# Copyright 2014-2017 The ODL contributors
#
# This file is part of ODL.
#
# This Source Code Form is subject to the terms of the Mozilla Public License,
# v. 2.0. If a copy of the MPL was not distributed with this file, You can
# obtain one at https://mozilla.org/MPL/2.0/.

"""Maximum Likelihood Expectation Maximization algorithm."""

from __future__ import print_function, division, absolute_import
import numpy as np

__all__ = ('mlem', 'osmlem', 'loglikelihood')


AVAILABLE_MLEM_NOISE = ('poisson',)


def mlem(op, x, data, niter, noise='poisson', callback=None, **kwargs):

    """Maximum Likelihood Expectation Maximation algorithm.

    Attempts to solve::

        max_x L(x | data)

    where ``L(x | data)`` is the likelihood of ``x`` given ``data``. The
    likelihood depends on the forward operator ``op`` such that
    (approximately)::

        op(x) = data

    With the precise form of *approximately* is determined by ``noise``.

    Parameters
    ----------
    op : `Operator`
        Forward operator in the inverse problem.
    x : ``op.domain`` element
        Vector to which the result is written. Its initial value is
        used as starting point of the iteration, and its values are
        updated in each iteration step.
    data : ``op.range`` `element-like`
        Right-hand side of the equation defining the inverse problem.
    niter : int
        Number of iterations.
    noise : {'poisson'}, optional
        Noise model determining the variant of MLEM.
        For ``'poisson'``, the initial value of ``x`` should be
        non-negative.
    callback : callable, optional
        Function called with the current iterate after each iteration.

    Other Parameters
    ----------------
    sensitivities : float or ``op.domain`` `element-like`, optional
        Usable with ``noise='poisson'``. The algorithm contains a ``A^T 1``
        term, if this parameter is given, it is replaced by it.
        Default: ``op.adjoint(op.range.one())``

    Notes
    -----
    Given a forward model :math:`A`, data :math:`g` and noise model :math:`X`,
    the algorithm attempts find an :math:`x` that maximizes:

    .. math::
        P(g | g \\text{ is } X(A(x)) \\text{ distributed}).

    With 'poisson' noise the algorithm is given by:

    .. math::
       x_{n+1} = \\frac{x_n}{A^* 1} A^* (g / A(x_n))

    See Also
    --------
    osmlem : Ordered subsets MLEM
    loglikelihood : Function for calculating the logarithm of the likelihood
    """
    osmlem([op], x, [data], niter=niter, noise=noise, callback=callback,
           **kwargs)


def osmlem(op, x, data, niter, noise='poisson', callback=None, **kwargs):
    """Ordered Subsets Maximum Likelihood Expectation Maximation algorithm.

    This solver attempts to solve::

        max_x L(x | data)

    where ``L(x, | data)`` is the likelihood of ``x`` given ``data``. The
    likelihood depends on the forward operators ``op[0], ..., op[n-1]`` such
    that (approximately)::

        op[i](x) = data[i]

    where the precise form of *approximately* is determined by ``noise``.

    Parameters
    ----------
    op : sequence of `Operator`
        Forward operators in the inverse problem.
    x : ``op.domain`` element
        Vector to which the result is written. Its initial value is
        used as starting point of the iteration, and its values are
        updated in each iteration step.
    data : sequence of ``op.range`` `element-like`
        Right-hand sides of the equation defining the inverse problem.
    niter : int
        Number of iterations.
    noise : {'poisson'}, optional
        Noise model determining the variant of MLEM.
        For ``'poisson'``, the initial value of ``x`` should be
        non-negative.
    callback : callable, optional
        Function called with the current iterate after each iteration.

    Other Parameters
    ----------------
    sensitivities : float or ``op.domain`` `element-like`, optional
        Usable with ``noise='poisson'``. The algorithm contains an ``A^T 1``
        term, if this parameter is given, it is replaced by it.
        Default: ``op[i].adjoint(op[i].range.one())``

    Notes
    -----
    Given a forward models :math:`A_i`, data :math:`g_i`, :math:`i = 1, ..., M`
    and noise model :math:`X`, the algorithm attempts find an :math:`x` that
    maximizes:

    .. math::
        \prod_{i=1}^M P(g_i | g_i \\text{ is } X(A_i(x)) \\text{ distributed}).

    With 'poisson' noise the algorithm is given by partial updates:

    .. math::
       x_{n + m/M} =
       \\frac{x_{n + (m - 1)/M}}{A_i^* 1} A_i^* (g_i / A_i(x_{n + (m - 1)/M}))

    for :math:`m = 1, ..., M` and :math:`x_{n+1} = x_{n + M/M}`.

    The algorithm is not guaranteed to converge, but works for many practical
    problems.

    References
    ----------
    Natterer, F. Mathematical Methods in Image Reconstruction, section 5.3.2.

    See Also
    --------
    mlem : Ordinary MLEM algorithm without subsets.
    loglikelihood : Function for calculating the logarithm of the likelihood
    """
    noise, noise_in = str(noise).lower(), noise
    if noise not in AVAILABLE_MLEM_NOISE:
        raise NotImplemented("noise '{}' not understood"
                             ''.format(noise_in))

    n_ops = len(op)
    if len(data) != n_ops:
        raise ValueError('number of data ({}) does not match number of '
                         'operators ({})'.format(len(data), n_ops))
    if not all(x in opi.domain for opi in op):
        raise ValueError('`x` not an element in the domains of all operators')

    # Convert data to range elements
    data = [op[i].range.element(data[i]) for i in range(len(op))]

    if noise == 'poisson':
        # Parameter used to enforce positivity.
        # TODO: let users give this.
        eps = 1e-8

        if np.any(np.less(x, 0)):
            raise ValueError('`x` must be non-negative')

        # Extract the sensitivites parameter
        sensitivities = kwargs.pop('sensitivities', None)
        if sensitivities is None:
            sensitivities = [np.maximum(opi.adjoint(opi.range.one()), eps)
                             for opi in op]
        else:
            # Make sure the sensitivities is a list of the correct size.
            try:
                list(sensitivities)
            except TypeError:
                sensitivities = [sensitivities] * n_ops

        tmp_dom = op[0].domain.element()
        tmp_ran = [opi.range.element() for opi in op]

        for _ in range(niter):
            for i in range(n_ops):
                op[i](x, out=tmp_ran[i])
                tmp_ran[i].ufuncs.maximum(eps, out=tmp_ran[i])
                data[i].divide(tmp_ran[i], out=tmp_ran[i])

                op[i].adjoint(tmp_ran[i], out=tmp_dom)
                tmp_dom /= sensitivities[i]

                x *= tmp_dom

                if callback is not None:
                    callback(x)
    else:
        raise RuntimeError('unknown noise model')


def loglikelihood(x, data, noise='poisson'):
    """log-likelihood of ``data`` given noise parametrized by ``x``.

    Parameters
    ----------
    x : ``op.domain`` element
        Value to condition the log-likelihood on.
    data : ``op.range`` element
        Data whose log-likelihood given ``x`` shall be calculated.
    noise : {'poisson'}, optional
        The type of noise.
    """
    noise, noise_in = str(noise).lower(), noise
    if noise not in AVAILABLE_MLEM_NOISE:
        raise NotImplemented("noise '{}' not understood"
                             ''.format(noise_in))

    if noise == 'poisson':
        if np.any(np.less(x, 0)):
            raise ValueError('`x` must be non-negative')

        return np.sum(data * np.log(x + 1e-8) - x)
    else:
        raise RuntimeError('unknown noise model')
