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

  • * Corresponding author: Deren Han

    * Corresponding author: Deren Han
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  • 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.

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


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  • Figure 1.  Cartoon images for retinex

    Figure 2.  Numerical results of retinex on cartoon images

    Figure 3.  Numerical results of retinex on cartoon images

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

    Figure 5.  Numerical results of retinex on color wheel image

    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

    Figure 7.  Numerical results on "Girl" image

    Figure 8.  Numerical results on "Wall" image

    Figure 9.  Numerical results on "Book" image

    Figure 10.  Numerical results on "Room" image

    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

  • [1] R. G. BaraniukT. GoldsteinA. C. SankaranarayananC. StuderA. Veeraraghavan and M. B. Wakin, Compressive video sensing: Algorithms, architectures, and applications, IEEE Signal Processing Magazine, 34 (2017), 52-66.  doi: 10.1109/MSP.2016.2602099.
    [2] H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, 2011. doi: 10.1007/978-1-4419-9467-7.
    [3] M. BertalmíoV. Caselles and E. Provenzi, Issues about retinex theory and contrast enhancement, International Journal of Computer Vision, 83 (2009), 101-119. 
    [4] M. Bertalmío and J. D. Cowan, Implementing the retinex algorithm with Wilson–Cowan equations, Journal of Physiology-Paris, 103 (2009), 69-72. 
    [5] A. Blake, Boundary conditions for lightness computation in Mondrian world, Computer Vision, Graphics, and Image Processing, 32 (1985), 314-327.  doi: 10.1016/0734-189X(85)90054-4.
    [6] A. BuadesB. Coll and J.-M. Morel, A review of image denoising algorithms, with a new one, Multiscale Modeling & Simulation, 4 (2005), 490-530.  doi: 10.1137/040616024.
    [7] H. Chang, M. K. Ng, W. Wang and T. Zeng, Retinex image enhancement via a learned dictionary, Optical Engineering, 54 (2015), 013107. doi: 10.1117/1.OE.54.1.013107.
    [8] C. ChenB. HeY. Ye and X. Yuan, The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent, Mathematical Programming, 155 (2016), 57-79.  doi: 10.1007/s10107-014-0826-5.
    [9] P. L. Combettes and J.-C. Pesquet, Proximal splitting methods in signal processing, in Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer, 49 (2011), 185–212. doi: 10.1007/978-1-4419-9569-8_10.
    [10] T. J. Cooper and F. A. Baqai, Analysis and extensions of the Frankle-McCann retinex algorithm, Journal of Electronic Imaging, 13 (2004), 85-92.  doi: 10.1117/1.1636182.
    [11] Y.-H. DaiD. HanX. Yuan and W. Zhang, A sequential updating scheme of the Lagrange multiplier for separable convex programming, Mathematics of Computation, 86 (2017), 315-343.  doi: 10.1090/mcom/3104.
    [12] M. Ebner, Color Constancy, Wiley, 2007.
    [13] M. Elad, Retinex by two bilateral filters, in International Conference on Scale-Space Theories in Computer Vision, Springer, (2005), 217–229. doi: 10.1007/11408031_19.
    [14] F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer-Verlag, New York, 2003.
    [15] J. A. Frankle and J. J. McCann, Method and Apparatus for Lightness Imaging, US Patent 4,384,336, 1983.
    [16] G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Modeling & Simulation, 7 (2009), 1005-1028.  doi: 10.1137/070698592.
    [17] R. Glowinski, J.-L. Lions and R. Trémolières, Analyse Numérique Des Inéquations Variationnelles, Dunod, Paris, 1976.
    [18] D. HanD. Sun and L. Zhang, Linear rate convergence of the alternating direction method of multipliers for convex composite programming, Mathematics of Operations Research, 43 (2018), 622-637.  doi: 10.1287/moor.2017.0875.
    [19] D. Han and X. Yuan, Local linear convergence of the alternating direction method of multipliers for quadratic programs, SIAM Journal on Numerical Analysis, 51 (2013), 3446-3457.  doi: 10.1137/120886753.
    [20] D. HanX. Yuan and W. Zhang, An augmented Lagrangian based parallel splitting method for separable convex minimization with applications to image processing, Mathematics of Computation, 83 (2014), 2263-2291.  doi: 10.1090/S0025-5718-2014-02829-9.
    [21] P. C. Hansen, J. G. Nagy and D. P. O'Leary, Deblurring Images: Matrices, Spectra, and Filtering, SIAM, 2006. doi: 10.1137/1.9780898718874.
    [22] H. He and D. Han, A distributed Douglas-Rachford splitting method for multi-block convex minimization problems, Advances in Computational Mathematics, 42 (2016), 27-53.  doi: 10.1007/s10444-015-9408-1.
    [23] B. K. P. Horn, Determining lightness from an image, Computer Graphics and Image Processing, 3 (1974), 277-299.  doi: 10.1016/0146-664X(74)90022-7.
    [24] D. J. JobsonZ. Rahman and G. A. Woodell, A multiscale retinex for bridging the gap between color images and the human observation of scenes, IEEE Transactions on Image Processing, 6 (1997), 965-976.  doi: 10.1109/83.597272.
    [25] D. J. JobsonZ. Rahman and G. A. Woodell, Properties and performance of a center/surround retinex, IEEE Transactions on Image Processing, 6 (1997), 451-462.  doi: 10.1109/83.557356.
    [26] R. KimmelM. EladD. ShakedR. Keshet and I. Sobel, A variational framework for retinex, International Journal of Computer Vision, 52 (2003), 7-23. 
    [27] E. H. Land, The retinex theory of color vision, Scientific American, 237 (1977), 108-129. 
    [28] E. H. Land, Recent advances in retinex theory and some implications for cortical computations: color vision and the natural image, Proceedings of the National Academy of Sciences of the United States of America, 80 (1983), 5163-5169. 
    [29] E. H. Land, An alternative technique for the computation of the designator in the retinex theory of color vision, Proceedings of the National Academy of Sciences of the United States of America, 83 (1986), 3078-3080. 
    [30] E. H. Land and J. J. McCann, Lightness and retinex theory, Journal of the Optical Society of America, 61 (1971), 1-11. 
    [31] L. LiuZ.-F. Pang and Y. Duan, Retinex based on exponent-type total variation scheme, Inverse Problems and Imaging, 12 (2018), 1199-1217.  doi: 10.3934/ipi.2018050.
    [32] W. Ma and S. Osher, A TV Bregman iterative model of retinex theory, Inverse Problems & Imaging, 6 (2012), 697-708.  doi: 10.3934/ipi.2012.6.697.
    [33] D. Marini and A. Rizzi, A computational approach to color adaptation effects, Image and Vision Computing, 18 (2000), 1005-1014.  doi: 10.1016/S0262-8856(00)00037-8.
    [34] J. McCann and I. Sobel, Experiments with retinex, in HPL Color Summit, Hewlett Packard Laboratories, 1998.
    [35] J.-M. Morel, A. B. Petro and C. Sbert, Fast implementation of color constancy algorithms, in Color Imaging XIV: Displaying, Processing, Hardcopy, and Applications, International Society for Optics and Photonics, 7241 (2009), 724106. doi: 10.1117/12.805474.
    [36] J.-M. MorelA. B. Petro and C. Sbert, A PDE formalization of retinex theory, IEEE Transactions on Image Processing, 19 (2010), 2825-2837.  doi: 10.1109/TIP.2010.2049239.
    [37] M. K. Ng and W. Wang, A total variation model for retinex, SIAM Journal on Imaging Sciences, 4 (2011), 345-365.  doi: 10.1137/100806588.
    [38] M. Nikolova, A variational approach to remove outliers and impulse noise, Journal of Mathematical Imaging and Vision, 20 (2004), 99-120. 
    [39] E. ProvenziL. De CarliA. Rizzi and D. Marini, Mathematical definition and analysis of the retinex algorithm, Journal of the Optical Society of America A, 22 (2005), 2613-2621.  doi: 10.1364/JOSAA.22.002613.
    [40] R. T. RockafellarConvex Analysis, Princeton University Press, Princeton, NJ, 1970. 
    [41] L. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.
    [42] O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, Springer, New York, 2009.
    [43] J. YangY. Zhang and W. Yin, An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise, SIAM Journal on Scientific Computing, 31 (2009), 2842-2865.  doi: 10.1137/080732894.
    [44] W. H. Yang and D. Han, Linear convergence of the alternating direction method of multipliers for a class of convex optimization problems, SIAM Journal on Numerical Analysis, 54 (2016), 625-640.  doi: 10.1137/140974237.
    [45] X. Zhang and B. A. Wandell, A spatial extension of CIELAB for digital color-image reproduction, Journal of the Society for Information Display, 5 (1997), 61-63. 
    [46] X. Y. Zheng and K. F. Ng, Metric subregularity of piecewise linear multifunctions and applications to piecewise linear multiobjective optimization, SIAM Journal on Optimization, 24 (2014), 154-174.  doi: 10.1137/120889502.
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