def construct_matrix_of_integrals(g, order, delta):
logging.debug("Constructing matrix of Basis-Monomial Integrals...")
# Unit vectors [321,3]
g2 = numpy.loadtxt(get_sphere("symmetric321"))
# Construct matrix of integrals
N = g.shape[0]
Bg = numpy.zeros((N, factorial(2 + order) / (2 * factorial(order))),
dtype=numpy.single)
for k in range(N):
c = 0
for i in range(order + 1):
for j in range(order + 1 - i):
Bg[k, c] = 0
for integr in range(321):
Bg[k, c] = (
Bg[k, c] + numpy.exp(- delta * numpy.dot(
g2[integr, :], g[k, :]) ** 2) *
(g2[integr, 0] ** i) * (g2[integr, 1] ** j) *
(g2[integr, 2] ** (order - i - j)))
Bg[k, c] = Bg[k, c] / 321
c = c + 1
return Bg
def tensor2odf(tensors, b, order):
""" Compute a Cartesian Tensor ODF from a given Higher-order diffusion
tensor
Parameters
----------
tensors: list of array [e,] (mandatory)
a list with tensor independent coefficients
b: float (mandatory)
the diffusion b-value
Returns
-------
odfs list of array [e,]
a list of tensor independant coefficients
Reference
Barmpoutnis
"""
# Unit vectors [321,3]
g = numpy.loadtxt(get_sphere("symmetric321"))
# Construct basis
G = construct_matrix_of_monomials(g, order)
C = construct_set_of_polynomials(order).T
BG = construct_matrix_of_integrals(g, order, 100)
B = numpy.dot(BG, C)
# loop over elements
odfs = []
for tensor in tensors:
x = scipy.optimize.nnls(B, numpy.exp(-b * numpy.dot(G, tensor)))[0]
odfs.append(numpy.dot(C, x))
return odfs
def tensor2odf(tensors, b, order):
""" Compute a Cartesian Tensor ODF from a given Higher-order diffusion
tensor
Parameters
----------
tensors: list of array [e,] (mandatory)
a list with tensor independent coefficients
b: float (mandatory)
the diffusion b-value
Returns
-------
odfs list of array [e,]
a list of tensor independant coefficients
Reference
Barmpoutnis
"""
# Unit vectors [321,3]
g = numpy.loadtxt(get_sphere("symmetric321"))
# Construct basis
G = construct_matrix_of_monomials(g, order)
C = construct_set_of_polynomials(order).T
BG = construct_matrix_of_integrals(g, order, 100)
B = numpy.dot(BG, C)
# loop over elements
odfs = []
for tensor in tensors:
x = scipy.optimize.nnls(B, numpy.exp(-b * numpy.dot(G, tensor)))[0]
odfs.append(numpy.dot(C, x))
return odfs
def construct_set_of_polynomials(order):
""" Construct the coefficients of homogenous polynomials in 3
variables of a given order, which correspond to squares of lower
(half) order polynomials.
Parameters
----------
order: odd int (mandatory)
reconstruction order
Returns
-------
C: array [M,(2+order)!/2(order)!]
the computed list of polynomial coefficients
"""
logging.debug("Constructing matrix of 321 polynomials...")
# first load the unit vectors [321,3]
g = numpy.loadtxt(get_sphere("symmetric321"))
M = 321
# Get the multiplicity of each independant element
multiplicity = numpy.zeros((order + 1, order + 1, order + 1),
dtype=numpy.single)
for i in range(order + 1):
for j in range(order - i + 1):
multiplicity[i, j, order - i - j] = get_coefficient_multiplicity(
order, i, j, order - i - j)
e = factorial(2 + order) / (2 * factorial(order))
C = numpy.zeros((M, e), dtype=numpy.single)
for k in range(M):
c = 0
for i in range(order + 1):
for j in range(order - i + 1):
C[k, c] = (multiplicity[i, j, order - i - j] *
(g[k, 0] ** i) * (g[k, 1] ** j) *
(g[k, 2] ** (order - i - j)))
c += 1
return C
def construct_set_of_polynomials(order):
""" Construct the coefficients of homogenous polynomials in 3
variables of a given order, which correspond to squares of lower
(half) order polynomials.
Parameters
----------
order: odd int (mandatory)
reconstruction order
Returns
-------
C: array [M,(2+order)!/2(order)!]
the computed list of polynomial coefficients
"""
logging.debug("Constructing matrix of 321 polynomials...")
# first load the unit vectors [321,3]
g = numpy.loadtxt(get_sphere("symmetric321"))
M = 321
# Get the multiplicity of each independant element
multiplicity = numpy.zeros((order + 1, order + 1, order + 1),
dtype=numpy.single)
for i in range(order + 1):
for j in range(order - i + 1):
multiplicity[i, j, order - i - j] = get_coefficient_multiplicity(
order, i, j, order - i - j)
e = factorial(2 + order) / (2 * factorial(order))
C = numpy.zeros((M, e), dtype=numpy.single)
for k in range(M):
c = 0
for i in range(order + 1):
for j in range(order - i + 1):
C[k, c] = (multiplicity[i, j, order - i - j] * (g[k, 0]**i) *
(g[k, 1]**j) * (g[k, 2]**(order - i - j)))
c += 1
return C
def construct_matrix_of_integrals(g, order, delta):
logging.debug("Constructing matrix of Basis-Monomial Integrals...")
# Unit vectors [321,3]
g2 = numpy.loadtxt(get_sphere("symmetric321"))
# Construct matrix of integrals
N = g.shape[0]
Bg = numpy.zeros((N, factorial(2 + order) / (2 * factorial(order))),
dtype=numpy.single)
for k in range(N):
c = 0
for i in range(order + 1):
for j in range(order + 1 - i):
Bg[k, c] = 0
for integr in range(321):
Bg[k, c] = (Bg[k, c] + numpy.exp(
-delta * numpy.dot(g2[integr, :], g[k, :])**2) *
(g2[integr, 0]**i) * (g2[integr, 1]**j) *
(g2[integr, 2]**(order - i - j)))
Bg[k, c] = Bg[k, c] / 321
c = c + 1
return Bg
