8 research outputs found
Epigraphical Projection for Solving Least Squares Anscombe Transformed Constrained Optimization Problems
This papers deals with the restoration of images corrupted by a non-invertible or ill-conditioned linear transform and Poisson noise. Poisson data typically occur in imaging processes where the images are obtained by counting particles, e.g., photons, that hit the image support. By using the Anscombe transform, the Poisson noise can be approximated by an additive Gaussian noise with zero mean and unit variance. Then, the least squares difference between the Anscombe transformed corrupted image and the original image can be estimated by the number of observations. We use this information by considering an Anscombe transformed constrained model to restore the image. The advantage with respect to corresponding penalized approaches lies in the existence of a simple model for parameter estimation. We solve the constrained minimization problem by applying a primal-dual algorithm together with a projection onto the epigraph of a convex function related to the Anscombe transform. We show that this epigraphical projection can be efficiently computed by Newton's methods with an appropriate initialization. Numerical examples demonstrate the good performance of our approach, in particular, its close behaviour with respect to the -divergence constrained model
Solving monotone inclusions involving parallel sums of linearly composed maximally monotone operators
The aim of this article is to present two different primal-dual methods for
solving structured monotone inclusions involving parallel sums of compositions
of maximally monotone operators with linear bounded operators. By employing
some elaborated splitting techniques, all of the operators occurring in the
problem formulation are processed individually via forward or backward steps.
The treatment of parallel sums of linearly composed maximally monotone
operators is motivated by applications in imaging which involve first- and
second-order total variation functionals, to which a special attention is
given.Comment: 25 page
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
Epigraphical Projection for Solving Least Squares Anscombe Transformed Constrained Optimization Problems
Abstract. This papers deals with the restoration of images corrupted by a non-invertible or ill-conditioned linear transform and Poisson noise. Poisson data typically occur in imaging processes where the images are obtained by counting particles, e.g., photons, that hit the image support. By using the Anscombe transform, the Poisson noise can be approximated by an additive Gaussian noise with zero mean and unit variance. Then, the least squares difference between the Anscombe transformed corrupted image and the original image can be estimated by the number of observations. We use this information by considering an Anscombe transformed constrained model to restore the image. The advantage with respect to corresponding penalized approaches lies in the existence of a simple model for parameter estimation. We solve the constrained minimization problem by applying a primal-dual algorithm together with a projection onto the epigraph of a convex function related to the Anscombe transform. We show that this epigraphical projection can be efficiently computed by Newtonâs methods with an appropriate initialization. Numerical examples demonstrate the good performance of our approach, in particular, its close behaviour with respect to the I-divergence constrained model.
Proximity Operators of Discrete Information Divergences
Information divergences allow one to assess how close two distributions are
from each other. Among the large panel of available measures, a special
attention has been paid to convex -divergences, such as
Kullback-Leibler, Jeffreys-Kullback, Hellinger, Chi-Square, Renyi, and
I divergences. While -divergences have been extensively
studied in convex analysis, their use in optimization problems often remains
challenging. In this regard, one of the main shortcomings of existing methods
is that the minimization of -divergences is usually performed with
respect to one of their arguments, possibly within alternating optimization
techniques. In this paper, we overcome this limitation by deriving new
closed-form expressions for the proximity operator of such two-variable
functions. This makes it possible to employ standard proximal methods for
efficiently solving a wide range of convex optimization problems involving
-divergences. In addition, we show that these proximity operators are
useful to compute the epigraphical projection of several functions of practical
interest. The proposed proximal tools are numerically validated in the context
of optimal query execution within database management systems, where the
problem of selectivity estimation plays a central role. Experiments are carried
out on small to large scale scenarios
A Class of Randomized Primal-Dual Algorithms for Distributed Optimization
Based on a preconditioned version of the randomized block-coordinate
forward-backward algorithm recently proposed in [Combettes,Pesquet,2014],
several variants of block-coordinate primal-dual algorithms are designed in
order to solve a wide array of monotone inclusion problems. These methods rely
on a sweep of blocks of variables which are activated at each iteration
according to a random rule, and they allow stochastic errors in the evaluation
of the involved operators. Then, this framework is employed to derive
block-coordinate primal-dual proximal algorithms for solving composite convex
variational problems. The resulting algorithm implementations may be useful for
reducing computational complexity and memory requirements. Furthermore, we show
that the proposed approach can be used to develop novel asynchronous
distributed primal-dual algorithms in a multi-agent context
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
First order algorithms in variational image processing
Variational methods in imaging are nowadays developing towards a quite
universal and flexible tool, allowing for highly successful approaches on tasks
like denoising, deblurring, inpainting, segmentation, super-resolution,
disparity, and optical flow estimation. The overall structure of such
approaches is of the form ; where the functional is a data fidelity term also
depending on some input data and measuring the deviation of from such
and is a regularization functional. Moreover is a (often linear)
forward operator modeling the dependence of data on an underlying image, and
is a positive regularization parameter. While is often
smooth and (strictly) convex, the current practice almost exclusively uses
nonsmooth regularization functionals. The majority of successful techniques is
using nonsmooth and convex functionals like the total variation and
generalizations thereof or -norms of coefficients arising from scalar
products with some frame system. The efficient solution of such variational
problems in imaging demands for appropriate algorithms. Taking into account the
specific structure as a sum of two very different terms to be minimized,
splitting algorithms are a quite canonical choice. Consequently this field has
revived the interest in techniques like operator splittings or augmented
Lagrangians. Here we shall provide an overview of methods currently developed
and recent results as well as some computational studies providing a comparison
of different methods and also illustrating their success in applications.Comment: 60 pages, 33 figure