# American Institute of Mathematical Sciences

June  2019, 13(3): 461-478. doi: 10.3934/ipi.2019023

## A variational gamma correction model for image contrast enhancement

 1 School of Mathematical Sciences, Tongji University, Shanghai, China 2 Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China

* Corresponding author: Michael K. Ng

Received  February 2018 Revised  November 2018 Published  March 2019

Fund Project: W. Wang is supported by Natural Science Foundation of Shanghai and Fundamental Research Funds for the Central Universities of China (22120180255, 22120180067), Michael K. Ng is supported in part by HKRGC GRF 12306616 and 12200317.

Image contrast enhancement plays an important role in computer vision and pattern recognition by improving image quality. The main aim of this paper is to propose and develop a variational model for contrast enhancement of color images based on local gamma correction. The proposed variational model contains an energy functional to determine a local gamma function such that the gamma values can be set according to the local information of the input image. A spatial regularization of the gamma function is incorporated into the functional so that the contrast in an image can be modified by using the information of each pixel and its neighboring pixels. Another regularization term is also employed to preserve the ordering of pixel values. Theoretically, the existence and uniqueness of the minimizer of the proposed model are established. A fast algorithm can be developed to solve the resulting minimization model. Experimental results on benchmark images are presented to show that the performance of the proposed model are better than that of the other testing methods.

Citation: Wei Wang, Na Sun, Michael K. Ng. A variational gamma correction model for image contrast enhancement. Inverse Problems & Imaging, 2019, 13 (3) : 461-478. doi: 10.3934/ipi.2019023
##### References:

show all references

##### References:
From left to right: the input image; the zooming part; the enhanced results by using LRC-AGC; the zooming part; the enhanced results by using GRC-AGC; the zooming part
First row (from left to right): the input image; the enhanced results by setting $\alpha_1 = 1000$, and $\alpha_2 = 10,100, 1000, 10000$; Second row: the enhanced results by setting $\alpha_2 = 1000$, and $\alpha_1 = 10,100, 1000, 10000$
First row (from left to right): the input low contrast image; the enhanced results by using GC with $\gamma = 1/2.2$; the enhanced results by using GC with $\gamma = 1/5$; the enhanced results by using GRC-AGC; Second row: the enhanced results by using LRC-AGC; the enhanced results by using GHE; the enhanced results by using the proposed model. The corresponding zooming parts are displayed in the last two rows respectively
First row (from left to right): the input low contrast image; the enhanced results by using GC with $\gamma = 1/2.2$; the enhanced results by using GC with $\gamma = 1/5$; the enhanced results by using GRC-AGC; Second row: the enhanced results by using LRC-AGC; the enhanced results by using GHE; the enhanced results by using the proposed model. The corresponding zooming parts are displayed in the last two rows respectively
First row (from left to right): the input low contrast image; the enhanced results by using GC with $\gamma = 1/2.2$; the enhanced results by using GC with $\gamma = 1/5$; the enhanced results by using GRC-AGC; Second row: the enhanced results by using LRC-AGC; the enhanced results by using GHE; the enhanced results by using the proposed model. The corresponding zooming parts are displayed in the last two rows respectively
First row (from left to right): the input low contrast image; the enhanced results by using GC with $\gamma = 1/2.2$; the enhanced results by using GC with $\gamma = 1/5$; the enhanced results by using GRC-AGC; Second row: the enhanced results by using LRC-AGC; the enhanced results by using GHE; the enhanced results by using the proposed model. The corresponding zooming parts are displayed in the last two rows respectively
The ground-truth image, the low contrast image, and the enhanced results by using different methods
The ground-truth image, the low contrast image, and the enhanced results by using different methods
ALC and DE values for enhanced results with different values of $\alpha_1$ and $\alpha_2$
 $\alpha_2 = 10$ $100$ $1000$ $10000$ $\alpha_1 = 10$ $100$ $1000$ $10000$ ALC 0.0770 0.0625 0.0569 0.0562 0.0659 0.0630 0.0569 0.0537 DE 7.8509 7.7954 7.7525 7.7454 7.7326 7.7525 7.7525 7.7228
 $\alpha_2 = 10$ $100$ $1000$ $10000$ $\alpha_1 = 10$ $100$ $1000$ $10000$ ALC 0.0770 0.0625 0.0569 0.0562 0.0659 0.0630 0.0569 0.0537 DE 7.8509 7.7954 7.7525 7.7454 7.7326 7.7525 7.7525 7.7228
ALC and DE values for enhanced results of different models
 measures figure GC1 GC2 GRC-AGC LRC-AGC GHE Proposed ALC 3 0.0402 0.0475 0.0805 0.1411 0.0540 0.0622 4 0.0300 0.0383 0.0536 0.0807 0.0751 0.0401 5 0.0052 0.0053 0.0511 0.0398 0.0375 0.0189 6 0.2315 0.2610 0.3438 0.2569 0.2643 0.2488 DE 3 6.0297 5.5654 6.0067 7.8225 5.9928 7.4358 4 6.0746 5.4781 6.1407 7.7833 6.1230 7.6986 5 6.8130 5.9722 6.9451 7.5226 7.1981 7.5682 6 4.9458 4.1280 4.5059 4.9025 5.3131 5.2316
 measures figure GC1 GC2 GRC-AGC LRC-AGC GHE Proposed ALC 3 0.0402 0.0475 0.0805 0.1411 0.0540 0.0622 4 0.0300 0.0383 0.0536 0.0807 0.0751 0.0401 5 0.0052 0.0053 0.0511 0.0398 0.0375 0.0189 6 0.2315 0.2610 0.3438 0.2569 0.2643 0.2488 DE 3 6.0297 5.5654 6.0067 7.8225 5.9928 7.4358 4 6.0746 5.4781 6.1407 7.7833 6.1230 7.6986 5 6.8130 5.9722 6.9451 7.5226 7.1981 7.5682 6 4.9458 4.1280 4.5059 4.9025 5.3131 5.2316
SSIM and PSNR values of the enhanced results by using different methods
 measures Figure GC1 GC2 GRC-AGC LRC-AGC GHE Proposed 7a 0.6823 0.5214 0.8370 0.7559 0.8372 0.8983 7b 0.7724 0.5202 0.9205 0.7806 0.8414 0.9609 SSIM 7c 0.7876 0.5168 0.9268 0.7769 0.8432 0.9408 8a 0.8189 0.6188 0.9699 0.8459 0.7927 0.9837 8b 0.8356 0.6746 0.9280 0.6213 0.8759 0.9559 8c 0.8351 0.6240 0.9511 0.7340 0.9006 0.9668 7a 12.8072 12.5648 20.9844 15.4181 18.2775 22.8673 7b 16.4475 11.0526 23.3122 16.4290 18.5631 26.2987 PSNR 7c 17.4108 9.9382 21.1454 16.0486 18.7101 21.2444 8a 15.5798 10.9477 26.6400 21.1583 16.7663 31.3392 8b 16.7286 11.0620 22.8416 14.6840 22.8626 29.2247 8c 18.5565 12.6350 23.7865 17.3074 20.9938 26.8890
 measures Figure GC1 GC2 GRC-AGC LRC-AGC GHE Proposed 7a 0.6823 0.5214 0.8370 0.7559 0.8372 0.8983 7b 0.7724 0.5202 0.9205 0.7806 0.8414 0.9609 SSIM 7c 0.7876 0.5168 0.9268 0.7769 0.8432 0.9408 8a 0.8189 0.6188 0.9699 0.8459 0.7927 0.9837 8b 0.8356 0.6746 0.9280 0.6213 0.8759 0.9559 8c 0.8351 0.6240 0.9511 0.7340 0.9006 0.9668 7a 12.8072 12.5648 20.9844 15.4181 18.2775 22.8673 7b 16.4475 11.0526 23.3122 16.4290 18.5631 26.2987 PSNR 7c 17.4108 9.9382 21.1454 16.0486 18.7101 21.2444 8a 15.5798 10.9477 26.6400 21.1583 16.7663 31.3392 8b 16.7286 11.0620 22.8416 14.6840 22.8626 29.2247 8c 18.5565 12.6350 23.7865 17.3074 20.9938 26.8890
 [1] Xuefeng Zhang, Hui Yan. Image enhancement algorithm using adaptive fractional differential mask technique. Mathematical Foundations of Computing, 2019, 2 (4) : 347-359. doi: 10.3934/mfc.2019022 [2] Wenzhong Zhu, Huanlong Jiang, Erli Wang, Yani Hou, Lidong Xian, Joyati Debnath. X-ray image global enhancement algorithm in medical image classification. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1297-1309. doi: 10.3934/dcdss.2019089 [3] Lori Badea, Marius Cocou. Approximation results and subspace correction algorithms for implicit variational inequalities. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1507-1524. doi: 10.3934/dcdss.2013.6.1507 [4] Elena Beretta, Markus Grasmair, Monika Muszkieta, Otmar Scherzer. A variational algorithm for the detection of line segments. Inverse Problems & Imaging, 2014, 8 (2) : 389-408. doi: 10.3934/ipi.2014.8.389 [5] Zhi-Feng Pang, Yu-Fei Yang. Semismooth Newton method for minimization of the LLT model. Inverse Problems & Imaging, 2009, 3 (4) : 677-691. doi: 10.3934/ipi.2009.3.677 [6] Yuan Shen, Xin Liu. An alternating minimization method for matrix completion problems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1757-1772. doi: 10.3934/dcdss.2020103 [7] Shui-Nee Chow, Ke Yin, Hao-Min Zhou, Ali Behrooz. Solving inverse source problems by the Orthogonal Solution and Kernel Correction Algorithm (OSKCA) with applications in fluorescence tomography. Inverse Problems & Imaging, 2014, 8 (1) : 79-102. doi: 10.3934/ipi.2014.8.79 [8] Hyeuknam Kwon, Yoon Mo Jung, Jaeseok Park, Jin Keun Seo. A new computer-aided method for detecting brain metastases on contrast-enhanced MR images. Inverse Problems & Imaging, 2014, 8 (2) : 491-505. doi: 10.3934/ipi.2014.8.491 [9] Sanming Liu, Zhijie Wang, Chongyang Liu. Proximal iterative Gaussian smoothing algorithm for a class of nonsmooth convex minimization problems. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 79-89. doi: 10.3934/naco.2015.5.79 [10] M. Montaz Ali. A recursive topographical differential evolution algorithm for potential energy minimization. Journal of Industrial & Management Optimization, 2010, 6 (1) : 29-46. doi: 10.3934/jimo.2010.6.29 [11] Jie Huang, Xiaoping Yang, Yunmei Chen. A fast algorithm for global minimization of maximum likelihood based on ultrasound image segmentation. Inverse Problems & Imaging, 2011, 5 (3) : 645-657. doi: 10.3934/ipi.2011.5.645 [12] Jie Sun, Honglei Xu, Min Zhang. A new interpretation of the progressive hedging algorithm for multistage stochastic minimization problems. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1655-1662. doi: 10.3934/jimo.2019022 [13] Zhengshan Dong, Jianli Chen, Wenxing Zhu. Homotopy method for matrix rank minimization based on the matrix hard thresholding method. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 211-224. doi: 10.3934/naco.2019015 [14] Qingzhi Yang. The revisit of a projection algorithm with variable steps for variational inequalities. Journal of Industrial & Management Optimization, 2005, 1 (2) : 211-217. doi: 10.3934/jimo.2005.1.211 [15] Hongxia Yin. An iterative method for general variational inequalities. Journal of Industrial & Management Optimization, 2005, 1 (2) : 201-209. doi: 10.3934/jimo.2005.1.201 [16] Luchuan Ceng, Qamrul Hasan Ansari, Jen-Chih Yao. Extragradient-projection method for solving constrained convex minimization problems. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 341-359. doi: 10.3934/naco.2011.1.341 [17] Yu-Ning Yang, Su Zhang. On linear convergence of projected gradient method for a class of affine rank minimization problems. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1507-1519. doi: 10.3934/jimo.2016.12.1507 [18] Lei Wu, Zhe Sun. A new spectral method for $l_1$-regularized minimization. Inverse Problems & Imaging, 2015, 9 (1) : 257-272. doi: 10.3934/ipi.2015.9.257 [19] Sandro Zagatti. Minimization of non quasiconvex functionals by integro-extremization method. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 625-641. doi: 10.3934/dcds.2008.21.625 [20] Tengfei Yan, Qunying Liu, Bowen Dou, Qing Li, Bowen Li. An adaptive dynamic programming method for torque ripple minimization of PMSM. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2019136

2018 Impact Factor: 1.469