def heat_finite_elt_2D_tensor2(x0, t_final, t_step, h, g):
"""
:param x0: vector x (at t = 0)
:param t_final: time
:param t_step: time step (must satisfy the CFL max(lambda) < 2)
:param h:
:param g: metric tensor
:return: vector x (at t = t_final)
"""
import clinica.pipelines.machine_learning_spatial_svm.spatial_svm_utils as utils
import numpy as np
# parameters
nb_step = np.ceil(t_final / t_step) # number of time step
t_step = t_final / nb_step
# tensors
detg = utils.tensor_determinant(g)
detg = np.sqrt(detg)
ginv = utils.tensor_inverse(g)
ginv = utils.tensor_scalar_product(detg, ginv)
detg2 = detg[1:-1, 1:-1]
# LOOP
x = x0
for i in range(nb_step):
m = t_step / h / h
x = np.sum(x, -(
utils.tensor_scalar_product(m,
np.divide(utils.tensor_scalar_product(h, utils.operateur(x, ginv, detg)), detg2, dtype=object))))
return x
def largest_eigenvalue_heat_3D_tensor2(g, h, epsilon):
"""
:param g: metric tensor
:param h: space step
:param epsilon: stop criterion
:return: lamba = the largest eigenvalues
"""
import clinica.pipelines.machine_learning_spatial_svm.spatial_svm_utils as utils
import numpy as np
import cmath
# parameters
if epsilon is None:
epsilon = 1e-6
erreur = 1 + epsilon
# tensors
detg = utils.tensor_determinant(g)
detg = np.array(detg, dtype=np.complex128) # complex tensor
detg = np.sqrt(detg)
detg = detg[0]
ginv = utils.tensor_inverse(g)
if len(ginv.shape) == 6:
ginv = ginv[:, :, 0, :, :, :]
ginv = utils.tensor_scalar_product(detg, ginv)
detg2 = detg[1:-1, 1:-1, 1:-1] # 141*121*141
detg2[detg2 != detg2] = 0
detg[detg != detg] = 0
ginv[ginv != ginv] = 0
# initialisation
s = [g[0][0].shape[0] - 2, g[0][0].shape[1] - 2, g[0][0].shape[2] - 2]
b1 = np.ones([s[0], s[1], s[2]])
b1 = np.divide(b1, np.array(cmath.sqrt(np.dot(b1.flatten('F').transpose(), b1.flatten('F'))), dtype=np.complex128))
print("Computation of the largest eigenvalue ...")
while erreur > epsilon:
b0 = b1
b2 = np.array(np.divide(np.array(utils.operateur(b1, ginv, detg)) * h, detg2) / h / h / h, dtype=np.complex128)
b1 = np.divide(b2, np.array(cmath.sqrt(np.dot(b2.flatten('F').transpose(), b2.flatten('F')))),
dtype=np.complex128)
erreur = np.linalg.norm(b1.flatten('F') - b0.flatten('F'))
print("done")
lam = (cmath.sqrt(np.dot(b2.flatten('F').transpose(), b2.flatten('F'))))
return lam
def heat_finite_elt_3D_tensor2(x0, t_final, t_step, h, g):
"""
:param x0: vector x (at t = 0)
:param t_final: time
:param t_step: time step (must satisfy the CFL max(lambda) < 2)
:param h:
:param g: metric tensor
:return: vector x (at t = t_final)
"""
import numpy as np
import clinica.pipelines.machine_learning_spatial_svm.spatial_svm_utils as utils
if len(x0.shape) == 4:
x0 = x0[0, :, :, :]
# parameters
nb_step = np.ceil(t_final / t_step) # number of time step
nb_step = nb_step.astype(int)
t_step = t_final / nb_step
# tensors
detg = utils.tensor_determinant(g)
detg = np.sqrt(detg)
ginv = utils.tensor_inverse(g)
ginv = utils.tensor_scalar_product(detg, ginv)
detg2 = detg[:, 1:-1, 1:-1, 1:-1]
if len(ginv.shape) == 6:
ginv = ginv[:, :, 0, :, :, :]
if len(detg.shape) == 4:
detg = detg[0, :, :, :]
ginv = np.array(ginv.real, dtype="float64")
detg = np.array(detg.real, dtype="float64")
detg2 = np.array(detg2.real, dtype="float64")
# LOOP
x = x0
for i in range(nb_step):
x = np.array(
x
- t_step
* (np.divide(np.array(utils.operateur(x, ginv, detg)) * h, detg2))
/ h
/ h
/ h
)
return x