Fast Nonoverlapping Block Jacobi Method for the Dual Rudin--Osher--Fatemi Model

Abstract

We consider nonoverlapping domain decomposition methods for the Rudin-Osher-Fatemi (ROF) model, which is one of the standard models in mathematical image processing. The image domain is partitioned into rectangular subdomains and local problems in subdomains are solved in parallel. Local problems can adopt existing state-of-the-art solvers for the ROF model. We show that the nonoverlapping relaxed block Jacobi method for a dual formulation of the ROF model has the O(1/n) convergence rate of the energy functional, where n is the number of iterations. Moreover, by exploiting the forward-backward splitting structure of the method, we propose an accelerated version whose convergence rate is O(1/n 2 ). The proposed method converges faster than existing domain decomposition methods both theoretically and practically, while the main computational cost of each iteration remains the same. We also provide the dependence of the convergence rates of the block Jacobi methods on the image size and the number of subdomains. Numerical results for comparisons with existing methods are presented.

Pseudo-linear convergence of an additive Schwarz method for dual total variation minimization

“…Compared to the methods in [10], the proposed method has the advantage that it does not depend on either a particular function decomposition or a constraint decomposition. We prove that the proposed method is O(1/n)-convergent similarly to the existing methods in [10,19]. In addition, we explicitly describe the dependency of the convergence rate on the condition number of F .…”

“…where λ > 0 and (−∆) −1 denotes the inverse of the negative Laplacian with homogeneous Dirichlet boundary conditions. Since (1.3) Schwarz methods primal decomposition: [12,13] dual decomposition: [16, 17] [10, 15] convergence to a global minimizer primal decomposition: not guaranteed [17] dual decomposition: convergent [16,17] sublinear [10,19] of higher-order variational models for imaging, such as the smooth connection of shapes, it has various applications in advanced imaging problems including image decomposition [23] and image inpainting [6]. One may refer to [9] for various other examples of (1.1).…”

“…The above mentioned difficulties for (1.2) and (1.5), which are special cases of (1.1) and (1.4), respectively, are summarized in Table 1.1 with comparisons to some related problems in structural mechanics: the obstacle problem and the grad-div problem. Despite these difficulties, several successful Schwarz methods for (1.5) have been developed [10,15,19]. In [15], subspace correction methods for (1.5) based on a nonoverlapping domain decomposition were proposed.…”

“…Since then, the O(1/n)-energy convergence of overlapping Schwarz methods for (1.5) in a continuous setting was derived in [10], where n is the number of iterations. In [19], it was shown that the methods proposed in [15] are also O(1/n)-convergent. In addition, an O(1/n 2 )-convergent additive method was designed using an idea of pre-relaxation.…”

“…In this paper, we propose an additive Schwarz method for (1.4) based on an overlapping domain decomposition. While the existing methods in [15,19] for (1.5) are based on finite difference discretizations, the proposed method is based on finite element discretizations that were recently proposed in [14,18,20]. Compared to the methods in [10], the proposed method has the advantage that it does not depend on either a particular function decomposition or a constraint decomposition.…”