# 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:
 [1] T. Arici, S. Dikbas and Y. Altunbasak, A histogram modification framework and its application for image contrast enhancement, IEEE Transactions on Image Processing, 18 (2009), 1921-1935. doi: 10.1109/TIP.2009.2021548. [2] A. Beghdadi and A. L. Negrate, Contrast enhancement technique based on local detection of edges, Comput. Vis, Graph., Image Process., 46 (1989), 162-174. [3] R. Chan, M. Nikolova and Y. Wen, A variational approach for exact histogram specification, Scale Space and Variational Methods in Computer Vision, (2012), 86–97. [4] S. D. Chen and A. R. Ramli, Contrast enhancement using recursive mean-separate histogram equalization for scalable brightness preservation, IEEE Transactions on Consumer Electronics, 49 (2003), 1301-1309. [5] H.-D. Cheng and H. J. Xu, A novel fuzzy logic approach to contrast enhancement, Pattern Recognition, 33 (2000), 809-819. [6] Y.-S. Chiu, F.-C. Cheng and S.-C. 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Woods, Digital Image Processing, 2nd Edition, Prentice Hall, 2002. [12] S.-C. Huang, F.-C. Cheng and Y.-S. Chiu, Efficient contrast enhancement using adaptive gamma correction with weighting distribution, IEEE Transactions on Image Processing, 22 (2013), 1032-1041. doi: 10.1109/TIP.2012.2226047. [13] A. Laine, J. Fan and W. Yang, Wavelets for contrast enhancement of digital mammography, IEEE Engineering in Medicine and Biology Magazine, 14 (1995), 536-550. [14] Y. Li, X. Liu and Y. Liu, Adaptive local gamma correction based on mean value adjustment, 2015 Fifth International Conference on Instrumentation and Measurement, Computer, Communication and Control (IMCCC), 2015, 1858–1863. [15] S. C. Matz and R. J. P. de Figueiredo, A nonlinear image contrast sharpening approach based on munsell's scale, IEEE Transactions on Image Processing, 15 (2016), 900-909. [16] M. Nikolova, A fast algorithm for exact histogram specification. simple extension to colour images, Scale Space and Variational Methods in Computer Vision, 2013,174–185. [17] M. Nikolova and G. Steidl, Fast hue and range preserving histogram specification: Theory and new algorithms for color image enhancement, IEEE Transactions on Image Processing, 23 (2014), 4087-4100. doi: 10.1109/TIP.2014.2337755. [18] M. Nikolova and G. Steidl, Fast ordering algorithm for exact histogram specification, IEEE Transactions on Image Processing, 23 (2014), 5274-5283. doi: 10.1109/TIP.2014.2364119. [19] M. Nikolova, Y. Wen and R. Chan, Exact histogram specification for digital images using a variational approach, Journal of Mathematical Imaging and Vision, 46 (2013), 309-325. doi: 10.1007/s10851-012-0401-8. [20] M. K. Ng and W. Wang, A total variation model for retinex, SIAM J. Imaging Sciences, 4 (2011), 345-365. doi: 10.1137/100806588. [21] F. Pierre, J. F. Aujol, A. Bugeau, G. Steidl and V. T. Ta, Variational contrast enhancement of gray-scale and RGB images, Journal of Mathematical Imaging and Vision, 57 (2017), 99-116. doi: 10.1007/s10851-016-0670-8. [22] F. Pierre, J. F. Aujol, A. Bugeau, G. Steidl and V. T. Ta, Hue-preserving perceptual contrast enhancement, Image Processing (ICIP), IEEE International Conference on, 2016, 4067–4071. [23] A. Polesel, G. Ramponi and V. Mathews, Image enhancement via adaptive unsharp masking, IEEE Trans. Image Process., 9 (2000), 505-510. [24] E. Provenzi and V. Caselles, A wavelet perspective on variational perceptually-inspired color enhancement, International Journal of Computer Vision, 106 (2014), 153-171. doi: 10.1007/s11263-013-0651-y. [25] A. Rizzi, C. Gatta and D. Marini, A new algorithm for unsupervised global and local color correction, Pattern Recognition Letters, 24 (2003), 1663-1677. [26] J. C. Russ, The Image Processing Handbook, Fifth edition. CRC Press, Boca Raton, FL, 2007. [27] C. E. Shannon, A mathematical theory of communication, Bell System Technical Jornal, 27 (1948), 379-423. doi: 10.1002/j.1538-7305.1948.tb01338.x. [28] R. Sherrier and G. Johnson, Regionally adaptive histogram equalization of the chest, IEEE Trans. Med. Imag., MI-6 (1987), 1-7. [29] J. F. Shi and Y. Cai, A novel image enhancement method using local Gamma correction with three-level thresholding, Proceedings of the 6th IEEE Joint International Information Technology and Artificial Intelligence Conference, 2011,374–378. [30] Y. H. Shi, J. F. Yang and R. B. Wu, Reducing illumination based on nonlinear Gamma correction, Proceedings of the IEEE International Conference on Image Processing, (2007), 529–532. [31] J. Tang, X. Liu and Q. Sun, A direct image contrast enhancement algorithm in the wavelet domain for screening mammograms, IEEE J.Sel. Topics Signal Process., 3 (2009), 74-80. [32] Z. Wang, A. Bovik, H. Sheikh and E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans. on Image Processing, 13 (2004), 600–612. [33] W. Wang, C. Chen, M. K. Ng, An Image Pixel Based Variational Model for Histogram Equalization, Journal of Visual Communication and Image Representation, vol. 34, pp. 118-134, January 2016. [34] W. Wang and M. K. Ng, A variational histogram equalization method for image contrast enhancement, SIAM J. Imaging Sciences, 6 (2013), 1823-1849. doi: 10.1137/130909196. [35] G. Xu, J. Su and H. D. Pan, An image enhancement method based on Gamma correction, Proceedings of the 2nd International Symposium on Computational Intelligence and Design, 2009, 60–63.

show all references

##### References:
 [1] T. Arici, S. Dikbas and Y. Altunbasak, A histogram modification framework and its application for image contrast enhancement, IEEE Transactions on Image Processing, 18 (2009), 1921-1935. doi: 10.1109/TIP.2009.2021548. [2] A. Beghdadi and A. L. Negrate, Contrast enhancement technique based on local detection of edges, Comput. Vis, Graph., Image Process., 46 (1989), 162-174. [3] R. Chan, M. Nikolova and Y. Wen, A variational approach for exact histogram specification, Scale Space and Variational Methods in Computer Vision, (2012), 86–97. [4] S. D. Chen and A. R. Ramli, Contrast enhancement using recursive mean-separate histogram equalization for scalable brightness preservation, IEEE Transactions on Consumer Electronics, 49 (2003), 1301-1309. [5] H.-D. Cheng and H. J. Xu, A novel fuzzy logic approach to contrast enhancement, Pattern Recognition, 33 (2000), 809-819. [6] Y.-S. Chiu, F.-C. Cheng and S.-C. Huang, Efficient contrast enhancement using adaptive gamma correction and cumulative intensity distribution, IEEE International Conference on Systems, Man, and Cybernetics, 22 (2013), 1032-1041. doi: 10.1109/TIP.2012.2226047. [7] S. Deivalakshmi, A. Saha and R. Pandeeswari, Raised Cosine Adaptive Gamma Correction for Efficient Image and Video Contrast Enhancement, TENCON 2017-2017 IEEE Region 10 Conference, Nov. 2017. [8] J. Eckstein and D. Bertsekas, On the douglas-rachford splitting method and the proximal point algorithm for maximal monotone operators, Mathematical Programming, 55 (1992), 293-318. doi: 10.1007/BF01581204. [9] L. C. Evans, Partial Differential Equations, AMS, Providence, RI, 1998. [10] M. FarshbafDoustar and H. Hassanpour, A locally-adaptive approach for image gamma correction, 10th International Conference on Information Science, Signal Processing and their Applications (ISSPA 2010), 2010, 73–76. [11] R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2nd Edition, Prentice Hall, 2002. [12] S.-C. Huang, F.-C. Cheng and Y.-S. Chiu, Efficient contrast enhancement using adaptive gamma correction with weighting distribution, IEEE Transactions on Image Processing, 22 (2013), 1032-1041. doi: 10.1109/TIP.2012.2226047. [13] A. Laine, J. Fan and W. Yang, Wavelets for contrast enhancement of digital mammography, IEEE Engineering in Medicine and Biology Magazine, 14 (1995), 536-550. [14] Y. Li, X. Liu and Y. Liu, Adaptive local gamma correction based on mean value adjustment, 2015 Fifth International Conference on Instrumentation and Measurement, Computer, Communication and Control (IMCCC), 2015, 1858–1863. [15] S. C. Matz and R. J. P. de Figueiredo, A nonlinear image contrast sharpening approach based on munsell's scale, IEEE Transactions on Image Processing, 15 (2016), 900-909. [16] M. Nikolova, A fast algorithm for exact histogram specification. simple extension to colour images, Scale Space and Variational Methods in Computer Vision, 2013,174–185. [17] M. Nikolova and G. Steidl, Fast hue and range preserving histogram specification: Theory and new algorithms for color image enhancement, IEEE Transactions on Image Processing, 23 (2014), 4087-4100. doi: 10.1109/TIP.2014.2337755. [18] M. Nikolova and G. Steidl, Fast ordering algorithm for exact histogram specification, IEEE Transactions on Image Processing, 23 (2014), 5274-5283. doi: 10.1109/TIP.2014.2364119. [19] M. Nikolova, Y. Wen and R. Chan, Exact histogram specification for digital images using a variational approach, Journal of Mathematical Imaging and Vision, 46 (2013), 309-325. doi: 10.1007/s10851-012-0401-8. [20] M. K. Ng and W. Wang, A total variation model for retinex, SIAM J. Imaging Sciences, 4 (2011), 345-365. doi: 10.1137/100806588. [21] F. Pierre, J. F. Aujol, A. Bugeau, G. Steidl and V. T. Ta, Variational contrast enhancement of gray-scale and RGB images, Journal of Mathematical Imaging and Vision, 57 (2017), 99-116. doi: 10.1007/s10851-016-0670-8. [22] F. Pierre, J. F. Aujol, A. Bugeau, G. Steidl and V. T. Ta, Hue-preserving perceptual contrast enhancement, Image Processing (ICIP), IEEE International Conference on, 2016, 4067–4071. [23] A. Polesel, G. Ramponi and V. Mathews, Image enhancement via adaptive unsharp masking, IEEE Trans. Image Process., 9 (2000), 505-510. [24] E. Provenzi and V. Caselles, A wavelet perspective on variational perceptually-inspired color enhancement, International Journal of Computer Vision, 106 (2014), 153-171. doi: 10.1007/s11263-013-0651-y. [25] A. Rizzi, C. Gatta and D. Marini, A new algorithm for unsupervised global and local color correction, Pattern Recognition Letters, 24 (2003), 1663-1677. [26] J. C. Russ, The Image Processing Handbook, Fifth edition. CRC Press, Boca Raton, FL, 2007. [27] C. E. Shannon, A mathematical theory of communication, Bell System Technical Jornal, 27 (1948), 379-423. doi: 10.1002/j.1538-7305.1948.tb01338.x. [28] R. Sherrier and G. Johnson, Regionally adaptive histogram equalization of the chest, IEEE Trans. Med. Imag., MI-6 (1987), 1-7. [29] J. F. Shi and Y. Cai, A novel image enhancement method using local Gamma correction with three-level thresholding, Proceedings of the 6th IEEE Joint International Information Technology and Artificial Intelligence Conference, 2011,374–378. [30] Y. H. Shi, J. F. Yang and R. B. Wu, Reducing illumination based on nonlinear Gamma correction, Proceedings of the IEEE International Conference on Image Processing, (2007), 529–532. [31] J. Tang, X. Liu and Q. Sun, A direct image contrast enhancement algorithm in the wavelet domain for screening mammograms, IEEE J.Sel. Topics Signal Process., 3 (2009), 74-80. [32] Z. Wang, A. Bovik, H. Sheikh and E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans. on Image Processing, 13 (2004), 600–612. [33] W. Wang, C. Chen, M. K. Ng, An Image Pixel Based Variational Model for Histogram Equalization, Journal of Visual Communication and Image Representation, vol. 34, pp. 118-134, January 2016. [34] W. Wang and M. K. Ng, A variational histogram equalization method for image contrast enhancement, SIAM J. Imaging Sciences, 6 (2013), 1823-1849. doi: 10.1137/130909196. [35] G. Xu, J. Su and H. D. Pan, An image enhancement method based on Gamma correction, Proceedings of the 2nd International Symposium on Computational Intelligence and Design, 2009, 60–63.
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
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