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# -*- coding: utf-8 -*-
# Copyright 2019 Science Technology Facilities Council
# Copyright 2019 University of Manchester
#
# This work is part of the Core Imaging Library developed by Science Technology
# Facilities Council and 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.txt
#
# 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.
from ccpi.optimisation.functions import Function
from ccpi.optimisation.functions.ScaledFunction import ScaledFunction
from ccpi.optimisation.operators import ShrinkageOperator
class L1Norm(Function):
r'''L1Norm function:
Cases considered (with/without data):
a) .. math:: f(x) = ||x||_{1}
b) .. math:: f(x) = ||x - b||_{1}
'''
def __init__(self, **kwargs):
super(L1Norm, self).__init__()
self.b = kwargs.get('b',None)
def __call__(self, x):
'''Evaluates L1Norm at x'''
y = x
if self.b is not None:
y = x - self.b
return y.abs().sum()
def gradient(self,x):
return ValueError('Not Differentiable')
def convex_conjugate(self,x):
'''Convex conjugate of L1Norm at x'''
y = 0
if self.b is not None:
y = 0 + self.b.dot(x)
return y
def proximal(self, x, tau, out=None):
r'''Proximal operator of L1Norm at x
..math:: prox_{\tau * f}(x)
'''
if out is None:
if self.b is not None:
return self.b + ShrinkageOperator.__call__(self, x - self.b, tau)
else:
return ShrinkageOperator.__call__(self, x, tau)
else:
if self.b is not None:
out.fill(self.b + ShrinkageOperator.__call__(self, x - self.b, tau))
else:
out.fill(ShrinkageOperator.__call__(self, x, tau))
def proximal_conjugate(self, x, tau, out=None):
r'''Proximal operator of the convex conjugate of L1Norm at x:
.. math:: prox_{\tau * f^{*}}(x)
'''
if out is None:
if self.b is not None:
return (x - tau*self.b).divide((x - tau*self.b).abs().maximum(1.0))
else:
return x.divide(x.abs().maximum(1.0))
else:
if self.b is not None:
out.fill((x - tau*self.b).divide((x - tau*self.b).abs().maximum(1.0)))
else:
out.fill(x.divide(x.abs().maximum(1.0)) )
def __rmul__(self, scalar):
'''Multiplication of L2NormSquared with a scalar
Returns: ScaledFunction
'''
return ScaledFunction(self, scalar)
if __name__ == '__main__':
from ccpi.framework import ImageGeometry
import numpy
N, M = 400,400
ig = ImageGeometry(N, M)
scalar = 10
b = ig.allocate('random')
u = ig.allocate('random')
f = L1Norm()
f_scaled = scalar * L1Norm()
f_b = L1Norm(b=b)
f_scaled_b = scalar * L1Norm(b=b)
# call
a1 = f(u)
a2 = f_scaled(u)
numpy.testing.assert_equal(scalar * a1, a2)
a3 = f_b(u)
a4 = f_scaled_b(u)
numpy.testing.assert_equal(scalar * a3, a4)
# proximal
tau = 0.4
b1 = f.proximal(u, tau*scalar)
b2 = f_scaled.proximal(u, tau)
numpy.testing.assert_array_almost_equal(b1.as_array(), b2.as_array(), decimal=4)
b3 = f_b.proximal(u, tau*scalar)
b4 = f_scaled_b.proximal(u, tau)
z1 = b + (u-b).sign() * ((u-b).abs() - tau * scalar).maximum(0)
numpy.testing.assert_array_almost_equal(b3.as_array(), b4.as_array(), decimal=4)
#
# #proximal conjugate
#
c1 = f_scaled.proximal_conjugate(u, tau)
c2 = u.divide((u.abs()/scalar).maximum(1.0))
numpy.testing.assert_array_almost_equal(c1.as_array(), c2.as_array(), decimal=4)
c3 = f_scaled_b.proximal_conjugate(u, tau)
c4 = (u - tau*b).divide( ((u-tau*b).abs()/scalar).maximum(1.0) )
numpy.testing.assert_array_almost_equal(c3.as_array(), c4.as_array(), decimal=4)
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