# 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.functions import Function
from cil.framework import DataContainer
from cil.optimisation.operators import DiagonalOperator
[docs]
class L2NormSquared(Function):
r""" L2NormSquared function: :math:`F(x) = \| x\|^{2}_{2} = \underset{i}{\sum}x_{i}^{2}`
Following cases are considered:
a) :math:`F(x) = \|x\|^{2}_{2}`
b) :math:`F(x) = \|x - b\|^{2}_{2}`
Parameters
----------
b:`DataContainer`, optional
Translation of the function
Note
-----
For case b) we can use :code:`F = L2NormSquared().centered_at(b)`, see *TranslateFunction*.
Example
-------
>>> F = L2NormSquared()
>>> F = L2NormSquared(b=b)
>>> F = L2NormSquared().centered_at(b)
"""
def __init__(self, **kwargs):
super(L2NormSquared, self).__init__(L=2)
self.b = kwargs.get('b', None)
def __call__(self, x):
y = x
if self.b is not None:
y = x - self.b
try:
return y.squared_norm()
except AttributeError as ae:
# added for compatibility with SIRF
return (y.norm()**2)
[docs]
def gradient(self, x, out=None):
r"""Returns the value of the gradient of the L2NormSquared function at x.
Following cases are considered:
a) :math:`F'(x) = 2x`
b) :math:`F'(x) = 2(x-b)`
"""
if self.b is None:
return x.multiply(2, out=out)
else:
return x.sapyb(2, self.b, -2, out=out)
[docs]
def convex_conjugate(self, x):
r"""Returns the value of the convex conjugate of the L2NormSquared function at x.
Consider the following cases:
a) .. math:: F^{*}(x^{*}) = \frac{1}{4}\|x^{*}\|^{2}_{2}
b) .. math:: F^{*}(x^{*}) = \frac{1}{4}\|x^{*}\|^{2}_{2} + \langle x^{*}, b\rangle
"""
tmp = 0
if self.b is not None:
tmp = x.dot(self.b)
return 0.25 * x.squared_norm() + tmp
[docs]
def proximal(self, x, tau, out=None):
r"""Returns the value of the proximal operator of the L2NormSquared function at x.
Consider the following cases:
a) .. math:: \text{prox}_{\tau F}(x) = \frac{x}{1+2\tau}
b) .. math:: \text{prox}_{\tau F}(x) = \frac{x-b}{1+2\tau} + b
"""
mult = 1/(1+2*tau)
if self.b is None:
return x.multiply(mult, out=out)
else:
return x.sapyb(mult, self.b, (1-mult), out=out)
[docs]
class WeightedL2NormSquared(Function):
r""" WeightedL2NormSquared function: :math:`F(x) = \|x\|_{W,2}^2 = \Sigma_iw_ix_i^2 = \langle x, Wx\rangle = x^TWx`
where :math:`W=\text{diag}(weight)` if `weight` is a `DataContainer` or :math:`W=\text{weight} I` if `weight` is a scalar.
Parameters
-----------
**kwargs
weight: a `scalar` or a `DataContainer` with the same shape as the intended domain of this `WeightedL2NormSquared` function
b: a `DataContainer` with the same shape as the intended domain of this `WeightedL2NormSquared` function
A shift so that the function becomes :math:`F(x) = \| x-b\|_{W,2}^2 = \Sigma_iw_i(x_i-b_i)^2 = \langle x-b, W(x-b) \rangle = (x-b)^TW(x-b)`
"""
def __init__(self, **kwargs):
# Weight can be either a scalar or a DataContainer
# Lispchitz constant L = 2 *||weight||
self.weight = kwargs.get('weight', 1.0)
self.b = kwargs.get('b', None)
tmp_norm = 1.0
self.tmp_space = self.weight*0.
if isinstance(self.weight, DataContainer):
self.operator_weight = DiagonalOperator(self.weight)
tmp_norm = self.operator_weight.norm()
self.tmp_space = self.operator_weight.domain_geometry().allocate()
if (self.weight < 0).any():
raise ValueError('Weight contains negative values')
super(WeightedL2NormSquared, self).__init__(L=2 * tmp_norm)
def __call__(self, x):
self.operator_weight.direct(x, out=self.tmp_space)
y = x.dot(self.tmp_space)
if self.b is not None:
self.operator_weight.direct(x - self.b, out=self.tmp_space)
y = (x - self.b).dot(self.tmp_space)
return y
[docs]
def gradient(self, x, out=None):
r""" Returns the value of :math:`F'(x) = 2Wx` or, if `b` is defined, :math:`F'(x) = 2W(x-b)`
where :math:`W=\text{diag}(weight)` if `weight` is a `DataContainer` or :math:`\text{weight}I` if `weight` is a scalar.
"""
if out is not None:
out.fill(x)
if self.b is not None:
out -= self.b
self.operator_weight.direct(out, out=out)
out *= 2
return out
else:
y = x
if self.b is not None:
y = x - self.b
return 2*self.weight*y
[docs]
def convex_conjugate(self, x):
r"""Returns the value of the convex conjugate of the WeightedL2NormSquared function at x."""
tmp = 0
if self.b is not None:
tmp = x.dot(self.b)
return (1./4) * (x/self.weight.sqrt()).squared_norm() + tmp
[docs]
def proximal(self, x, tau, out=None):
r"""Returns the value of the proximal operator of the WeightedL2NormSquared function at x."""
if self.b is not None:
ret = x.subtract(self.b, out=out)
ret /= (1+2*tau*self.weight)
ret += self.b
else:
ret = x.divide((1+2*tau*self.weight), out=out)
return ret
