Retinex based on exponent-type total variation scheme

Lu Liu; Zhi-Feng Pang; Yuping Duan

doi:10.3934/ipi.2018050

Article Contents

2018, Volume 12, Issue 5: 1199-1217. Doi: 10.3934/ipi.2018050

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Retinex based on exponent-type total variation scheme

1.
Center for Applied Mathematics, Tianjin University, Tianjin, China
2.
School of Mathematics and Statistics, and Laboratory of Data Analysis Technology, Henan University, Kaifeng 475004, China
3.
Center for Applied Mathematics, Tianjin University, Tianjin, China

* Corresponding author: Yuping Duan. yuping.duan@tju.edu.cn
* Corresponding author: Yuping Duan. yuping.duan@tju.edu.cn

Received: July 31, 2017

Revised: February 28, 2018

Published: September 2018

The work is supported by the 1000 Talents Program for Young Scientists of China, the Ministry of Science and Technology of China ("863" Program: 2015AA020101), and NSFC 11701418 and 11526208. Dr. Z.-F. Pang was partially supported by National Basic Research Program of China (973 Program No.2015CB856003) and NSFC (Nos.U1304610 and 11401170), and also gratefully acknowledges financial support from China Scholarship Council(CSC) as a research scholar to visit the University of Liverpool from August 2017 to August 2018.

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Abstract

Retinex theory deals with compensation for illumination effects in images, which has a number of applications including Retinex illusion, medical image intensity inhomogeneity and color image shadow effect etc.. Such ill-posed problem has been studied by researchers for decades. However, most exiting methods paid little attention to the noises contained in the images and lost effectiveness when the noises increase. The main aim of this paper is to present a general Retinex model to effectively and robustly restore images degenerated by both illusion and noises. We propose a novel variational model by incorporating appropriate regularization technique for the reflectance component and illumination component accordingly. Although the proposed model is non-convex, we prove the existence of the minimizers theoretically. Furthermore, we design a fast and efficient alternating minimization algorithm for the proposed model, where all subproblems have the closed-form solutions. Applications of the algorithm to various gray images and color images with noises of different distributions yield promising results.

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Mathematics Subject Classification: 80M30, 80M50, 68U10.

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Figure 1. Performances of four methods on $T_1$-weighted brain MR images with different levels of noises

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Figure 2. Comparison among the four methods in terms of PSNR and MSSIM for images with different intensity inhomogeneities

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Figure 3. Performances of four methods on $T_1$-weighted brain images with different intensity inhomogeneities

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Figure 4. Ground truth and estimated bias field of the proposed method with examples in FIGURE 3

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Figure 5. Comparison of the performance in terms of CV(%)

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Figure 6. The relative errors of $r$ and $l$ and numerical energy of our model for the first image in FIGURE 3

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Figure 7. Tests on real MR images. The parameters used in our model are $\alpha = 0.03$ and $\beta = 0.015$

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Figure 8. Tests on color images of HoTVL1 model and our model

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Figure 9. Decomposition of the checkboard image

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Figure 10. Decomposition of the logvi image

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Figure 11. Denoising and decomposition of the images containing Poisson noise

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Table 1. PSNR and MSSIM of $T_1$-weighted brain MR images with different levels of noises


		$3\%$		$5\%$		$7\%$		$9\%$
		PSNR	MSSIM	PSNR	MSSIM	PSNR	MSSIM	PSNR	MSSIM
Test Image1	TVH1	26.8369	0.9436	25.2879	0.9174	24.4622	0.8933	23.2509	0.8629
	HoTVL1	29.7605	0.9515	27.4964	0.9342	26.6143	0.9214	25.2707	0.9063
	L0MS	29.6193	0.9198	27.4895	0.9033	26.1466	0.8915	24.3937	0.8650
	ETV	32.6749	0.9904	31.1170	0.9844	29.1291	0.9767	28.4244	0.9686
Test Image2	TVH1	27.3897	0.9229	26.4466	0.8932	25.4457	0.8655	23.7962	0.8334
	HoTVL1	29.5698	0.9321	28.7948	0.9142	27.1248	0.8988	25.4362	0.8833
	L0MS	31.8155	0.9227	28.5140	0.8950	27.1074	0.8746	24.8900	0.8374
	ETV	33.4328	0.9911	31.4092	0.9842	29.8200	0.9764	28.6543	0.9698

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