def derivative(self, displacement):
"""Derivative of the operator at ``displacement``.
Parameters
----------
displacement : `domain` `element-like`
Point at which the derivative is computed.
Returns
-------
derivative : `PointwiseInner`
The derivative evaluated at ``displacement``.
"""
# To implement the complex case we need to be able to embed the real
# vector field space into the range of the gradient. Issue #59.
if not self.range.is_rn:
raise NotImplementedError('derivative not implemented for complex '
'spaces.')
displacement = self.domain.element(displacement)
# TODO: allow users to select what method to use here.
grad = Gradient(domain=self.range, method='central',
pad_mode='symmetric')
grad_templ = grad(self.template)
def_grad = self.domain.element(
[_linear_deform(gf, displacement) for gf in grad_templ])
return PointwiseInner(self.domain, def_grad)
def __init__(self, DomainField, Ntrans, kernel, partialderdim1kernel,
partialder2kernel):
"""Initialize a new instance.
DomainField : space on wich vector fields will be defined
Ntrans : number of translations
Kernel : kernel
partialderdim1kernel : partial derivative with respect to one componenet of
the 1st component,
to be used as partialder2kernel(x, y, d) where d is in [|0, dim-1|]
partialder2kernel : partial derivative with respect to 2nd component,
to be used as partialder2kernel(x, y, u) with x and y points, and u
vectors for differentiation (same number of vectors as the number of
points for y)
"""
self.Ntrans = Ntrans
self.kernel = kernel
self.partialderdim1kernel = partialderdim1kernel
self.partialder2kernel = partialder2kernel
self.dim = DomainField.ndim
self.get_unstructured_op = usefun.get_from_structured_to_unstructured(
DomainField, kernel)
self.gradient_op = Gradient(DomainField)
GDspace = odl.ProductSpace(odl.space.rn(self.dim), self.Ntrans)
Contspace = odl.ProductSpace(odl.space.rn(self.dim), self.Ntrans)
self.dimCont = self.Ntrans * self.dim
super().__init__(GDspace, Contspace, DomainField)
def _call(self, X):
""" Shooting equations and Integration
Args: X = [I0, p] with I0 template image and p scalarvalued momentum
Returns:
I: deformed Image over time (N+1 timesteps)
M: vectorvalued momentum in eulerian coordinates over time
V: vectorfields over time
"""
template = X[0]
P0 = X[1]
series_image_space_integration = ProductSpace(template.space,
self.N + 1)
series_vector_space_integration = ProductSpace(
template.space.tangent_bundle, self.N + 1)
inv_N = 1 / self.N
I = series_image_space_integration.element()
I[0] = template.copy()
P = series_image_space_integration.element()
P[0] = P0.copy()
U = series_vector_space_integration.element()
# Create the gradient op
grad_op = Gradient(domain=self.space,
method='forward',
pad_mode='symmetric')
# Create the divergence op
div_op = -grad_op.adjoint
for i in range(self.N):
# compute vectorvalued momentum from scalarvalued momentum
gradI = grad_op(I[i])
m = self.space.tangent_bundle.element(-P[i] * gradI)
# Kernel convolution to obtain velocity fields from momentum
# u = K*m
U[i] = (2 * np.pi)**(
self.dim / 2.0) * self.vectorial_ft_fit_op.inverse(
self.vectorial_ft_fit_op(m) * self.ft_kernel_fitting)
# Integration step
dtI = -sum(gradI * U[i])
dtP = -div_op(P[i] * U[i])
I[i + 1] = I[i] + inv_N * dtI
P[i + 1] = P[i] + inv_N * dtP
U[self.N] = (2 * np.pi)**(
self.dim / 2.0) * self.vectorial_ft_fit_op.inverse(
self.vectorial_ft_fit_op(
self.space.tangent_bundle.element(-P[self.N] * gradI)) *
self.ft_kernel_fitting)
return I, P, U
def ConvolveIntegrate(self,grad_S_init,H,j0,vector_field_list,zeta_list):
dim = self.image_domain.ndim
k_j=self.k_j_list[j0]
h=odl.ProductSpace(self.image_domain.tangent_bundle,k_j+1).zero()
eta=odl.ProductSpace(self.image_domain,k_j+1).zero()
grad_op = Gradient(domain=self.image_domain, method='forward',
pad_mode='symmetric')
# Create the divergence op
div_op = -grad_op.adjoint
delta0=self.data_time_points[j0] -(k_j/self.N)
detDphi=self.image_domain.element(
1+delta0 *
div_op(vector_field_list[k_j])).copy()
grad_S=self.image_domain.element(
linear_deform(grad_S_init,
delta0 * vector_field_list[k_j])).copy()
if (not delta0==0):
tmp=H[k_j].copy()
tmp1=(grad_S * detDphi).copy()
for d in range(dim):
tmp[d] *= tmp1
tmp3=(2 * np.pi) ** (dim / 2.0) * self.vectorial_ft_fit_op.inverse(self.vectorial_ft_fit_op(tmp) * self.ft_kernel_fitting)
h[k_j]+=tmp3.copy()
eta[k_j]+=detDphi*grad_S
delta_t= self.inv_N
for u in range(k_j):
k=k_j -u-1
detDphi=self.image_domain.element(
linear_deform(detDphi,
delta_t*vector_field_list[k])).copy()
detDphi=self.image_domain.element(detDphi*
self.image_domain.element(1+delta_t *
div_op(vector_field_list[k]))).copy()
grad_S=self.image_domain.element(
linear_deform(grad_S,
delta_t * vector_field_list[k])).copy()
tmp=H[k].copy()
tmp1=(grad_S * detDphi).copy()
for d in range(dim):
tmp[d] *= tmp1.copy()
tmp3=(2 * np.pi) ** (dim / 2.0) * self.vectorial_ft_fit_op.inverse(self.vectorial_ft_fit_op(tmp) * self.ft_kernel_fitting)
h[k]+=tmp3.copy()
eta[k]+=detDphi*grad_S
return [h,eta]
def __init__(self, DomainField, GDspace, NbGD, kernel,
partialderdim1kernel, partialder2kernel, pointfunctionlist,
pointfunctiondifflist, vectorfunctionlist,
vectorfunctiondifflist):
"""Initialize a new instance.
DomainField : space on wich vector fields will be defined
dimGD = dimension of GDspace
Kernel : kernel
partialderdim1kernel : partial derivative with respect to one componenet of
the 1st component,
to be used as partialder2kernel(x, y, d) where d is in [|0, dim-1|]
partialder2kernel : partial derivative with respect to 2nd component,
to be used as partialder2kernel(x, y, u) with x and y points, and u
vectors for differentiation (same number of vectors as the number of
points for y)
get_columnvector_gd : function that takes a GD and return a vector
so that when we multiply it by a point/vectorfunctiondiff we
obtain its differential
get_spaceelement_gd : function, inverse of get_columnvector_gd
(takes a vector and return the corresponding GD)
pointfunctionlist : list of functions, each one associating a list of
points to a geometrical descriptor
pointfunctiondifflist : list of functions, each one associating a list of
matrix to a geometrical descriptor, the k-th matrix is the differential
of the k-th point with respect to the geometrical descriptor
vectorfunctionlist : list of functions, each one associating a list of
vectors to a geometrical descriptor
vectorfunctiondifflist : list of functions, each one associating a list of
matrix to a geometrical descriptor, the k-th matrix is the differential
of the k-th vector with respect to the geometrical descriptor
"""
self.kernel = kernel
self.partialderdim1kernel = partialderdim1kernel
self.partialder2kernel = partialder2kernel
self.dim = DomainField.ndim
self.NbGD = NbGD
self.dimGD = NbGD * self.dim
self.get_unstructured_op = usefun.get_from_structured_to_unstructured(
DomainField, kernel)
self.gradient_op = Gradient(DomainField)
self.ptfun = pointfunctionlist
self.vecfun = vectorfunctionlist
self.ptfundiff = pointfunctiondifflist
self.vecfundiff = vectorfunctiondifflist
self.dimCont = len(pointfunctionlist)
Contspace = odl.space.rn(self.dimCont)
super().__init__(GDspace, Contspace, DomainField)
def __init__(self, DomainField, inv_N):
"""Initialize a new instance.
DomainField : space to which GD belongs
invN : used for small differences for applying vector field
(supposed to be stepsize for integration)
"""
self.dim = DomainField.ndim
self.inv_N = inv_N
self.grad_op = Gradient(DomainField)
GDspace = DomainField
Contspace = odl.space.rn(0)
self.dimCont = 0
super().__init__(GDspace, Contspace, DomainField)
def _call(self, X):
"""
Compute Gradient of the Energy functional
Gradient of regularization term norm(v): K*m
Gradient of data term: mhat
"""
dim=2
Imv = self.operator.Shooting(X)
I = Imv[0]
m = Imv[1]
v = Imv[2]
N = I.shape[0] - 1
mhat = m[0].space.zero()
Ihat = self.operator.reg_param * self.operator.attach.gradient(I[-1])
# Create the gradient op
grad_op = Gradient(domain=self.operator.space, method='forward',pad_mode='constant', pad_const = 0)
# Create the divergence op
div_op = -grad_op.adjoint
for i in range(N):
gradmhat0 = grad_op(mhat[0])
gradmhat1 = grad_op(mhat[1])
gradmhat = [gradmhat0, gradmhat1]
gradI = grad_op(I[N-i])
coad0 = sum(grad_op(m[N-i][0])*mhat) + sum([gradmhat[j][0] * m[N-i][j] for j in range(dim)]) + div_op(mhat)*m[N-i][0]
coad1 = sum(grad_op(m[N-i][1])*mhat) + sum([gradmhat[j][1] * m[N-i][j] for j in range(dim)]) + div_op(mhat)*m[N-i][1]
coad_mhat_m = mhat.space.element([coad0, coad1])
vhat = (2 * np.pi) ** (dim / 2.0) * self.vectorial_ft_fit_op.inverse(self.vectorial_ft_fit_op(-coad_mhat_m + Ihat * gradI) * self.ft_kernel_fitting)
Ihat = Ihat + 1/N * div_op(Ihat * v[N-i])
advm0 = sum(grad_op(v[N-i][0])*mhat) - sum(grad_op(mhat[0])*v[N-i])
advm1 = sum(grad_op(v[N-i][1])*mhat) - sum(grad_op(mhat[1])*v[N-i])
advm = mhat.space.element([advm0, advm1])
mhat = mhat - 1/N*(advm + vhat)
Km = (2 * np.pi) ** (dim / 2.0) * self.vectorial_ft_fit_op.inverse(self.vectorial_ft_fit_op(m[0]) * self.ft_kernel_fitting)
return Km + mhat
def _call(self, X):
vector_field_list=X[0]
zeta_list=X[1]
h=vector_field_list.space.zero()
eta=zeta_list.space.zero()
# Compute the metamorphosis at the data time points
# zeta_transp [k] = zeta_list [k] \circ \varphi_{\tau_k}
zeta_transp=ShootSourceTermBackwardlist(vector_field_list, zeta_list).copy()
image_list=functional.ComputeMetamorphosisFromZetaTransp(vector_field_list,zeta_transp).copy()
template_deformation=ShootTemplateFromVectorFields(vector_field_list,functional.template).copy()
# Computation of G_k=\nabla (template \circ \phi_{\tau_k}^-1) for each k
G=odl.ProductSpace(functional.image_domain.tangent_bundle, functional.N +1).element()
grad_op = Gradient(domain=functional.image_domain, method='forward',
pad_mode='symmetric')
for k in range(functional.N +1):
G[k]=grad_op(template_deformation[k]).copy()
# Addition of \int_0^\tau_k \nabla (\zeta(\tau) \circ \phi_{\tau_k , \tau}) d\tau
for k in range(functional.N +1):
#G[k]+= (functional.inv_N * grad_op(zeta_list[k])).copy()
for q in range(k):
p=k-1-q
temp=ShootTemplateFromVectorFieldsFinal(vector_field_list,zeta_transp[p],0,k).copy()
G[k]+= (functional.inv_N * grad_op(temp)).copy()
for j in range(functional.nb_data):
conv=functional.ConvolveIntegrate((functional.Norm_list[j]*(functional.forward_op[j] - functional.data[j])).gradient(image_list[j]),G,j,vector_field_list,zeta_list).copy()
#conv=functional.ConvolveIntegrate(functional.S[j].gradient(image_list[j]),G,j,vector_field_list,zeta_list).copy()
for k in range(functional.k_j_list[j]):
h[k]-=conv[0][k].copy()
eta[k]+=conv[1][k].copy()
for k in range(functional.N):
h[k]+=2*functional.lamb*vector_field_list[k]
eta[k]+=2*functional.tau*zeta_list[k]
return [h,eta]
def LDDMM_Beg_solver(template,
reference,
time_pts,
niter,
eps,
lamb,
kernel,
callback=None):
"""
Solver for the shape-based reconstruction using LDDMM_Beg.
Notes
-----
The model is:
.. math:: \min_{v(t) \in V, \phi_{0,1}^v \in G_v} \lambda \int_0^1 \|v(t)\|_V^2 dt + \int_\Omega |(\phi_{0,1}^v.I_0) - I_1|^2 dx
Here :math:`I_0` is the template. :math:`v(t)` is the
velocity vector. :math:`V` is a reproducing kernel Hilbert space for
the velocity vector. :math:`I_1` is the reference. :math:`\lambda` is
the regularization parameter. :math:`\phi_{0, 1}^v.I
:= I \circ \phi_{1, 0}^v` is for geometric deformation,
:math:`\phi_{1, 0}^v` is the inverse of
:math:`\phi_{0, 1}^v` i.e. the solution at :math:`t=1` of flow of
diffeomorphisms. :math:`|D\phi_{1, 0}^v|` is the Jacobian determinant
of :math:`\phi_{1, 0}^v`.
Parameters
----------
template : `DiscreteLpElement`
Fixed template.
reference: `DiscreteLpElement`
Fixed reference.
time_pts : `int`
The number of time intervals
niter : `int`
The given maximum iteration number.
eps : `float`
The given step size.
lamb : `float`
The given regularization parameter. It's a weighted value on the
regularization-term side.
kernel : `function`
Kernel function in reproducing kernel Hilbert space.
callback : `class`, optional
Show the intermediate results of iteration.
Returns
-------
image_N0 : `ProductSpaceElement`
The series of images produced by template and velocity field.
E : `numpy.array`
Storage of the energy values for iterations.
"""
# Give the number of time intervals
N = time_pts
# Get the inverse of time intervals
inv_N = 1.0 / N
# Create the space of images
image_space = template.space
#image_space= rec_space
# Get the dimension of the space of images
dim = image_space.ndim
# Fourier transform setting for data matching term
# The padded_size is the size of the padded domain
padded_size = 2 * image_space.shape[0]
# Create operator of Fourier transform composing with padded operator
pad_ft_op = _padded_ft_op(image_space, padded_size)
# Create vectorial Fourier transform operator
# Construct the diagnal element of a matrix operator
vectorial_ft_op = DiagonalOperator(*([pad_ft_op] * dim))
# Compute the FT of kernel in fitting term
discretized_kernel = _vectorized_kernel(image_space, kernel)
ft_kernel_fitting = vectorial_ft_op(discretized_kernel)
# Create the space for series deformations and series Jacobian determinant
pspace = image_space.tangent_bundle
series_pspace = ProductSpace(pspace, N + 1)
series_image_space = ProductSpace(image_space, N + 1)
# Initialize vector fileds at different time points
vector_fields = series_pspace.zero()
# Give the initial two series deformations and series Jacobian determinant
image_N0 = series_image_space.element()
image_N1 = series_image_space.element()
detDphi_N1 = series_image_space.element()
for i in range(N + 1):
image_N0[i] = image_space.element(template)
image_N1[i] = image_space.element(reference)
detDphi_N1[i] = image_space.one()
# Create the gradient operator
grad_op = Gradient(domain=image_space,
method='forward',
pad_mode='symmetric')
# Create the divergence operator, which can be obtained from
# the adjoint of gradient operator
# div_op = Divergence(domain=pspace, method='forward', pad_mode='symmetric')
div_op = -grad_op.adjoint
# Store energy
E = []
kE = len(E)
E = np.hstack((E, np.zeros(niter)))
# Begin iteration
for k in range(niter):
# Update the velocity field
for i in range(N + 1):
#first term
term1_tmp1 = 2 * np.abs(image_N0[i] -
image_N1[i]) * (detDphi_N1[i])
term1_tmp = grad_op(image_N0[i])
for j in range(dim):
term1_tmp[j] *= term1_tmp1
tmp3 = (2 * np.pi)**(dim / 2.0) * vectorial_ft_op.inverse(
vectorial_ft_op(term1_tmp) * ft_kernel_fitting)
vector_fields[i] = (vector_fields[i] - eps *
(lamb * vector_fields[i] - tmp3))
# Update image_N0,image_N1 and detDphi_N0 detDphi_N1
for i in range(N):
# Update image_N0[i+1] by image_N0[i] and vector_fields[i+1]
image_N0[i + 1] = image_space.element(
_linear_deform(image_N0[i], -inv_N * vector_fields[i + 1]))
# Update image_N1[N-i-1] by image_N1[N-i] and vector_fields[N-i-1]
image_N1[N - i - 1] = image_space.element(
_linear_deform(image_N1[N - i],
inv_N * vector_fields[N - i - 1]))
# # Update detDphi_N1[N-i-1] by detDphi_N1[N-i]
# jacobian_det = image_domain.element(
# np.exp(inv_N * div_o p(vector_fields[N-i-1])))
jacobian_det = image_space.element(
1.0 + inv_N * div_op(vector_fields[N - i - 1]))
detDphi_N1[N - i - 1] = (jacobian_det * image_space.element(
_linear_deform(detDphi_N1[N - i],
inv_N * vector_fields[N - i - 1])))
# Update the deformed template
PhiStarI = image_N0[N]
# Show intermediate result
if callback is not None:
callback(PhiStarI)
# Compute the energy of the data fitting term
E[k + kE] += np.asarray((PhiStarI - reference)**2).sum()
return image_N0, E
def _call(self, X):
""" Shooting equations and Integration
Args: X = [I0, m] with I0 template image and m vectorvalued momentum
Returns:
I: deformed Image over time (N+1 timesteps)
M: vectorvalued momentum in eulerian coordinates over time
V: vectorfields over time
"""
template = X[0].copy()
m0 = X[1].copy()
# create spaces for time-dependent Image and vectorfields
series_image_space_integration = ProductSpace(template.space,
self.N + 1)
series_vector_space_integration = ProductSpace(m0.space, self.N + 1)
inv_N = 1 / self.N
I = series_image_space_integration.element()
I[0] = template.copy()
M = series_vector_space_integration.element()
M[0] = m0.copy()
U = series_vector_space_integration.element()
# Create the gradient op
grad_op = Gradient(domain=self.space,
method='forward',
pad_mode='symmetric')
# Create the divergence op
div_op = -grad_op.adjoint
for i in range(self.N):
m = M[i]
# Kernel convolution to obtain vectorfields from momentum
# v = K*m
U[i] = (2 * np.pi)**(
self.dim / 2.0) * self.vectorial_ft_fit_op.inverse(
self.vectorial_ft_fit_op(m) * self.ft_kernel_fitting)
u = U[i]
# shooting equation
# d/dt m = - ad*_v m
gradu = [grad_op(u[0]), grad_op(u[1])]
dtM0 = sum(grad_op(m[0]) * u) + sum(
[gradu[i][0] * m[i]
for i in range(self.dim)]) + div_op(u) * m[0]
dtM1 = sum(grad_op(m[1]) * u) + sum(
[gradu[i][1] * m[i]
for i in range(self.dim)]) + div_op(u) * m[1]
# Integration step
M[i + 1] = M[i] - inv_N * m0.space.element([dtM0, dtM1])
I[i + 1] = odl.deform.linear_deform(I[i], -inv_N * u)
U[self.N] = (2 * np.pi)**(
self.dim / 2.0) * self.vectorial_ft_fit_op.inverse(
self.vectorial_ft_fit_op(M[self.N]) * self.ft_kernel_fitting)
return I, M, U
def infinitesimal_action_withgrad(v, I):
grad_op = Gradient(I.space)
grad = grad_op(I)
return -sum(v * grad)
def LDDMM_gradient_descent_solver_spatiotemporal(forward_op,
noise_proj_data,
template,
vector_fields,
gate_pts,
discr_deg,
niter,
in_niter1,
in_inter2,
stepsize1,
stepsize2,
mu_1,
mu_2,
lamb,
kernel,
impl1='geom',
impl2='least_square',
callback=None):
"""
Solver for spatiotemporal image reconstruction using LDDMM.
Notes
-----
The model is:
.. math:: \min_{v} \lambda * \int_0^1 \|v(t)\|_V^2 dt + \|T(\phi_1.I) - g\|_2^2,
where :math:`\phi_1.I := |D\phi_1^{-1}| I(\phi_1^{-1})` is for
mass-preserving deformation, instead, :math:`\phi_1.I := I(\phi_1^{-1})`
is for geometric deformation. :math:`\phi_1^{-1}` is the inverse of
the solution at :math:`t=1` of flow of doffeomorphisms.
:math:`|D\phi_1^{-1}|` is the Jacobian determinant of :math:`\phi_1^{-1}`.
:math:`T` is the forward operator. If :math:`T` is an identity operator,
the above model reduces to image matching. If :math:`T` is a non-identity
forward operator, the above model is for shape-based image reconstrction.
:math:`g` is the detected data, `data_elem`. :math:`I` is the `template`.
:math:`v(t)` is the velocity vector. :math:`V` is a reproducing kernel
Hilbert space for velocity vector. :math:`lamb` is the regularization
parameter.
Parameters
----------
forward_op : `Operator`
The forward operator of imaging.
data_elem : `DiscreteLpElement`
The given data.
template : `DiscreteLpElement`
Fixed template deformed by the deformation.
time_pts : `int`
The number of time intervals
iter : `int`
The given maximum iteration number.
eps : `float`
The given step size.
lamb : `float`
The given regularization parameter. It's a weighted value on the
regularization-term side.
kernel : `function`
Kernel function in reproducing kernel Hilbert space.
impl1 : {'geom', 'mp'}, optional
The given implementation method for group action.
The impl1 chooses 'mp' or 'geom', where 'mp' means using
mass-preserving method, and 'geom' means using
non-mass-preserving geometric method. Its defalt choice is 'geom'.
impl2 : {'least square'}, optional
The given implementation method for data matching term.
Here the implementation only supports the case of least square.
callback : `class`, optional
Show the intermediate results of iteration.
Returns
-------
image_N0 : `ProductSpaceElement`
The series of images produced by template and velocity field.
mp_deformed_image_N0 : `ProductSpaceElement`
The series of mass-preserving images produced by template
and velocity field.
E : `numpy.array`
Storage of the energy values for iterations.
"""
# Max index of discretized points
N = gate_pts
# Discretized degree
M = discr_deg
# Compute the max index of discretized points
MN = M * N
MN1 = MN + 1
# Get the inverse of the number of discretized points
inv_MN = 1.0 / MN
N1 = N + 1
N2 = 2. / N
ss1 = stepsize1 * N2
ss2 = stepsize2 * N2
ss3 = stepsize1 * mu_1
# Create the gradient operator for the squared L2 functional
if impl2 == 'least_square':
gradS = [forward_op[0].adjoint *
(forward_op[0] - noise_proj_data[0])] * N1
for i in range(N):
j = i + 1
gradS[j] = forward_op[j].adjoint * (forward_op[j] -
noise_proj_data[j])
else:
raise NotImplementedError('now only support least square')
# Create the space of images
image_space = template.space
# Get the dimension of the space of images
dim = image_space.ndim
# Fourier transform setting for data matching term
# The padded_size is the size of the padded domain
padded_size = 2 * image_space.shape[0]
# The pad_ft_op is the operator of Fourier transform
# composing with padded operator
pad_ft_op = padded_ft_op(image_space, padded_size)
# The vectorial_ft_op is a vectorial Fourier transform operator,
# which constructs the diagnal element of a matrix.
vectorial_ft_op = DiagonalOperator(*([pad_ft_op] * dim))
# Compute the FT of kernel in fitting term
discretized_kernel = fitting_kernel(image_space, kernel)
ft_kernel_fitting = vectorial_ft_op(discretized_kernel)
# Create the space for series deformations and series Jacobian determinant
series_image_space = ProductSpace(image_space, MN1)
series_backprojection_space = ProductSpace(image_space, N1)
series_bp_all = [image_space.element()] * N1
for i in range(N):
j = i + 1
series_bp_all[j] = [image_space.element()] * (j * M + 1)
# Initialize vector fileds at different time points
vector_fields = vector_fields
# Initialize two series deformations and series Jacobian determinant
image_MN0 = series_image_space.element()
if impl1 == 'geom':
eta_tt = series_backprojection_space.element()
else:
raise NotImplementedError('unknown group action')
for j in range(MN1):
image_MN0[j] = image_space.element(template)
eta_tt[0] = gradS[0](image_MN0[0])
series_bp_all[0] = eta_tt[0]
for i in range(1, N1):
iM = i * M
eta_tt[i] = gradS[i](image_MN0[iM])
for j in range(iM + 1):
series_bp_all[i][j] = eta_tt[i]
# Create the gradient operator
grad_op = Gradient(domain=image_space,
method='forward',
pad_mode='symmetric')
# Create the divergence operator, which can be obtained from
# the adjoint of gradient operator
# div_op = Divergence(domain=pspace, method='forward', pad_mode='symmetric')
grad_op_adjoint = grad_op.adjoint
div_op = -grad_op.adjoint
# Begin iteration for non-mass-preserving case
if impl1 == 'geom':
print(impl1)
# Outer iteration
for k in range(niter):
print('iter = {!r}'.format(k))
#%%%Setting for getting a proper initial template
# Inner iteration for updating template
if k == 0:
niter1 = 50
# niter1 = in_niter1
else:
niter1 = in_niter1
#%%%Solving TV-L2 by Gradient Descent
# Store energy
E = []
E = np.hstack((E, np.zeros(niter1)))
for k1 in range(niter1):
image_MN0[0] = template
# Update partial of template
grad_template = grad_op(template)
grad_template_norm = np.sqrt(grad_template[0]**2 +
grad_template[1]**2 + 1.0e-12)
E[k1] += mu_1 * np.asarray(
grad_template_norm).sum() * template.space.cell_volume
for i in range(1, N1):
E[k1] += 1. / N * np.asarray((forward_op[i](image_MN0[i*M]) - noise_proj_data[i])**2).sum() \
* noise_proj_data[0].space.cell_volume
template = template - \
ss3 * grad_op_adjoint(grad_template/grad_template_norm)
for j in range(MN):
temp1 = j + 1
# Update image_MN0
image_MN0[temp1] = image_space.element(
_linear_deform(image_MN0[j],
-inv_MN * vector_fields[temp1]))
if temp1 % M == 0:
temp2 = temp1 // M
# print(temp1)
# print(temp2)
# Update eta_tt
eta_tt[temp2] = gradS[temp2](image_MN0[temp1])
# eta_tt[temp2].show('eta_tt[{!r}]'.format(temp2))
series_bp_all[temp2][temp1] = eta_tt[temp2]
# the above two lines can be combined into one
# series_bp_all[temp2][temp1] = gradS[temp2](image_MN0[temp1])
for l in range(temp1):
jacobian_det = image_space.element(
1.0 +
inv_MN * div_op(vector_fields[temp1 - l - 1]))
# Update eta_tau_tnp
series_bp_all[temp2][temp1-l-1] = \
jacobian_det * image_space.element(
_linear_deform(
series_bp_all[temp2][temp1-l],
inv_MN * vector_fields[temp1-l-1]))
# Update partial of template
template = template - \
ss1 * series_bp_all[temp2][0]
for k2 in range(in_inter2):
image_MN0[0] = template
series_bp_all[0] = gradS[0](image_MN0[0])
for j in range(MN):
temp1 = j + 1
# Update image_MN0
image_MN0[temp1] = image_space.element(
_linear_deform(image_MN0[j],
-inv_MN * vector_fields[temp1]))
if temp1 % M == 0:
temp2 = temp1 // M
# Update eta_tt
eta_tt[temp2] = gradS[temp2](image_MN0[temp1])
series_bp_all[temp2][temp1] = eta_tt[temp2]
for l in range(temp1):
jacobian_det = image_space.element(
1.0 +
inv_MN * div_op(vector_fields[temp1 - l - 1]))
# Update eta_tau_t
series_bp_all[temp2][temp1-l-1] = \
jacobian_det * image_space.element(
_linear_deform(
series_bp_all[temp2][temp1-l],
inv_MN * vector_fields[temp1-l-1]))
for j in range(MN1):
tmp1 = grad_op(image_MN0[j])
tmp2 = int(np.ceil(j * 1. / M))
tmp0 = tmp2 + 1
# print(tmp2)
if tmp2 == 0:
tmp3 = image_space.zero()
tmp4 = image_space.tangent_bundle.zero()
else:
tmp3 = series_bp_all[tmp2][j]
tmp4 = vector_fields[j]
for i in range(tmp0, N1):
tmp3 = tmp3 + series_bp_all[i][j]
tmp4 = tmp4 + vector_fields[j]
for i in range(dim):
tmp1[i] *= tmp3
tmp5 = (2 * np.pi)**(dim / 2.0) * vectorial_ft_op.inverse(
vectorial_ft_op(tmp1) * ft_kernel_fitting)
# Update vector_fields
vector_fields[j] = vector_fields[j] + ss2 * (tmp5 -
mu_2 * tmp4)
return template, vector_fields, image_MN0
else:
raise NotImplementedError('unknown group action')