Convergent Non-overlapping Domain Decomposition Methods for Variational Image Segmentation

Abstract

This paper concerns with the non-overlapping domain decomposition methods (DDMs) for the Chan–Vese model in variational image segmentation. We work with a saddle point formulation for the non-overlapping decompositions, which leads to independent subproblems decoupled by the primal–dual algorithm. With the non-overlapping DDMs, only the interfaces of the adjacent subdomains are coupled, which means the information transmission takes place on such interfaces and therefore makes the proposed DDMs flexible and efficient. Moreover, we consider both the stripe-type and checkerboard-type decomposition methods and provide the rigorous proof of the convergence. Our numerical experiments demonstrate that the proposed DDMs are convergent, efficient, and quite robust with respect to the model parameters and image resolutions.

Appendix

There are many solvers to minimize (2.1) including the primal–dual algorithm [57]. The saddle point problem of (2.1) is built up as follows:

$$\begin{aligned} \max _{\varvec{p}}\min _{h} \int _\Omega \nabla h \cdot \varvec{p}+\alpha \int _\Omega s h-I_{K_p}(\varvec{p})+I_{K_h}(h), \end{aligned}$$

(5.1)

where

By defining $G_1(h)=\alpha \int _\Omega s h +I_{K_h}(h)$ and $G_2({\varvec{p}})=I_{K_p}({\varvec{p}})$, Eq. (5.1) reduces to the following form:

The primal–dual algorithm by Chambolle and Pock [31] can be used to solve the above problem.

Both subproblems of $\varvec{p}$ and h in Eq. (5.3) have the closed-form forms, which are

$$\begin{aligned} \varvec{p}^{n+1}(x)=\dfrac{\varvec{\hat{p}^n}(x)}{\max \{1,|\varvec{\hat{p}^n}(x)|\}}, \end{aligned}$$

and

$$\begin{aligned} h^{n+1}(x)=\min \left\{ 1,\max \{0, \hat{h}^n(x)-\tau \alpha s(x)\}\right\} ,\quad \forall x \in \Omega . \end{aligned}$$

Theorem 5.1

([31]) The primal–dual algorithm is convergent if $\tau \sigma <1/8.$