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 and Imaging, 2016, 10 (2) : 461-497. doi: 10.3934/ipi.2016008
References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, Cambridge, UK, 2000. [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), Philadelphia, PA, USA, 2006. [3] G. Aubert and J. F. Aujol, Modelling very oscillating signals. Application to image processing, Applied Mathematics and Optimization, 51 (2005), 163-182. doi: 10.1007/s00245-004-0812-z. [4] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equation and Calculus of Variations, Springer-Verlag, New York, USA, 2006. [5] J.-F. Aujol and A. Chambolle, Dual norms and image decomposition models, International Journal of Computer Vision, 63 (2005), 85-104. doi: 10.1007/s11263-005-4948-3. [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), 1004-1023. doi: 10.1016/j.jvcir.2006.02.001. [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), 111-136. doi: 10.1007/s11263-006-4331-z. [8] J.-F. Aujol and S. H. Kang, Color image decomposition and restoration, Journal of Visual Communication and Image Representation, 17 (2006), 916-928. doi: 10.1016/j.jvcir.2005.02.001. [9] L. Bar, N. Sochen and N. Kiryati, Image deblurring in the presence of salt-and-pepper noise, in Lecture Notes in Computer Science, Germany, 3459 (2005), 107-118. doi: 10.1007/11408031_10. [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), 2419-2434. doi: 10.1109/TIP.2009.2028250. [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), 304-309. doi: 10.1109/83.661180. [12] G. Boccignone, M. Ferraro and T. Caelli, Generalized spatio-chromatic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 24 (2002), 1298-1309. doi: 10.1109/TPAMI.2002.1039202. [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), 455-484. doi: 10.3934/ipi.2008.2.455. [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), 151-167. doi: 10.1007/s10851-007-0002-0. [15] A. Brook, R. Kimmel and N. Sochen, Variational restoration and edge detection for color images, Journal of Mathematical Imaging and Vision, 18 (2003), 247-268. doi: 10.1023/A:1022895410391. [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), 490-530. doi: 10.1137/040616024. [17] V. Caselles, F. Catté, T. Coll and F. Dibos, A geometric model for active contours in image processing, Numerische Mathematik, 66 (1993), 1-31. doi: 10.1007/BF01385685. [18] V. Caselles, B. Coll and J.-M. Morel, Geometry and color in natural images, Journal of Mathematical Imaging and Vision, 16 (2002), 89-105. doi: 10.1023/A:1013943314097. [19] V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours, International Journal of Computer Vision, (1995), 694-699. doi: 10.1109/ICCV.1995.466871. [20] A. Chambolle, An algorithm for total variation minimization and applications, Journal of Mathematical Imaging and Vision, 20 (2004), 89-97. doi: 10.1023/B:JMIV.0000011321.19549.88. [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), 120-145. doi: 10.1007/s10851-010-0251-1. [22] T. F. Chan and S. Esedoglu, Aspects of total variation regularized $L^{1}$ function approximation, SIAM Journal on Applied Mathematics, 65 (2005), 1817-1837. doi: 10.1137/040604297. [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), 1632-1648. doi: 10.1137/040615286. [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), 1964-1967. doi: 10.1137/S1064827596299767. [25] T. F. Chan and J. Shen, Variational image inpainting, Communications on Pure and Applied Mathematics, 58 (2005), 579-619. doi: 10.1002/cpa.20075. [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, San Antonio, TX, USA, 1 (2007), 313-316. doi: 10.1109/ICIP.2007.4378954. [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), Zagreb, Croatia, 2005, 221-226. doi: 10.1109/ISPA.2005.195413. [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), 296-307. doi: 10.1007/s11263-010-0359-1. [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), 201-226. [30] V. Duval, J.-F. Aujol and Y. Gousseau, The TVL1 model: A geometrical point of view, Multiscale Modelling & Simulation, 8 (2009), 154-189. doi: 10.1137/090757083. [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), 232-248. doi: 10.1007/s10851-010-0203-9. [32] M. J. Ehrhardt and S. R. Arridge, Vector-valued image processing by parallel level sets, IEEE Transactions on Image Processing, 23 (2014), 9-18. doi: 10.1109/TIP.2013.2277775. [33] G. Gilboa, A total variation spectral framework for scale and texture analysis, Multiscale Modelling and Simulation, 7 (2014), 1937-1961. doi: 10.1137/130930704. [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), 285-295. doi: 10.1007/s10851-007-0020-y. [35] J. Gilles, Multiscale texture separation, Multiscale Modeling & Simulation, 10 (2012), 1409-1427. doi: 10.1137/120881579. [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), 537-564. doi: 10.1137/110823766. [37] T. Goldstein and S. Osher, The split Bregman algorithm for L1 regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891. [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), 38-68. doi: 10.1137/S0036141003427373. [39] M. Kass, A. Witkin and D. Terzopoulos, Snakes: Active contour models, International Journal of Computer Vision, 1 (1988), 321-331. doi: 10.1007/BF00133570. [40] T. M. Le and L. A. Vese, Image decomposition using total variation and div(BMO), Multiscale Modelling & Simulation, 4 (2005), 390-423. doi: 10.1137/040610052. [41] X. Liu, L. Huang and Z. Guo, Adaptive fourth-order partial differential equation filter for image denoising, Applied Mathematics Letters, 24 (2011), 1282-1288. doi: 10.1016/j.aml.2011.01.028. [42] Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, American Mathematical Society, Boston, MA, USA, 2001, The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures, Vol. 22 of University Lecture Series. doi: 10.1090/ulect/022. [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, ArXiv, 2015, \urlprefixhttp://arxiv.org/abs/1505.00866. [44] Y. Nesterov, Smooth minimization of non-smooth functions, Mathematical Programming, 103 (2005), 127-152. doi: 10.1007/s10107-004-0552-5. [45] M. Nikolova, Minimizers of cost-function involving nonsmooth data-fidelity terms, SIAM Journal on Numerical Analysis, 40 (2002), 965-994. doi: 10.1137/S0036142901389165. [46] M. Nikolova, A variational approach to remove outliers and impulse noise, Journal of Mathematical Imaging and Vision, 20 (2004), 99-120, Special Issue on Mathematics and Image Analysis. doi: 10.1023/B:JMIV.0000011920.58935.9c. [47] M. Nikolova, Weakly constrained minimization: application to the estimation of images and signals involving constant regions, Journal of Mathematical Imaging and Vision, 21 (2004), 155-175. doi: 10.1023/B:JMIV.0000035180.40477.bd. [48] S. Ono and I. Yamada, Decorrelated vectorial total variation, in IEEE Conference on Computer Vision and Pattern Recognition, Columbus, OH, USA, 2014, 4090-4097. doi: 10.1109/CVPR.2014.521. [49] S. Osher and O. Scherzer, G-norm properties of bounded variation regularization, Communications in Mathematical Sciences, 2 (2004), 237-254. doi: 10.4310/CMS.2004.v2.n2.a6. [50] N. Paragios, O. Mellina-Gottardo and V. Ramesh, Gradient vector flow fast geodesic active contours, Computer Vision, 2001. ICCV 2001. Proceedings. Eighth IEEE International Conference on, 1 (2001), 67-73. doi: 10.1109/ICCV.2001.937500. [51] V. B. S. Prasath, Weighted Laplacian differences based multispectral anisotropic diffusion, in IEEE International Geoscience and Remote Sensing Symposium (IGARSS), Vancouver BC, Canada, 2011, 4042-4045. doi: 10.1109/IGARSS.2011.6050119. [52] V. B. S. Prasath, A well-posed multiscale regularization scheme for digital image denoising, International Journal of Applied Mathematics and Computer Science, 21 (2011), 769-777. doi: 10.2478/v10006-011-0061-7. [53] V. B. S. Prasath, J. C. Moreno and K. Palaniappan, Color image denoising by chromatic edges based vector valued diffusion, Technical Report 1304.5587, ArXiv, 2013, URL http://arxiv.org/abs/1304.5587. [54] V. B. S. Prasath and A. Singh, Multichannel image restoration using combined channel information and robust M-estimator approach, International Journal Tomography and Statistics, 12 (2010), 9-22. [55] V. B. S. Prasath and A. Singh, Multispectral image denoising by well-posed anisotropic diffusion scheme with channel coupling, International Journal of Remote Sensing, 31 (2010), 2091-2099. doi: 10.1080/01431160903260965. [56] V. B. S. Prasath and A. Singh, Well-posed inhomogeneous nonlinear diffusion scheme for digital image denoising, Journal of Applied Mathematics, 2010 (2010), 14pp, Article {ID} 763847. [57] V. B. S. Prasath and A. Singh, An adaptive anisotropic diffusion scheme for image restoration and selective smoothing, International Journal of Image and Graphics, 12 (2012), 1250003, 18 pp. doi: 10.1142/S0219467812500039. [58] L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F. [59] G. Sapiro and D. L. Ringach, Anisotropic diffusion of multivalued images with application to color filtering, IEEE Transactions on Image Processing, 5 (1996), 1582-1586. doi: 10.1109/83.541429. [60] O. Scherzer, W. Yin and S. Osher, Slope and $G$-set characterization of set-valued functions and applications to non-differentiable optimization problems, Communications in Mathematical Sciences, 3 (2005), 479-492. doi: 10.4310/CMS.2005.v3.n4.a1. [61] Y. Shi, L.-L. Wang and X.-C. Tai, Geometry of total variation regularized $L^p$-model, Journal of Computational and Applied Mathematics, 236 (2012), 2223-2234. doi: 10.1016/j.cam.2011.09.043. [62] D. Strong, Adaptive Total Variation Minimizing Image Restoration, PhD thesis, UCLA Mathematics Department, USA, 1997. [63] D. Strong and T. Chan, Edge-preserving and scale-dependent properties of total variation regularization, Inverse Problems, 19 (2003), 165-187. doi: 10.1088/0266-5611/19/6/059. [64] D. M. Strong and T. F. Chan, Spatially and Scale Adaptive Total Variation Based Regularization and Anisotropic Diffusion in Image Processing, Technical Report 96-46, UCLA CAM, 1996. [65] E. Tadmor, S. Nezzar and L. Vese, A multiscale image representation using hierarchical ($BV,L^{2}$) decomposition, Multiscale Modeling and Simulation, 2 (2004), 554-579. doi: 10.1137/030600448. [66] B. Tang, G. Sapiro and V. Caselles, Color image enhancement via chromaticity diffusion, IEEE Transactions on Pattern Analysis Machine Intelligence, 10 (2001), 701-707. doi: 10.1109/83.918563. [67] L. Tang and C. He, Multiscale texture extraction with hierarchical $(BV,G_p,L^{2})$, Journal of Mathematical Imaging and Vision, 45 (2013), 148-163. doi: 10.1007/s10851-012-0351-1. [68] D. Tschumperle and R. Deriche, Vector-valued image regularization with PDE's: A common framework for different applications, Computer Vision and Pattern Recognition, 2003. Proceedings. 2003 IEEE Computer Society Conference on, 1 (2003), 651-656. doi: 10.1109/CVPR.2003.1211415. [69] L. A. Vese and S. J. Osher, Modeling textures with total variation minimization and oscillating patterns in image processing, Journal of Scientific Computing, 19 (2003), 553-572. doi: 10.1023/A:1025384832106. [70] L. A. Vese and S. J. Osher, Color texture modeling and color image decomposition in variational PDE approach, in Proceedings of the Eighth International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC '06), New York, USA, 2006, 103-110. doi: 10.1109/SYNASC.2006.24. [71] Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: from error visibility to structural similarity, IEEE Transactions on Image Processing, 13 (2004), 600-612, URL http://ece.uwaterloo.ca/~z70wang/research/ssim/. doi: 10.1109/TIP.2003.819861. [72] W. Yin, D. Goldfarb and S. Osher, Image cartoon-texture decomposition and feature selection using total variation regularized $L^1$ functional, in Variational, Geometric, and Level Set Methods in Computer Vision, 3752 (2005), 73-84. doi: 10.1007/11567646_7. [73] W. Yin, D. Goldfarb and S. Osher, The total variation regularized $L^1$ model for multiscale decomposition, Multiscale Modeling & Simulation, 6 (2007), 190-211. doi: 10.1137/060663027.

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References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, Cambridge, UK, 2000. [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), Philadelphia, PA, USA, 2006. [3] G. Aubert and J. F. Aujol, Modelling very oscillating signals. Application to image processing, Applied Mathematics and Optimization, 51 (2005), 163-182. doi: 10.1007/s00245-004-0812-z. [4] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equation and Calculus of Variations, Springer-Verlag, New York, USA, 2006. [5] J.-F. Aujol and A. Chambolle, Dual norms and image decomposition models, International Journal of Computer Vision, 63 (2005), 85-104. doi: 10.1007/s11263-005-4948-3. [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), 1004-1023. doi: 10.1016/j.jvcir.2006.02.001. [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), 111-136. doi: 10.1007/s11263-006-4331-z. [8] J.-F. Aujol and S. H. Kang, Color image decomposition and restoration, Journal of Visual Communication and Image Representation, 17 (2006), 916-928. doi: 10.1016/j.jvcir.2005.02.001. [9] L. Bar, N. Sochen and N. Kiryati, Image deblurring in the presence of salt-and-pepper noise, in Lecture Notes in Computer Science, Germany, 3459 (2005), 107-118. doi: 10.1007/11408031_10. [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), 2419-2434. doi: 10.1109/TIP.2009.2028250. [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), 304-309. doi: 10.1109/83.661180. [12] G. Boccignone, M. Ferraro and T. Caelli, Generalized spatio-chromatic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 24 (2002), 1298-1309. doi: 10.1109/TPAMI.2002.1039202. [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), 455-484. doi: 10.3934/ipi.2008.2.455. [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), 151-167. doi: 10.1007/s10851-007-0002-0. [15] A. Brook, R. Kimmel and N. Sochen, Variational restoration and edge detection for color images, Journal of Mathematical Imaging and Vision, 18 (2003), 247-268. doi: 10.1023/A:1022895410391. [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), 490-530. doi: 10.1137/040616024. [17] V. Caselles, F. Catté, T. Coll and F. Dibos, A geometric model for active contours in image processing, Numerische Mathematik, 66 (1993), 1-31. doi: 10.1007/BF01385685. [18] V. Caselles, B. Coll and J.-M. Morel, Geometry and color in natural images, Journal of Mathematical Imaging and Vision, 16 (2002), 89-105. doi: 10.1023/A:1013943314097. [19] V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours, International Journal of Computer Vision, (1995), 694-699. doi: 10.1109/ICCV.1995.466871. [20] A. Chambolle, An algorithm for total variation minimization and applications, Journal of Mathematical Imaging and Vision, 20 (2004), 89-97. doi: 10.1023/B:JMIV.0000011321.19549.88. [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), 120-145. doi: 10.1007/s10851-010-0251-1. [22] T. F. Chan and S. Esedoglu, Aspects of total variation regularized $L^{1}$ function approximation, SIAM Journal on Applied Mathematics, 65 (2005), 1817-1837. doi: 10.1137/040604297. [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), 1632-1648. doi: 10.1137/040615286. [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), 1964-1967. doi: 10.1137/S1064827596299767. [25] T. F. Chan and J. Shen, Variational image inpainting, Communications on Pure and Applied Mathematics, 58 (2005), 579-619. doi: 10.1002/cpa.20075. [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, San Antonio, TX, USA, 1 (2007), 313-316. doi: 10.1109/ICIP.2007.4378954. [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), Zagreb, Croatia, 2005, 221-226. doi: 10.1109/ISPA.2005.195413. [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), 296-307. doi: 10.1007/s11263-010-0359-1. [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), 201-226. [30] V. Duval, J.-F. Aujol and Y. Gousseau, The TVL1 model: A geometrical point of view, Multiscale Modelling & Simulation, 8 (2009), 154-189. doi: 10.1137/090757083. [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), 232-248. doi: 10.1007/s10851-010-0203-9. [32] M. J. Ehrhardt and S. R. Arridge, Vector-valued image processing by parallel level sets, IEEE Transactions on Image Processing, 23 (2014), 9-18. doi: 10.1109/TIP.2013.2277775. [33] G. Gilboa, A total variation spectral framework for scale and texture analysis, Multiscale Modelling and Simulation, 7 (2014), 1937-1961. doi: 10.1137/130930704. [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), 285-295. doi: 10.1007/s10851-007-0020-y. [35] J. Gilles, Multiscale texture separation, Multiscale Modeling & Simulation, 10 (2012), 1409-1427. doi: 10.1137/120881579. [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), 537-564. doi: 10.1137/110823766. [37] T. Goldstein and S. Osher, The split Bregman algorithm for L1 regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891. [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), 38-68. doi: 10.1137/S0036141003427373. [39] M. Kass, A. Witkin and D. Terzopoulos, Snakes: Active contour models, International Journal of Computer Vision, 1 (1988), 321-331. doi: 10.1007/BF00133570. [40] T. M. Le and L. A. Vese, Image decomposition using total variation and div(BMO), Multiscale Modelling & Simulation, 4 (2005), 390-423. doi: 10.1137/040610052. [41] X. Liu, L. Huang and Z. Guo, Adaptive fourth-order partial differential equation filter for image denoising, Applied Mathematics Letters, 24 (2011), 1282-1288. doi: 10.1016/j.aml.2011.01.028. [42] Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, American Mathematical Society, Boston, MA, USA, 2001, The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures, Vol. 22 of University Lecture Series. doi: 10.1090/ulect/022. [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, ArXiv, 2015, \urlprefixhttp://arxiv.org/abs/1505.00866. [44] Y. Nesterov, Smooth minimization of non-smooth functions, Mathematical Programming, 103 (2005), 127-152. doi: 10.1007/s10107-004-0552-5. [45] M. Nikolova, Minimizers of cost-function involving nonsmooth data-fidelity terms, SIAM Journal on Numerical Analysis, 40 (2002), 965-994. doi: 10.1137/S0036142901389165. [46] M. Nikolova, A variational approach to remove outliers and impulse noise, Journal of Mathematical Imaging and Vision, 20 (2004), 99-120, Special Issue on Mathematics and Image Analysis. doi: 10.1023/B:JMIV.0000011920.58935.9c. [47] M. Nikolova, Weakly constrained minimization: application to the estimation of images and signals involving constant regions, Journal of Mathematical Imaging and Vision, 21 (2004), 155-175. doi: 10.1023/B:JMIV.0000035180.40477.bd. [48] S. Ono and I. Yamada, Decorrelated vectorial total variation, in IEEE Conference on Computer Vision and Pattern Recognition, Columbus, OH, USA, 2014, 4090-4097. doi: 10.1109/CVPR.2014.521. [49] S. Osher and O. Scherzer, G-norm properties of bounded variation regularization, Communications in Mathematical Sciences, 2 (2004), 237-254. doi: 10.4310/CMS.2004.v2.n2.a6. [50] N. Paragios, O. Mellina-Gottardo and V. Ramesh, Gradient vector flow fast geodesic active contours, Computer Vision, 2001. ICCV 2001. Proceedings. Eighth IEEE International Conference on, 1 (2001), 67-73. doi: 10.1109/ICCV.2001.937500. [51] V. B. S. Prasath, Weighted Laplacian differences based multispectral anisotropic diffusion, in IEEE International Geoscience and Remote Sensing Symposium (IGARSS), Vancouver BC, Canada, 2011, 4042-4045. doi: 10.1109/IGARSS.2011.6050119. [52] V. B. S. Prasath, A well-posed multiscale regularization scheme for digital image denoising, International Journal of Applied Mathematics and Computer Science, 21 (2011), 769-777. doi: 10.2478/v10006-011-0061-7. [53] V. B. S. Prasath, J. C. Moreno and K. Palaniappan, Color image denoising by chromatic edges based vector valued diffusion, Technical Report 1304.5587, ArXiv, 2013, URL http://arxiv.org/abs/1304.5587. [54] V. B. S. Prasath and A. Singh, Multichannel image restoration using combined channel information and robust M-estimator approach, International Journal Tomography and Statistics, 12 (2010), 9-22. [55] V. B. S. Prasath and A. Singh, Multispectral image denoising by well-posed anisotropic diffusion scheme with channel coupling, International Journal of Remote Sensing, 31 (2010), 2091-2099. doi: 10.1080/01431160903260965. [56] V. B. S. Prasath and A. Singh, Well-posed inhomogeneous nonlinear diffusion scheme for digital image denoising, Journal of Applied Mathematics, 2010 (2010), 14pp, Article {ID} 763847. [57] V. B. S. Prasath and A. Singh, An adaptive anisotropic diffusion scheme for image restoration and selective smoothing, International Journal of Image and Graphics, 12 (2012), 1250003, 18 pp. doi: 10.1142/S0219467812500039. [58] L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F. [59] G. Sapiro and D. L. Ringach, Anisotropic diffusion of multivalued images with application to color filtering, IEEE Transactions on Image Processing, 5 (1996), 1582-1586. doi: 10.1109/83.541429. [60] O. Scherzer, W. Yin and S. Osher, Slope and $G$-set characterization of set-valued functions and applications to non-differentiable optimization problems, Communications in Mathematical Sciences, 3 (2005), 479-492. doi: 10.4310/CMS.2005.v3.n4.a1. [61] Y. Shi, L.-L. Wang and X.-C. Tai, Geometry of total variation regularized $L^p$-model, Journal of Computational and Applied Mathematics, 236 (2012), 2223-2234. doi: 10.1016/j.cam.2011.09.043. [62] D. Strong, Adaptive Total Variation Minimizing Image Restoration, PhD thesis, UCLA Mathematics Department, USA, 1997. [63] D. Strong and T. Chan, Edge-preserving and scale-dependent properties of total variation regularization, Inverse Problems, 19 (2003), 165-187. doi: 10.1088/0266-5611/19/6/059. [64] D. M. Strong and T. F. Chan, Spatially and Scale Adaptive Total Variation Based Regularization and Anisotropic Diffusion in Image Processing, Technical Report 96-46, UCLA CAM, 1996. [65] E. Tadmor, S. Nezzar and L. Vese, A multiscale image representation using hierarchical ($BV,L^{2}$) decomposition, Multiscale Modeling and Simulation, 2 (2004), 554-579. doi: 10.1137/030600448. [66] B. Tang, G. Sapiro and V. Caselles, Color image enhancement via chromaticity diffusion, IEEE Transactions on Pattern Analysis Machine Intelligence, 10 (2001), 701-707. doi: 10.1109/83.918563. [67] L. Tang and C. He, Multiscale texture extraction with hierarchical $(BV,G_p,L^{2})$, Journal of Mathematical Imaging and Vision, 45 (2013), 148-163. doi: 10.1007/s10851-012-0351-1. [68] D. Tschumperle and R. Deriche, Vector-valued image regularization with PDE's: A common framework for different applications, Computer Vision and Pattern Recognition, 2003. Proceedings. 2003 IEEE Computer Society Conference on, 1 (2003), 651-656. doi: 10.1109/CVPR.2003.1211415. [69] L. A. Vese and S. J. Osher, Modeling textures with total variation minimization and oscillating patterns in image processing, Journal of Scientific Computing, 19 (2003), 553-572. doi: 10.1023/A:1025384832106. [70] L. A. Vese and S. J. Osher, Color texture modeling and color image decomposition in variational PDE approach, in Proceedings of the Eighth International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC '06), New York, USA, 2006, 103-110. doi: 10.1109/SYNASC.2006.24. [71] Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: from error visibility to structural similarity, IEEE Transactions on Image Processing, 13 (2004), 600-612, URL http://ece.uwaterloo.ca/~z70wang/research/ssim/. doi: 10.1109/TIP.2003.819861. [72] W. Yin, D. Goldfarb and S. Osher, Image cartoon-texture decomposition and feature selection using total variation regularized $L^1$ functional, in Variational, Geometric, and Level Set Methods in Computer Vision, 3752 (2005), 73-84. doi: 10.1007/11567646_7. [73] W. Yin, D. Goldfarb and S. Osher, The total variation regularized $L^1$ model for multiscale decomposition, Multiscale Modeling & Simulation, 6 (2007), 190-211. doi: 10.1137/060663027.
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