Domain Decomposition Methods Using Dual Conversion for the Total Variation Minimization with L-1 Fidelity Term

Abstract

Nowadays, as large scale images become available, the necessity of parallel algorithms for image processing has been arisen. In this paper, we propose domain decomposition methods as parallel solvers for solving total variation minimization problems with L1 fidelity term. The image domain is decomposed into rectangular subdomains, where the local total variation problems are solved. We introduce the notion of dual conversion, which generalizes the framework of Chambolle-Pock primal-dual algorithm (J Math Imaging Vis 40:120-145, 2011). By the dual conversion, the TV-L1 model is transformed into an equivalent saddle point problem which has a natural parallel structure. The primal problem of the resulting saddle point problem is decoupled in the sense that each local problem can be solved independently. Convergence analysis of the proposed algorithms is provided. We apply the proposed algorithms for image denoising, inpainting, and deblurring problems. Our numerical results ensure that the proposed algorithms have good performance as parallel solvers.