def adjoint_scaling_factor(self):
"""Compute scaling factor of adjoint projector. Consider A x = y,
the adjoint A* of A is defined as:
<A x, y>_D = <x, A* y>_I
Assume A* = s B with B being the ASTRA backprojector, then:
s = <A x, A x> / <B A x, x>
Returns
-------
:rtype: float
:returns: s
"""
vol_rn = Rn(self.geom.vol_size)
proj_rn = Rn(self.geom.proj_size)
vol_rn_ones = vol_rn.element(1)
proj_rn_ones = proj_rn.element(1)
# projector = Projector(self.geom, vol_rn, proj_rn)
projector = Projector(self.geom)
proj = projector.forward(vol_rn_ones)
vol = projector.backward(proj_rn_ones)
# print vol.data.min(), vol.data.max()
# print proj.data.min(), proj.data.max()
self.adj_scal_fac = proj.inner(proj_rn_ones) / vol_rn_ones.inner(vol)
# self.adj_scal_fac = proj.norm()**2 / vol_rn.inner(vol, vol_rn_ones)
# return proj.norm()**2 / vol_rn._inner(vol, vol_rn_ones)
projector.clear_astra_memory()
def adjoint_scaling_factor(self):
"""Compute scaling factor of adjoint projector. Consider A x = y,
the adjoint A* of A is defined as:
<A x, y>_D = <x, A* y>_I
Assume A* = s B with B being the ASTRA backprojector, then:
s = <A x, A x> / <B A x, x>
Returns
-------
:rtype: float
:returns: s
"""
vol_rn = Rn(self.geom.vol_size)
proj_rn = Rn(self.geom.proj_size)
vol_rn_ones = vol_rn.element(1)
proj_rn_ones = proj_rn.element(1)
# projector = Projector(self.geom, vol_rn, proj_rn)
projector = Projector(self.geom)
proj = projector.forward(vol_rn_ones)
vol = projector.backward(proj_rn_ones)
# print vol.data.min(), vol.data.max()
# print proj.data.min(), proj.data.max()
self.adj_scal_fac = proj.inner(proj_rn_ones) / vol_rn_ones.inner(vol)
# self.adj_scal_fac = proj.norm()**2 / vol_rn.inner(vol, vol_rn_ones)
# return proj.norm()**2 / vol_rn._inner(vol, vol_rn_ones)
projector.clear_astra_memory()
def matrix_norm(self,
iterations,
vol_init=1.0,
tv_norm=False,
return_volume=False,
intermediate_results=False):
"""The matrix norm || K ||_2 of 'K' defined here as largest
singular value of 'K'. Employs the generic power method to obtain a
scalar 's' which tends to || K ||_2 as the iterations N increase.
To be implemented: optionally return volume 'x', such that it can be
re-used as initializer to continue the iteration.
Parameters
----------
:type iterations: int
:param iterations: Number of iterations of the generic power method.
:type vol_init: float | ndarray (default 1.0)
:param vol_init: in I, initial image to start with.
:type intermediate_results: bool
:param intermediate_results: Returns list of intermediate results
instead of scalar.
:type return_volume: bool
:param return_volume: Return volume in order to resume iteration via
passing it over as initial volume.
Returns
-------
:rtype: float | numpy.ndarray, numpay.array (optional)
:returns: s, vol
s: Scalar of final iteration or numpy.ndarray containing all
results during iteration.
vol: Volume vector
"""
geom = self.geom
vol = self.recon_space.element(vol_init)
proj = Rn(geom.proj_size).zero()
# projector = Projector(geom, vol.space, proj.space)
projector = Projector(geom)
# print 'projector scaling factor', projector.scal_fac
tmp = None
if intermediate_results:
s = np.zeros(iterations)
else:
s = 0
# Power method loop
for n in range(iterations):
# step 4: x_{n+1} <- K^T K x_n
if tv_norm:
# K = (A, grad) instead of K = A
# Compute: - div grad x_n
# use sum over generator expression
tmp = -reduce(add, (partial(
partial(vol.data.reshape(geom.vol_shape), dim,
geom.voxel_width[dim]), dim, geom.voxel_width[dim])
for dim in range(geom.vol_ndim)))
# x_n <- A^T (A x_n)
vol = projector.backward(projector.forward(vol))
vol *= self.adj_scal_fac
if tv_norm:
# x_n <- x_n - div grad x_n
# print 'n: {2}. vol: min = {0}, max = {1}'.format(
# vol.data.min(), vol.data.max(), n)
# print 'n: {2}. tv: min = {0}, max = {1}'.format(tmp.min(),
# tmp.max(), n)
vol.data[:] += tmp.ravel()
# step 5:
# x_n <- x_n/||x_n||_2
vol /= vol.norm()
# step 6:
# s_n <-|| K x ||_2
if intermediate_results:
# proj <- A^T x_n
proj = projector.forward(vol)
s[n] = proj.norm()
if tv_norm:
s[n] = np.sqrt(
s[n]**2 +
reduce(add, (np.linalg.norm(
partial(vol.data.reshape(geom.vol_shape), dim,
geom.voxel_width[dim]))**2
for dim in range(geom.vol_ndim))))
# step 6: || K x ||_2
if not intermediate_results:
proj = projector.forward(vol)
s = proj.norm()
if tv_norm:
s = np.sqrt(s**2 +
reduce(add, (np.linalg.norm(
partial(vol.data.reshape(geom.vol_shape), dim,
geom.voxel_width[dim]))**2
for dim in range(geom.vol_ndim))))
# Clear ASTRA memory
projector.clear_astra_memory()
# Returns
if not return_volume:
return s
else:
return s, vol.data
def least_squares(self,
iterations=1,
L=None,
tau=None,
sigma=None,
theta=None,
non_negativiy_constraint=False,
tv_norm=False,
verbose=True):
"""Least-squares problem with optional TV-regularisation and/or
non-negativity constraint.
Parameters
----------
:type iterations: int (default 1)
:param iterations: Number of iterations the optimization should
run for.
:type L: float (defaul: None)
:param L: Matrix norm of forward projector. If 'None' matrix_norm is
called with 20 iterations.
:type tau: float (default 1/L)
:param tau:
:type sigma: float (default 1/L)
:param sigma:
:type theta: float (default 1)
:param theta:
:type non_negativiy_constraint: bool (default False)
:param non_negativiy_constraint: Add non-negativity constraint to
optimization problem (via indicator function).
:type tv_norm: bool | float (default False)
:param tv_norm: Unless False, coincides with the numerical value of
the parameter lambda for TV-Regularisation.
:type verbose: bool (default False)
:param verbose: Show intermediate reconstructions and
convergence measures during iteration.
Returns
-------
:rtype: odl.Vector, odl.Vector, numpy.ndarray, numpy.ndarray
:returns: u, p, cpd, l2_du
u: vector of reconstructed volume
p: vector of dual projection variable
cpd: condition primal-dual gap (convergence measure)
l2_du: l2-norm of constraint-induced convergence measure
"""
# step 1:
if L is None:
L = self.matrix_norm(20)
if tau is None:
tau = 1 / L
if sigma is None:
sigma = 1 / L
if theta is None:
theta = 1
# print 'tau:', tau
# print 'sigma:', sigma
# print 'theta:', theta
geom = self.geom
g = self.proj # domain: D
# l2-norm of (volume update / tau)
l2_du = np.zeros(iterations)
# conditional primal-dual gap
cpd = np.zeros(iterations)
# step 2: initialize u and p with zeros
u = self.recon_space.zero() # domain: I
p = g.space.zero() # domain: D
# q: spatial vector = list of ndarrays in I (not Rn vectors)
if tv_norm:
ndim = geom.vol_ndim
# domain of q: V = [I, I, ...]
q = [
np.zeros(geom.vol_shape, dtype=u.data.dtype)
for _ in range(ndim)
]
# step 3: ub <- u
ub = u.copy() # domain: I
# initialize projector
# A = Projector(geom, u.space, p.space)
A = Projector(geom)
# visual output instance
disp = DisplayIntermediates(verbose=verbose,
vol=u.data.reshape(geom.vol_shape),
cpd=cpd,
l2_du=l2_du)
# step 4: repeat
for n in range(iterations):
# step 5: p_{n+1} <- (p_n + sigma(A^T ub_n - g)) / (1 + sigma)
if n >= 0:
# with(Timer('proj:')):
# # p_tmp <- A ub
# p_tmp = A.forward(ub)
# # p_tmp <- p_tmp - g
# p_tmp -= g
# # p <- p + sigma * p_tmp
# p += sigma * p_tmp
# p_n <- p_n + sigma(A ub -g )
tmp = A.forward(ub)
# print 'p:', p.data.shape, 'Au:', tmp.data.shape, 'g:', \
# g.data.shape
p += sigma * (A.forward(ub) - g)
else:
p -= sigma * g
# p <- p / (1 + sigma)
p /= 1 + sigma
# TV step 6: q_{n+1} <- lambda(q_n + sigma grad ub_n) /
# max(lambda 1_I, |q_n + sigma grad ub_n|)
if tv_norm:
for dim in range(ndim):
# q_n <- q_n + sigma * grad ub_n
q[dim] += sigma * partial(
ub.data.reshape(self.geom.vol_shape), dim,
geom.voxel_width[dim])
# |q_n|: isotropic TV
# use div_q to save memory, q = [qi] where qi are ndarrays
div_q = np.sqrt(reduce(add, (qi**2 for qi in q)))
# max(lambda 1_I, |q_n + sigma diff ub_n|)
# print 'q_mag:', div_q.min(), div_q.max()
div_q[div_q < tv_norm] = tv_norm
# q_n <- lambda * q_n / |q_n|
for dim in range(ndim):
q[dim] /= div_q
q[dim] *= tv_norm
# div q_{n+1}
div_q = reduce(add, (partial(qi, dim, geom.voxel_width[dim])
for (dim, qi) in enumerate(q)))
div_q *= tau
# step 6: u_{n+1} <- u_{n} - tau * A^T p_{n+1}
# TV step 7: u_{n+1} <- u_{n} - tau * A^T p_{n+1} + div q_{n+1}
# ub_tmp <- A^T p
ub_tmp = A.backward(p)
ub_tmp *= tau
ub_tmp *= self.adj_scal_fac
# l2-norm per voxel of ub_tmp = A^T p
l2_du[n:] = ub_tmp.norm() # / u.data.size
if tv_norm:
l2_du[n:] += np.linalg.norm(div_q.ravel()) # / u.data.size
# store current u_n temporarily in ub_n
ub = -u.copy()
# u <- u - tau ub_tmp
u -= ub_tmp
# TV: u <- u + tau div q
if tv_norm:
print('{0}: u - A^T p: min = {1}, max = {2}'.format(
n, u.data.min(), u.data.max()))
print('{0}: div q: min = {1}, max = {2}'.format(
n, div_q.min(), div_q.max()))
u.data[:] += div_q.ravel()
# Positivity constraint
if non_negativiy_constraint:
u.data[u.data < 0] = 0
# print '\nu:', u.data.min(), u.data.max()
# conditional primal-dual gap for current u and p
# 1/2||A u - g||_2^2 + 1/2||p||_2^2 + <p,g>_D
# p_tmp <- A u
# p_tmp = A.forward(u)
# p_tmp -= g
# cpd[n:] = (0.5 * p_tmp.norm() ** 2 +
cpd[n:] = (0.5 * p.space.norm(A.forward(u) - g)**2 +
0.5 * p.norm()**2 + p.inner(g)) # / p.data.size
if tv_norm:
cpd[n:] += tv_norm * np.linalg.norm(reduce(
add, (partial(u.data.reshape(geom.vol_shape), dim,
geom.voxel_width[dim])
for dim in range(geom.vol_ndim))).ravel(),
ord=1) # / u.data.size
# step 7 / TV step 8: ub_{n+1} <- u_{n+1} + theta(u_{n+1} - u_n)
# ub <- ub + u_{n+1}, remember ub = -u_n
ub += u
# ub <- theta * ub
ub *= theta
# ub <- ub + u_{n+1}
ub += u
# visual output
disp.update()
A.clear_astra_memory()
# Should avoid window freezing
disp.show()
return u, p, cpd, l2_du
def matrix_norm(self, iterations, vol_init=1.0,
tv_norm=False, return_volume=False,
intermediate_results=False):
"""The matrix norm || K ||_2 of 'K' defined here as largest
singular value of 'K'. Employs the generic power method to obtain a
scalar 's' which tends to || K ||_2 as the iterations N increase.
To be implemented: optionally return volume 'x', such that it can be
re-used as initializer to continue the iteration.
Parameters
----------
:type iterations: int
:param iterations: Number of iterations of the generic power method.
:type vol_init: float | ndarray (default 1.0)
:param vol_init: in I, initial image to start with.
:type intermediate_results: bool
:param intermediate_results: Returns list of intermediate results
instead of scalar.
:type return_volume: bool
:param return_volume: Return volume in order to resume iteration via
passing it over as initial volume.
Returns
-------
:rtype: float | numpy.ndarray, numpay.array (optional)
:returns: s, vol
s: Scalar of final iteration or numpy.ndarray containing all
results during iteration.
vol: Volume vector
"""
geom = self.geom
vol = self.recon_space.element(vol_init)
proj = Rn(geom.proj_size).zero()
# projector = Projector(geom, vol.space, proj.space)
projector = Projector(geom)
# print 'projector scaling factor', projector.scal_fac
tmp = None
if intermediate_results:
s = np.zeros(iterations)
else:
s = 0
# Power method loop
for n in range(iterations):
# step 4: x_{n+1} <- K^T K x_n
if tv_norm:
# K = (A, grad) instead of K = A
# Compute: - div grad x_n
# use sum over generator expression
tmp = -reduce(add,
(partial(
partial(vol.data.reshape(geom.vol_shape),
dim, geom.voxel_width[dim]),
dim, geom.voxel_width[dim]) for dim in
range(geom.vol_ndim)))
# x_n <- A^T (A x_n)
vol = projector.backward(projector.forward(vol))
vol *= self.adj_scal_fac
if tv_norm:
# x_n <- x_n - div grad x_n
# print 'n: {2}. vol: min = {0}, max = {1}'.format(
# vol.data.min(), vol.data.max(), n)
# print 'n: {2}. tv: min = {0}, max = {1}'.format(tmp.min(),
# tmp.max(), n)
vol.data[:] += tmp.ravel()
# step 5:
# x_n <- x_n/||x_n||_2
vol /= vol.norm()
# step 6:
# s_n <-|| K x ||_2
if intermediate_results:
# proj <- A^T x_n
proj = projector.forward(vol)
s[n] = proj.norm()
if tv_norm:
s[n] = np.sqrt(s[n] ** 2 +
reduce(add,
(np.linalg.norm(
partial(vol.data.reshape(
geom.vol_shape), dim,
geom.voxel_width[dim])) ** 2
for dim in range(geom.vol_ndim))))
# step 6: || K x ||_2
if not intermediate_results:
proj = projector.forward(vol)
s = proj.norm()
if tv_norm:
s = np.sqrt(s ** 2 + reduce(add,
(np.linalg.norm(partial(
vol.data.reshape(
geom.vol_shape), dim,
geom.voxel_width[dim])) ** 2
for dim in range(geom.vol_ndim))))
# Clear ASTRA memory
projector.clear_astra_memory()
# Returns
if not return_volume:
return s
else:
return s, vol.data
def least_squares(self, iterations=1, L=None, tau=None, sigma=None,
theta=None, non_negativiy_constraint=False,
tv_norm=False,
verbose=True):
"""Least-squares problem with optional TV-regularisation and/or
non-negativity constraint.
Parameters
----------
:type iterations: int (default 1)
:param iterations: Number of iterations the optimization should
run for.
:type L: float (defaul: None)
:param L: Matrix norm of forward projector. If 'None' matrix_norm is
called with 20 iterations.
:type tau: float (default 1/L)
:param tau:
:type sigma: float (default 1/L)
:param sigma:
:type theta: float (default 1)
:param theta:
:type non_negativiy_constraint: bool (default False)
:param non_negativiy_constraint: Add non-negativity constraint to
optimization problem (via indicator function).
:type tv_norm: bool | float (default False)
:param tv_norm: Unless False, coincides with the numerical value of
the parameter lambda for TV-Regularisation.
:type verbose: bool (default False)
:param verbose: Show intermediate reconstructions and
convergence measures during iteration.
Returns
-------
:rtype: odl.Vector, odl.Vector, numpy.ndarray, numpy.ndarray
:returns: u, p, cpd, l2_du
u: vector of reconstructed volume
p: vector of dual projection variable
cpd: condition primal-dual gap (convergence measure)
l2_du: l2-norm of constraint-induced convergence measure
"""
# step 1:
if L is None:
L = self.matrix_norm(20)
if tau is None:
tau = 1 / L
if sigma is None:
sigma = 1 / L
if theta is None:
theta = 1
# print 'tau:', tau
# print 'sigma:', sigma
# print 'theta:', theta
geom = self.geom
g = self.proj # domain: D
# l2-norm of (volume update / tau)
l2_du = np.zeros(iterations)
# conditional primal-dual gap
cpd = np.zeros(iterations)
# step 2: initialize u and p with zeros
u = self.recon_space.zero() # domain: I
p = g.space.zero() # domain: D
# q: spatial vector = list of ndarrays in I (not Rn vectors)
if tv_norm:
ndim = geom.vol_ndim
# domain of q: V = [I, I, ...]
q = [np.zeros(geom.vol_shape, dtype=u.data.dtype) for _ in range(
ndim)]
# step 3: ub <- u
ub = u.copy() # domain: I
# initialize projector
# A = Projector(geom, u.space, p.space)
A = Projector(geom)
# visual output instance
disp = DisplayIntermediates(verbose=verbose, vol=u.data.reshape(
geom.vol_shape), cpd=cpd, l2_du=l2_du)
# step 4: repeat
for n in range(iterations):
# step 5: p_{n+1} <- (p_n + sigma(A^T ub_n - g)) / (1 + sigma)
if n >= 0:
# with(Timer('proj:')):
# # p_tmp <- A ub
# p_tmp = A.forward(ub)
# # p_tmp <- p_tmp - g
# p_tmp -= g
# # p <- p + sigma * p_tmp
# p += sigma * p_tmp
# p_n <- p_n + sigma(A ub -g )
tmp = A.forward(ub)
# print 'p:', p.data.shape, 'Au:', tmp.data.shape, 'g:', \
# g.data.shape
p += sigma * (A.forward(ub) - g)
else:
p -= sigma * g
# p <- p / (1 + sigma)
p /= 1 + sigma
# TV step 6: q_{n+1} <- lambda(q_n + sigma grad ub_n) /
# max(lambda 1_I, |q_n + sigma grad ub_n|)
if tv_norm:
for dim in range(ndim):
# q_n <- q_n + sigma * grad ub_n
q[dim] += sigma * partial(ub.data.reshape(
self.geom.vol_shape), dim, geom.voxel_width[dim])
# |q_n|: isotropic TV
# use div_q to save memory, q = [qi] where qi are ndarrays
div_q = np.sqrt(reduce(add, (qi ** 2 for qi in q)))
# max(lambda 1_I, |q_n + sigma diff ub_n|)
# print 'q_mag:', div_q.min(), div_q.max()
div_q[div_q < tv_norm] = tv_norm
# q_n <- lambda * q_n / |q_n|
for dim in range(ndim):
q[dim] /= div_q
q[dim] *= tv_norm
# div q_{n+1}
div_q = reduce(add, (partial(qi, dim, geom.voxel_width[dim])
for (dim, qi) in enumerate(q)))
div_q *= tau
# step 6: u_{n+1} <- u_{n} - tau * A^T p_{n+1}
# TV step 7: u_{n+1} <- u_{n} - tau * A^T p_{n+1} + div q_{n+1}
# ub_tmp <- A^T p
ub_tmp = A.backward(p)
ub_tmp *= tau
ub_tmp *= self.adj_scal_fac
# l2-norm per voxel of ub_tmp = A^T p
l2_du[n:] = ub_tmp.norm() # / u.data.size
if tv_norm:
l2_du[n:] += np.linalg.norm(div_q.ravel()) # / u.data.size
# store current u_n temporarily in ub_n
ub = -u.copy()
# u <- u - tau ub_tmp
u -= ub_tmp
# TV: u <- u + tau div q
if tv_norm:
print('{0}: u - A^T p: min = {1}, max = {2}'.format(
n, u.data.min(), u.data.max()))
print('{0}: div q: min = {1}, max = {2}'.format(
n, div_q.min(), div_q.max()))
u.data[:] += div_q.ravel()
# Positivity constraint
if non_negativiy_constraint:
u.data[u.data < 0] = 0
# print '\nu:', u.data.min(), u.data.max()
# conditional primal-dual gap for current u and p
# 1/2||A u - g||_2^2 + 1/2||p||_2^2 + <p,g>_D
# p_tmp <- A u
# p_tmp = A.forward(u)
# p_tmp -= g
# cpd[n:] = (0.5 * p_tmp.norm() ** 2 +
cpd[n:] = (0.5 * p.space.norm(A.forward(u) - g) ** 2 +
0.5 * p.norm() ** 2 +
p.inner(g)) # / p.data.size
if tv_norm:
cpd[n:] += tv_norm * np.linalg.norm(
reduce(add, (partial(u.data.reshape(geom.vol_shape),
dim, geom.voxel_width[dim]) for dim
in range(geom.vol_ndim))
).ravel(), ord=1) # / u.data.size
# step 7 / TV step 8: ub_{n+1} <- u_{n+1} + theta(u_{n+1} - u_n)
# ub <- ub + u_{n+1}, remember ub = -u_n
ub += u
# ub <- theta * ub
ub *= theta
# ub <- ub + u_{n+1}
ub += u
# visual output
disp.update()
A.clear_astra_memory()
# Should avoid window freezing
disp.show()
return u, p, cpd, l2_du