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

show all references

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
 [1] Sören Bartels, Marijo Milicevic. Iterative finite element solution of a constrained total variation regularized model problem. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1207-1232. doi: 10.3934/dcdss.2017066 [2] Zhiwei Tian, Yanyan Shi, Meng Wang, Xiaolong Kong, Lei Li, Feng Fu. A wavelet frame constrained total generalized variation model for imaging conductivity distribution. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021074 [3] Lu Liu, Zhi-Feng Pang, Yuping Duan. Retinex based on exponent-type total variation scheme. Inverse Problems & Imaging, 2018, 12 (5) : 1199-1217. doi: 10.3934/ipi.2018050 [4] Liejune Shiau, Roland Glowinski. Operator splitting method for friction constrained dynamical systems. Conference Publications, 2005, 2005 (Special) : 806-815. doi: 10.3934/proc.2005.2005.806 [5] Adriana González, Laurent Jacques, Christophe De Vleeschouwer, Philippe Antoine. Compressive optical deflectometric tomography: A constrained total-variation minimization approach. Inverse Problems & Imaging, 2014, 8 (2) : 421-457. doi: 10.3934/ipi.2014.8.421 [6] Yazheng Dang, Fanwen Meng, Jie Sun. Convergence analysis of a parallel projection algorithm for solving convex feasibility problems. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 505-519. doi: 10.3934/naco.2016023 [7] Zhili Ge, Gang Qian, Deren Han. Global convergence of an inexact operator splitting method for monotone variational inequalities. Journal of Industrial & Management Optimization, 2011, 7 (4) : 1013-1026. doi: 10.3934/jimo.2011.7.1013 [8] Wei Wang, Ling Pi, Michael K. Ng. Saturation-Value Total Variation model for chromatic aberration correction. Inverse Problems & Imaging, 2020, 14 (4) : 733-755. doi: 10.3934/ipi.2020034 [9] Liyan Ma, Lionel Moisan, Jian Yu, Tieyong Zeng. A stable method solving the total variation dictionary model with $L^\infty$ constraints. Inverse Problems & Imaging, 2014, 8 (2) : 507-535. doi: 10.3934/ipi.2014.8.507 [10] Zhengmeng Jin, Chen Zhou, Michael K. Ng. A coupled total variation model with curvature driven for image colorization. Inverse Problems & Imaging, 2016, 10 (4) : 1037-1055. doi: 10.3934/ipi.2016031 [11] Sudeb Majee, Subit K. Jain, Rajendra K. Ray, Ananta K. Majee. A fuzzy edge detector driven telegraph total variation model for image despeckling. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021054 [12] Ran Ma, Jiping Tao. An improved 2.11-competitive algorithm for online scheduling on parallel machines to minimize total weighted completion time. Journal of Industrial & Management Optimization, 2018, 14 (2) : 497-510. doi: 10.3934/jimo.2017057 [13] Yanxing Cui, Chuanlong Wang, Ruiping Wen. On the convergence of generalized parallel multisplitting iterative methods for semidefinite linear systems. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 863-873. doi: 10.3934/naco.2012.2.863 [14] Yves Bourgault, Damien Broizat, Pierre-Emmanuel Jabin. Convergence rate for the method of moments with linear closure relations. Kinetic & Related Models, 2015, 8 (1) : 1-27. doi: 10.3934/krm.2015.8.1 [15] Xiaojing Ye, Haomin Zhou. Fast total variation wavelet inpainting via approximated primal-dual hybrid gradient algorithm. Inverse Problems & Imaging, 2013, 7 (3) : 1031-1050. doi: 10.3934/ipi.2013.7.1031 [16] Yaonan Ma, Li-Zhi Liao. The Glowinski–Le Tallec splitting method revisited: A general convergence and convergence rate analysis. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1681-1711. doi: 10.3934/jimo.2020040 [17] Berat Karaagac. Numerical treatment of Gray-Scott model with operator splitting method. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2373-2386. doi: 10.3934/dcdss.2020143 [18] Matthias Gerdts, Martin Kunkel. Convergence analysis of Euler discretization of control-state constrained optimal control problems with controls of bounded variation. Journal of Industrial & Management Optimization, 2014, 10 (1) : 311-336. doi: 10.3934/jimo.2014.10.311 [19] Xiaoqun Zhang, Tony F. Chan. Wavelet inpainting by nonlocal total variation. Inverse Problems & Imaging, 2010, 4 (1) : 191-210. doi: 10.3934/ipi.2010.4.191 [20] Baoli Shi, Zhi-Feng Pang, Jing Xu. Image segmentation based on the hybrid total variation model and the K-means clustering strategy. Inverse Problems & Imaging, 2016, 10 (3) : 807-828. doi: 10.3934/ipi.2016022

2020 Impact Factor: 1.639