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

Leyu Hu; Wenxing Zhang; Xingju Cai; Deren Han

doi:10.3934/ipi.2020058

Article Contents

2020, Volume 14, Issue 6: 1135-1156. Doi: 10.3934/ipi.2020058

This issue Previous Article Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations Next Article Spatial-Frequency domain nonlocal total variation for image denoising

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
^* Corresponding author: Deren Han

Received: May 2019

Revised: May 2020

Early access: October 2020

Published: December 2020

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Abstract

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.

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Mathematics Subject Classification: 65K15, 68U10, 90C30, 90C90.

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Figure 1. Cartoon images for retinex

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Figure 2. Numerical results of retinex on cartoon images

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Figure 3. Numerical results of retinex on cartoon images

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Figure 4. RGB image for retinex. (a) ideal color wheel image. (b) color wheel image with illumination

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Figure 5. Numerical results of retinex on color wheel image

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Figure 6. 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

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Figure 7. Numerical results on "Girl" image

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Figure 8. Numerical results on "Wall" image

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Figure 9. Numerical results on "Book" image

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Figure 10. Numerical results on "Room" image

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Figure 11. 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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