Domain decomposition methods for convex optimization in image processing

Abstract

This dissertation is concerned with domain decomposition methods for convex optimization in image processing. In particular, we focus on total variation minimization which is one of the standard techniques in image processing. Domain decomposition methods are suitable for parallel computation since they solve a large-scale problem by dividing it into smaller problems and treating them in parallel. Since total variation is nonlinear, nonsmooth, and nonseparable, it is quite challenging to design an efficient domain decomposition method for total variation minimization.

In this dissertation, two fast iterative substructuring methods for total variation minimization are proposed. We show that the proposed iterative substructuring methods outperform the existing methods both theoretically and numerically. In addition, we present a convergence rate analysis for a nonoverlapping Schwarz method for total variation minimization, and propose its accelerated variant which has superior convergence properties. We present another novel domain decomposition method which can be applied to not only total variation minimization but also more general optimization problems in image processing such as higher models and nonconvex models. Numerical results for all of the proposed domain decomposition methods applied to various problems of image processing such as image denoising, inpainting, deconvolution, and segmentation are provided.