# American Institute of Mathematical Sciences

December  2020, 14(6): 1135-1156. doi: 10.3934/ipi.2020058

## A parallel operator splitting algorithm for solving constrained total-variation retinex

 1 School of Mathematical Sciences, Key Laboratory for NSLSCS of Jiangsu Province, Nanjing Normal University, Nanjing, 210023, China 2 School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China 3 LMIB, School of Mathematical Sciences, Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing, 100191, China

* Corresponding author: Deren Han

Received  May 2019 Revised  May 2020 Published  December 2020 Early access  October 2020

An ideal image is desirable to faithfully represent the real-world scene. However, the observed images from imaging system are typically involved in the illumination of light. As the human visual system (HVS) is capable of perceiving identical color with spatially varying illumination, retinex theory is established to probe the rationale of HVS on perceiving color. In the realm of image processing, the retinex models are devoted to diminishing illumination effects from observed images. In this paper, following the recent work by Ng and Wang (SIAM J. Imaging Sci. 4:345-356, 2011), we develop a parallel operator splitting algorithm tailored for the constrained total-variation retinex model, in which all the resulting subproblems admit closed form solutions or can be tractably solved by some subroutines without any internally nested iterations. The global convergence of the novel algorithm is analysed on the perspective of variational inequality in optimization community. Preliminary numerical simulations demonstrate the promising performance of the proposed algorithm.

Citation: Leyu Hu, Wenxing Zhang, Xingju Cai, Deren Han. A parallel operator splitting algorithm for solving constrained total-variation retinex. Inverse Problems and Imaging, 2020, 14 (6) : 1135-1156. doi: 10.3934/ipi.2020058
##### References:

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##### References:
Cartoon images for retinex
Numerical results of retinex on cartoon images
Numerical results of retinex on cartoon images
RGB image for retinex. (a) ideal color wheel image. (b) color wheel image with illumination
Numerical results of retinex on color wheel image
Test RGB images for retinex. (a) $501\times328$ "Girl" image. (b) $324\times323$ "Wall" image. (c) $400\times224$ "Book" image. (d) $281\times375$ "Room" image
Numerical results on "Girl" image
Numerical results on "Wall" image
Numerical results on "Book" image
Numerical results on "Room" image
The evolutions of merits $\|u^k-\hat{u}\|_2$ and $\frac{\|u^{k+1}-\hat{u}\|_2}{\|u^k-\hat{u}\|_2}$ w.r.t. iterations
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