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Mathematical Problems in Engineering
Volume 2017 (2017), Article ID 4012767, 14 pages
Research Article

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.


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.