import time
from pyproximal.ProxOperator import _check_tau
from pyproximal import ProxOperator
from pyproximal.optimization.primal import ADMM
class _Denoise(ProxOperator):
r"""Denoiser of choice
Parameters
----------
denoiser : :obj:`func`
Denoiser (must be a function with two inputs, the first is the signal
to be denoised, the second is the `tau` constant of the y-update in
the PnP optimization
dims : :obj:`tuple`
Dimensions used to reshape the vector ``x`` in the ``prox`` method
prior to calling the ``denoiser``
"""
def __init__(self, denoiser, dims):
super().__init__(None, False)
self.denoiser = denoiser
self.dims = dims
def __call__(self, x):
return 0.
@_check_tau
def prox(self, x, tau):
x = x.reshape(self.dims)
xden = self.denoiser(x, tau)
return xden.ravel()
[docs]def PlugAndPlay(proxf, denoiser, dims, x0, solver=ADMM, **kwargs_solver):
r"""Plug-and-Play Priors with any proximal algorithm of choice
Solves the following minimization problem using any proximal a
lgorithm of choice:
.. math::
\mathbf{x},\mathbf{z} = \argmin_{\mathbf{x}}
f(\mathbf{x}) + \lambda g(\mathbf{x})
where :math:`f(\mathbf{x})` is a function that has a known gradient or
proximal operator and :math:`g(\mathbf{x})` is a function acting as implicit
prior. Implicit means that no explicit function should be defined: instead,
a denoising algorithm of choice is used. See Notes for details.
Parameters
----------
proxf : :obj:`pyproximal.ProxOperator`
Proximal operator of f function
denoiser : :obj:`func`
Denoiser (must be a function with two inputs, the first is the signal
to be denoised, the second is the tau constant of the y-update in
PlugAndPlay)
dims : :obj:`tuple`
Dimensions used to reshape the vector ``x`` in the ``prox`` method
prior to calling the ``denoiser``
x0 : :obj:`numpy.ndarray`
Initial vector
solver : :func:`pyproximal.optimization.primal` or :func:`pyproximal.optimization.primaldual`
Solver of choice
kwargs_solver : :obj:`dict`
Additonal parameters required by the selected solver
Returns
-------
out : :obj:`numpy.ndarray` or :obj:`tuple`
Output of the solver of choice
Notes
-----
Plug-and-Play Priors [1]_ can be used with any proximal algorithm of choice. For example, when
ADMM is selected, the resulting scheme can be expressed by the following recursion:
.. math::
\mathbf{x}^{k+1} = \prox_{\tau f}(\mathbf{z}^{k} - \mathbf{u}^{k})\\
\mathbf{z}^{k+1} = \operatorname{Denoise}(\mathbf{x}^{k+1} + \mathbf{u}^{k}, \tau \lambda)\\
\mathbf{u}^{k+1} = \mathbf{u}^{k} + \mathbf{x}^{k+1} - \mathbf{z}^{k+1}
where :math:`\operatorname{Denoise}` is a denoising algorithm of choice. This rather peculiar step originates
from the intuition that the optimization process associated with the z-update can be interpreted as a denoising
inverse problem, or more specifically a MAP denoiser where the noise is gaussian with zero mean and variance
equal to :math:`\tau \lambda`. For this reason any denoising of choice can be used instead of a function with
known proximal operator.
Finally, whilst the :math:`\tau \lambda` denoising parameter should be chosen to
represent an estimate of the noise variance (of the denoiser, not the data of the problem we wish to solve!),
special care must be taken when setting up the denoiser and calling this optimizer. More specifically,
:math:`\lambda` should not be passed to the optimizer, rather set directly in the denoiser.
On the other hand :math:`\tau` must be passed to the optimizer as it is also affecting the x-update;
when defining the denoiser, ensure that :math:`\tau` is multiplied to :math:`\lambda` as shown in the tutorial.
Alternative, as suggested in [2]_, the :math:`\tau` could be set to 1. The parameter :math:`\lambda` can then be set
to maximize the value of the denoiser and a second tuning parameter can be added directly to :math:`f`.
.. [1] Venkatakrishnan, S. V., Bouman, C. A. and Wohlberg, B.
"Plug-and-Play priors for model based reconstruction",
IEEE. 2013.
.. [2] Meinhardt, T., Moeller, M, Hazirbas, C., and Cremer, D.
"Learning Proximal Operators: Using Denoising Networks for Regularizing Inverse Imaging Problems",
arXiv. 2017.
"""
# Denoiser
proxpnp = _Denoise(denoiser, dims=dims)
return solver(proxf, proxpnp, x0=x0, **kwargs_solver)
