# A Finite Element Nonoverlapping Domain Decomposition Method with Lagrange Multipliers for the Dual Total Variation Minimizations

## TL;DR

This paper introduces a primal-dual domain decomposition method using Lagrange multipliers for total variation regularized image processing, demonstrating superior performance in denoising, inpainting, and segmentation tasks.

## Contribution

It develops a novel finite element domain decomposition approach with Lagrange multipliers for dual total variation minimization, improving computational efficiency and accuracy.

## Key findings

- Outperforms existing state-of-the-art methods in image denoising, inpainting, and segmentation.
- Effective handling of interface constraints via Lagrange multipliers.
- Applicable to problems with $L^2$- and $L^1$-fidelity.

## Abstract

In this paper, we consider a primal-dual domain decomposition method for total variation regularized problems appearing in mathematical image processing. The model problem is transformed into an equivalent constrained minimization problem by tearing-and-interconnecting domain decomposition. Then, the continuity constraints on the subdomain interfaces are treated by introducing Lagrange multipliers. The resulting saddle point problem is solved by the first order primal-dual algorithm. We apply the proposed method to image denoising, inpainting, and segmentation problems with either $L^2$-fidelity or $L^1$-fidelity. Numerical results show that the proposed method outperforms the existing state-of-the-art methods.