Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 4012767, 14 pages

https://doi.org/10.1155/2017/4012767

## A Convex Optimization Model and Algorithm for Retinex

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China

Correspondence should be addressed to Ting-Zhu Huang; moc.621@gnauhuhzgnit and Xi-Le Zhao; moc.361@300221oahzlx

Received 27 March 2017; Revised 14 May 2017; Accepted 23 May 2017; Published 24 July 2017

Academic Editor: Francesco Marotti de Sciarra

Copyright © 2017 Qing-Nan Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Retinex is a theory on simulating and explaining how human visual system perceives colors under different illumination conditions. The main contribution of this paper is to put forward a new convex optimization model for Retinex. Different from existing methods, the main idea is to rewrite a multiplicative form such that the illumination variable and the reflection variable are decoupled in spatial domain. The resulting objective function involves three terms including the Tikhonov regularization of the illumination component, the total variation regularization of the reciprocal of the reflection component, and the data-fitting term among the input image, the illumination component, and the reciprocal of the reflection component. We develop an alternating direction method of multipliers (ADMM) to solve the convex optimization model. Numerical experiments demonstrate the advantages of the proposed model which can decompose an image into the illumination and the reflection components.

#### 1. Introduction

The idea of Retinex was introduced and pioneered by Land and McCann [1] to explain how a combination of processes occurs both in the “retina” and in the “cortex.” Retinex theory tells us that human visual system can ensure that the perceived colors of objects remain relatively constant under varying illumination conditions. That is to say our visual system is robust when it comes to color perception. Retinex theory can deal with the compensation for illumination effect. Therefore, we would like to reduce the influence of nonuniform illumination to enhance an image.

Usually, we consider an input image as a two-dimensional function which can be decomposed into the illumination function and the reflection function . Generally speaking, the input image is assumed to have the following relation with these two functions:where represents the element-wise multiplication.

Removing the illumination effect means to decompose the input image into the illumination component and the reflection component. This problem is known to be mathematically ill-posed [2], and many methods have been proposed to solve it in the literature [3–5]. Retinex methods can be classified into random walk methods, recursive methods, center/surround methods, PDE-based methods, and variational methods. Firstly, the original method of Land and McCann was proposed relying on a random walk. The random walk is a discrete time random-process in which the next pixel position is chosen randomly from the neighbors of the current pixel position; see, for example, [6–9]. It needs to regulate many parameters and has high computational complexity. In [10–12], the researchers perform recursive matrix operations to develop recursive methods. The computational efficiency of the recursive methods is improved significantly than that of the random walk methods. However, it is difficult to know how many iterations should be executed in the process. Then Land and McCann put forward the center/surround methods [1]. Later Jobson et al. proposed the single scale Retinex (SSR) and the multiscale Retinex (MSR) [13]. It is easy to implement the SSR and the MSR; both of them need many parameters. In Poisson equation-based methods [14–16], they often convert the original formulation into the logarithmic formulation. Their methods rely on the Mondrian world model which boils down to the assumption on the reflection as a piecewise constant image.

Recently, many variational methods were proposed in [17, 18]. The fundamental assumptions are that the illumination component is spatially smooth and the reflection component is piecewise constant. Based on the above assumptions Kimmel et al*.* presented a variational Retinex formulation [4]. In their model the piecewise constant assumption of the reflection component is not considered. Total Variation (TV) had been widely used in image processing [19–23]. Ma and Osher [24] applied TV and nonlocal TV regularization to Retinex theory. The Bregman iteration was employed to solve their models. It is difficult to set up existence results for their models. Ma et al. further proposed a -based variational model to recover the reflection component [25]. In [26], Zosso et al. proposed a unifying framework for Retinex theory. In [27], Liang and Zhang proposed a decomposition model for the Retinex problem via high-order TV.

Ng and Wang proposed a TV model for image enhancement [28]. They consider both the illumination component and the reflection component in the objective function. They proposed transforming the multiplicative form (1) into . Different from the Ma and Osher model [24], they added some constraints and a fidelity term which ensure that the theoretical analysis can be performed and established. An energy functional was proposed for Retinex as follows:where , , , , , and are positive regularization parameters, and the term is only used for the theoretical proof which has no practical sense. is the total variation term of [29] and it is employed to characterize the reflection function which makes the model more reasonable. The motivation behind is that we have noticed that once the input image is converted to the logarithmic domain, the small differences between image pixels tend to be ignored. For example, in Figure 1 there is a signal with quite different values in two parts; if we convert the signal to the logarithmic domain, the difference between the two parts is significantly reduced. So we rewrite (1) in spatial domain to avoid the loss of these small textures and details, and at the same time we can also ensure the convexity of the model.