# Copyright 2019 United Kingdom Research and Innovation
# Copyright 2019 The University of Manchester
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
#
# Authors:
# CIL Developers, listed at: https://github.com/TomographicImaging/CIL/blob/master/NOTICE.txt
from cil.optimisation.operators import LinearOperator
from cil.framework import BlockGeometry, BlockDataContainer
from cil.optimisation.operators import FiniteDifferenceOperator
[docs]
class SymmetrisedGradientOperator(LinearOperator):
r''' The symmetrised gradient is the operator, :math:`E`, defined by :math:`E: V \rightarrow W` where `V` is `BlockGeometry` and `W` is the range of the Symmetrised Gradient and
.. math::
E(v) = 0.5 ( \nabla v + (\nabla v)^{T} ) \\
In 2 dimensions, let :math:`v(x,y)=(v_1(x,y),v_2(x,y))` which gives
.. math::
\nabla v =\left( \begin{matrix}
\partial_{x} v_1 & \partial_x v_2\\
\partial_{y}v_1 & \partial_y v_2
\end{matrix}\right)
and thus
.. math::
E(v) = 0.5 ( \nabla v + (\nabla v)^{T} )
=\left( \begin{matrix}
\partial_{x} v_1 & 0.5 (\partial_{y} v_1 + \partial_{x} v_2) \\
0.5 (\partial_{x} v_1 + \partial_{y} v_2) & \partial_{y} v_2
\end{matrix}\right)
Parameters
----------
domain_geometry: `BlockGeometry` with shape (2,1) or (3,1)
Set up the domain of the function.
bnd_cond: str, optional, default :code:`Neumann`
Boundary condition either :code:`Neumann` or :code:`Periodic`
correlation: str, optional, default :code:`Channel`
Correlation either :code:`SpaceChannel` or :code:`Channel`
'''
CORRELATION_SPACE = "Space"
CORRELATION_SPACECHANNEL = "SpaceChannels"
def __init__(self, domain_geometry, bnd_cond = 'Neumann', **kwargs):
self.bnd_cond = bnd_cond
self.correlation = kwargs.get('correlation',SymmetrisedGradientOperator.CORRELATION_SPACE)
tmp_gm = len(domain_geometry.geometries)*domain_geometry.geometries
# Define FD operator. We need one geometry from the BlockGeometry of the domain
self.FD = FiniteDifferenceOperator(domain_geometry.get_item(0), direction = 0,
bnd_cond = self.bnd_cond)
if domain_geometry.shape[0]==2:
self.order_ind = [0,2,1,3]
else:
self.order_ind = [0,3,6,1,4,7,2,5,8]
super(SymmetrisedGradientOperator, self).__init__(
domain_geometry=domain_geometry,
range_geometry=BlockGeometry(*tmp_gm))
[docs]
def direct(self, x, out=None):
r'''Returns :math:`E(v) = 0.5 * ( \nabla v + (\nabla v)^{T} )`
Parameters:
-------------
x: BlockDataContainer
out: BlockDataContainer, default None
If out is not None the output of direct will be filled in out, otherwise a new object is instantiated and returned.
'''
if out is None:
tmp = []
for i in range(self.domain_geometry().shape[0]):
for j in range(x.shape[0]):
self.FD.direction = i
tmp.append(self.FD.adjoint(x.get_item(j)))
tmp1 = [tmp[i] for i in self.order_ind]
res = [0.5 * sum(x) for x in zip(tmp, tmp1)]
return BlockDataContainer(*res)
else:
ind = 0
for i in range(self.domain_geometry().shape[0]):
for j in range(x.shape[0]):
self.FD.direction = i
self.FD.adjoint(x.get_item(j), out=out[ind])
ind+=1
out1 = BlockDataContainer(*[out[i] for i in self.order_ind])
out.fill( 0.5 * (out + out1) )
return out
[docs]
def adjoint(self, x, out=None):
r'''Returns the adjoint of the symmetrised gradient operator
Parameters:
-------------
x: BlockDataContainer
out: BlockDataContainer, default None
If out is not None the output of adjoint will be filled in out, otherwise a new object is instantiated and returned.
'''
if out is None:
tmp = [None]*self.domain_geometry().shape[0]
i = 0
for k in range(self.domain_geometry().shape[0]):
tmp1 = 0
for j in range(self.domain_geometry().shape[0]):
self.FD.direction = j
tmp1 += self.FD.direct(x[i])
i+=1
tmp[k] = tmp1
return BlockDataContainer(*tmp)
else:
tmp = self.domain_geometry().allocate()
i = 0
for k in range(self.domain_geometry().shape[0]):
tmp1 = 0
for j in range(self.domain_geometry().shape[0]):
self.FD.direction = j
self.FD.direct(x[i], out=tmp[j])
i+=1
tmp1+=tmp[j]
out[k].fill(tmp1)
return out
