December  2020, 14(6): 1135-1156. 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  December 2020 Early access  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 and Imaging, 2020, 14 (6) : 1135-1156. 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.

[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. Rockafellar, Convex 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.

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.

[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. Rockafellar, Convex 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.

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]

Sören Bartels, Marijo Milicevic. Iterative finite element solution of a constrained total variation regularized model problem. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1207-1232. doi: 10.3934/dcdss.2017066

[2]

Zhiwei Tian, Yanyan Shi, Meng Wang, Xiaolong Kong, Lei Li, Feng Fu. A wavelet frame constrained total generalized variation model for imaging conductivity distribution. Inverse Problems and Imaging, 2022, 16 (4) : 753-769. doi: 10.3934/ipi.2021074

[3]

Lu Liu, Zhi-Feng Pang, Yuping Duan. Retinex based on exponent-type total variation scheme. Inverse Problems and Imaging, 2018, 12 (5) : 1199-1217. doi: 10.3934/ipi.2018050

[4]

Liejune Shiau, Roland Glowinski. Operator splitting method for friction constrained dynamical systems. Conference Publications, 2005, 2005 (Special) : 806-815. doi: 10.3934/proc.2005.2005.806

[5]

Adriana González, Laurent Jacques, Christophe De Vleeschouwer, Philippe Antoine. Compressive optical deflectometric tomography: A constrained total-variation minimization approach. Inverse Problems and Imaging, 2014, 8 (2) : 421-457. doi: 10.3934/ipi.2014.8.421

[6]

Yazheng Dang, Fanwen Meng, Jie Sun. Convergence analysis of a parallel projection algorithm for solving convex feasibility problems. Numerical Algebra, Control and Optimization, 2016, 6 (4) : 505-519. doi: 10.3934/naco.2016023

[7]

Zhili Ge, Gang Qian, Deren Han. Global convergence of an inexact operator splitting method for monotone variational inequalities. Journal of Industrial and Management Optimization, 2011, 7 (4) : 1013-1026. doi: 10.3934/jimo.2011.7.1013

[8]

Wei Wang, Ling Pi, Michael K. Ng. Saturation-Value Total Variation model for chromatic aberration correction. Inverse Problems and Imaging, 2020, 14 (4) : 733-755. doi: 10.3934/ipi.2020034

[9]

Liyan Ma, Lionel Moisan, Jian Yu, Tieyong Zeng. A stable method solving the total variation dictionary model with $L^\infty$ constraints. Inverse Problems and Imaging, 2014, 8 (2) : 507-535. doi: 10.3934/ipi.2014.8.507

[10]

Zhengmeng Jin, Chen Zhou, Michael K. Ng. A coupled total variation model with curvature driven for image colorization. Inverse Problems and Imaging, 2016, 10 (4) : 1037-1055. doi: 10.3934/ipi.2016031

[11]

Sudeb Majee, Subit K. Jain, Rajendra K. Ray, Ananta K. Majee. A fuzzy edge detector driven telegraph total variation model for image despeckling. Inverse Problems and Imaging, 2022, 16 (2) : 367-396. doi: 10.3934/ipi.2021054

[12]

Ran Ma, Jiping Tao. An improved 2.11-competitive algorithm for online scheduling on parallel machines to minimize total weighted completion time. Journal of Industrial and Management Optimization, 2018, 14 (2) : 497-510. doi: 10.3934/jimo.2017057

[13]

Yanxing Cui, Chuanlong Wang, Ruiping Wen. On the convergence of generalized parallel multisplitting iterative methods for semidefinite linear systems. Numerical Algebra, Control and Optimization, 2012, 2 (4) : 863-873. doi: 10.3934/naco.2012.2.863

[14]

Yves Bourgault, Damien Broizat, Pierre-Emmanuel Jabin. Convergence rate for the method of moments with linear closure relations. Kinetic and Related Models, 2015, 8 (1) : 1-27. doi: 10.3934/krm.2015.8.1

[15]

Xiaojing Ye, Haomin Zhou. Fast total variation wavelet inpainting via approximated primal-dual hybrid gradient algorithm. Inverse Problems and Imaging, 2013, 7 (3) : 1031-1050. doi: 10.3934/ipi.2013.7.1031

[16]

Yaonan Ma, Li-Zhi Liao. The Glowinski–Le Tallec splitting method revisited: A general convergence and convergence rate analysis. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1681-1711. doi: 10.3934/jimo.2020040

[17]

Berat Karaagac. Numerical treatment of Gray-Scott model with operator splitting method. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2373-2386. doi: 10.3934/dcdss.2020143

[18]

Matthias Gerdts, Martin Kunkel. Convergence analysis of Euler discretization of control-state constrained optimal control problems with controls of bounded variation. Journal of Industrial and Management Optimization, 2014, 10 (1) : 311-336. doi: 10.3934/jimo.2014.10.311

[19]

Harun Karsli, Purshottam Narain Agrawal. Rate of convergence of Stancu type modified $ q $-Gamma operators for functions with derivatives of bounded variation. Mathematical Foundations of Computing, 2022  doi: 10.3934/mfc.2022002

[20]

Xiaoqun Zhang, Tony F. Chan. Wavelet inpainting by nonlocal total variation. Inverse Problems and Imaging, 2010, 4 (1) : 191-210. doi: 10.3934/ipi.2010.4.191

2020 Impact Factor: 1.639

Metrics

  • PDF downloads (386)
  • HTML views (145)
  • Cited by (0)

Other articles
by authors

[Back to Top]