# American Institute of Mathematical Sciences

May  2016, 10(2): 461-497. doi: 10.3934/ipi.2016008

## Color image processing by vectorial total variation with gradient channels coupling

 1 Department of Computer Science, University of Beira Interior, 6201-001 Covilhã, Portugal, Portugal 2 Computational Imaging and VisAnalysis (CIVA) Lab, Department of Computer Science, University of Missouri-Columbia, Columbia MO 65211, United States

Received  December 2014 Revised  August 2015 Published  May 2016

We study a regularization method for color images based on the vectorial total variation approach along with channel coupling for color image processing, which facilitates the modeling of inter channel relations in multidimensional image data. We focus on penalizing channel gradient magnitude similarities by using $L^{2}$ differences, which allow us to explicitly couple all the channels along with a vectorial total variation regularization for edge preserving smoothing of multichannel images. By using matched gradients to align edges from different channels we obtain multichannel edge preserving smoothing and decomposition. A detailed mathematical analysis of the vectorial total variation with penalized gradient channels coupling is provided. We characterize some important properties of the minimizers of the model as well as provide geometrical results regarding the regularization parameter. We are interested in applying our model to color image processing and in particular to denoising and decomposition. A fast global minimization based on the dual formulation of the total variation is used and convergence of the iterative scheme is provided. Extensive experiments are given to show that our approach obtains good decomposition and denoising results in natural images. Comparison with previous color image decomposition and denoising methods demonstrate the advantages of our approach.
Citation: Juan C. Moreno, V. B. Surya Prasath, João C. Neves. Color image processing by vectorial total variation with gradient channels coupling. Inverse Problems & Imaging, 2016, 10 (2) : 461-497. doi: 10.3934/ipi.2016008
##### References:
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Chan, Combining geometrical and textured information to perform image classification,, Journal of Visual Communication and Image Representation, 17 (2006), 1004. doi: 10.1016/j.jvcir.2006.02.001. Google Scholar [7] J.-F. Aujol, G. Gilboa, T. Chan and S. Osher, Structure-texture image decomposition - modeling, algorithms and parameter selection,, International Journal of Computer Vision, 67 (2006), 111. doi: 10.1007/s11263-006-4331-z. Google Scholar [8] J.-F. Aujol and S. H. Kang, Color image decomposition and restoration,, Journal of Visual Communication and Image Representation, 17 (2006), 916. doi: 10.1016/j.jvcir.2005.02.001. Google Scholar [9] L. Bar, N. Sochen and N. Kiryati, Image deblurring in the presence of salt-and-pepper noise,, in Lecture Notes in Computer Science, 3459 (2005), 107. doi: 10.1007/11408031_10. Google Scholar [10] A. Beck and M. Teboulle, Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,, IEEE Transactions on Image Processing, 18 (2009), 2419. doi: 10.1109/TIP.2009.2028250. Google Scholar [11] P. Blomgren and T. F. Chan, Color TV: Total variation methods for restoration of vector valued images,, IEEE Transactions on Image Processing, 7 (1998), 304. doi: 10.1109/83.661180. Google Scholar [12] G. Boccignone, M. Ferraro and T. Caelli, Generalized spatio-chromatic diffusion,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 24 (2002), 1298. doi: 10.1109/TPAMI.2002.1039202. Google Scholar [13] X. Bresson and T. F. Chan, Fast dual minimization of the vectorial total variation norm and applications to color image processing,, Inverse Problem and Imaging, 2 (2008), 455. doi: 10.3934/ipi.2008.2.455. Google Scholar [14] X. Bresson, S. Esedoglu, P. Vandergheynst, J. Thiran and S. Osher, Fast global minimization of the active contour/snake model,, Journal of Mathematical Imaging and Vision, 28 (2007), 151. doi: 10.1007/s10851-007-0002-0. Google Scholar [15] A. Brook, R. Kimmel and N. Sochen, Variational restoration and edge detection for color images,, Journal of Mathematical Imaging and Vision, 18 (2003), 247. doi: 10.1023/A:1022895410391. Google Scholar [16] A. Buades, B. Coll and J. M. Morel, A review of image denoising methods, with a new one,, Multiscale Modeling and Simulation, 4 (2005), 490. doi: 10.1137/040616024. Google Scholar [17] V. Caselles, F. Catté, T. Coll and F. Dibos, A geometric model for active contours in image processing,, Numerische Mathematik, 66 (1993), 1. doi: 10.1007/BF01385685. Google Scholar [18] V. Caselles, B. Coll and J.-M. Morel, Geometry and color in natural images,, Journal of Mathematical Imaging and Vision, 16 (2002), 89. doi: 10.1023/A:1013943314097. Google Scholar [19] V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours,, International Journal of Computer Vision, (1995), 694. doi: 10.1109/ICCV.1995.466871. Google Scholar [20] A. Chambolle, An algorithm for total variation minimization and applications,, Journal of Mathematical Imaging and Vision, 20 (2004), 89. doi: 10.1023/B:JMIV.0000011321.19549.88. Google Scholar [21] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging,, Journal of Mathematical Imaging and Vision, 40 (2011), 120. doi: 10.1007/s10851-010-0251-1. Google Scholar [22] T. F. Chan and S. Esedoglu, Aspects of total variation regularized $L^{1}$ function approximation,, SIAM Journal on Applied Mathematics, 65 (2005), 1817. doi: 10.1137/040604297. Google Scholar [23] T. F. Chan, S. Esedoglu and M. Nikolova, Algorithms for finding global minimizers of image segmentation and denoising models,, SIAM Journal on Applied Mathematics, 66 (2006), 1632. doi: 10.1137/040615286. Google Scholar [24] T. F. Chan, G. Golub and P. Mulet, A nonlinear primal-dual method for total variation-based image restoration,, SIAM Journal on Scientific Computing, 20 (1999), 1964. doi: 10.1137/S1064827596299767. Google Scholar [25] T. F. Chan and J. Shen, Variational image inpainting,, Communications on Pure and Applied Mathematics, 58 (2005), 579. doi: 10.1002/cpa.20075. Google Scholar [26] K. Dabov, A. Foi, V. Katkovnik and K. Egiazarian, Color image denoising via sparse 3d collaborative filtering with grouping constraint in luminance-chrominance space,, in IEEE International Conference on Image Processing, 1 (2007), 313. doi: 10.1109/ICIP.2007.4378954. Google Scholar [27] J. Darbon, Total variation minimization with $L^{1}$ data fidelity as a contrast invariant filter,, in 4th Symposium on Image and Signal Processing and Analysis (ISPA), (2005), 221. doi: 10.1109/ISPA.2005.195413. Google Scholar [28] Y. Dong, M. Hintermuller and M. M. Rincon-Camacho, A multi-scale vectorial $L^\tau$-TV framework for color image restoration,, International Journal of Computer Vision, 92 (2011), 296. doi: 10.1007/s11263-010-0359-1. Google Scholar [29] S. Durand, J. Fadili and M. Nikolova, Multiplicative noise removal using $L^1$ fidelity on frame coefficients,, Journal of Mathematical Imaging and Vision, 36 (2010), 201. Google Scholar [30] V. Duval, J.-F. Aujol and Y. Gousseau, The TVL1 model: A geometrical point of view,, Multiscale Modelling & Simulation, 8 (2009), 154. doi: 10.1137/090757083. Google Scholar [31] V. Duval, J.-F. Aujol and L. Vese, Mathematical modelling of textures: Application to color image decomposition with a projected gradient algorithm,, Journal of Mathematical Imaging and Vision, 37 (2010), 232. doi: 10.1007/s10851-010-0203-9. Google Scholar [32] M. J. Ehrhardt and S. R. Arridge, Vector-valued image processing by parallel level sets,, IEEE Transactions on Image Processing, 23 (2014), 9. doi: 10.1109/TIP.2013.2277775. Google Scholar [33] G. Gilboa, A total variation spectral framework for scale and texture analysis,, Multiscale Modelling and Simulation, 7 (2014), 1937. doi: 10.1137/130930704. Google Scholar [34] J. Gilles, Noisy image decomposition: A new structure, texture and noise model based on local adaptivity,, Journal of Mathematical Imaging and Vision, 28 (2007), 285. doi: 10.1007/s10851-007-0020-y. Google Scholar [35] J. Gilles, Multiscale texture separation,, Multiscale Modeling & Simulation, 10 (2012), 1409. doi: 10.1137/120881579. Google Scholar [36] B. Goldluecke, E. Strekalovskiy and D. Cremers, The natural vectorial total variation which arises from geometric measure theory,, SIAM Journal on Imaging Sciences, 5 (2012), 537. doi: 10.1137/110823766. Google Scholar [37] T. Goldstein and S. Osher, The split Bregman algorithm for L1 regularized problems,, SIAM Journal on Imaging Sciences, 2 (2009), 323. doi: 10.1137/080725891. Google Scholar [38] J. B. Greer and A. L. Bertozzi, Traveling wave solutions of fourth order PDEs for image processing,, SIAM Journal on Mathematical Analysis, 36 (2004), 38. doi: 10.1137/S0036141003427373. Google Scholar [39] M. Kass, A. Witkin and D. Terzopoulos, Snakes: Active contour models,, International Journal of Computer Vision, 1 (1988), 321. doi: 10.1007/BF00133570. Google Scholar [40] T. M. Le and L. A. Vese, Image decomposition using total variation and div(BMO),, Multiscale Modelling & Simulation, 4 (2005), 390. doi: 10.1137/040610052. Google Scholar [41] X. Liu, L. Huang and Z. Guo, Adaptive fourth-order partial differential equation filter for image denoising,, Applied Mathematics Letters, 24 (2011), 1282. doi: 10.1016/j.aml.2011.01.028. Google Scholar [42] Y. 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##### References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems,, Oxford University Press, (2000). Google Scholar [2] H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization,, Society for Industrial and Applied Mathematics (SIAM), (2006). Google Scholar [3] G. Aubert and J. F. Aujol, Modelling very oscillating signals. Application to image processing,, Applied Mathematics and Optimization, 51 (2005), 163. doi: 10.1007/s00245-004-0812-z. Google Scholar [4] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equation and Calculus of Variations,, Springer-Verlag, (2006). Google Scholar [5] J.-F. Aujol and A. Chambolle, Dual norms and image decomposition models,, International Journal of Computer Vision, 63 (2005), 85. doi: 10.1007/s11263-005-4948-3. Google Scholar [6] J.-F. Aujol and T. F. Chan, Combining geometrical and textured information to perform image classification,, Journal of Visual Communication and Image Representation, 17 (2006), 1004. doi: 10.1016/j.jvcir.2006.02.001. Google Scholar [7] J.-F. Aujol, G. Gilboa, T. Chan and S. Osher, Structure-texture image decomposition - modeling, algorithms and parameter selection,, International Journal of Computer Vision, 67 (2006), 111. doi: 10.1007/s11263-006-4331-z. Google Scholar [8] J.-F. Aujol and S. H. Kang, Color image decomposition and restoration,, Journal of Visual Communication and Image Representation, 17 (2006), 916. doi: 10.1016/j.jvcir.2005.02.001. Google Scholar [9] L. Bar, N. Sochen and N. Kiryati, Image deblurring in the presence of salt-and-pepper noise,, in Lecture Notes in Computer Science, 3459 (2005), 107. doi: 10.1007/11408031_10. Google Scholar [10] A. Beck and M. Teboulle, Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,, IEEE Transactions on Image Processing, 18 (2009), 2419. doi: 10.1109/TIP.2009.2028250. Google Scholar [11] P. Blomgren and T. F. Chan, Color TV: Total variation methods for restoration of vector valued images,, IEEE Transactions on Image Processing, 7 (1998), 304. doi: 10.1109/83.661180. Google Scholar [12] G. Boccignone, M. Ferraro and T. Caelli, Generalized spatio-chromatic diffusion,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 24 (2002), 1298. doi: 10.1109/TPAMI.2002.1039202. Google Scholar [13] X. Bresson and T. F. Chan, Fast dual minimization of the vectorial total variation norm and applications to color image processing,, Inverse Problem and Imaging, 2 (2008), 455. doi: 10.3934/ipi.2008.2.455. Google Scholar [14] X. Bresson, S. Esedoglu, P. Vandergheynst, J. Thiran and S. Osher, Fast global minimization of the active contour/snake model,, Journal of Mathematical Imaging and Vision, 28 (2007), 151. doi: 10.1007/s10851-007-0002-0. Google Scholar [15] A. Brook, R. Kimmel and N. Sochen, Variational restoration and edge detection for color images,, Journal of Mathematical Imaging and Vision, 18 (2003), 247. doi: 10.1023/A:1022895410391. Google Scholar [16] A. Buades, B. Coll and J. M. Morel, A review of image denoising methods, with a new one,, Multiscale Modeling and Simulation, 4 (2005), 490. doi: 10.1137/040616024. Google Scholar [17] V. Caselles, F. Catté, T. Coll and F. Dibos, A geometric model for active contours in image processing,, Numerische Mathematik, 66 (1993), 1. doi: 10.1007/BF01385685. Google Scholar [18] V. Caselles, B. Coll and J.-M. Morel, Geometry and color in natural images,, Journal of Mathematical Imaging and Vision, 16 (2002), 89. doi: 10.1023/A:1013943314097. Google Scholar [19] V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours,, International Journal of Computer Vision, (1995), 694. doi: 10.1109/ICCV.1995.466871. Google Scholar [20] A. Chambolle, An algorithm for total variation minimization and applications,, Journal of Mathematical Imaging and Vision, 20 (2004), 89. doi: 10.1023/B:JMIV.0000011321.19549.88. Google Scholar [21] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging,, Journal of Mathematical Imaging and Vision, 40 (2011), 120. doi: 10.1007/s10851-010-0251-1. Google Scholar [22] T. F. Chan and S. Esedoglu, Aspects of total variation regularized $L^{1}$ function approximation,, SIAM Journal on Applied Mathematics, 65 (2005), 1817. doi: 10.1137/040604297. Google Scholar [23] T. F. Chan, S. Esedoglu and M. Nikolova, Algorithms for finding global minimizers of image segmentation and denoising models,, SIAM Journal on Applied Mathematics, 66 (2006), 1632. doi: 10.1137/040615286. Google Scholar [24] T. F. Chan, G. Golub and P. Mulet, A nonlinear primal-dual method for total variation-based image restoration,, SIAM Journal on Scientific Computing, 20 (1999), 1964. doi: 10.1137/S1064827596299767. Google Scholar [25] T. F. Chan and J. Shen, Variational image inpainting,, Communications on Pure and Applied Mathematics, 58 (2005), 579. doi: 10.1002/cpa.20075. Google Scholar [26] K. Dabov, A. Foi, V. Katkovnik and K. Egiazarian, Color image denoising via sparse 3d collaborative filtering with grouping constraint in luminance-chrominance space,, in IEEE International Conference on Image Processing, 1 (2007), 313. doi: 10.1109/ICIP.2007.4378954. Google Scholar [27] J. Darbon, Total variation minimization with $L^{1}$ data fidelity as a contrast invariant filter,, in 4th Symposium on Image and Signal Processing and Analysis (ISPA), (2005), 221. doi: 10.1109/ISPA.2005.195413. Google Scholar [28] Y. Dong, M. Hintermuller and M. M. Rincon-Camacho, A multi-scale vectorial $L^\tau$-TV framework for color image restoration,, International Journal of Computer Vision, 92 (2011), 296. doi: 10.1007/s11263-010-0359-1. Google Scholar [29] S. Durand, J. Fadili and M. Nikolova, Multiplicative noise removal using $L^1$ fidelity on frame coefficients,, Journal of Mathematical Imaging and Vision, 36 (2010), 201. Google Scholar [30] V. Duval, J.-F. Aujol and Y. Gousseau, The TVL1 model: A geometrical point of view,, Multiscale Modelling & Simulation, 8 (2009), 154. doi: 10.1137/090757083. Google Scholar [31] V. Duval, J.-F. Aujol and L. Vese, Mathematical modelling of textures: Application to color image decomposition with a projected gradient algorithm,, Journal of Mathematical Imaging and Vision, 37 (2010), 232. doi: 10.1007/s10851-010-0203-9. Google Scholar [32] M. J. Ehrhardt and S. R. Arridge, Vector-valued image processing by parallel level sets,, IEEE Transactions on Image Processing, 23 (2014), 9. doi: 10.1109/TIP.2013.2277775. Google Scholar [33] G. Gilboa, A total variation spectral framework for scale and texture analysis,, Multiscale Modelling and Simulation, 7 (2014), 1937. doi: 10.1137/130930704. Google Scholar [34] J. Gilles, Noisy image decomposition: A new structure, texture and noise model based on local adaptivity,, Journal of Mathematical Imaging and Vision, 28 (2007), 285. doi: 10.1007/s10851-007-0020-y. Google Scholar [35] J. Gilles, Multiscale texture separation,, Multiscale Modeling & Simulation, 10 (2012), 1409. doi: 10.1137/120881579. Google Scholar [36] B. Goldluecke, E. Strekalovskiy and D. Cremers, The natural vectorial total variation which arises from geometric measure theory,, SIAM Journal on Imaging Sciences, 5 (2012), 537. doi: 10.1137/110823766. Google Scholar [37] T. Goldstein and S. Osher, The split Bregman algorithm for L1 regularized problems,, SIAM Journal on Imaging Sciences, 2 (2009), 323. doi: 10.1137/080725891. Google Scholar [38] J. B. Greer and A. L. Bertozzi, Traveling wave solutions of fourth order PDEs for image processing,, SIAM Journal on Mathematical Analysis, 36 (2004), 38. doi: 10.1137/S0036141003427373. Google Scholar [39] M. Kass, A. Witkin and D. Terzopoulos, Snakes: Active contour models,, International Journal of Computer Vision, 1 (1988), 321. doi: 10.1007/BF00133570. Google Scholar [40] T. M. Le and L. A. Vese, Image decomposition using total variation and div(BMO),, Multiscale Modelling & Simulation, 4 (2005), 390. doi: 10.1137/040610052. Google Scholar [41] X. Liu, L. Huang and Z. Guo, Adaptive fourth-order partial differential equation filter for image denoising,, Applied Mathematics Letters, 24 (2011), 1282. doi: 10.1016/j.aml.2011.01.028. Google Scholar [42] Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations,, American Mathematical Society, (2001). doi: 10.1090/ulect/022. Google Scholar [43] J. C. Moreno, V. B. S. Prasath, D. Vorotnikov, H. Proenca and K. Palaniappan, Adaptive Diffusion Constrained Total Variation Scheme with Application to Cartoon + Texture + Edge Image Decomposition,, Technical Report 1505.00866, (1505). Google Scholar [44] Y. Nesterov, Smooth minimization of non-smooth functions,, Mathematical Programming, 103 (2005), 127. doi: 10.1007/s10107-004-0552-5. Google Scholar [45] M. Nikolova, Minimizers of cost-function involving nonsmooth data-fidelity terms,, SIAM Journal on Numerical Analysis, 40 (2002), 965. doi: 10.1137/S0036142901389165. Google Scholar [46] M. Nikolova, A variational approach to remove outliers and impulse noise,, Journal of Mathematical Imaging and Vision, 20 (2004), 99. doi: 10.1023/B:JMIV.0000011920.58935.9c. Google Scholar [47] M. 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