def tv_reconstruction(self, y, param=def_lambda):
"""
NOTE: I'll follow the example from the odl github: https://github.com/odlgroup/odl/blob/master/examples/solvers/pdhg_tomography.py
NOTE: The only thing I changed was swap what g and functional were supposed to be. That's it.
"""
# internal method to evaluate tv on a single element y with shape [width, height]
# the operators
gradients = odl.Gradient(self.space, method='forward')
broad_op = odl.BroadcastOperator(self.operator, gradients)
# define empty functional to fit the chambolle_pock framework
functional = odl.solvers.ZeroFunctional(broad_op.domain)
# the norms
l1_norm = param * odl.solvers.L1Norm(gradients.range)
l2_norm_squared = odl.solvers.L2NormSquared(self.range).translated(y)
g = odl.solvers.SeparableSum(l2_norm_squared, l1_norm)
# Find parameters
op_norm = 1.1 * odl.power_method_opnorm(broad_op)
tau = 10.0 / op_norm
sigma = 0.1 / op_norm
niter = 200
# find starting point
x = self.space.element(self.model.inverse(np.expand_dims(y, axis=-1))[...,0])
# Run the optimization algoritm
# odl.solvers.chambolle_pock_solver(x, functional, g, broad_op, tau = tau, sigma = sigma, niter=niter)
odl.solvers.pdhg(x, functional, g, broad_op, tau=tau, sigma=sigma, niter=niter)
return x
def reconstruction(proj_data, lam):
lam = float(lam)
print('lam = {}'.format(lam))
# We do not allow negative paramters, so return a bogus result
if lam <= 0:
return np.inf * space.one()
# Construct operators and functionals
gradient = odl.Gradient(space)
op = odl.BroadcastOperator(ray_trafo, gradient)
f = odl.solvers.ZeroFunctional(op.domain)
l2_norm = odl.solvers.L2NormSquared(
ray_trafo.range).translated(proj_data)
l1_norm = lam * odl.solvers.GroupL1Norm(gradient.range)
g = odl.solvers.SeparableSum(l2_norm, l1_norm)
# Select solver parameters
op_norm = 1.5 * odl.power_method_opnorm(op, maxiter=10)
# Run the algorithm
x = op.domain.zero()
odl.solvers.pdhg(x,
f,
g,
op,
niter=200,
tau=1.0 / op_norm,
sigma=1.0 / op_norm)
return x
def tv_reconstruction(self, y, param=def_lambda):
# internal method to evaluate tv on a single element y with shape [width, height]
# the operators
gradients = odl.Gradient(self.space, method='forward')
broad_op = odl.BroadcastOperator(self.operator, gradients)
# define empty functional to fit the chambolle_pock framework
g = odl.solvers.ZeroFunctional(broad_op.domain)
# the norms
l1_norm = param * odl.solvers.L1Norm(gradients.range)
l2_norm_squared = odl.solvers.L2NormSquared(self.range).translated(y)
functional = odl.solvers.SeparableSum(l2_norm_squared, l1_norm)
# Find parameters
op_norm = 1.1 * odl.power_method_opnorm(broad_op)
tau = 10.0 / op_norm
sigma = 0.1 / op_norm
niter = 200
# find starting point
x = self.space.element(
self.model.inverse(np.expand_dims(y, axis=-1))[..., 0])
# Run the optimization algoritm
# odl.solvers.chambolle_pock_solver(x, functional, g, broad_op, tau = tau, sigma = sigma, niter=niter)
odl.solvers.pdhg(x,
functional,
g,
broad_op,
tau=tau,
sigma=sigma,
niter=niter)
return x
def tgv(operator, data, alpha, beta, grad=None, nonneg=True, datafit=None):
space = operator.domain
if grad is None:
grad = gradient(space)
E = symm_derivative(space)
I = odl.IdentityOperator(grad.range)
A1 = odl.ReductionOperator(operator,
odl.ZeroOperator(grad.range, operator.range))
A2 = odl.ReductionOperator(grad, -I)
A3 = odl.ReductionOperator(odl.ZeroOperator(space, E.range), E)
A = odl.BroadcastOperator(*[A1, A2, A3])
F1 = get_data_fit(datafit, data)
F2 = alpha * odl.solvers.GroupL1Norm(grad.range)
F3 = alpha * beta * odl.solvers.GroupL1Norm(E.range)
F = odl.solvers.SeparableSum(F1, F2, F3)
if nonneg:
G = odl.solvers.SeparableSum(odl.solvers.ZeroFunctional(space),
odl.solvers.ZeroFunctional(E.domain))
else:
G = odl.solvers.SeparableSum(odl.solvers.IndicatorNonnegativity(space),
odl.solvers.ZeroFunctional(E.domain))
return G, F, A
def test_chambolle_pock_solver_produce_space():
"""Test the Chambolle-Pock algorithm using a product space operator."""
# Create a discretized image space
space = odl.uniform_discr(0, 1, DATA.size)
# Operator
identity = odl.IdentityOperator(space)
# Create broadcasting operator
prod_op = odl.BroadcastOperator(identity, -2 * identity)
# Starting point for explicit computation
discr_vec_0 = prod_op.domain.element(DATA)
# Copy to be overwritten by the algorithm
discr_vec = discr_vec_0.copy()
# Proximal operator using the same factory function for F^* and G
g = odl.solvers.ZeroFunctional(prod_op.domain)
f = odl.solvers.ZeroFunctional(prod_op.range).convex_conj
# Run the algorithm
chambolle_pock_solver(discr_vec, f, g, prod_op, tau=TAU, sigma=SIGMA,
theta=THETA, niter=1)
vec_expl = discr_vec_0 - TAU * SIGMA * prod_op.adjoint(
prod_op(discr_vec_0))
assert all_almost_equal(discr_vec, vec_expl, PLACES)
def _call(self, x, out):
xi = vfield
Id = IdentityOperator(domain)
xiT = odl.PointwiseInner(domain, xi)
xixiT = odl.BroadcastOperator(*[x * xiT for x in xi])
gamma = 1
P = (Id - gamma * xixiT)
out.assign(P(x))
def transl_op_fixed_im(self, im):
if isinstance(self.image_space, odl.ProductSpace):
deform_op = odl.BroadcastOperator(defm.LinDeformFixedTempl(im[0]),
defm.LinDeformFixedTempl(im[1]))
else:
deform_op = defm.LinDeformFixedTempl(im)
return deform_op * self.embedding
def gradient(space,
sinfo=None,
mode=None,
gamma=1,
eta=1e-2,
show_sinfo=False,
prefix=None):
grad = odl.Gradient(space, method='forward', pad_mode='symmetric')
if sinfo is not None:
if mode == 'direction':
norm = odl.PointwiseNorm(grad.range)
grad_sinfo = grad(sinfo)
ngrad_sinfo = norm(grad_sinfo)
for i in range(len(grad_sinfo)):
grad_sinfo[i] /= ngrad_sinfo.ufuncs.max()
ngrad_sinfo = norm(grad_sinfo)
ngrad_sinfo_eta = np.sqrt(ngrad_sinfo**2 + eta**2)
xi = grad.range.element([g / ngrad_sinfo_eta
for g in grad_sinfo]) # UGLY
Id = odl.operator.IdentityOperator(grad.range)
xiT = odl.PointwiseInner(grad.range, xi)
xixiT = odl.BroadcastOperator(*[x * xiT for x in xi])
grad = (Id - gamma * xixiT) * grad
if show_sinfo:
misc.save_image(ngrad_sinfo, prefix + '_sinfo_norm')
misc.save_vfield(xi.asarray(), filename=prefix + '_sinfo_xi')
misc.save_vfield_cmap(filename=prefix + '_sinfo_xi_cmap')
elif mode == 'location':
norm = odl.PointwiseNorm(grad.range)
ngrad_sinfo = norm(grad(sinfo))
ngrad_sinfo /= ngrad_sinfo.ufuncs.max()
w = eta / np.sqrt(ngrad_sinfo**2 + eta**2)
grad = odl.DiagonalOperator(odl.MultiplyOperator(w), 2) * grad
if show_sinfo:
misc.save_image(ngrad_sinfo, prefix + '_sinfo_norm')
misc.save_image(w, prefix + '_w')
else:
grad = None
return grad
def jtv(operator, data, alpha, sinfo, eta, nonneg=True, datafit=None):
space = operator.domain
Dx = odl.PartialDerivative(space, 0)
Dy = odl.PartialDerivative(space, 1)
Z = odl.ZeroOperator(space)
D = odl.BroadcastOperator(Dx, Dy, Z, Z)
A = odl.BroadcastOperator(operator, D)
F1 = get_data_fit(datafit, data)
Q = odl.BroadcastOperator(Z, Z, Dx, Dy)
N = odl.solvers.GroupL1Norm(D.range)
F2 = alpha * N.translated(-eta * Q(sinfo))
F = odl.solvers.SeparableSum(F1, F2)
if nonneg:
G = odl.solvers.IndicatorNonnegativity(space)
else:
G = odl.solvers.ZeroFunctional(space)
return G, F, A
def get_operators(space):
# Create the forward operator
filter_width = 4 # standard deviation of the Gaussian filter
ft = odl.trafos.FourierTransform(space)
c = filter_width**2 / 4.0**2
gaussian = ft.range.element(lambda x: np.exp(-(x[0]**2 + x[1]**2) * c))
operator = ft.inverse * gaussian * ft
# Normalize the operator and create pseudo-inverse
opnorm = odl.power_method_opnorm(operator)
operator = (1 / opnorm) * operator
# Do not need good pseudo-inverse, but keep to have same interface.
pseudoinverse = odl.ZeroOperator(space)
# Create gradient operator and normalize it
part_grad_0 = odl.PartialDerivative(space,
0,
method='forward',
pad_mode='order0')
part_grad_1 = odl.PartialDerivative(space,
1,
method='forward',
pad_mode='order0')
grad_norm = odl.power_method_opnorm(
odl.BroadcastOperator(part_grad_0, part_grad_1),
xstart=odl.util.testutils.noise_element(space))
part_grad_0 = (1 / grad_norm) * part_grad_0
part_grad_1 = (1 / grad_norm) * part_grad_1
# Create tensorflow layer from odl operator
with tf.name_scope('odl_layers'):
odl_op_layer = odl.contrib.tensorflow.as_tensorflow_layer(
operator, 'RayTransform')
odl_op_layer_adjoint = odl.contrib.tensorflow.as_tensorflow_layer(
operator.adjoint, 'RayTransformAdjoint')
odl_grad0_layer = odl.contrib.tensorflow.as_tensorflow_layer(
part_grad_0, 'PartialGradientDim0')
odl_grad0_layer_adjoint = odl.contrib.tensorflow.as_tensorflow_layer(
part_grad_0.adjoint, 'PartialGradientDim0Adjoint')
odl_grad1_layer = odl.contrib.tensorflow.as_tensorflow_layer(
part_grad_1, 'PartialGradientDim1')
odl_grad1_layer_adjoint = odl.contrib.tensorflow.as_tensorflow_layer(
part_grad_1.adjoint, 'PartialGradientDim1Adjoint')
return (odl_op_layer, odl_op_layer_adjoint, odl_grad0_layer,
odl_grad0_layer_adjoint, odl_grad1_layer, odl_grad1_layer_adjoint,
part_grad_0, part_grad_1, operator, pseudoinverse)
def h1(operator, data, alpha, grad=None, nonneg=True, datafit=None):
space = operator.domain
if grad is None:
grad = gradient(space)
A = odl.BroadcastOperator(operator, grad)
F1 = get_data_fit(datafit, data)
F2 = alpha * odl.solvers.L2NormSquared(grad.range)
F = odl.solvers.SeparableSum(F1, F2)
if nonneg:
G = odl.solvers.IndicatorNonnegativity(space)
else:
G = odl.solvers.ZeroFunctional(space)
return G, F, A
def get_operators(space, geometry):
# Create the forward operator
operator = odl.tomo.RayTransform(space, geometry)
pseudoinverse = odl.tomo.fbp_op(operator)
# Normalize the operator and create pseudo-inverse
opnorm = odl.power_method_opnorm(operator)
operator = (1 / opnorm) * operator
pseudoinverse = pseudoinverse * opnorm
# Create gradient operator and normalize it
part_grad_0 = odl.PartialDerivative(space, 0, method='forward',
pad_mode='order0')
part_grad_1 = odl.PartialDerivative(space, 1, method='forward',
pad_mode='order0')
grad_norm = odl.power_method_opnorm(
odl.BroadcastOperator(part_grad_0, part_grad_1),
xstart=odl.util.testutils.noise_element(space))
part_grad_0 = (1 / grad_norm) * part_grad_0
part_grad_1 = (1 / grad_norm) * part_grad_1
# Create tensorflow layer from odl operator
with tf.name_scope('odl_layers'):
odl_op_layer = odl.contrib.tensorflow.as_tensorflow_layer(
operator, 'RayTransform')
odl_op_layer_adjoint = odl.contrib.tensorflow.as_tensorflow_layer(
operator.adjoint, 'RayTransformAdjoint')
odl_grad0_layer = odl.contrib.tensorflow.as_tensorflow_layer(
part_grad_0, 'PartialGradientDim0')
odl_grad0_layer_adjoint = odl.contrib.tensorflow.as_tensorflow_layer(
part_grad_0.adjoint, 'PartialGradientDim0Adjoint')
odl_grad1_layer = odl.contrib.tensorflow.as_tensorflow_layer(
part_grad_1, 'PartialGradientDim1')
odl_grad1_layer_adjoint = odl.contrib.tensorflow.as_tensorflow_layer(
part_grad_1.adjoint, 'PartialGradientDim1Adjoint')
return (odl_op_layer, odl_op_layer_adjoint, odl_grad0_layer,
odl_grad0_layer_adjoint, odl_grad1_layer, odl_grad1_layer_adjoint,
operator, pseudoinverse)
def tnv(operator, data, alpha, sinfo, eta, nonneg=True, datafit=None):
space = operator.domain
grad = odl.Gradient(space)
P = odl.ComponentProjection(grad.range**2, 0)
D = P.adjoint * grad
Q = odl.ComponentProjection(grad.range**2, 1)
A = odl.BroadcastOperator(operator, D)
F1 = get_data_fit(datafit, data)
N = odl.solvers.NuclearNorm(D.range, outer_exp=1, singular_vector_exp=1)
F2 = alpha * N.translated(-Q.adjoint(eta * grad(sinfo)))
F = odl.solvers.SeparableSum(F1, F2)
if nonneg:
G = odl.solvers.IndicatorNonnegativity(space)
else:
G = odl.solvers.ZeroFunctional(space)
return G, F, A
def tv_reconsruction(self, y, param=1000000):
# the operators
gradients = odl.Gradient(self.space, method='forward')
operator = odl.BroadcastOperator(self.ray_transf, gradients)
# define empty functional to fit the chambolle_pock framework
g = odl.solvers.ZeroFunctional(operator.domain)
# compute transformed data
# ensure y stays away from 0
y_cut = np.maximum(y, 0.03)
data = -(np.log(y_cut)) / self.attenuation_coeff
# the norms
l1_norm = param * odl.solvers.L1Norm(gradients.range)
l2_norm_squared = odl.solvers.L2NormSquared(
self.ray_transf.range).translated(data)
functional = odl.solvers.SeparableSum(l2_norm_squared, l1_norm)
# Find parameters
op_norm = 1.1 * odl.power_method_opnorm(operator)
tau = 10.0 / op_norm
sigma = 0.1 / op_norm
niter = 5000
# find starting point
x = self.fbp(data)
# Run the optimization algoritm
odl.solvers.chambolle_pock_solver(x,
functional,
g,
operator,
tau=tau,
sigma=sigma,
niter=niter)
# plot results
plt.figure(1)
plt.imshow(x)
plt.show()
# Initialize gradient operator
gradient = odl.Gradient(reco_space, method='forward')
gradient_back = odl.Gradient(reco_space, method='backward')
eps = odl.DiagonalOperator(gradient_back, reco_space.ndim)
# Create the domain of the problem, given by the reconstruction space and the
# range of the gradient on the reconstruction space.
domain = odl.ProductSpace(reco_space, gradient.range)
# Column vector of three operators defined as:
# 1. Computes ``A(x)``
# 2. Computes ``grad(x) - y``
# 3. Computes ``eps(y)``
op = odl.BroadcastOperator(
ray_trafo * odl.ComponentProjection(domain, 0),
odl.ReductionOperator(gradient, odl.ScalingOperator(gradient.range, -1)),
eps * odl.ComponentProjection(domain, 1))
# Do not use the g functional, set it to zero.
g = odl.solvers.ZeroFunctional(op.domain)
# Create functionals for the dual variable
# l2-squared data matching
l2_norm = odl.solvers.L2NormSquared(ray_trafo.range).translated(data)
# The l1-norms scaled by regularization paramters
l1_norm_1 = 0.001 * odl.solvers.L1Norm(gradient.range)
l1_norm_2 = 1e-4 * odl.solvers.L1Norm(eps.range)
# Combine functionals, order must correspond to the operator K
# create data
data = odl.phantom.white_noise(X, mean=groundtruth, stddev=0.1, seed=1807)
# save images and data
if not os.path.exists('{}/groundtruth.png'.format(folder_main)):
misc.save_image(groundtruth, 'groundtruth', folder_main, 1, clim=clim)
misc.save_image(data, 'data', folder_main, 2, clim=clim)
alpha = .12 # set regularisation parameter
gamma = 0.99 # gamma^2 is upper bound of step size constraint
# create forward operators
Dx = odl.PartialDerivative(X, 0, pad_mode='symmetric')
Dy = odl.PartialDerivative(X, 1, pad_mode='symmetric')
A = odl.BroadcastOperator(Dx, Dy)
Y = A.range
# set up functional f
f = odl.solvers.SeparableSum(*[odl.solvers.L1Norm(Yi) for Yi in Y])
# set up functional g
g = 1 / (2 * alpha) * odl.solvers.L2NormSquared(X).translated(data)
obj_fun = f * A + g # define objective function
mu_g = 1 / alpha # define strong convexity constants
# create target / compute a saddle point
file_target = '{}/target.npy'.format(folder_main)
if not os.path.exists(file_target):
# compute a saddle point with PDHG and time the reconstruction
# Adjoint currently bugged, needs to be fixed
proj._adjoint *= proj(phantom0).inner(proj(phantom0)) / phantom0.inner(proj.adjoint(proj(phantom0)))
# Not scaled correctly
proj = proj/5.0
# Create product space
proj_op = odl.diagonal_operator([proj, proj])
energies = np.linspace(0.5, 1.0, 5)
spectrum_low = np.exp(-((energies-0.5) * 2)**2)
spectrum_high = np.exp(-((energies-1.0) * 2)**2)
A_low = SpectralDetector(proj.range, energies, spectrum_low)
A_high = SpectralDetector(proj.range, energies, spectrum_high)
detector_op = odl.BroadcastOperator(A_low, A_high)
# Compose operators
spectral_proj = detector_op * proj_op
# Create data
phantom = spectral_proj.domain.element([phantom0, phantom1])
proj_op(phantom).show(title='materials', clim=[0, 5])
projections = spectral_proj(phantom)
projections.show(title='spectral', clim=[0, 5])
# Reconstruct with big op
if 0:
partial = (odl.solvers.util.ShowPartial(indices=np.s_[0, :, :, :, n//2], clim=[0, 5]) &
odl.solvers.util.ShowPartial(indices=np.s_[1, :, n//2, :, n//2]) &
odl.solvers.util.ShowPartial(indices=np.s_[1, :, :, :, n//2], clim=[0, 1]) &
G, F, A = models.tgv(operator,
data,
regparam,
beta,
grad=grad,
datafit='l1')
else:
D = None
norm_As = []
for Ai in A:
xs = odl.phantom.white_noise(Ai.domain, seed=1807)
norm_As.append(Ai.norm(estimate=True, xstart=xs))
Atilde = odl.BroadcastOperator(
*[Ai / norm_Ai for Ai, norm_Ai in zip(A, norm_As)])
Ftilde = odl.solvers.SeparableSum(
*[Fi * norm_Ai for Fi, norm_Ai in zip(F, norm_As)])
obj_fun = Ftilde * Atilde + G
Atilde_norm = Atilde.norm(estimate=True)
x = Atilde.domain.zero()
scaling = 1
sigma = scaling / Atilde_norm
tau = 0.999 / (scaling * Atilde_norm)
cb = (odl.solvers.CallbackPrintIteration(step=step,
end=', ')
& odl.solvers.CallbackPrintTiming(
# --- Generate artificial data --- # # Create phantom discr_phantom = odl.phantom.shepp_logan(reco_space, modified=True) # Create sinogram of forward projected phantom with noise data = ray_trafo(discr_phantom) data += odl.phantom.white_noise(ray_trafo.range) * np.mean(data) * 0.1 # --- Set up the inverse problem --- # # Initialize gradient operator gradient = odl.Gradient(reco_space) # Column vector of two operators op = odl.BroadcastOperator(ray_trafo, gradient) # Do not use the f functional, set it to zero. f = odl.solvers.ZeroFunctional(op.domain) # Create functionals for the dual variable # l2-squared data matching l2_norm = odl.solvers.L2NormSquared(ray_trafo.range).translated(data) # Isotropic TV-regularization i.e. the l1-norm l1_norm = 0.015 * odl.solvers.L1Norm(gradient.range) # Combine functionals, order must correspond to the operator K g = odl.solvers.SeparableSum(l2_norm, l1_norm)
ray_trafo = odl.tomo.RayTransform(reco_space, geometry) # --- Generate artificial data --- # # Create phantom and noisy projection data phantom = odl.phantom.shepp_logan(reco_space, modified=True) data = ray_trafo(phantom) data += odl.phantom.white_noise(ray_trafo.range) * np.mean(data) * 0.1 # --- Set up the inverse problem --- # # Gradient operator for the TV part grad = odl.Gradient(reco_space) # Stacking of the two operators L = odl.BroadcastOperator(ray_trafo, grad) # Data matching and regularization functionals data_fit = odl.solvers.L2NormSquared(ray_trafo.range).translated(data) reg_func = 0.015 * odl.solvers.L1Norm(grad.range) g = odl.solvers.SeparableSum(data_fit, reg_func) # We don't use the f functional, setting it to zero f = odl.solvers.ZeroFunctional(L.domain) # --- Select parameters and solve using ADMM --- # # Estimated operator norm, add 10 percent for some safety margin op_norm = 1.1 * odl.power_method_opnorm(L, maxiter=20) niter = 200 # Number of iterations
# create ground truth
X = odl.uniform_discr([0, 0], simage, simage)
groundtruth = 100 * X.element(image_raw)
clim = [0, 100]
tol_norm = 1.05
# create forward operators
Dx = odl.PartialDerivative(X, 0, pad_mode='symmetric')
Dy = odl.PartialDerivative(X, 1, pad_mode='symmetric')
kernel = images.blurring_kernel(shape=[15, 15])
convolution = misc.Blur2D(X, kernel)
K = odl.uniform_discr([0, 0], kernel.shape, kernel.shape)
kernel = K.element(kernel)
scale = 1e+3
A = odl.BroadcastOperator(Dx, Dy, scale / clim[1] * convolution)
Y = A.range
# create data
background = 200 * Y[2].one()
data = odl.phantom.poisson_noise(A[2](groundtruth) + background, seed=1807)
# save images and data
if not os.path.exists('{}/groundtruth.png'.format(folder_main)):
misc.save_image(groundtruth, 'groundtruth', folder_main, 1, clim=clim)
misc.save_image(data - background, 'data', folder_main, 2, clim=[0, scale])
misc.save_image(kernel, 'kernel', folder_main, 3)
alpha = 0.1 # set regularisation parameter
gamma = 0.99 # auxiliary step size parameter < 1
#sym_gradient = odl.operator.ProductSpaceOperator(
# [[Dx, 0], [0, Dy], [0.5*Dy, 0.5*Dx]], range=W)
E = odl.operator.ProductSpaceOperator(
[[Dx, 0], [0, Dy], [0.5*Dy, 0.5*Dx], [0.5*Dy, 0.5*Dx]])
W = E.range
# Create the domain of the problem, given by the reconstruction space and the
# range of the gradient on the reconstruction space.
domain = odl.ProductSpace(U, V)
# Column vector of three operators defined as:
# 1. Computes ``Ax``
# 2. Computes ``Gx - y``
# 3. Computes ``Ey``
op = odl.BroadcastOperator(
A * odl.ComponentProjection(domain, 0),
odl.ReductionOperator(G, odl.ScalingOperator(V, -1)),
E * odl.ComponentProjection(domain, 1))
# Do not use the g functional, set it to zero.
g = odl.solvers.ZeroFunctional(domain)
# l2-squared data matching
l2_norm = odl.solvers.L2NormSquared(A.range).translated(data)
# parameters
alpha = 1e-1
beta = 1
# The l1-norms scaled by regularization paramters
l1_norm_1 = alpha * odl.solvers.L1Norm(V)
l1_norm_2 = alpha * beta * odl.solvers.L1Norm(W)
def pdhg(x, f, g, A, tau, sigma, niter, **kwargs):
"""Computes a saddle point with PDHG.
This algorithm is the same as "algorithm 1" in [CP2011a] but with
extrapolation on the dual variable.
Parameters
----------
x : primal variable
This variable is both input and output of the method.
f : function
Functional Y -> IR_infty that has a convex conjugate with a
proximal operator, i.e. f.convex_conj.proximal(sigma) : Y -> Y.
g : function
Functional X -> IR_infty that has a proximal operator, i.e.
g.proximal(tau) : X -> X.
A : function
Operator A : X -> Y that possesses an adjoint: A.adjoint
tau : scalar / vector / matrix
Step size for primal variable. Note that the proximal operator of g
has to be well-defined for this input.
sigma : scalar
Scalar / vector / matrix used as step size for dual variable. Note that
the proximal operator related to f (see above) has to be well-defined
for this input.
niter : int
Number of iterations
Other Parameters
----------------
y: dual variable
Dual variable is part of a product space
z: variable
Adjoint of dual variable, z = A^* y.
theta : scalar
Extrapolation factor.
callback : callable
Function called with the current iterate after each iteration.
References
----------
[CP2011a] Chambolle, A and Pock, T. *A First-Order
Primal-Dual Algorithm for Convex Problems with Applications to
Imaging*. Journal of Mathematical Imaging and Vision, 40 (2011),
pp 120-145.
"""
def fun_select(k):
return [0]
f = odl.solvers.SeparableSum(f)
A = odl.BroadcastOperator(A, 1)
# Dual variable
y = kwargs.pop('y', None)
if y is None:
y_new = None
else:
y_new = A.range.element([y])
spdhg_generic(x, f, g, A, tau, [sigma], niter, fun_select, y=y_new,
**kwargs)
if y is not None:
y.assign(y_new[0])
# Create phantom phantom = odl.phantom.shepp_logan(space, modified=True) # Create sinogram of forward projected phantom with noise data = operator(phantom) data += odl.phantom.white_noise(operator.range) * np.mean(np.abs(data)) * 0.05 # --- Set up the inverse problem --- # # Initialize gradient operator gradient = odl.Gradient(space) # Column vector of two operators op = odl.BroadcastOperator(operator, gradient) # Do not use the g functional, set it to zero. g = odl.solvers.ZeroFunctional(op.domain) # Create functionals for the dual variable # l2-squared data matching l2_norm = odl.solvers.L2NormSquared(operator.range).translated(data) # Isotropic TV-regularization i.e. the l1-norm l1_norm = 0.3 * odl.solvers.L1Norm(gradient.range) # Combine functionals, order must correspond to the operator K f = odl.solvers.SeparableSum(l2_norm, l1_norm)
tmp = X.element()
tmp_op = mMR.operator_mmr()
tmp_op.toodl(image, tmp)
fldr = '{}/pics'.format(folder_param)
misc.save_image(tmp.asarray(), 'image_pet', fldr, planes=planes)
tmp_op.toodl(image_mr, tmp)
misc.save_image(tmp.asarray(), 'image_mr', fldr, planes=planes)
tmp_op.toodl(image_ct, tmp)
misc.save_image(tmp.asarray(), 'image_ct', fldr, planes=planes)
# --- get target --- BE CAREFUL, THIS TAKES TIME
file_target = '{}/target.npy'.format(folder_param)
if not os.path.exists(file_target):
print('file {} does not exist. Compute it.'.format(file_target))
A = odl.BroadcastOperator(K, D)
f = odl.solvers.SeparableSum(KL, L1)
norm_A = misc.norm(A, '{}/norm_tv.npy'.format(folder_main))
sigma = rho / norm_A
tau = rho / norm_A
niter_target = nepoch_target
step = 10
cb = (CallbackPrintIteration(step=step, end=', ')
& CallbackPrintTiming(step=step, cumulative=False, end=', ')
& CallbackPrintTiming(
step=step, cumulative=True, fmt='total={:.3f} s'))
x_opt = X.zero()
# Create phantom
phantom = odl.phantom.shepp_logan(space, modified=True)
# Create the convolved version of the phantom
data = convolution(phantom)
data += odl.phantom.white_noise(convolution.range) * np.mean(data) * 0.1
data.show('Convolved data')
# Set up PDHG:
# Initialize gradient operator
gradient = odl.Gradient(space, method='forward')
# Column vector of two operators
op = odl.BroadcastOperator(convolution, gradient)
# Create the functional for unconstrained primal variable
g = odl.solvers.ZeroFunctional(op.domain)
# l2-squared data matching
l2_norm_squared = odl.solvers.L2NormSquared(space).translated(data)
# Isotropic TV-regularization i.e. the l1-norm
l1_norm = 0.01 * odl.solvers.L1Norm(gradient.range)
# Make separable sum of functionals, order must be the same as in `op`
f = odl.solvers.SeparableSum(l2_norm_squared, l1_norm)
# --- Select solver parameters and solve using PDHG --- #
os.makedirs(folder_main)
folder_today = '{}/{}'.format(folder_main, subfolder)
if not os.path.exists(folder_today):
os.makedirs(folder_today)
folder_npy = '{}/npy'.format(folder_today)
if not os.path.exists(folder_npy):
os.makedirs(folder_npy)
# create geometry of operator
X = odl.uniform_discr(min_pt=[-1, -1], max_pt=[1, 1], shape=[nvoxelx, nvoxelx])
geometry = odl.tomo.parallel_beam_geometry(X, num_angles=200, det_shape=250)
G = odl.BroadcastOperator(
*[odl.tomo.RayTransform(X, g, impl='astra_cpu') for g in geometry])
# create ground truth
Y = G.range
groundtruth = X.element(images.resolution_phantom(shape=X.shape))
clim = [0, 1]
tol_norm = 1.05
# save images and data
file_data = '{}/data.npy'.format(folder_main)
if not os.path.exists(file_data):
sino = G(groundtruth)
support = X.element(groundtruth.ufuncs.greater(0))
factors = -G(0.005 / X.cell_sides[0] * support)
factors.ufuncs.exp(out=factors)
), -1))
phantom /= 1000.0 # convert go g/cm^3
data = nonlinear_operator(phantom)
noisy_data = odl.phantom.poisson_noise(
data * photons_per_pixel) / photons_per_pixel
# --- Set up the inverse problem --- #
# Initialize gradient operator
gradient = odl.Gradient(space)
# Column vector of two operators
# scaling the operator acts as a pre-conditioner, improving convergence.
op = odl.BroadcastOperator(nonlinear_operator, gradient)
# Do not use the g functional, set it to zero.
g = odl.solvers.ZeroFunctional(op.domain)
# Create functionals for the dual variable
# l2-squared data matching
data_discr = odl.solvers.KullbackLeibler(operator.range, noisy_data)
# Isotropic TV-regularization i.e. the l1-norm
l1_norm = 0.00011 * odl.solvers.GroupL1Norm(gradient.range)
# Combine functionals, order must correspond to the operator K
f = odl.solvers.SeparableSum(data_discr, l1_norm)
space = odl.uniform_discr([0, 0], shape, shape) # Original image orig = space.element(image) # Add noise image += np.random.normal(0, 0.1, shape) # Data of noisy image noisy = space.element(image) # Gradient operator gradient = odl.Gradient(space, method='forward') # Matrix of operators op = odl.BroadcastOperator(odl.IdentityOperator(space), gradient) # Set up the functionals # l2-squared data matching l2_norm = odl.solvers.L2NormSquared(space).translated(noisy) # Isotropic TV-regularization: l1-norm of grad(x) l1_norm = 0.15 * odl.solvers.L1Norm(gradient.range) # Make separable sum of functionals, order must correspond to the operator K f = odl.solvers.SeparableSum(l2_norm, l1_norm) # Non-negativity constraint g = odl.solvers.IndicatorNonnegativity(op.domain)
vol = odl.uniform_partition([-v / sqrt(128) for v in vol],
[v / sqrt(128) for v in vol], vol)
vol = odl.uniform_discr_frompartition(vol[:, :, 500:564], dtype='float32')
data = ascontiguousarray(data[:, :, 500:564], dtype='float32')
PSpace = (angles,
odl.uniform_partition([-v / sqrt(128) for v in data.shape[1:]],
[v / sqrt(128) for v in data.shape[1:]],
data.shape[1:]))
PSpace = odl.tomo.Parallel3dAxisGeometry(*PSpace, axis=[0, 0, 1])
# Operators
Radon = odl.tomo.RayTransform(vol, PSpace)
grad = odl.Gradient(vol)
op = odl.BroadcastOperator(Radon, grad)
g = odl.solvers.functional.default_functionals.IndicatorNonnegativity(
op.domain)
fidelity = odl.solvers.L2NormSquared(Radon.range).translated(data)
TV = .0003 * odl.solvers.L1Norm(grad.range)
# fidelity = odl.solvers.L1Norm(Radon.range).translated(data)
# TV = .03 * odl.solvers.L1Norm(grad.range)
f = odl.solvers.SeparableSum(fidelity, TV)
data /= abs(Radon.adjoint(data).__array__()).max()
# fbp = odl.tomo.fbp_op(Radon)
# fbp = fbp(data)
# fbp.show(title='FBP', force_show='True')
# exit()
# op_norm = [odl.power_method_opnorm(Radon), odl.power_method_opnorm(grad)]