Fast Non-overlapping Domain Decomposition Methods for Continuous Multi-phase Labeling Problem

This paper presents the domain decomposition methods (DDMs) for achieving fast parallel computing on multi-core computers when dealing with the multi-phase labeling problem. To handle the non-smooth multi-phase labeling model, we introduce a quadratic proximal term, resulting in a strongly convex model. This paper provides theoretical evidence supporting the convergence of the proposed non-overlapping DDMs. Specifically, it is demonstrated that the non-overlapping DDMs for the non-smooth labeling model exhibits an O (1/ n ) convergence rate of the energy functional, where n is the number of iterations. Moreover, the fast iterative shrinkage-thresholding algorithm (Beck and Teboulle in SIAM J Imaging Sci 2(1):183–202, 2009) is applied to achieve an O(1/n^2) convergence rate. Numerical experiments are evaluated to demonstrate the convergence and efficiency of the proposed DDMs in solving multi-phase labeling problems.