def test_check_directions():
# Testing spherical angles for two principal coordinate axis
angles = [(0, 0)] # axis z
sticks = _check_directions(angles)
assert_array_almost_equal(sticks, [[0, 0, 1]])
angles = [(0, 90)] # axis z again (phi can be anything it theta is zero)
sticks = _check_directions(angles)
assert_array_almost_equal(sticks, [[0, 0, 1]])
angles = [(90, 0)] # axis x
sticks = _check_directions(angles)
assert_array_almost_equal(sticks, [[1, 0, 0]])
# Testing if directions are already given in cartesian coordinates
angles = [(0, 0, 1)]
sticks = _check_directions(angles)
assert_array_almost_equal(sticks, [[0, 0, 1]])
# Testing more than one direction simultaneously
angles = np.array([[90, 0], [30, 0]])
sticks = _check_directions(angles)
ref_vec = [np.sin(np.pi * 30 / 180), 0, np.cos(np.pi * 30 / 180)]
assert_array_almost_equal(sticks, [[1, 0, 0], ref_vec])
# Testing directions not aligned to planes x = 0, y = 0, or z = 0
the1 = 0
phi1 = 90
the2 = 30
phi2 = 45
angles = np.array([(the1, phi1), (the2, phi2)])
sticks = _check_directions(angles)
ref_vec1 = (np.sin(np.pi * the1 / 180) * np.cos(np.pi * phi1 / 180),
np.sin(np.pi * the1 / 180) * np.sin(np.pi * phi1 / 180),
np.cos(np.pi * the1 / 180))
ref_vec2 = (np.sin(np.pi * the2 / 180) * np.cos(np.pi * phi2 / 180),
np.sin(np.pi * the2 / 180) * np.sin(np.pi * phi2 / 180),
np.cos(np.pi * the2 / 180))
assert_array_almost_equal(sticks, [ref_vec1, ref_vec2])
def test_check_directions():
# Testing spherical angles for two principal coordinate axis
angles = [(0, 0)] # axis z
sticks = _check_directions(angles)
assert_array_almost_equal(sticks, [[0, 0, 1]])
angles = [(0, 90)] # axis z again (phi can be anything it theta is zero)
sticks = _check_directions(angles)
assert_array_almost_equal(sticks, [[0, 0, 1]])
angles = [(90, 0)] # axis x
sticks = _check_directions(angles)
assert_array_almost_equal(sticks, [[1, 0, 0]])
# Testing if directions are already given in cartesian coordinates
angles = [(0, 0, 1)]
sticks = _check_directions(angles)
assert_array_almost_equal(sticks, [[0, 0, 1]])
# Testing more than one direction simultaneously
angles = np.array([[90, 0], [30, 0]])
sticks = _check_directions(angles)
ref_vec = [np.sin(np.pi*30/180), 0, np.cos(np.pi*30/180)]
assert_array_almost_equal(sticks, [[1, 0, 0], ref_vec])
# Testing directions not aligned to planes x = 0, y = 0, or z = 0
the1 = 0
phi1 = 90
the2 = 30
phi2 = 45
angles = np.array([(the1, phi1), (the2, phi2)])
sticks = _check_directions(angles)
ref_vec1 = (np.sin(np.pi*the1/180) * np.cos(np.pi*phi1/180),
np.sin(np.pi*the1/180) * np.sin(np.pi*phi1/180),
np.cos(np.pi*the1/180))
ref_vec2 = (np.sin(np.pi*the2/180) * np.cos(np.pi*phi2/180),
np.sin(np.pi*the2/180) * np.sin(np.pi*phi2/180),
np.cos(np.pi*the2/180))
assert_array_almost_equal(sticks, [ref_vec1, ref_vec2])
def test_kurtosis_elements():
""" Testing symmetry of the elements of the KT
As an 4th order tensor, KT has 81 elements. However, due to diffusion
symmetry the KT is fully characterized by 15 independent elements. This
test checks for this property.
"""
# two fiber not aligned to planes x = 0, y = 0, or z = 0
mevals = np.array([[0.00099, 0, 0], [0.00226, 0.00087, 0.00087],
[0.00099, 0, 0], [0.00226, 0.00087, 0.00087]])
angles = [(80, 10), (80, 10), (20, 30), (20, 30)]
fie = 0.49 # intra axonal water fraction
frac = [fie * 50, (1 - fie) * 50, fie * 50, (1 - fie) * 50]
sticks = _check_directions(angles)
mD = np.zeros((len(frac), 3, 3))
for i in range(len(frac)):
R = all_tensor_evecs(sticks[i])
mD[i] = np.dot(np.dot(R, np.diag(mevals[i])), R.T)
# compute global DT
D = np.zeros((3, 3))
for i in range(len(frac)):
D = D + frac[i] * mD[i]
# compute voxel's MD
MD = (D[0][0] + D[1][1] + D[2][2]) / 3
# Reference dictionary with the 15 independent elements.
# Note: The multiplication of the indexes (i+1) * (j+1) * (k+1) * (l+1)
# for of an elements is only equal to this multiplication for another
# element if an only if the element corresponds to an symmetry element.
# Thus indexes multiplication is used as key of the reference dictionary
kt_ref = {
1: kurtosis_element(mD, frac, 0, 0, 0, 0),
16: kurtosis_element(mD, frac, 1, 1, 1, 1),
81: kurtosis_element(mD, frac, 2, 2, 2, 2),
2: kurtosis_element(mD, frac, 0, 0, 0, 1),
3: kurtosis_element(mD, frac, 0, 0, 0, 2),
8: kurtosis_element(mD, frac, 0, 1, 1, 1),
24: kurtosis_element(mD, frac, 1, 1, 1, 2),
27: kurtosis_element(mD, frac, 0, 2, 2, 2),
54: kurtosis_element(mD, frac, 1, 2, 2, 2),
4: kurtosis_element(mD, frac, 0, 0, 1, 1),
9: kurtosis_element(mD, frac, 0, 0, 2, 2),
36: kurtosis_element(mD, frac, 1, 1, 2, 2),
6: kurtosis_element(mD, frac, 0, 0, 1, 2),
12: kurtosis_element(mD, frac, 0, 1, 1, 2),
18: kurtosis_element(mD, frac, 0, 1, 2, 2)
}
# Testing all 81 possible elements
xyz = [0, 1, 2]
for i in xyz:
for j in xyz:
for k in xyz:
for l in xyz:
key = (i + 1) * (j + 1) * (k + 1) * (l + 1)
assert_almost_equal(kurtosis_element(mD, frac, i, k, j, l),
kt_ref[key])
# Testing optional function inputs
assert_almost_equal(
kurtosis_element(mD, frac, i, k, j, l),
kurtosis_element(mD, frac, i, k, j, l, D, MD))
def test_single_fiber_model():
# single fiber simulate (which is the assumption of our model)
fie = 0.49
ADi = 0.00099
ADe = 0.00226
RDi = 0
RDe = 0.00087
# 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)
# DKI fit
dkiM = dki_micro.DiffusionKurtosisModel(gtab_2s, fit_method="WLS")
dkiF = dkiM.fit(signal)
# Axonal Water Fraction
sphere = get_sphere('symmetric724')
AWF = dki_micro.axonal_water_fraction(dkiF.model_params,
sphere,
mask=None,
gtol=1e-5)
assert_almost_equal(AWF, fie)
# Extra-cellular and intra-cellular components
edt, idt = dki_micro.diffusion_components(dkiF.model_params, sphere)
EDT = eig_from_lo_tri(edt)
IDT = eig_from_lo_tri(idt)
# check eigenvalues
assert_array_almost_equal(EDT[0:3], np.array([ADe, RDe, RDe]))
assert_array_almost_equal(IDT[0:3], np.array([ADi, RDi, RDi]))
# first eigenvalue should be the direction of the fibers
fiber_direction = _check_directions([(theta, phi)])
f_norm = abs(np.dot(fiber_direction, np.array((EDT[3], EDT[6], EDT[9]))))
assert_almost_equal(f_norm, 1.)
f_norm = abs(np.dot(fiber_direction, np.array((IDT[3], IDT[6], IDT[9]))))
assert_almost_equal(f_norm, 1.)
# Test model and fit objects
wmtiM = dki_micro.KurtosisMicrostructureModel(gtab_2s, fit_method="WLS")
wmtiF = wmtiM.fit(signal)
assert_almost_equal(wmtiF.awf, AWF)
assert_array_almost_equal(wmtiF.hindered_evals, np.array([ADe, RDe, RDe]))
assert_array_almost_equal(wmtiF.restricted_evals, np.array([ADi, RDi,
RDi]))
assert_almost_equal(wmtiF.hindered_ad, ADe)
assert_almost_equal(wmtiF.hindered_rd, RDe)
assert_almost_equal(wmtiF.axonal_diffusivity, ADi)
assert_almost_equal(wmtiF.tortuosity, ADe / RDe, decimal=4)
# Test diffusion_components when a kurtosis tensors is associated with
# negative kurtosis values. E.g of this cases is given below:
dkiparams = np.array([
1.67135726e-03, 5.03651205e-04, 9.35365328e-05, -7.11167583e-01,
6.23186820e-01, -3.25390313e-01, -1.75247376e-02, -4.78415563e-01,
-8.77958674e-01, 7.02804064e-01, 6.18673368e-01, -3.51154825e-01,
2.18384153, -2.76378153e-02, 2.22893297, -2.68306546e-01, -1.28411610,
-1.56557645e-01, -1.80850619e-01, -8.33152110e-01, -3.62410766e-01,
1.57775442e-01, 8.73775381e-01, 2.77188975e-01, -3.67415502e-02,
-1.56330984e-01, -1.62295407e-02
])
edt, idt = dki_micro.diffusion_components(dkiparams)
assert_(np.all(np.isfinite(edt)))
def test_kurtosis_elements():
""" Testing symmetry of the elements of the KT
As an 4th order tensor, KT has 81 elements. However, due to diffusion
symmetry the KT is fully characterized by 15 independent elements. This
test checks for this property.
"""
# two fiber not aligned to planes x = 0, y = 0, or z = 0
mevals = np.array([[0.00099, 0, 0], [0.00226, 0.00087, 0.00087],
[0.00099, 0, 0], [0.00226, 0.00087, 0.00087]])
angles = [(80, 10), (80, 10), (20, 30), (20, 30)]
fie = 0.49 # intra axonal water fraction
frac = [fie * 50, (1-fie) * 50, fie * 50, (1-fie) * 50]
sticks = _check_directions(angles)
mD = np.zeros((len(frac), 3, 3))
for i in range(len(frac)):
R = all_tensor_evecs(sticks[i])
mD[i] = np.dot(np.dot(R, np.diag(mevals[i])), R.T)
# compute global DT
D = np.zeros((3, 3))
for i in range(len(frac)):
D = D + frac[i]*mD[i]
# compute voxel's MD
MD = (D[0][0] + D[1][1] + D[2][2]) / 3
# Reference dictionary with the 15 independent elements.
# Note: The multiplication of the indexes (i+1) * (j+1) * (k+1) * (l+1)
# for of an elements is only equal to this multiplication for another
# element if an only if the element corresponds to an symmetry element.
# Thus indexes multiplication is used as key of the reference dictionary
kt_ref = {1: kurtosis_element(mD, frac, 0, 0, 0, 0),
16: kurtosis_element(mD, frac, 1, 1, 1, 1),
81: kurtosis_element(mD, frac, 2, 2, 2, 2),
2: kurtosis_element(mD, frac, 0, 0, 0, 1),
3: kurtosis_element(mD, frac, 0, 0, 0, 2),
8: kurtosis_element(mD, frac, 0, 1, 1, 1),
24: kurtosis_element(mD, frac, 1, 1, 1, 2),
27: kurtosis_element(mD, frac, 0, 2, 2, 2),
54: kurtosis_element(mD, frac, 1, 2, 2, 2),
4: kurtosis_element(mD, frac, 0, 0, 1, 1),
9: kurtosis_element(mD, frac, 0, 0, 2, 2),
36: kurtosis_element(mD, frac, 1, 1, 2, 2),
6: kurtosis_element(mD, frac, 0, 0, 1, 2),
12: kurtosis_element(mD, frac, 0, 1, 1, 2),
18: kurtosis_element(mD, frac, 0, 1, 2, 2)}
# Testing all 81 possible elements
xyz = [0, 1, 2]
for i in xyz:
for j in xyz:
for k in xyz:
for l in xyz:
key = (i+1) * (j+1) * (k+1) * (l+1)
assert_almost_equal(kurtosis_element(mD, frac, i, k, j, l),
kt_ref[key])
# Testing optional funtion inputs
assert_almost_equal(kurtosis_element(mD, frac, i, k, j, l),
kurtosis_element(mD, frac, i, k, j, l,
D, MD))
def test_single_fiber_model():
# single fiber simulate (which is the assumption of our model)
fie = 0.49
ADi = 0.00099
ADe = 0.00226
RDi = 0
RDe = 0.00087
# 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)
# DKI fit
dkiM = dki_micro.DiffusionKurtosisModel(gtab_2s, fit_method="WLS")
dkiF = dkiM.fit(signal)
# Axonal Water Fraction
sphere = get_sphere('symmetric724')
AWF = dki_micro.axonal_water_fraction(dkiF.model_params, sphere, mask=None,
gtol=1e-5)
assert_almost_equal(AWF, fie)
# Extra-cellular and intra-cellular components
edt, idt = dki_micro.diffusion_components(dkiF.model_params, sphere)
EDT = eig_from_lo_tri(edt)
IDT = eig_from_lo_tri(idt)
# check eigenvalues
assert_array_almost_equal(EDT[0:3], np.array([ADe, RDe, RDe]))
assert_array_almost_equal(IDT[0:3], np.array([ADi, RDi, RDi]))
# first eigenvalue should be the direction of the fibers
fiber_direction = _check_directions([(theta, phi)])
f_norm = abs(np.dot(fiber_direction, np.array((EDT[3], EDT[6], EDT[9]))))
assert_almost_equal(f_norm, 1.)
f_norm = abs(np.dot(fiber_direction, np.array((IDT[3], IDT[6], IDT[9]))))
assert_almost_equal(f_norm, 1.)
# Test model and fit objects
wmtiM = dki_micro.KurtosisMicrostructureModel(gtab_2s, fit_method="WLS")
wmtiF = wmtiM.fit(signal)
assert_almost_equal(wmtiF.awf, AWF)
assert_array_almost_equal(wmtiF.hindered_evals,
np.array([ADe, RDe, RDe]))
assert_array_almost_equal(wmtiF.restricted_evals,
np.array([ADi, RDi, RDi]))
assert_almost_equal(wmtiF.hindered_ad, ADe)
assert_almost_equal(wmtiF.hindered_rd, RDe)
assert_almost_equal(wmtiF.axonal_diffusivity, ADi)
assert_almost_equal(wmtiF.tortuosity, ADe/RDe, decimal=4)
# Test diffusion_components when a kurtosis tensors is associated with
# negative kurtosis values. E.g of this cases is given below:
dkiparams = np.array([1.67135726e-03, 5.03651205e-04, 9.35365328e-05,
-7.11167583e-01, 6.23186820e-01, -3.25390313e-01,
-1.75247376e-02, -4.78415563e-01, -8.77958674e-01,
7.02804064e-01, 6.18673368e-01, -3.51154825e-01,
2.18384153, -2.76378153e-02, 2.22893297,
-2.68306546e-01, -1.28411610, -1.56557645e-01,
-1.80850619e-01, -8.33152110e-01, -3.62410766e-01,
1.57775442e-01, 8.73775381e-01, 2.77188975e-01,
-3.67415502e-02, -1.56330984e-01, -1.62295407e-02])
edt, idt = dki_micro.diffusion_components(dkiparams)
assert_(np.all(np.isfinite(edt)))
