/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

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Principles Of Optics Electromagnetic Theory Of Propagation Interference And Diffraction Of Light

The proximity operator of a convex function is a natural extension of the notion of a projection operator onto a convex set. This tool, which plays a central role in the analysis and the numerical solution of convex optimization problems, has recently been introduced in the arena of signal processing, where it has become increasingly important. In this paper, we review the basic properties of proximity operators which are relevant to signal processing and present optimization methods based on these operators. These proximal splitting methods are shown to capture and extend several well-known algorithms in a unifying framework. Applications of proximal methods in signal recovery and synthesis are discussed.

Proximal Splitting Methods in Signal Processing

The proximity operator of a convex function is a natural extension of the notion of a projection operator onto a convex set. This tool, which plays a central role in the analysis and the numerical solution of convex optimization problems, has recently been introduced in the arena of inverse problems and, especially, in signal processing, where it has become increasingly important. In this paper, we review the basic properties of proximity operators which are relevant to signal processing and present optimization methods based on these operators. These proximal splitting methods are shown to capture and extend several well-known algorithms in a unifying framework. Applications of proximal methods in signal recovery and synthesis are discussed.

/pdf/proximal-splitting-methods-in-signal-processing-1ra5ajtjtn.pdf

Matrix Differential Calculus with Applications in Statistics and Econometrics

Alternating direction methods are a common tool for general mathematical programming and optimization. These methods have become particularly important in the field of variational image processing, which frequently requires the minimization of nondifferentiable objectives. This paper considers accelerated (i.e., fast) variants of two common alternating direction methods: the alternating direction method of multipliers (ADMM) and the alternating minimization algorithm (AMA). The proposed acceleration is of the form first proposed by Nesterov for gradient descent methods. In the case that the objective function is strongly convex, global convergence bounds are provided for both classical and accelerated variants of the methods. Numerical examples are presented to demonstrate the superior performance of the fast methods for a wide variety of problems.

/pdf/fast-alternating-direction-optimization-methods-363hs8r2zw.pdf

Fast Alternating Direction Optimization Methods

https://www.mia.uni-saarland.de/setzer/publications/poisson_deblurring_elsevier_revised2.pdf

Deblurring Poissonian images by split Bregman techniques

We examine the underlying structure of popular algorithms for variational methods used in image processing. We focus here on operator splittings and Bregman methods based on a unified approach via fixed point iterations and averaged operators. In particular, the recently proposed alternating split Bregman method can be interpreted from different points of view--as a Bregman, as an augmented Lagrangian and as a Douglas-Rachford splitting algorithm which is a classical operator splitting method. We also study similarities between this method and the forward-backward splitting method when applied to two frequently used models for image denoising which employ a Besov-norm and a total variation regularization term, respectively. In the first setting, we show that for a discretization based on Parseval frames the gradient descent reprojection and the alternating split Bregman algorithm are equivalent and turn out to be a frame shrinkage method. For the total variation regularizer, we also present a numerical comparison with multistep methods.

https://www.mia.uni-saarland.de/Publications/setzer-ijcv11.pdf

Operator Splittings, Bregman Methods and Frame Shrinkage in Image Processing

We examine relations between popular variational methods in image processing and classical operator splitting methods in convex analysis. We focus on a gradient descent reprojection algorithm for image denoising and the recently proposed Split Bregman and alternating Split Bregman methods. By identifying the latter with the so-called Douglas-Rachford splitting algorithm we can guarantee its convergence. We show that for a special setting based on Parseval frames the gradient descent reprojection and the alternating Split Bregman algorithm are equivalent and turn out to be a frame shrinkage method.

https://www.mia.uni-saarland.de/setzer/publications/setzer_fba_fbs_frames08.pdf

Split Bregman Algorithm, Douglas-Rachford Splitting and Frame Shrinkage

As first demonstrated by Chambolle and Lions the staircasing effect of the RudinOsher-Fatemi model can be reduced by using infimal convolutions of functionals containing higher order derivatives. In this paper, we examine a modification of such infimal convolutions in a general discrete setting. For the special case of finite difference matrices, we show the relation of our approach to the continuous total generalized variation approach recently developed by Bredies, Kunisch and Pock. We present splitting methods to compute the minimizers of the l 2 (modified) infimal convolution functionals which are superior to previously applied second order cone programming methods. Moreover, we illustrate the differences between the ordinary and the modified infimal convolution approach by numerical examples.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cms/2011/0009/0003/CMS-2011-0009-0003-a007.pdf

Simon Setzer

Papers

Fast Alternating Direction Optimization Methods

Deblurring Poissonian images by split Bregman techniques

Operator Splittings, Bregman Methods and Frame Shrinkage in Image Processing

Split Bregman Algorithm, Douglas-Rachford Splitting and Frame Shrinkage

Infimal convolution regularizations with discrete ℓ1-type functionals