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

Abstract

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.

Appendix

Now we introduce the relaxed preconditioned splitting method of Douglas Rachford type (PDRQ) [4, 36, 37] for solving the following primal-dual convex optimization problem

$$\begin{aligned} \min _{x\in X}\max _{y\in Y} G_1(x)+\langle Kx,y \rangle -G_2(y), \end{aligned}$$

(26)

where $G_1(x)=\langle \frac{1}{2} Qx+f,x\rangle$ and Q is linear, continuous, and positive semidefinite. Here, X and Y are real Hilbert spaces, and $K: X\rightarrow Y$ is a continuous linear mapping. The functionals $G_1: X\rightarrow {\textbf{R}}_\infty$ (where ${\textbf{R}}_\infty$ are extended real numbers) and $G_2: Y\rightarrow {\textbf{R}}_\infty$ are proper, convex, and lower semicontinuous.

The optimality system for (26) is given as $0\in A_1z+B_1z$ for $z=(x,y)^T$, $z\in H=X\times Y$, where $A_1$ and $B_1$ are provided that

$$\begin{aligned}A_1=\begin{pmatrix} \begin{array}{ll} 0 &{} 0 \\ 0&{} \partial G_2 \end{array} \end{pmatrix},~ B_1=\begin{pmatrix} \begin{array}{ll} Q+1_f&{} K^* \\ -K&{} 0 \end{array} \end{pmatrix}. \end{aligned}$$

Here, $1_f$ denotes the constant mapping which yields f for each argument. By setting $\sigma >0$ and introducing an auxiliary variable $w\in H$ with $w\in \sigma A_1z$, the optimality system can be equivalently written as

$$\begin{aligned} \begin{pmatrix} \begin{array}{l} 0 \\ 0 \end{array} \end{pmatrix}\in \begin{pmatrix} \begin{array}{l} \sigma B_1z +w\\ -z+(\sigma A_1)^{-1}w \end{array} \end{pmatrix}. \end{aligned}$$

Let ${\mathcal {U}}\in H\times H$, $u=(z,w)$ and ${\mathcal {A}}:{\mathcal {U}}\rightarrow 2^{\mathcal {U}}$. Then this problem becomes $0\in {\mathcal {A}} u$. By introducing a linear and continuous preconditioner ${\mathcal {M}}:{\mathcal {U}}\rightarrow {\mathcal {U}}$, the latter results in the iteration becomes

$$\begin{aligned} 0\in {\mathcal {M}}(u^{m+1}-u^m)+{\mathcal {A}}u^{m+1} \end{aligned}$$

with

$$\begin{aligned} {\mathcal {M}}=\begin{pmatrix} \begin{array}{ll} I &{} -I \\ -I&{} I \end{array} \end{pmatrix},~ {\mathcal {A}}=\begin{pmatrix} \begin{array}{ll} \sigma B_1 &{} I \\ -I&{} (\sigma A_1)^{-1} \end{array} \end{pmatrix}. \end{aligned}$$

Therefore, the iteration can be defined as follows

$$\begin{aligned} \left\{ \begin{aligned} z^{m+1}&=(I+\sigma B_1)^{-1}(z^m-w^m);\\ v^{m+1}&=v^m+(I+\sigma A_1)^{-1}(2z^{m+1}-v^m)-z^{m+1}; \end{aligned}\right. \end{aligned}$$

where $v^{m}:=z^{m}-w^m$. Hence this iteration becomes the Douglas-Rachford iteration.

In [36, 37], the preconditioner ${\mathcal {M}}$ can be defined more flexibility. By introducing the operator as follows

$$\begin{aligned} N_1: X\rightarrow X,~\text{ linear, } \text{ continuous, } \text{ self-adjoint } \text{ and }~N_1-I\ge 0, \end{aligned}$$

we define ${\mathcal {M}}:{\mathcal {U}}\rightarrow {\mathcal {U}}$ according to

$$\begin{aligned}{\mathcal {M}}:=\begin{pmatrix} \begin{array}{llll} N_1 &{} 0&{}-I&{}0 \\ 0 &{} I&{}0 &{}-I \\ -I &{} 0&{}I&{}0 \\ 0 &{} -I&{}0&{}I \\ \end{array} \end{pmatrix}, \end{aligned}$$

and $w=\sigma Az=(0,\tilde{y} )$ with $\tilde{y} \in \sigma \partial G_2(y)$. By introducing $\bar{y}^m=y^m-\tilde{y}^m$, the preconditioned splitting method of Douglas Rachford can be formulated as follows

$$\begin{aligned} \left\{ \begin{aligned} x^{m+1}&=(N_1+\sigma Q)^{-1}(N_1 x^m-\sigma K^*y^{m+1}-\sigma f);\\ y^{m+1}&=\bar{y}^m+\sigma K x^{m+1}\\ \bar{y}^{m+1}&=\bar{y}^{m}+(I+\sigma \partial G_2)^{-1}(2y^{m+1}-\bar{y}^m)-y^{m+1}; \end{aligned}\right. \end{aligned}$$

(27)

Note that the updates w.r.t. $x^{m+1}$ and $y^{m+1}$ are implicit. One possibility is to compute $x^{m+1}$ by plugging $y^{m+1}$ to obtain

$$\begin{aligned} (N_1+\sigma Q+\sigma ^2 K^*K)x^{m+1}= N_1 x^m -\sigma K^*\bar{y}^{m}-\sigma f. \end{aligned}$$

Let $M_Q$ be a positive definite operator such that

$$\begin{aligned} N_1=M_Q-\sigma Q-\sigma ^2 K^*K. \end{aligned}$$

Then we have

$$\begin{aligned} x^{m+1}=x^m+M_Q^{-1}[-\sigma f -\sigma K^*\bar{y}^{m}-T_Q x^m], \end{aligned}$$

with $T_Q=\sigma Q+\sigma ^2 K^*K$.

By combining the second and third equations of (27), the relaxed iteration (27) can be written as

$$\begin{aligned} \left\{ \begin{aligned} x^{m+1}&=x^m+M_Q^{-1}[\sigma f-\sigma K^*\bar{y}^m-T_Qx^m];\\ \bar{y}^{m+1}&=(1-\rho ) \bar{y}^{m} + \rho (I+\tau \partial G_2)^{-1}(2\bar{y}^{m}+2\sigma K x^{m+1})-\rho \tau K x^{m+1}, \end{aligned}\right. \end{aligned}$$

(28)

with $T_Q=\sigma Q+\sigma \tau K^*K$ for $\tau >0$. The convergence of PDRQ for the case $\rho \in (0,2)$ can be guaranteed; see [36, 37] for more details.

Theorem 4

If a solution to the saddle point problem (26) exists, then the iteration (28) converges weakly to a fixed point $(x^*,y^*)$. The pair $(x^*,y^*)$ is a solution to the saddle-point problem.