  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.
