# American Institute of Mathematical Sciences

November  2012, 6(4): 697-708. doi: 10.3934/ipi.2012.6.697

## A TV Bregman iterative model of Retinex theory

 1 Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, United States, United States

Received  April 2010 Revised  October 2012 Published  November 2012

A feature of the human visual system (HVS) is color constancy, namely, the ability to determine the color under varying illumination conditions. Retinex theory, formulated by Edwin H. Land, aimed to simulate and explain how the HVS perceives color. In this paper, we establish a total variation (TV) and nonlocal TV regularized model of Retinex theory that can be solved by a fast computational approach based on Bregman iteration. We demonstrate the performance of our method by numerical results.
Citation: Wenye Ma, Stanley Osher. A TV Bregman iterative model of Retinex theory. Inverse Problems and Imaging, 2012, 6 (4) : 697-708. doi: 10.3934/ipi.2012.6.697
##### References:
 [1] A. Blake, Boundary conditions of lightness computation in mondrian world, Computer Vision Graphics and Image Processing, 32 (1985), 314-327. doi: 10.1016/0734-189X(85)90054-4. [2] D. Brainard and B. Wandell, Analysis of the Retinex theory of color vision, J. Opt. Soc. of Am. A, 3 (1986), 1651-1661. doi: 10.1364/JOSAA.3.001651. [3] L. Bregman, A relaxation method of finding the common points of convex sets and its application to the solution of problems in convex programming, USSR Comput Math and Math. Phys., 7 (1967), 200-217. [4] A. Buades, B. Coll and J. M. Morel, A review of image denoising algorithms, with a new one, Multiscale Model. Simul., 4 (2005), 490-530. [5] J. Frankle and J. McCann, Method and apparatus for lightness imaging, US Patent no. 4384336, (1983). [6] B. Funt, F. Ciuera and J. McCann, Retinex in Matlab, Journal of Electronic Imaging, 13 (2004), 48-57. doi: 10.1117/1.1636761. [7] G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028. [8] T. Goldstein and S. Osher, The split Bregman algorithm for L1 regularized problems, SIAM J. Imaging Sci., 2 (2009), 323-343. [9] T. Goldstein, X. Bresson and S. Osher, Geometric applications of the split Bregman method: Segmentation and surface reconstruction, J. Sci. Comput., 45 (2010), 272-293. doi: 10.1007/s10915-009-9331-z. [10] B. K. P. Horn, Determining lightness from an image, Computer Graphics and Image Processing, 3 (1974), 277-299. [11] D. J. Jobson, Z. Rahman and G. A. Woodell, A multiscale retinex for bridging the gap between color images and the human observation of scenes, IEEE Trans. on Image Processing, 6 (1997), 965-976. [12] R. Kimmel, M. Elad, D. Shaked, R. Keshet and I. Sobel, A variational framework for retinex, Int. Journal of Computer Vision, 52 (2003), 7-23. doi: 10.1023/A:1022314423998. [13] E. H. Land and J. J. McCann, Lightness and the retinex theory, J. Opt. Soc. Am., 61 (1971), 1-11. doi: 10.1364/JOSA.61.000001. [14] E. H. Land, The retinex theory of color vision, Sci. Amer., 237 (1977), 108-128. doi: 10.1038/scientificamerican1277-108. [15] E. H. Land, Recent advances in the Retinex theory and some implications for cortical computations: Color vision and the natural image, Proc. Nat. Acad. Sci. USA, 80 (1983), 5163-5169. doi: 10.1073/pnas.80.16.5163. [16] E. H. Land, An alternative technique for computation of the designator in the Retinex of color vision, Proc. Nat. Acad. Sci. USA, 83 (1986), 3078-3080. doi: 10.1073/pnas.83.10.3078. [17] Y. Lei, Y. Zhou and J. Li, An investigation of retinex algorithms for image enhancement, Jouranl of Electron, China, 24 (2007), 696-700. [18] W. Ma, J. M. Morel, A. Chien and S. Osher, An L1-based variational model for Retinex theory and its application to medical images, IEEE Conference on Computer Vision and Pattern Recognition, (2011), 153-160. [19] J. M. Morel, A. B. Petro and C. Sbert, Fast implementation of color constancy algorithms, Proc. SPIE, 7241 (2009). [20] J. M. Morel, A. B. Petro and C. Sbert, A PDE formalization of the retinex theory, IEEE Transaction on Image Processing, 19 (2010), 2825-2837. [21] S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An iterative regularization method for total variation based image restoration, SIAM Multiscale Model. and Simu., 4 (2005), 460-489. [22] E. Provenzi, M. Fierro, A. Rizzi, L. De Carli, D. Gadia and D. Marini, Random spray retinex: A new retinex implementation to investigate the local properties of the model, IEEE Transactions on Image Processing, 16 (2007), 162-171. doi: 10.1109/TIP.2006.884946. [23] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica, D(60) (1992), 259-268. [24] X. Zhang, M. Burger, X. Bresson and S. Osher, Bregmanized nonlocal regularization for deconvolution and sparse reconstruction, SIAM J. Imaging Sci., 3 (2010), 253-276.

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##### References:
 [1] A. Blake, Boundary conditions of lightness computation in mondrian world, Computer Vision Graphics and Image Processing, 32 (1985), 314-327. doi: 10.1016/0734-189X(85)90054-4. [2] D. Brainard and B. Wandell, Analysis of the Retinex theory of color vision, J. Opt. Soc. of Am. A, 3 (1986), 1651-1661. doi: 10.1364/JOSAA.3.001651. [3] L. Bregman, A relaxation method of finding the common points of convex sets and its application to the solution of problems in convex programming, USSR Comput Math and Math. Phys., 7 (1967), 200-217. [4] A. Buades, B. Coll and J. M. Morel, A review of image denoising algorithms, with a new one, Multiscale Model. Simul., 4 (2005), 490-530. [5] J. Frankle and J. McCann, Method and apparatus for lightness imaging, US Patent no. 4384336, (1983). [6] B. Funt, F. Ciuera and J. McCann, Retinex in Matlab, Journal of Electronic Imaging, 13 (2004), 48-57. doi: 10.1117/1.1636761. [7] G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028. [8] T. Goldstein and S. Osher, The split Bregman algorithm for L1 regularized problems, SIAM J. Imaging Sci., 2 (2009), 323-343. [9] T. Goldstein, X. Bresson and S. Osher, Geometric applications of the split Bregman method: Segmentation and surface reconstruction, J. Sci. Comput., 45 (2010), 272-293. doi: 10.1007/s10915-009-9331-z. [10] B. K. P. Horn, Determining lightness from an image, Computer Graphics and Image Processing, 3 (1974), 277-299. [11] D. J. Jobson, Z. Rahman and G. A. Woodell, A multiscale retinex for bridging the gap between color images and the human observation of scenes, IEEE Trans. on Image Processing, 6 (1997), 965-976. [12] R. Kimmel, M. Elad, D. Shaked, R. Keshet and I. Sobel, A variational framework for retinex, Int. Journal of Computer Vision, 52 (2003), 7-23. doi: 10.1023/A:1022314423998. [13] E. H. Land and J. J. McCann, Lightness and the retinex theory, J. Opt. Soc. Am., 61 (1971), 1-11. doi: 10.1364/JOSA.61.000001. [14] E. H. Land, The retinex theory of color vision, Sci. Amer., 237 (1977), 108-128. doi: 10.1038/scientificamerican1277-108. [15] E. H. Land, Recent advances in the Retinex theory and some implications for cortical computations: Color vision and the natural image, Proc. Nat. Acad. Sci. USA, 80 (1983), 5163-5169. doi: 10.1073/pnas.80.16.5163. [16] E. H. Land, An alternative technique for computation of the designator in the Retinex of color vision, Proc. Nat. Acad. Sci. USA, 83 (1986), 3078-3080. doi: 10.1073/pnas.83.10.3078. [17] Y. Lei, Y. Zhou and J. Li, An investigation of retinex algorithms for image enhancement, Jouranl of Electron, China, 24 (2007), 696-700. [18] W. Ma, J. M. Morel, A. Chien and S. Osher, An L1-based variational model for Retinex theory and its application to medical images, IEEE Conference on Computer Vision and Pattern Recognition, (2011), 153-160. [19] J. M. Morel, A. B. Petro and C. Sbert, Fast implementation of color constancy algorithms, Proc. SPIE, 7241 (2009). [20] J. M. Morel, A. B. Petro and C. Sbert, A PDE formalization of the retinex theory, IEEE Transaction on Image Processing, 19 (2010), 2825-2837. [21] S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An iterative regularization method for total variation based image restoration, SIAM Multiscale Model. and Simu., 4 (2005), 460-489. [22] E. Provenzi, M. Fierro, A. Rizzi, L. De Carli, D. Gadia and D. Marini, Random spray retinex: A new retinex implementation to investigate the local properties of the model, IEEE Transactions on Image Processing, 16 (2007), 162-171. doi: 10.1109/TIP.2006.884946. [23] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica, D(60) (1992), 259-268. [24] X. Zhang, M. Burger, X. Bresson and S. Osher, Bregmanized nonlocal regularization for deconvolution and sparse reconstruction, SIAM J. Imaging Sci., 3 (2010), 253-276.
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