ~gtab.b0s_mask] = 0
# Masked voxel, all data set to zero
data[2] = 0.
tensor_model = dti.TensorModel(gtab)
fit = tensor_model.fit(data)
assert_array_almost_equal(fit[0].evals, evals)
# Return S0_test
tensor_model = dti.TensorModel(gtab, return_S0_hat=True)
fit = tensor_model.fit(data)
assert_array_almost_equal(fit[0].evals, evals)
assert_array_almost_equal(fit[0].S0_hat, b0)
# Evals should be high for high diffusion voxel
assert_(all(fit[1].evals > evals[0] * .9))
# Evals should be zero where data is masked
assert_array_almost_equal(fit[2].evals, 0.)
def wls_fit_tensor(gtab, data, Diso=3e-3, mask=None, min_signal=1.0e-6,
piterations=3, mdreg=2.7e-3):
r""" Computes weighted least squares (WLS) fit to calculate self-diffusion
tensor using a linear regression model [1]_.
Parameters
----------
gtab : a GradientTable class instance
The gradient table containing diffusion acquisition parameters.
data : ndarray ([X, Y, Z, ...], g)
Data or response variables holding the data. Note that the last
dimension should contain the data. It makes no copies of data.
Diso : float, optional
Value of the free water isotropic diffusion. Default is set to 3e-3
$mm^{2}.s^{-1}$. Please ajust this value if you are assuming different
units of diffusion.
mask : array, optional
A boolean array used to mark the coordinates in the data that should
be analyzed that has the shape data.shape[:-1]
min_signal : float
The minimum signal value. Needs to be a strictly positive
number. Default: 1.0e-6.
piterations : inter, optional
Number of iterations used to refine the precision of f. Default is set
to 3 corresponding to a precision of 0.01.
mdreg : float, optimal
DTI's mean diffusivity regularization threshold. If standard DTI
diffusion tensor's mean diffusivity is almost near the free water
diffusion value, the diffusion signal is assumed to be only free water
diffusion (i.e. volume fraction will be set to 1 and tissue's diffusion
parameters are set to zero). Default md_reg is 2.7e-3 $mm^{2}.s^{-1}$
(corresponding to 90% of the free water diffusion value).
Returns
-------
fw_params : ndarray (x, y, z, 13)
Matrix containing in the last dimention the free water model parameters
in the following order:
1) Three diffusion tensor's eigenvalues
2) Three lines of the eigenvector matrix each containing the
first, second and third coordinates of the eigenvector
3) The volume fraction of the free water compartment.
References
----------
.. [1] Hoy, A.R., Koay, C.G., Kecskemeti, S.R., Alexander, A.L., 2014.
Optimization of a free water elimination two-compartmental model
for diffusion tensor imaging. NeuroImage 103, 323-333.
doi: 10.1016/j.neuroimage.2014.09.053
"""
fw_params = np.zeros(data.shape[:-1] + (13,))
W = design_matrix(gtab)
# Prepare mask
if mask is None:
mask = np.ones(data.shape[:-1], dtype=bool)
else:
if mask.shape != data.shape[:-1]:
raise ValueError("Mask is not the same shape as data.")
mask = np.array(mask, dtype=bool, copy=False)
# Prepare S0
S0 = np.mean(data[:, :, :, gtab.b0s_mask], axis=-1)
index = ndindex(mask.shape)
for v in index:
if mask[v]:
params = wls_iter(W, data[v], S0[v], min_signal=min_signal,
Diso=3e-3, piterations=piterations, mdreg=mdreg)
fw_params[v] = params
return fw_params
def nls_fit_tensor(gtab, data, mask=None, Diso=3e-3, mdreg=2.7e-3,
min_signal=1.0e-6, f_transform=True, cholesky=False,
jac=True):
"""
Fit the water elimination tensor model using the non-linear least-squares.
Parameters
----------
gtab : a GradientTable class instance
The gradient table containing diffusion acquisition parameters.
data : ndarray ([X, Y, Z, ...], g)
Data or response variables holding the data. Note that the last
dimension should contain the data. It makes no copies of data.
mask : array, optional
A boolean array used to mark the coordinates in the data that should
be analyzed that has the shape data.shape[:-1]
Diso : float, optional
Value of the free water isotropic diffusion. Default is set to 3e-3
$mm^{2}.s^{-1}$. Please ajust this value if you are assuming different
units of diffusion.
mdreg : float, optimal
DTI's mean diffusivity regularization threshold. If standard DTI
diffusion tensor's mean diffusivity is almost near the free water
diffusion value, the diffusion signal is assumed to be only free water
diffusion (i.e. volume fraction will be set to 1 and tissue's diffusion
parameters are set to zero). Default md_reg is 2.7e-3 $mm^{2}.s^{-1}$
(corresponding to 90% of the free water diffusion value).
min_signal : float
The minimum signal value. Needs to be a strictly positive
number. Default: 1.0e-6.
f_transform : bool, optional
If true, the water volume fractions is converted during the convergence
procedure to ft = arcsin(2*f - 1) + pi/2, insuring f estimates between
0 and 1.
Default: True
cholesky : bool, optional
If true it uses cholesky decomposition to insure that diffusion tensor
is positive define.
Default: False
jac : bool
Use the Jacobian? Default: False
Returns
-------
fw_params : ndarray (x, y, z, 13)
Matrix containing in the dimention the free water model parameters in
the following order:
1) Three diffusion tensor's eigenvalues
2) Three lines of the eigenvector matrix each containing the
first, second and third coordinates of the eigenvector
3) The volume fraction of the free water compartment
"""
# Analyse compatible input cases
if jac is True and cholesky is True:
raise ValueError("Cholesky decomposition is not compatible with jac.")
fw_params = np.zeros(data.shape[:-1] + (13,))
W = design_matrix(gtab)
# Prepare mask
if mask is None:
mask = np.ones(data.shape[:-1], dtype=bool)
else:
if mask.shape != data.shape[:-1]:
raise ValueError("Mask is not the same shape as data.")
mask = np.array(mask, dtype=bool, copy=False)
# Prepare S0
S0 = np.mean(data[:, :, :, gtab.b0s_mask], axis=-1)
index = ndindex(mask.shape)
for v in index:
if mask[v]:
params = nls_iter(W, data[v], S0[v],
Diso=Diso, mdreg=mdreg,
min_signal=min_signal,
f_transform=f_transform, cholesky=cholesky,
jac=jac)
fw_params[v] = params
return fw_params
def nls_fit_tensor(gtab, data, mask=None, Diso=3e-3, mdreg=2.7e-3,
min_signal=1.0e-6, f_transform=True, cholesky=False,
jac=False, weighting=None, sigma=None):
"""
Fit the water elimination tensor model using the non-linear least-squares.
Parameters
----------
gtab : a GradientTable class instance
The gradient table containing diffusion acquisition parameters.
data : ndarray ([X, Y, Z, ...], g)
Data or response variables holding the data. Note that the last
dimension should contain the data. It makes no copies of data.
mask : array, optional
A boolean array used to mark the coordinates in the data that should
be analyzed that has the shape data.shape[:-1]
Diso : float, optional
Value of the free water isotropic diffusion. Default is set to 3e-3
$mm^{2}.s^{-1}$. Please ajust this value if you are assuming different
units of diffusion.
mdreg : float, optimal
DTI's mean diffusivity regularization threshold. If standard DTI
diffusion tensor's mean diffusivity is almost near the free water
diffusion value, the diffusion signal is assumed to be only free water
diffusion (i.e. volume fraction will be set to 1 and tissue's diffusion
parameters are set to zero). Default md_reg is 2.7e-3 $mm^{2}.s^{-1}$
(corresponding to 90% of the free water diffusion value).
min_signal : float
The minimum signal value. Needs to be a strictly positive
number. Default: 1.0e-6.
f_transform : bool, optional
If true, the water volume fractions is converted during the convergence
procedure to ft = arcsin(2*f - 1) + pi/2, insuring f estimates between
0 and 1.
Default: True
cholesky : bool, optional
If true it uses cholesky decomposition to insure that diffusion tensor
is positive define.
Default: False
jac : bool
Use the Jacobian? Default: False
weighting: str, optional
the weighting scheme to use in considering the
squared-error. Default behavior is to use uniform weighting. Other
options: 'sigma' 'gmm'
sigma: float, optional
If the 'sigma' weighting scheme is used, a value of sigma needs to be
provided here. According to [Chang2005]_, a good value to use is
1.5267 * std(background_noise), where background_noise is estimated
from some part of the image known to contain no signal (only noise).
Returns
-------
fw_params : ndarray (x, y, z, 13)
Matrix containing in the dimention the free water model parameters in
the following order:
1) Three diffusion tensor's eigenvalues
2) Three lines of the eigenvector matrix each containing the
first, second and third coordinates of the eigenvector
3) The volume fraction of the free water compartment
"""
fw_params = np.zeros(data.shape[:-1] + (13,))
W = design_matrix(gtab)
# Prepare mask
if mask is None:
mask = np.ones(data.shape[:-1], dtype=bool)
else:
if mask.shape != data.shape[:-1]:
raise ValueError("Mask is not the same shape as data.")
mask = np.array(mask, dtype=bool, copy=False)
# Prepare S0
S0 = np.mean(data[:, :, :, gtab.b0s_mask], axis=-1)
index = ndindex(mask.shape)
for v in index:
if mask[v]:
params = nls_iter(W, data[v], S0[v], Diso=Diso, mdreg=mdreg,
min_signal=min_signal, f_transform=f_transform,
cholesky=cholesky, jac=jac, weighting=weighting,
sigma=sigma)
fw_params[v] = params
return fw_params
import nibabel as nib
import dipy.reconst.dti as dti
import dipy.data as dpd
import dipy.core.gradients as grad
b0 = 1000.
bvecs, bval = dpd.read_bvec_file(dpd.get_data('55dir_grad.bvec'))
gtab = grad.gradient_table(bval, bvecs)
B = bval[1]
D = np.array([1., 1., 1., 0., 0., 1., -np.log(b0) * B]) / B
evals = np.array([2., 1., 0.]) / B
md = evals.mean()
tensor = dti.from_lower_triangular(D)
X = dti.design_matrix(bvecs, bval)
data = np.exp(np.dot(X,D))
data.shape = (-1,) + data.shape
dti_wls = dti.TensorModel(gtab)
fit_wls = dti_wls.fit(data)
fa1 = fit_wls.fa
noisy_data = np.copy(data)
noisy_data[..., -1] = 1.0
fit_wls_noisy = dti_wls.fit(noisy_data)
fa2 = fit_wls_noisy.fa
dti_restore = dti.TensorModel(gtab, fit_method='RESTORE', sigma=67.)
def test_tensor_model():
fdata, fbval, fbvec = get_data('small_25')
data1 = nib.load(fdata).get_data()
gtab1 = grad.gradient_table(fbval, fbvec)
data2, gtab2 = dsi_voxels()
for data, gtab in zip([data1, data2], [gtab1, gtab2]):
dm = dti.TensorModel(gtab, 'LS')
dtifit = dm.fit(data[0, 0, 0])
assert_equal(dtifit.fa < 0.9, True)
dm = dti.TensorModel(gtab, 'WLS')
dtifit = dm.fit(data[0, 0, 0])
assert_equal(dtifit.fa < 0.9, True)
assert_equal(dtifit.fa > 0, True)
sphere = create_unit_sphere(4)
assert_equal(len(dtifit.odf(sphere)), len(sphere.vertices))
# Check that the multivoxel case works:
dtifit = dm.fit(data)
# Check that it works on signal that has already been normalized to S0:
dm_to_relative = dti.TensorModel(gtab)
if np.any(gtab.b0s_mask):
relative_data = (data[0, 0, 0] /
np.mean(data[0, 0, 0, gtab.b0s_mask]))
dtifit_to_relative = dm_to_relative.fit(relative_data)
npt.assert_almost_equal(dtifit.fa[0, 0, 0],
dtifit_to_relative.fa,
decimal=3)
# And smoke-test that all these operations return sensibly-shaped arrays:
assert_equal(dtifit.fa.shape, data.shape[:3])
assert_equal(dtifit.ad.shape, data.shape[:3])
assert_equal(dtifit.md.shape, data.shape[:3])
assert_equal(dtifit.rd.shape, data.shape[:3])
assert_equal(dtifit.trace.shape, data.shape[:3])
assert_equal(dtifit.mode.shape, data.shape[:3])
assert_equal(dtifit.linearity.shape, data.shape[:3])
assert_equal(dtifit.planarity.shape, data.shape[:3])
assert_equal(dtifit.sphericity.shape, data.shape[:3])
# Test for the shape of the mask
assert_raises(ValueError, dm.fit, np.ones((10, 10, 3)), np.ones((3, 3)))
# Make some synthetic data
b0 = 1000.
bvecs, bvals = read_bvec_file(get_data('55dir_grad.bvec'))
gtab = grad.gradient_table_from_bvals_bvecs(bvals, bvecs.T)
# The first b value is 0., so we take the second one:
B = bvals[1]
# Scale the eigenvalues and tensor by the B value so the units match
D = np.array([1., 1., 1., 0., 0., 1., -np.log(b0) * B]) / B
evals = np.array([2., 1., 0.]) / B
md = evals.mean()
tensor = from_lower_triangular(D)
A_squiggle = tensor - (1 / 3.0) * np.trace(tensor) * np.eye(3)
mode = 3 * np.sqrt(6) * np.linalg.det(
A_squiggle / np.linalg.norm(A_squiggle))
evecs = np.linalg.eigh(tensor)[1]
# Design Matrix
X = dti.design_matrix(gtab)
# Signals
Y = np.exp(np.dot(X, D))
assert_almost_equal(Y[0], b0)
Y.shape = (-1, ) + Y.shape
# Test fitting with different methods:
for fit_method in ['OLS', 'WLS', 'NLLS']:
tensor_model = dti.TensorModel(gtab, fit_method=fit_method)
tensor_fit = tensor_model.fit(Y)
assert_true(tensor_fit.model is tensor_model)
assert_equal(tensor_fit.shape, Y.shape[:-1])
assert_array_almost_equal(tensor_fit.evals[0], evals)
assert_array_almost_equal(tensor_fit.quadratic_form[0], tensor,
err_msg=\
"Calculation of tensor from Y does not compare to analytical solution")
assert_almost_equal(tensor_fit.md[0], md)
assert_array_almost_equal(tensor_fit.mode, mode, decimal=5)
assert_equal(tensor_fit.directions.shape[-2], 1)
assert_equal(tensor_fit.directions.shape[-1], 3)
# Test error-handling:
assert_raises(ValueError, dti.TensorModel, gtab, fit_method='crazy_method')
# Test multi-voxel data
data = np.zeros((3, Y.shape[1]))
# Normal voxel
data[0] = Y
# High diffusion voxel, all diffusing weighted signal equal to zero
data[1, gtab.b0s_mask] = b0
data[1, ~gtab.b0s_mask] = 0
# Masked voxel, all data set to zero
data[2] = 0.
tensor_model = dti.TensorModel(gtab)
fit = tensor_model.fit(data)
assert_array_almost_equal(fit[0].evals, evals)
# Evals should be high for high diffusion voxel
assert_(all(fit[1].evals > evals[0] * .9))
# Evals should be zero where data is masked
assert_array_almost_equal(fit[2].evals, 0.)
def test_tensor_model():
fdata, fbval, fbvec = get_data('small_25')
data1 = nib.load(fdata).get_data()
gtab1 = grad.gradient_table(fbval, fbvec)
data2, gtab2 = dsi_voxels()
for data, gtab in zip([data1, data2], [gtab1, gtab2]):
dm = dti.TensorModel(gtab, 'LS')
dtifit = dm.fit(data[0, 0, 0])
assert_equal(dtifit.fa < 0.9, True)
dm = dti.TensorModel(gtab, 'WLS')
dtifit = dm.fit(data[0, 0, 0])
assert_equal(dtifit.fa < 0.9, True)
assert_equal(dtifit.fa > 0, True)
sphere = create_unit_sphere(4)
assert_equal(len(dtifit.odf(sphere)), len(sphere.vertices))
# Check that the multivoxel case works:
dtifit = dm.fit(data)
# Check that it works on signal that has already been normalized to S0:
dm_to_relative = dti.TensorModel(gtab)
if np.any(gtab.b0s_mask):
relative_data = (data[0, 0, 0]/np.mean(data[0, 0, 0,
gtab.b0s_mask]))
dtifit_to_relative = dm_to_relative.fit(relative_data)
npt.assert_almost_equal(dtifit.fa[0, 0, 0], dtifit_to_relative.fa,
decimal=3)
# And smoke-test that all these operations return sensibly-shaped arrays:
assert_equal(dtifit.fa.shape, data.shape[:3])
assert_equal(dtifit.ad.shape, data.shape[:3])
assert_equal(dtifit.md.shape, data.shape[:3])
assert_equal(dtifit.rd.shape, data.shape[:3])
assert_equal(dtifit.trace.shape, data.shape[:3])
assert_equal(dtifit.mode.shape, data.shape[:3])
assert_equal(dtifit.linearity.shape, data.shape[:3])
assert_equal(dtifit.planarity.shape, data.shape[:3])
assert_equal(dtifit.sphericity.shape, data.shape[:3])
# Test for the shape of the mask
assert_raises(ValueError, dm.fit, np.ones((10, 10, 3)), np.ones((3, 3)))
# Make some synthetic data
b0 = 1000.
bvecs, bvals = read_bvec_file(get_data('55dir_grad.bvec'))
gtab = grad.gradient_table_from_bvals_bvecs(bvals, bvecs.T)
# The first b value is 0., so we take the second one:
B = bvals[1]
# Scale the eigenvalues and tensor by the B value so the units match
D = np.array([1., 1., 1., 0., 0., 1., -np.log(b0) * B]) / B
evals = np.array([2., 1., 0.]) / B
md = evals.mean()
tensor = from_lower_triangular(D)
A_squiggle = tensor - (1 / 3.0) * np.trace(tensor) * np.eye(3)
mode = (3 * np.sqrt(6) * np.linalg.det(A_squiggle /
np.linalg.norm(A_squiggle)))
evals_eigh, evecs_eigh = np.linalg.eigh(tensor)
# Sort according to eigen-value from large to small:
evecs = evecs_eigh[:, np.argsort(evals_eigh)[::-1]]
# Check that eigenvalues and eigenvectors are properly sorted through
# that previous operation:
for i in range(3):
assert_array_almost_equal(np.dot(tensor, evecs[:, i]),
evals[i] * evecs[:, i])
# Design Matrix
X = dti.design_matrix(gtab)
# Signals
Y = np.exp(np.dot(X, D))
assert_almost_equal(Y[0], b0)
Y.shape = (-1,) + Y.shape
# Test fitting with different methods:
for fit_method in ['OLS', 'WLS', 'NLLS']:
tensor_model = dti.TensorModel(gtab,
fit_method=fit_method,
return_S0_hat=True)
tensor_fit = tensor_model.fit(Y)
assert_true(tensor_fit.model is tensor_model)
assert_equal(tensor_fit.shape, Y.shape[:-1])
assert_array_almost_equal(tensor_fit.evals[0], evals)
assert_array_almost_equal(tensor_fit.S0_hat, b0, decimal=3)
# Test that the eigenvectors are correct, one-by-one:
for i in range(3):
# Eigenvectors have intrinsic sign ambiguity
# (see
# http://prod.sandia.gov/techlib/access-control.cgi/2007/076422.pdf)
# so we need to allow for sign flips. One of the following should
# always be true:
assert_(
np.all(np.abs(tensor_fit.evecs[0][:, i] -
evecs[:, i]) < 10e-6) or
np.all(np.abs(-tensor_fit.evecs[0][:, i] -
evecs[:, i]) < 10e-6))
# We set a fixed tolerance of 10e-6, similar to array_almost_equal
err_msg = "Calculation of tensor from Y does not compare to "
err_msg += "analytical solution"
assert_array_almost_equal(tensor_fit.quadratic_form[0], tensor,
err_msg=err_msg)
assert_almost_equal(tensor_fit.md[0], md)
assert_array_almost_equal(tensor_fit.mode, mode, decimal=5)
assert_equal(tensor_fit.directions.shape[-2], 1)
assert_equal(tensor_fit.directions.shape[-1], 3)
# Test error-handling:
assert_raises(ValueError,
dti.TensorModel,
gtab,
fit_method='crazy_method')
# Test custom fit tensor method
try:
model = dti.TensorModel(gtab, fit_method=lambda *args, **kwargs: 42)
fit = model.fit_method()
except Exception as exc:
assert False, "TensorModel should accept custom fit methods: %s" % exc
assert fit == 42, "Custom fit method for TensorModel returned %s." % fit
# Test multi-voxel data
data = np.zeros((3, Y.shape[1]))
# Normal voxel
data[0] = Y
# High diffusion voxel, all diffusing weighted signal equal to zero
data[1, gtab.b0s_mask] = b0
data[1, ~gtab.b0s_mask] = 0
# Masked voxel, all data set to zero
data[2] = 0.
tensor_model = dti.TensorModel(gtab)
fit = tensor_model.fit(data)
assert_array_almost_equal(fit[0].evals, evals)
# Return S0_test
tensor_model = dti.TensorModel(gtab, return_S0_hat=True)
fit = tensor_model.fit(data)
assert_array_almost_equal(fit[0].evals, evals)
assert_array_almost_equal(fit[0].S0_hat, b0)
# Evals should be high for high diffusion voxel
assert_(all(fit[1].evals > evals[0] * .9))
# Evals should be zero where data is masked
assert_array_almost_equal(fit[2].evals, 0.)
def nls_fit_tensor(gtab,
data,
mask=None,
Diso=3e-3,
min_signal=1.0e-6,
f_transform=True,
cholesky=False,
jac=True):
"""
Fit the water elimination tensor model using the non-linear least-squares.
Parameters
----------
gtab : a GradientTable class instance
The gradient table containing diffusion acquisition parameters.
data : ndarray ([X, Y, Z, ...], g)
Data or response variables holding the data. Note that the last
dimension should contain the data. It makes no copies of data.
mask : array, optional
A boolean array used to mark the coordinates in the data that should
be analyzed that has the shape data.shape[:-1]
Diso : float, optional
Value of the free water isotropic diffusion. Default is set to 3e-3
$mm^{2}.s^{-1}$. Please ajust this value if you are assuming different
units of diffusion.
min_signal : float
The minimum signal value. Needs to be a strictly positive
number. Default: 1.0e-6.
f_transform : bool, optional
If true, the water volume fractions is converted during the convergence
procedure to ft = arcsin(2*f - 1) + pi/2, insuring f estimates between
0 and 1.
Default: True
cholesky : bool, optional
If true it uses cholesky decomposition to insure that diffusion tensor
is positive define.
Default: False
jac : bool
Use the Jacobian? Default: False
Returns
-------
fw_params : ndarray (x, y, z, 13)
Matrix containing in the last dimension the free water model parameters
in the following order:
1) Three diffusion tensor's eigenvalues
2) Three lines of the eigenvector matrix each containing the
first, second and third coordinates of the eigenvector
3) The volume fraction of the free water compartment
References
----------
.. [1] Henriques, R.N., Rokem, A., Garyfallidis, E., St-Jean, S., Peterson,
E.T., Correia, M.M., 2017. Re: Optimization of a free water
elimination two-compartmental model for diffusion tensor imaging.
ReScience
"""
# Analyse compatible input cases
if jac is True and cholesky is True:
raise ValueError("Cholesky decomposition is not compatible with jac.")
fw_params = np.zeros(data.shape[:-1] + (13, ))
W = design_matrix(gtab)
# Prepare mask
if mask is None:
mask = np.ones(data.shape[:-1], dtype=bool)
else:
if mask.shape != data.shape[:-1]:
raise ValueError("Mask is not the same shape as data.")
mask = np.array(mask, dtype=bool, copy=False)
# Prepare S0
S0 = np.mean(data[..., gtab.b0s_mask], axis=-1)
# Loop data fitting through all voxels
index = ndindex(mask.shape)
for v in index:
if mask[v]:
params = nls_iter(W,
data[v],
S0[v],
Diso=Diso,
min_signal=min_signal,
f_transform=f_transform,
cholesky=cholesky,
jac=jac)
fw_params[v] = params
return fw_params