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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 __future__ import absolute_import
from __future__ import division
from __future__ import print_function
import numpy
from ccpi.optimisation.functions import Function
import functools
import scipy.special
class KullbackLeibler(Function):
r""" Kullback Leibler divergence function is defined as:
.. math:: F(u, v)
= \begin{cases}
u \log(\frac{u}{v}) - u + v & \mbox{ if } u > 0, v > 0\\
v & \mbox{ if } u = 0, v \ge 0 \\
\infty, & \mbox{otherwise}
\end{cases}
where we use the :math:`0\log0 := 0` convention.
At the moment, we use build-in implemention of scipy, see
https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.kl_div.html
The Kullback-Leibler function is used as a fidelity term in minimisation problems where the
acquired data follow Poisson distribution. If we denote the acquired data with :math:`b`
then, we write
.. math:: \underset{i}{\sum} F(b_{i}, (v + \eta)_{i})
where, :math:`\eta` is an additional noise.
Example: In the case of Positron Emission Tomography reconstruction :math:`\eta` represents
scatter and random events contribution during the PET acquisition. Hence, in that case the KullbackLeibler
fidelity measures the distance between :math:`\mathcal{A}v + \eta` and acquisition data :math:`b`, where
:math:`\mathcal{A}` is the projection operator.
This is related to PoissonLogLikelihoodWithLinearModelForMean definition that is used in PET reconstruction
in the PET-MR software , see https://github.com/CCPPETMR and for more details in
http://stir.sourceforge.net/documentation/doxy/html/classstir_1_1PoissonLogLikelihoodWithLinearModelForMean.html
"""
def __init__(self, **kwargs):
super(KullbackLeibler, self).__init__(L = None)
self.b = kwargs.get('b', None)
if self.b is None:
raise ValueError('Please define data, as b = ...')
if (self.b.as_array() < 0).any():
raise ValueError('Data should be larger or equal to 0')
self.eta = kwargs.get('eta',self.b * 0.0)
def __call__(self, x):
r"""Returns the value of the KullbackLeibler function at :math:`(b, x + \eta)`.
To avoid infinity values, we consider only pixels/voxels for :math:`x+\eta\geq0`.
"""
tmp_sum = (x + self.eta).as_array()
ind = tmp_sum >= 0
tmp = scipy.special.kl_div(self.b.as_array()[ind], tmp_sum[ind])
return numpy.sum(tmp)
def log(self, datacontainer):
'''calculates the in-place log of the datacontainer'''
if not functools.reduce(lambda x,y: x and y>0, datacontainer.as_array().ravel(), True):
raise ValueError('KullbackLeibler. Cannot calculate log of negative number')
datacontainer.fill( numpy.log(datacontainer.as_array()) )
def gradient(self, x, out=None):
r"""Returns the value of the gradient of the KullbackLeibler function at :math:`(b, x + \eta)`.
.. math:: F'(b, x + \eta) = 1 - \frac{b}{x+\eta}
We require the :math:`x+\eta>0` otherwise we have inf values.
"""
tmp_sum_array = (x + self.eta).as_array()
if out is None:
tmp_out = x.geometry.allocate()
tmp_out.as_array()[tmp_sum_array>0] = 1 - self.b.as_array()[tmp_sum_array>0]/tmp_sum_array[tmp_sum_array>0]
return tmp_out
else:
x.add(self.eta, out=out)
out.as_array()[tmp_sum_array>0] = 1 - self.b.as_array()[tmp_sum_array>0]/tmp_sum_array[tmp_sum_array>0]
def convex_conjugate(self, x):
r"""Returns the value of the convex conjugate of the KullbackLeibler function at :math:`(b, x + \eta)`.
.. math:: F^{*}(b, x + \eta) = - b \log(1-x^{*}) - <x^{*}, \eta>
"""
tmp = 1 - x.as_array()
ind = tmp>0
xlogy = - scipy.special.xlogy(self.b.as_array()[ind], tmp[ind])
return numpy.sum(xlogy) - self.eta.dot(x)
def proximal(self, x, tau, out=None):
r"""Returns the value of the proximal operator of the KullbackLeibler function at :math:`(b, x + \eta)`.
.. math:: \mathrm{prox}_{\tau F}(x) = \frac{1}{2}\bigg( (x - \eta - \tau) + \sqrt{ (x + \eta - \tau)^2 + 4\tau b} \bigg)
The proximal for the convex conjugate of :math:`F` is
.. math:: \mathrm{prox}_{\tau F^{*}}(x) = 0.5*((z + 1) - \sqrt{(z-1)^2 + 4 * \tau b})
where :math:`z = x + \tau \eta`
"""
if out is None:
return 0.5 *( (x - self.eta - tau) + ( (x + self.eta - tau)**2 + 4*tau*self.b ) .sqrt() )
else:
x.add(self.eta, out=out)
out -= tau
out *= out
out.add(self.b * (4 * tau), out=out)
out.sqrt(out=out)
out.subtract(tau, out = out)
out.add(x, out=out)
out *= 0.5
def proximal_conjugate(self, x, tau, out=None):
r'''Proximal operator of the convex conjugate of KullbackLeibler at x:
.. math:: prox_{\tau * f^{*}}(x)
'''
if out is None:
z = x + tau * self.eta
return 0.5*((z + 1) - ((z-1)**2 + 4 * tau * self.b).sqrt())
else:
tmp = tau * self.eta
tmp += x
tmp -= 1
self.b.multiply(4*tau, out=out)
out.add((tmp)**2, out=out)
out.sqrt(out=out)
out *= -1
tmp += 2
out += tmp
out *= 0.5
if __name__ == '__main__':
from ccpi.framework import ImageGeometry
import numpy as np
M, N, K = 30, 30, 20
ig = ImageGeometry(N, M, K)
u1 = ig.allocate('random_int', seed = 500)
g1 = ig.allocate('random_int', seed = 1000)
b1 = ig.allocate('random', seed = 5000)
# with no data
try:
f = KullbackLeibler()
except ValueError:
print('Give data b=...\n')
print('With negative data, no background\n')
try:
f = KullbackLeibler(b=-1*g1)
except ValueError:
print('We have negative data\n')
f = KullbackLeibler(b=g1)
print('Check KullbackLeibler(x,x)=0\n')
numpy.testing.assert_equal(0.0, f(g1))
print('Check gradient .... is OK \n')
res_gradient = f.gradient(u1)
res_gradient_out = u1.geometry.allocate()
f.gradient(u1, out = res_gradient_out)
numpy.testing.assert_array_almost_equal(res_gradient.as_array(), \
res_gradient_out.as_array(),decimal = 4)
print('Check proximal ... is OK\n')
tau = 0.4
res_proximal = f.proximal(u1, tau)
res_proximal_out = u1.geometry.allocate()
f.proximal(u1, tau, out = res_proximal_out)
numpy.testing.assert_array_almost_equal(res_proximal.as_array(), \
res_proximal_out.as_array(), decimal =5)
print('Check conjugate ... is OK\n')
res_conj = f.convex_conjugate(u1)
if (1 - u1.as_array()).all():
print('If 1-x<=0, Convex conjugate returns 0.0')
numpy.testing.assert_equal(0.0, f.convex_conjugate(u1))
print('Check KullbackLeibler with background\n')
f1 = KullbackLeibler(b=g1, eta=b1)
tmp_sum = (u1 + f1.eta).as_array()
ind = tmp_sum >= 0
tmp = scipy.special.kl_div(f1.b.as_array()[ind], tmp_sum[ind])
numpy.testing.assert_equal(f1(u1), numpy.sum(tmp) )
print('Check proximal KL without background\n')
tau = [0.1, 1, 10, 100, 10000]
for t1 in tau:
proxc = f.proximal_conjugate(u1,t1)
proxc_out = ig.allocate()
f.proximal_conjugate(u1, t1, out = proxc_out)
print('tau = {} is OK'.format(t1) )
numpy.testing.assert_array_almost_equal(proxc.as_array(),
proxc_out.as_array(),
decimal = 4)
print('\nCheck proximal KL with background\n')
for t1 in tau:
proxc1 = f1.proximal_conjugate(u1,t1)
proxc_out1 = ig.allocate()
f1.proximal_conjugate(u1, t1, out = proxc_out1)
numpy.testing.assert_array_almost_equal(proxc1.as_array(),
proxc_out1.as_array(),
decimal = 4)
print('tau = {} is OK'.format(t1) )
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