2,823 research outputs found
Stochastic forward-backward and primal-dual approximation algorithms with application to online image restoration
Stochastic approximation techniques have been used in various contexts in
data science. We propose a stochastic version of the forward-backward algorithm
for minimizing the sum of two convex functions, one of which is not necessarily
smooth. Our framework can handle stochastic approximations of the gradient of
the smooth function and allows for stochastic errors in the evaluation of the
proximity operator of the nonsmooth function. The almost sure convergence of
the iterates generated by the algorithm to a minimizer is established under
relatively mild assumptions. We also propose a stochastic version of a popular
primal-dual proximal splitting algorithm, establish its convergence, and apply
it to an online image restoration problem.Comment: 5 Figure
A forward-backward view of some primal-dual optimization methods in image recovery
A wide array of image recovery problems can be abstracted into the problem of
minimizing a sum of composite convex functions in a Hilbert space. To solve
such problems, primal-dual proximal approaches have been developed which
provide efficient solutions to large-scale optimization problems. The objective
of this paper is to show that a number of existing algorithms can be derived
from a general form of the forward-backward algorithm applied in a suitable
product space. Our approach also allows us to develop useful extensions of
existing algorithms by introducing a variable metric. An illustration to image
restoration is provided
Playing with Duality: An Overview of Recent Primal-Dual Approaches for Solving Large-Scale Optimization Problems
Optimization methods are at the core of many problems in signal/image
processing, computer vision, and machine learning. For a long time, it has been
recognized that looking at the dual of an optimization problem may drastically
simplify its solution. Deriving efficient strategies which jointly brings into
play the primal and the dual problems is however a more recent idea which has
generated many important new contributions in the last years. These novel
developments are grounded on recent advances in convex analysis, discrete
optimization, parallel processing, and non-smooth optimization with emphasis on
sparsity issues. In this paper, we aim at presenting the principles of
primal-dual approaches, while giving an overview of numerical methods which
have been proposed in different contexts. We show the benefits which can be
drawn from primal-dual algorithms both for solving large-scale convex
optimization problems and discrete ones, and we provide various application
examples to illustrate their usefulness
A Non-Local Structure Tensor Based Approach for Multicomponent Image Recovery Problems
Non-Local Total Variation (NLTV) has emerged as a useful tool in variational
methods for image recovery problems. In this paper, we extend the NLTV-based
regularization to multicomponent images by taking advantage of the Structure
Tensor (ST) resulting from the gradient of a multicomponent image. The proposed
approach allows us to penalize the non-local variations, jointly for the
different components, through various matrix norms with .
To facilitate the choice of the hyper-parameters, we adopt a constrained convex
optimization approach in which we minimize the data fidelity term subject to a
constraint involving the ST-NLTV regularization. The resulting convex
optimization problem is solved with a novel epigraphical projection method.
This formulation can be efficiently implemented thanks to the flexibility
offered by recent primal-dual proximal algorithms. Experiments are carried out
for multispectral and hyperspectral images. The results demonstrate the
interest of introducing a non-local structure tensor regularization and show
that the proposed approach leads to significant improvements in terms of
convergence speed over current state-of-the-art methods
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