Difference of convex functions algorithms (DCA) for image restoration via a Markov random field model

Abstract

In this paper, we introduce a novel approach in the nonconvex optimization framework for image restoration via a Markov random field (MRF) model. While image restoration is elegantly expressed in the language of MRF’s, the resulting energy minimization problem was widely viewed as intractable: it exhibits a highly nonsmooth nonconvex energy function with many local minima, and is known to be NP-hard. The main goal of this paper is to develop fast and scalable approximation optimization approaches to a nonsmooth nonconvex MRF model which corresponds to an MRF with a truncated quadratic (also known as half-quadratic) prior. For this aim, we use the difference of convex functions (DC) programming and DC algorithm (DCA), a fast and robust approach in smooth/nonsmooth nonconvex programming, which have been successfully applied in various fields in recent years. We propose two DC formulations and investigate the two corresponding versions of DCA. Numerical simulations show the efficiency, reliability and robustness of our customized DCAs with respect to the standard GNC algorithm and the Graph-Cut based method—a more recent and efficient approach to image analysis.

Appendices

Appendix 1: Subdifferential of a function being the supremum of a family of convex functions

Lemma 2

Let $f=\sup \left\{ f_{i}:i\in \varUpsilon \right\} ,$ where $\left( f_{i}\right) _{i\in \varUpsilon }$ is a family of proper convex functions on $\mathbb {R}^{n}$ . Then

• (i) $\partial f(x)\supset \overline{co}\left\{ \cup \partial f_{i}(x):i\in \varUpsilon (x)\right\}$ where $\varUpsilon (x)=\left\{ i\in \varUpsilon :f_{i}(x)=f(x)\right\} .$

• (ii) If $\varUpsilon$ is compact and there exists an open set $\varOmega$ in $\mathrm { I\!R^{n}}$ such that the map

  $$\begin{aligned} (i,x)\in \varUpsilon \times \varOmega \rightarrow f_{i}(x) \end{aligned}$$

  is finite and continuous on $\varUpsilon \times \varOmega$ , then f is continuous on $\varOmega$ and

  $$\begin{aligned} \partial f(x)=\overline{co}\left\{ \cup \partial f_{i}(x):i\in \varUpsilon (x)\right\} {\text { for\,all }}x\in \varOmega . \end{aligned}$$

Appendix 2: The formulation of the matrix B(0, 0)

$$\begin{aligned} B(0,0)= \begin{pmatrix} T_{0} & -I_{n+1} & 0 & 0 & \cdots & 0 & 0 & 0 \\ -I_{n+1} & T_{1} & -I_{n+1} & 0 & \cdots & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \cdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & -I_{n+1} & T_{m-1} & -I_{n+1} \\ 0 & 0 & 0 & 0 & \cdots & 0 & -I_{n+1} & T_{m} \end{pmatrix} \end{aligned}$$

(47)

where

$$\begin{aligned} T_{i}= {\left\{ \begin{array}{ll} \begin{array}{ll} P+I_{n+1} & {\text { if i }}= 0\,{\text { or i }}\,={\text { m;}} \\ P+2I_{n+1} & {\text { if i} }= 1, 2, \ldots , {\text { m}} - 1. \end{array} \end{array}\right. }. \end{aligned}$$

(48)

and

$$\begin{aligned} P= \begin{pmatrix} 1 & -1 & 0 & \cdots & 0 & 0 & 0 \\ -1 & 2 & -1 & \cdots & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \cdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & -1 & 2 & -1 \\ 0 & 0 & 0 & \cdots & 0 & -1 & 1 \end{pmatrix} . \end{aligned}$$

(49)