Primal-Dual Decomposition by Operator Splitting and Applications to Image Deblurring

Abstract

Professor Lieven Vandenberghe, ChairWe present primal-dual decomposition algorithms for convex optimization problems with cost functions f (x)+g(Ax), where f and g have inexpensive proximal operators and A can be decomposed as a sum of structured matrices. The methods are based on the Douglas-Rachford splitting algorithm applied to various splittings of the primal-dual optimality conditions. We discuss applications to image deblurring problems with non-quadratic data fidelity terms, different types of convex regularization, and simple convex constraints. In these applications, the primal-dual splitting approach allows us to handle general boundary conditions for the blurring operator. We also give a domain decomposition approach to image deblurring, where the blur operator may use a different blur kernel for each image patch, and at each iteration all patches are processed in parallel. Numerical results indicate that the primal-dual splitting methods compare favorably with the alternating direction method of multipliers, the Douglas-Rachford algorithm applied to a reformulated primal problem, and the Chambolle-Pock primal-dual algorithm.A key property exploited in most image deblurring methods is spatial invariance of the blurring operator, which makes it possible to use the fast Fourier transform (FFT) when solving linear equations involving the operator. In this thesis we extend this approach to two popular models for space-varying blurring operators, the Nagy-O'Leary model and the Efficient Filter Flow model. We show how splitting methods derived from the Douglasii Rachford algorithm can be implemented with a low complexity per iteration, dominated by a small number of FFTs.iii The dissertation of Daniel Verity O'Connor is approved.

Proximal Activation of Smooth Functions in Splitting Algorithms for Convex Image Recovery

“…x − y 2 , and that it reduces to (1.2) when ϕ = ι C i . We refer the reader to [9,Chapter 24] for a detailed account of the properties of proximity operators with various examples, to [31] for a tutorial on proximal methods in signal processing, and to [5,11,19,21,30,52,53,54] for specific applications to image recovery. Current proximal splitting methods can handle highly structured convex minimization problems such as the following, which will be the focus of our discussion (see below for notation).…”

Splitting with Near-Circulant Linear Systems: Applications to Total Variation CT and PET

“…O'Connor and Vandenberghe [48,47] presented splitting methods for imaging problems with linear systems of the form of circulant plus sparse, which are similar to, but not the same as, the near-circulant linear systems considered in this work. They presented several methods that utilize this problem structure and empirically compared their performances.…”

Classifying Partially Labeled Networked Data VIA Logistic Network Lasso

“…However, the objective function is highly structured since it is the sum of a smooth convex function h(w) and a non-smooth convex function g(Dw), which can be optimized efficiently when considered separately. This suggests to use a proximal splitting method [12], [22], [23] for solving (9). One particular such method is the preconditioned primal-dual method [24] which is based on reformulating the problem (9) as a saddle-point problem…”