doi: 10.3934/ipi.2020058

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

1. 

School of Mathematical Sciences, Key Laboratory for NSLSCS of Jiangsu Province, Nanjing Normal University, Nanjing, 210023, China

2. 

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China

3. 

LMIB, School of Mathematical Sciences, Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing, 100191, China

* Corresponding author: Deren Han

Received  May 2019 Revised  May 2020 Published  October 2020

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.

Citation: Leyu Hu, Wenxing Zhang, Xingju Cai, Deren Han. A parallel operator splitting algorithm for solving constrained total-variation retinex. Inverse Problems & Imaging, doi: 10.3934/ipi.2020058
References:
[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.  Google Scholar

[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.  Google Scholar

[3]

M. BertalmíoV. Caselles and E. Provenzi, Issues about retinex theory and contrast enhancement, International Journal of Computer Vision, 83 (2009), 101-119.   Google Scholar

[4]

M. Bertalmío and J. D. Cowan, Implementing the retinex algorithm with Wilson–Cowan equations, Journal of Physiology-Paris, 103 (2009), 69-72.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Ebner, Color Constancy, Wiley, 2007. Google Scholar

[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.  Google Scholar

[14]

F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer-Verlag, New York, 2003.  Google Scholar

[15]

J. A. Frankle and J. J. McCann, Method and Apparatus for Lightness Imaging, US Patent 4,384,336, 1983. Google Scholar

[16]

G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Modeling & Simulation, 7 (2009), 1005-1028.  doi: 10.1137/070698592.  Google Scholar

[17]

R. Glowinski, J.-L. Lions and R. Trémolières, Analyse Numérique Des Inéquations Variationnelles, Dunod, Paris, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

R. KimmelM. EladD. ShakedR. Keshet and I. Sobel, A variational framework for retinex, International Journal of Computer Vision, 52 (2003), 7-23.   Google Scholar

[27]

E. H. Land, The retinex theory of color vision, Scientific American, 237 (1977), 108-129.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[30]

E. H. Land and J. J. McCann, Lightness and retinex theory, Journal of the Optical Society of America, 61 (1971), 1-11.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

J. McCann and I. Sobel, Experiments with retinex, in HPL Color Summit, Hewlett Packard Laboratories, 1998. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

M. Nikolova, A variational approach to remove outliers and impulse noise, Journal of Mathematical Imaging and Vision, 20 (2004), 99-120.   Google Scholar

[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.  Google Scholar

[40] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.   Google Scholar
[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.  Google Scholar

[42]

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, Springer, New York, 2009.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[3]

M. BertalmíoV. Caselles and E. Provenzi, Issues about retinex theory and contrast enhancement, International Journal of Computer Vision, 83 (2009), 101-119.   Google Scholar

[4]

M. Bertalmío and J. D. Cowan, Implementing the retinex algorithm with Wilson–Cowan equations, Journal of Physiology-Paris, 103 (2009), 69-72.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Ebner, Color Constancy, Wiley, 2007. Google Scholar

[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.  Google Scholar

[14]

F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer-Verlag, New York, 2003.  Google Scholar

[15]

J. A. Frankle and J. J. McCann, Method and Apparatus for Lightness Imaging, US Patent 4,384,336, 1983. Google Scholar

[16]

G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Modeling & Simulation, 7 (2009), 1005-1028.  doi: 10.1137/070698592.  Google Scholar

[17]

R. Glowinski, J.-L. Lions and R. Trémolières, Analyse Numérique Des Inéquations Variationnelles, Dunod, Paris, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

R. KimmelM. EladD. ShakedR. Keshet and I. Sobel, A variational framework for retinex, International Journal of Computer Vision, 52 (2003), 7-23.   Google Scholar

[27]

E. H. Land, The retinex theory of color vision, Scientific American, 237 (1977), 108-129.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[30]

E. H. Land and J. J. McCann, Lightness and retinex theory, Journal of the Optical Society of America, 61 (1971), 1-11.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

J. McCann and I. Sobel, Experiments with retinex, in HPL Color Summit, Hewlett Packard Laboratories, 1998. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

M. Nikolova, A variational approach to remove outliers and impulse noise, Journal of Mathematical Imaging and Vision, 20 (2004), 99-120.   Google Scholar

[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.  Google Scholar

[40] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.   Google Scholar
[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.  Google Scholar

[42]

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, Springer, New York, 2009.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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
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