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May  2015, 9(2): 415-430. doi: 10.3934/ipi.2015.9.415

## A new nonlocal variational setting for image processing

 1 College of Information Engineering, Zhejiang University of Technology, Hangzhou 310023, China 2 Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig 3 Mathematisches Institut, Albert-Ludwigs-Universitat Freiburg, Eckerstr.1, D-79103, Freiburg, Germany

Received  July 2011 Revised  June 2013 Published  March 2015

We introduce a new nonlocal variational scheme for image denoising. This scheme is motivated by, but different from the nonlocal means filter of Buades et al [9] and the nonlocal TV model proposed by Gilboa-Osher by using nonlocal operators. Our approach is based on general geometric considerations. Experiments show that the corresponding TV model yields denoising results that can compare favorably with those obtained by other methods.
Citation: Yan Jin, Jürgen Jost, Guofang Wang. A new nonlocal variational setting for image processing. Inverse Problems and Imaging, 2015, 9 (2) : 415-430. doi: 10.3934/ipi.2015.9.415
##### References:
 [1] G. Aubert and J.-F. Aujol, Modeling very oscillating signals, application to image processing, Appl. Math. Optim, 51 (2005), 163-182. doi: 10.1007/s00245-004-0812-z. [2] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing. Partial Differential Equations and the Calculus of Variations, Appl. Math. Sci., 147, Springer, Second Edition, 2006. [3] J.-F. Aujol, G. Aubert, L. Blanc-Féraud and A. Chambolle, Image decomposition into a bounded variation component and an oscillating component, J. Math. Imaging Vision, 22 (2005), 71-88. doi: 10.1007/s10851-005-4783-8. [4] J. F. Aujol and A. Chambolle, Dual norms and image decomposition models, International Journal of Computer Vision, 63 (2005), 85-104. [5] J.-F. Aujol and G. Gilboa, Constrained and SNR-based solutions for TV-Hilbert space image denoising, J. Math. Imaging Vis., 26 (2006), 217-237. doi: 10.1007/s10851-006-7801-6. [6] J.-F. Aujol, G. Gilboa, T. Chan and S. Osher, Structure-texture image decomposition-modeling, algorithms, and parameter selection, Int. J. Comput. Vis., 67 (2006), 111-136. [7] L. Bartholdi, T. Schick, N. Smale and S. Smale, Hodge theory on metric spaces, arXiv:0912.0284v1, 2009. [8] X. Bresson and T. F. Chan, Nonlocal Unsupervised Variational Image Segmentation Models, UCLA C.A.M. Report 08-67, 2008. [9] A. Buades, B. Coll and J.-M. Morel, A review of image denoising algorithms, with a new one, Multiscale Model. Simul., 4 (2005), 490-530. doi: 10.1137/040616024. [10] A. Buades, B. Coll and J.-M. Morel, Image Enhancement by Nonlocal Reverse Heat Equation, Technical Report 22, CMLA, 2006. [11] A. Buades, B. Coll and J.-M. Morel, Non-local means denoising, Image Processing On Line, 1 (2011). doi: 10.5201/ipol.2011.bcm_nlm. [12] A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vis., 20 (2004), 89-97. doi: 10.1023/B:JMIV.0000011321.19549.88. [13] A. Chambolle and P. L. Lions, Image recovery via total variational minimization and related problems, Numer. Math., 76 (1997), 167-188. doi: 10.1007/s002110050258. [14] T. F. Chan and S. Esedoglu, Aspects of total variation regularized L1 function approximation, SIAM J. Appl. Math., 65 (2005), 1817-1837. doi: 10.1137/040604297. [15] T. F. Chan, A. Marquina and P. Mulet, High-order total variation based image restoration, SIAM J. Sci. Comput., 22 (2000), 503-516. doi: 10.1137/S1064827598344169. [16] T. F. Chan and J. Shen, Image Processing and Analysis. Variational, PDE, Wavelet, and Stochastic Methods, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005. doi: 10.1137/1.9780898717877. [17] F. R. K. Chung, Spectral Graph Theory, CBMS Regional Conference Series in Mathematics, 92, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1997. [18] P. Getreuer and R.-O. Fatemi, Total Variation Denoising Using Split Bregman, Image Processing On Line, 2012. doi: 10.5201/ipol.2012.g-tvd. [19] G. Gilboa, J. Darbon, S. Osher and T. Chan, Nonlocal Convex Functionals for Image Regularization, Technical Report 06-57, UCLA CAM Report, 2006. [20] G. Gilboa and S. Osher, Nonlocal linear image regularization and supervised segmentation, Multiscale Model. Simul., 6 (2007), 595-630. doi: 10.1137/060669358. [21] G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028. doi: 10.1137/070698592. [22] G. Gilboa and S. Osher, Nonlocal evolutions for image regularization, Proceedings of SPIE, 6498, Computational Imaging V, 64980U, 2007. doi: 10.1117/12.714701. [23] G. Gilboa, N. Sochen and Y. Y. Zeevi., Estimation of optimal PDE-based denoising in the SNR sense, IEEE Trans. on Image Processing, 15 (2006), 2269-2280. doi: 10.1109/TIP.2006.875248. [24] Y. Jin, J. Jost and G. Wang, Nonlocal version of Osher-Solé-Vese model, J. Math. Imaging Vision, 44 (2012), 99-113. doi: 10.1007/s10851-011-0313-z. [25] Y. Jin, J. Jost and G. Wang, A new nonlocal $H^1$ model for image denoising, J. Math. Imaging Vision, 48 (2014), 93-105. doi: 10.1007/s10851-012-0395-2. [26] J. Jost, Equilibrium maps between metric spaces, Calc. Var., 2 (1994), 173-204. doi: 10.1007/BF01191341. [27] J. Jost, Riemannian Geometry and Geometric Analysis, Sixth edition, Universitext, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-21298-7. [28] M. Jung and L. A. Vese, Nonlocal variational image deblurring models in the presence of gaussian or impulse noise, Lecture Notes in Computer Science, 5567 (2009), 401-412. doi: 10.1007/978-3-642-02256-2_34. [29] S. Kindermann, S. Osher and P. W. Jones, Deblurring and denoising of images by nonlocal functionals, Multiscale Model. Simul., 4 (2005), 1091-1115. doi: 10.1137/050622249. [30] M. Lebrun, An analysis and implementation of the BM3D image denoising method, Image Processing On Line, 2012. doi: 10.5201/ipol.2012.l-bm3d. [31] M. Lebrun and A. Leclaire, An implementation and detailed analysis of the K-SVD image denoising algorithm, Image Processing On Line, 2 (2012), 96-133. doi: 10.5201/ipol.2012.llm-ksvd. [32] Y. Lou, X. Zhang, S. Osher and A. Bertozzi, Image recovery via nonlocal operators, Journal of Scientific Computing, 42 (2010), 185-197. doi: 10.1007/s10915-009-9320-2. [33] Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, University Lecture Series, 22, American Mathematical Society, Providence, RI, 2001. [34] S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, Using geometry and iterated refinement for inverse problems (1): Total variation based image restoration, preprint. [35] S. J. Osher and S. Esedoglu, Decomposition of images by the anisotropic Rudin-Osher-Fatemi model, Comm. Pure Appl. Math., 57 (2004), 1609-1626. doi: 10.1002/cpa.20045. [36] S. Osher and R. P. Fedkiw, Level set methods and dynamic implicit surfaces, Appl. Mech. Rev., 57 (2004), B15. doi: 10.1115/1.1760520. [37] S. Osher, A. Solé and L. Vese, Image decomposition and restoration using total variation minimizaiton and the $H^{-1}$ norm, Multiscale Model. Simul., 1 (2003), 349-370. doi: 10.1137/S1540345902416247. [38] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, PAMI, 12 (1990), 629-639. [39] G. Peyré, Image processing with nonlocal spectral bases, SIAM Multiscale Modeling and Simulation, 7 (2008), 703-730. doi: 10.1137/07068881X. [40] G. Peyré, S. Bougleux and L. Cohen, Nonlocal regularization of inverse problems, in ECCV 8: European Conference on Computer Vision, Springer, Berlin, 2008, p578. [41] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F. [42] G. Sapiro, Geometric Partial Differential Equations and Image Analysis, Cambridge University Press, 2009. doi: 10.1017/CBO9780511626319. [43] O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, Appl. Math. Sci., 167, Springer, New York, 2009. [44] A. Tikhonov and V. Arsenin, Solution of Ill-Posed Problems, Wiley, New York, 1977. [45] C. Tomasi and R. Manduchi, Bilateral filtering for gray and color images, in Proceedings of the 6th IEEE International Conference on Computer Vision (ICCV'98), 1998, 839-846. doi: 10.1109/ICCV.1998.710815. [46] L. Vese and S. Osher, Modeling textures with total variation minimization and oscillating patterns in image processing, J. Sci. Comput., 19 (2003), 553-572. doi: 10.1023/A:1025384832106. [47] J. Weickert, Anisotropic Diffusion in Image Processing, Teubner, Stuttgart. 1998. [48] L. P. Yaroslavsky, Digital Picture Processing. An Introduction, Springer Seriesin Information Sciences, 9, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-642-81929-2. [49] G. Yu and G. Sapiro, DCT image denoising: A simple and effective image denoising algorithm, Image Processing On Line, 2011. doi: 10.5201/ipol.2011.ys-dct. [50] J. Yuan, C. Schnörr and G. Steidl, Convex Hodge decomposition and regularization of image flows, J. Math. Imaging Vis., 33 (2009), 169-177. doi: 10.1007/s10851-008-0122-1. [51] X. Zhang, M. Burger, X. Bresson and S. Osher, Bregmanized nonlocal regularization for deconvolution and sparse reconstruction, SIAM J. Imaging Sci., 3 (2010), 253-276. doi: 10.1137/090746379. [52] X. Zhang and T. Chan, Wavelet inpainting by nonlocal total variation, Inverse Problems and Imaging, 4 (2010), 191-210. doi: 10.3934/ipi.2010.4.191. [53] D. Zhou and B. Schölkopf, A regularization framework for learning from graph data, in ICML Workshop on Stat. Relational Learning and Its Connections to Other Fields, 2004. [54] D. Zhou and B. Schölkopf, Regularization on discrete spaces, in Pattern Recognition, Proceedings of the 27th DAGM Symposium, Berlin, Germany, 2005, 361-369.

show all references

##### References:
 [1] G. Aubert and J.-F. Aujol, Modeling very oscillating signals, application to image processing, Appl. Math. Optim, 51 (2005), 163-182. doi: 10.1007/s00245-004-0812-z. [2] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing. Partial Differential Equations and the Calculus of Variations, Appl. Math. Sci., 147, Springer, Second Edition, 2006. [3] J.-F. Aujol, G. Aubert, L. Blanc-Féraud and A. Chambolle, Image decomposition into a bounded variation component and an oscillating component, J. Math. Imaging Vision, 22 (2005), 71-88. doi: 10.1007/s10851-005-4783-8. [4] J. F. Aujol and A. Chambolle, Dual norms and image decomposition models, International Journal of Computer Vision, 63 (2005), 85-104. [5] J.-F. Aujol and G. Gilboa, Constrained and SNR-based solutions for TV-Hilbert space image denoising, J. Math. Imaging Vis., 26 (2006), 217-237. doi: 10.1007/s10851-006-7801-6. [6] J.-F. Aujol, G. Gilboa, T. Chan and S. Osher, Structure-texture image decomposition-modeling, algorithms, and parameter selection, Int. J. Comput. Vis., 67 (2006), 111-136. [7] L. Bartholdi, T. Schick, N. Smale and S. Smale, Hodge theory on metric spaces, arXiv:0912.0284v1, 2009. [8] X. Bresson and T. F. Chan, Nonlocal Unsupervised Variational Image Segmentation Models, UCLA C.A.M. Report 08-67, 2008. [9] A. Buades, B. Coll and J.-M. Morel, A review of image denoising algorithms, with a new one, Multiscale Model. Simul., 4 (2005), 490-530. doi: 10.1137/040616024. [10] A. Buades, B. Coll and J.-M. Morel, Image Enhancement by Nonlocal Reverse Heat Equation, Technical Report 22, CMLA, 2006. [11] A. Buades, B. Coll and J.-M. Morel, Non-local means denoising, Image Processing On Line, 1 (2011). doi: 10.5201/ipol.2011.bcm_nlm. [12] A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vis., 20 (2004), 89-97. doi: 10.1023/B:JMIV.0000011321.19549.88. [13] A. Chambolle and P. L. Lions, Image recovery via total variational minimization and related problems, Numer. Math., 76 (1997), 167-188. doi: 10.1007/s002110050258. [14] T. F. Chan and S. Esedoglu, Aspects of total variation regularized L1 function approximation, SIAM J. Appl. Math., 65 (2005), 1817-1837. doi: 10.1137/040604297. [15] T. F. Chan, A. Marquina and P. Mulet, High-order total variation based image restoration, SIAM J. Sci. Comput., 22 (2000), 503-516. doi: 10.1137/S1064827598344169. [16] T. F. Chan and J. Shen, Image Processing and Analysis. Variational, PDE, Wavelet, and Stochastic Methods, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005. doi: 10.1137/1.9780898717877. [17] F. R. K. Chung, Spectral Graph Theory, CBMS Regional Conference Series in Mathematics, 92, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1997. [18] P. Getreuer and R.-O. Fatemi, Total Variation Denoising Using Split Bregman, Image Processing On Line, 2012. doi: 10.5201/ipol.2012.g-tvd. [19] G. Gilboa, J. Darbon, S. Osher and T. Chan, Nonlocal Convex Functionals for Image Regularization, Technical Report 06-57, UCLA CAM Report, 2006. [20] G. Gilboa and S. Osher, Nonlocal linear image regularization and supervised segmentation, Multiscale Model. Simul., 6 (2007), 595-630. doi: 10.1137/060669358. [21] G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028. doi: 10.1137/070698592. [22] G. Gilboa and S. Osher, Nonlocal evolutions for image regularization, Proceedings of SPIE, 6498, Computational Imaging V, 64980U, 2007. doi: 10.1117/12.714701. [23] G. Gilboa, N. Sochen and Y. Y. Zeevi., Estimation of optimal PDE-based denoising in the SNR sense, IEEE Trans. on Image Processing, 15 (2006), 2269-2280. doi: 10.1109/TIP.2006.875248. [24] Y. Jin, J. Jost and G. Wang, Nonlocal version of Osher-Solé-Vese model, J. Math. Imaging Vision, 44 (2012), 99-113. doi: 10.1007/s10851-011-0313-z. [25] Y. Jin, J. Jost and G. Wang, A new nonlocal $H^1$ model for image denoising, J. Math. Imaging Vision, 48 (2014), 93-105. doi: 10.1007/s10851-012-0395-2. [26] J. Jost, Equilibrium maps between metric spaces, Calc. Var., 2 (1994), 173-204. doi: 10.1007/BF01191341. [27] J. Jost, Riemannian Geometry and Geometric Analysis, Sixth edition, Universitext, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-21298-7. [28] M. Jung and L. A. Vese, Nonlocal variational image deblurring models in the presence of gaussian or impulse noise, Lecture Notes in Computer Science, 5567 (2009), 401-412. doi: 10.1007/978-3-642-02256-2_34. [29] S. Kindermann, S. Osher and P. W. Jones, Deblurring and denoising of images by nonlocal functionals, Multiscale Model. Simul., 4 (2005), 1091-1115. doi: 10.1137/050622249. [30] M. Lebrun, An analysis and implementation of the BM3D image denoising method, Image Processing On Line, 2012. doi: 10.5201/ipol.2012.l-bm3d. [31] M. Lebrun and A. Leclaire, An implementation and detailed analysis of the K-SVD image denoising algorithm, Image Processing On Line, 2 (2012), 96-133. doi: 10.5201/ipol.2012.llm-ksvd. [32] Y. Lou, X. Zhang, S. Osher and A. Bertozzi, Image recovery via nonlocal operators, Journal of Scientific Computing, 42 (2010), 185-197. doi: 10.1007/s10915-009-9320-2. [33] Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, University Lecture Series, 22, American Mathematical Society, Providence, RI, 2001. [34] S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, Using geometry and iterated refinement for inverse problems (1): Total variation based image restoration, preprint. [35] S. J. Osher and S. Esedoglu, Decomposition of images by the anisotropic Rudin-Osher-Fatemi model, Comm. Pure Appl. Math., 57 (2004), 1609-1626. doi: 10.1002/cpa.20045. [36] S. Osher and R. P. Fedkiw, Level set methods and dynamic implicit surfaces, Appl. Mech. Rev., 57 (2004), B15. doi: 10.1115/1.1760520. [37] S. Osher, A. Solé and L. Vese, Image decomposition and restoration using total variation minimizaiton and the $H^{-1}$ norm, Multiscale Model. Simul., 1 (2003), 349-370. doi: 10.1137/S1540345902416247. [38] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, PAMI, 12 (1990), 629-639. [39] G. Peyré, Image processing with nonlocal spectral bases, SIAM Multiscale Modeling and Simulation, 7 (2008), 703-730. doi: 10.1137/07068881X. [40] G. Peyré, S. Bougleux and L. Cohen, Nonlocal regularization of inverse problems, in ECCV 8: European Conference on Computer Vision, Springer, Berlin, 2008, p578. [41] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F. [42] G. Sapiro, Geometric Partial Differential Equations and Image Analysis, Cambridge University Press, 2009. doi: 10.1017/CBO9780511626319. [43] O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, Appl. Math. Sci., 167, Springer, New York, 2009. [44] A. Tikhonov and V. Arsenin, Solution of Ill-Posed Problems, Wiley, New York, 1977. [45] C. Tomasi and R. Manduchi, Bilateral filtering for gray and color images, in Proceedings of the 6th IEEE International Conference on Computer Vision (ICCV'98), 1998, 839-846. doi: 10.1109/ICCV.1998.710815. [46] L. Vese and S. Osher, Modeling textures with total variation minimization and oscillating patterns in image processing, J. Sci. Comput., 19 (2003), 553-572. doi: 10.1023/A:1025384832106. [47] J. Weickert, Anisotropic Diffusion in Image Processing, Teubner, Stuttgart. 1998. [48] L. P. Yaroslavsky, Digital Picture Processing. An Introduction, Springer Seriesin Information Sciences, 9, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-642-81929-2. [49] G. Yu and G. Sapiro, DCT image denoising: A simple and effective image denoising algorithm, Image Processing On Line, 2011. doi: 10.5201/ipol.2011.ys-dct. [50] J. Yuan, C. Schnörr and G. Steidl, Convex Hodge decomposition and regularization of image flows, J. Math. Imaging Vis., 33 (2009), 169-177. doi: 10.1007/s10851-008-0122-1. [51] X. Zhang, M. Burger, X. Bresson and S. Osher, Bregmanized nonlocal regularization for deconvolution and sparse reconstruction, SIAM J. Imaging Sci., 3 (2010), 253-276. doi: 10.1137/090746379. [52] X. Zhang and T. Chan, Wavelet inpainting by nonlocal total variation, Inverse Problems and Imaging, 4 (2010), 191-210. doi: 10.3934/ipi.2010.4.191. [53] D. Zhou and B. Schölkopf, A regularization framework for learning from graph data, in ICML Workshop on Stat. Relational Learning and Its Connections to Other Fields, 2004. [54] D. Zhou and B. Schölkopf, Regularization on discrete spaces, in Pattern Recognition, Proceedings of the 27th DAGM Symposium, Berlin, Germany, 2005, 361-369.
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