def test_kurtosis_maximum():
# TEST 1
# simulate a crossing fibers interserting at 70 degrees. The first fiber
# is aligned to the x-axis while the second fiber is aligned to the x-z
# plane with an angular deviation of 70 degrees from the first one.
# According to Neto Henriques et al, 2015 (NeuroImage 111: 85-99), the
# kurtosis tensor of this simulation will have a maxima aligned to axis y
angles = [(90, 0), (90, 0), (20, 0), (20, 0)]
signal_70, dt_70, kt_70 = multi_tensor_dki(gtab_2s, mevals_cross, S0=100,
angles=angles,
fractions=frac_cross, snr=None)
# prepare inputs
dkiM = dki.DiffusionKurtosisModel(gtab_2s, fit_method="WLS")
dkiF = dkiM.fit(signal_70)
MD = dkiF.md
kt = dkiF.kt
R = dkiF.evecs
evals = dkiF.evals
dt = lower_triangular(np.dot(np.dot(R, np.diag(evals)), R.T))
sphere = get_sphere('symmetric724')
# compute maxima
k_max_cross, max_dir = dki._voxel_kurtosis_maximum(dt, MD, kt, sphere,
gtol=1e-5)
yaxis = np.array([0., 1., 0.])
cos_angle = np.abs(np.dot(max_dir[0], yaxis))
assert_almost_equal(cos_angle, 1.)
# TEST 2
# test the function on cases of well aligned fibers oriented in a random
# defined direction. According to Neto Henriques et al, 2015 (NeuroImage
# 111: 85-99), the maxima of kurtosis is any direction perpendicular to the
# fiber direction. Moreover, according to multicompartmetal simulations,
# kurtosis in this direction has to be equal to:
fie = 0.49
ADi = 0.00099
ADe = 0.00226
RDi = 0
RDe = 0.00087
RD = fie*RDi + (1-fie)*RDe
RK = 3 * fie * (1-fie) * ((RDi-RDe) / RD) ** 2
# prepare simulation:
theta = random.uniform(0, 180)
phi = random.uniform(0, 320)
angles = [(theta, phi), (theta, phi)]
mevals = np.array([[ADi, RDi, RDi], [ADe, RDe, RDe]])
frac = [fie*100, (1 - fie)*100]
signal, dt, kt = multi_tensor_dki(gtab_2s, mevals, angles=angles,
fractions=frac, snr=None)
# prepare inputs
dkiM = dki.DiffusionKurtosisModel(gtab_2s, fit_method="WLS")
dkiF = dkiM.fit(signal)
MD = dkiF.md
kt = dkiF.kt
R = dkiF.evecs
evals = dkiF.evals
dt = lower_triangular(np.dot(np.dot(R, np.diag(evals)), R.T))
# compute maxima
k_max, max_dir = dki._voxel_kurtosis_maximum(dt, MD, kt, sphere, gtol=1e-5)
# check if max direction is perpendicular to fiber direction
fdir = np.array([sphere2cart(1., np.deg2rad(theta), np.deg2rad(phi))])
cos_angle = np.abs(np.dot(max_dir[0], fdir[0]))
assert_almost_equal(cos_angle, 0., decimal=5)
# check if max direction is equal to expected value
assert_almost_equal(k_max, RK)
# According to Neto Henriques et al., 2015 (NeuroImage 111: 85-99),
# e.g. see figure 1 of this article, kurtosis maxima for the first test is
# also equal to the maxima kurtosis value of well-aligned fibers, since
# simulations parameters (apart from fiber directions) are equal
assert_almost_equal(k_max_cross, RK)
# Test 3 - Test performance when kurtosis is spherical - this case, can be
# problematic since a spherical kurtosis does not have an maximum
k_max, max_dir = dki._voxel_kurtosis_maximum(dt_sph, np.mean(evals_sph),
kt_sph, sphere, gtol=1e-2)
assert_almost_equal(k_max, Kref_sphere)
# Test 4 - Test performance when kt have all elements zero - this case, can
# be problematic this case does not have an maximum
k_max, max_dir = dki._voxel_kurtosis_maximum(dt_sph, np.mean(evals_sph),
np.zeros(15), sphere,
gtol=1e-2)
assert_almost_equal(k_max, 0.0)
def test_kurtosis_maximum():
# TEST 1
# simulate a crossing fibers interserting at 70 degrees. The first fiber
# is aligned to the x-axis while the second fiber is aligned to the x-z
# plane with an angular deviation of 70 degrees from the first one.
# According to Neto Henriques et al, 2015 (NeuroImage 111: 85-99), the
# kurtosis tensor of this simulation will have a maxima aligned to axis y
angles = [(90, 0), (90, 0), (20, 0), (20, 0)]
signal_70, dt_70, kt_70 = multi_tensor_dki(gtab_2s, mevals_cross, S0=100,
angles=angles,
fractions=frac_cross, snr=None)
# prepare inputs
dkiM = dki.DiffusionKurtosisModel(gtab_2s, fit_method="WLS")
dkiF = dkiM.fit(signal_70)
MD = dkiF.md
kt = dkiF.kt
R = dkiF.evecs
evals = dkiF.evals
dt = lower_triangular(np.dot(np.dot(R, np.diag(evals)), R.T))
sphere = get_sphere('symmetric724')
# compute maxima
k_max_cross, max_dir = dki._voxel_kurtosis_maximum(dt, MD, kt, sphere,
gtol=1e-5)
yaxis = np.array([0., 1., 0.])
cos_angle = np.abs(np.dot(max_dir[0], yaxis))
assert_almost_equal(cos_angle, 1.)
# TEST 2
# test the function on cases of well aligned fibers oriented in a random
# defined direction. According to Neto Henriques et al, 2015 (NeuroImage
# 111: 85-99), the maxima of kurtosis is any direction perpendicular to the
# fiber direction. Moreover, according to multicompartmetal simulations,
# kurtosis in this direction has to be equal to:
fie = 0.49
ADi = 0.00099
ADe = 0.00226
RDi = 0
RDe = 0.00087
RD = fie*RDi + (1-fie)*RDe
RK = 3 * fie * (1-fie) * ((RDi-RDe) / RD) ** 2
# prepare simulation:
theta = random.uniform(0, 180)
phi = random.uniform(0, 320)
angles = [(theta, phi), (theta, phi)]
mevals = np.array([[ADi, RDi, RDi], [ADe, RDe, RDe]])
frac = [fie*100, (1 - fie)*100]
signal, dt, kt = multi_tensor_dki(gtab_2s, mevals, angles=angles,
fractions=frac, snr=None)
# prepare inputs
dkiM = dki.DiffusionKurtosisModel(gtab_2s, fit_method="WLS")
dkiF = dkiM.fit(signal)
MD = dkiF.md
kt = dkiF.kt
R = dkiF.evecs
evals = dkiF.evals
dt = lower_triangular(np.dot(np.dot(R, np.diag(evals)), R.T))
# compute maxima
k_max, max_dir = dki._voxel_kurtosis_maximum(dt, MD, kt, sphere, gtol=1e-5)
# check if max direction is perpendicular to fiber direction
fdir = np.array([sphere2cart(1., np.deg2rad(theta), np.deg2rad(phi))])
cos_angle = np.abs(np.dot(max_dir[0], fdir[0]))
assert_almost_equal(cos_angle, 0., decimal=5)
# check if max direction is equal to expected value
assert_almost_equal(k_max, RK)
# According to Neto Henriques et al., 2015 (NeuroImage 111: 85-99),
# e.g. see figure 1 of this article, kurtosis maxima for the first test is
# also equal to the maxima kurtosis value of well-aligned fibers, since
# simulations parameters (apart from fiber directions) are equal
assert_almost_equal(k_max_cross, RK)
# Test 3 - Test performance when kurtosis is spherical - this case, can be
# problematic since a spherical kurtosis does not have an maximum
k_max, max_dir = dki._voxel_kurtosis_maximum(dt_sph, np.mean(evals_sph),
kt_sph, sphere, gtol=1e-2)
assert_almost_equal(k_max, Kref_sphere)
# Test 4 - Test performance when kt have all elements zero - this case, can
# be problematic this case does not have an maximum
k_max, max_dir = dki._voxel_kurtosis_maximum(dt_sph, np.mean(evals_sph),
np.zeros(15), sphere,
gtol=1e-2)
assert_almost_equal(k_max, 0.0)
def dki_prediction(dki_params, gtab, S0=150):
"""
In Dipy versions < 0.12, there is a bug in DKI prediction, that doesn't
allow using volumes of S0. This is temporary fix until the bug is fixed
upstream. See: https://github.com/nipy/dipy/pull/1028
For now, we provide this as a fix, to monkey-patch into dipy.reconst.dki
Parameters
----------
dki_params : ndarray (x, y, z, 27) or (n, 27)
All parameters estimated from the diffusion kurtosis model.
Parameters are ordered as follows:
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) Fifteen elements of the kurtosis tensor
gtab : a GradientTable class instance
The gradient table for this prediction
S0 : float or ndarray (optional)
The non diffusion-weighted signal in every voxel, or across all
voxels. Default: 150
Returns
--------
S : (..., N) ndarray
Simulated signal based on the DKI model:
.. math::
S=S_{0}e^{-bD+\frac{1}{6}b^{2}D^{2}K}
"""
evals, evecs, kt = dki.split_dki_param(dki_params)
# Define DKI design matrix according to given gtab
A = dki.design_matrix(gtab)
# Flat parameters and initialize pred_sig
fevals = evals.reshape((-1, evals.shape[-1]))
fevecs = evecs.reshape((-1,) + evecs.shape[-2:])
fkt = kt.reshape((-1, kt.shape[-1]))
pred_sig = np.zeros((len(fevals), len(gtab.bvals)))
if isinstance(S0, np.ndarray):
S0_vol = np.reshape(S0, (len(fevals)))
else:
S0_vol = S0
# looping for all voxels
for v in range(len(pred_sig)):
DT = np.dot(np.dot(fevecs[v], np.diag(fevals[v])), fevecs[v].T)
dt = dki.lower_triangular(DT)
MD = (dt[0] + dt[2] + dt[5]) / 3
if isinstance(S0_vol, np.ndarray):
this_S0 = S0_vol[v]
else:
this_S0 = S0_vol
X = np.concatenate((dt, fkt[v] * MD * MD,
np.array([np.log(this_S0)])),
axis=0)
pred_sig[v] = np.exp(np.dot(A, X))
# Reshape data according to the shape of dki_params
pred_sig = pred_sig.reshape(dki_params.shape[:-1] + (pred_sig.shape[-1],))
return pred_sig
