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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 & 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. doi: 10.1007/s00245-004-0812-z. Google Scholar [2] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing. Partial Differential Equations and the Calculus of Variations,, Appl. Math. Sci., (2006). Google Scholar [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. doi: 10.1007/s10851-005-4783-8. Google Scholar [4] J. F. Aujol and A. Chambolle, Dual norms and image decomposition models,, International Journal of Computer Vision, 63 (2005), 85. Google Scholar [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. doi: 10.1007/s10851-006-7801-6. Google Scholar [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. Google Scholar [7] L. Bartholdi, T. Schick, N. Smale and S. Smale, Hodge theory on metric spaces,, , (2009). Google Scholar [8] X. Bresson and T. F. Chan, Nonlocal Unsupervised Variational Image Segmentation Models,, UCLA C.A.M. Report 08-67, (2008), 08. Google Scholar [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. doi: 10.1137/040616024. Google Scholar [10] A. Buades, B. Coll and J.-M. Morel, Image Enhancement by Nonlocal Reverse Heat Equation,, Technical Report 22, (2006). Google Scholar [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. Google Scholar [12] A. Chambolle, An algorithm for total variation minimization and applications,, J. Math. Imaging Vis., 20 (2004), 89. doi: 10.1023/B:JMIV.0000011321.19549.88. Google Scholar [13] A. Chambolle and P. L. Lions, Image recovery via total variational minimization and related problems,, Numer. Math., 76 (1997), 167. doi: 10.1007/s002110050258. Google Scholar [14] T. F. Chan and S. Esedoglu, Aspects of total variation regularized L1 function approximation,, SIAM J. Appl. Math., 65 (2005), 1817. doi: 10.1137/040604297. Google Scholar [15] T. F. Chan, A. Marquina and P. Mulet, High-order total variation based image restoration,, SIAM J. Sci. Comput., 22 (2000), 503. doi: 10.1137/S1064827598344169. Google Scholar [16] T. F. Chan and J. Shen, Image Processing and Analysis. Variational, PDE, Wavelet, and Stochastic Methods,, Society for Industrial and Applied Mathematics (SIAM), (2005). doi: 10.1137/1.9780898717877. Google Scholar [17] F. R. K. Chung, Spectral Graph Theory,, CBMS Regional Conference Series in Mathematics, (1997). Google Scholar [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. Google Scholar [19] G. Gilboa, J. Darbon, S. Osher and T. Chan, Nonlocal Convex Functionals for Image Regularization,, Technical Report 06-57, (2006), 06. Google Scholar [20] G. Gilboa and S. Osher, Nonlocal linear image regularization and supervised segmentation,, Multiscale Model. Simul., 6 (2007), 595. doi: 10.1137/060669358. Google Scholar [21] G. Gilboa and S. Osher, Nonlocal operators with applications to image processing,, Multiscale Model. Simul., 7 (2008), 1005. doi: 10.1137/070698592. Google Scholar [22] G. Gilboa and S. Osher, Nonlocal evolutions for image regularization,, Proceedings of SPIE, (6498). doi: 10.1117/12.714701. Google Scholar [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. doi: 10.1109/TIP.2006.875248. Google Scholar [24] Y. Jin, J. Jost and G. Wang, Nonlocal version of Osher-Solé-Vese model,, J. Math. Imaging Vision, 44 (2012), 99. doi: 10.1007/s10851-011-0313-z. Google Scholar [25] Y. Jin, J. Jost and G. Wang, A new nonlocal $H^1$ model for image denoising,, J. Math. Imaging Vision, 48 (2014), 93. doi: 10.1007/s10851-012-0395-2. Google Scholar [26] J. Jost, Equilibrium maps between metric spaces,, Calc. Var., 2 (1994), 173. doi: 10.1007/BF01191341. Google Scholar [27] J. Jost, Riemannian Geometry and Geometric Analysis,, Sixth edition, (2011). doi: 10.1007/978-3-642-21298-7. Google Scholar [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. doi: 10.1007/978-3-642-02256-2_34. Google Scholar [29] S. Kindermann, S. Osher and P. W. Jones, Deblurring and denoising of images by nonlocal functionals,, Multiscale Model. Simul., 4 (2005), 1091. doi: 10.1137/050622249. Google Scholar [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. Google Scholar [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. doi: 10.5201/ipol.2012.llm-ksvd. Google Scholar [32] Y. Lou, X. Zhang, S. Osher and A. Bertozzi, Image recovery via nonlocal operators,, Journal of Scientific Computing, 42 (2010), 185. doi: 10.1007/s10915-009-9320-2. Google Scholar [33] Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations,, University Lecture Series, (2001). Google Scholar [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., (). Google Scholar [35] S. J. Osher and S. Esedoglu, Decomposition of images by the anisotropic Rudin-Osher-Fatemi model,, Comm. Pure Appl. Math., 57 (2004), 1609. doi: 10.1002/cpa.20045. Google Scholar [36] S. Osher and R. P. Fedkiw, Level set methods and dynamic implicit surfaces,, Appl. Mech. Rev., 57 (2004). doi: 10.1115/1.1760520. Google Scholar [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. doi: 10.1137/S1540345902416247. Google Scholar [38] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion,, PAMI, 12 (1990), 629. Google Scholar [39] G. Peyré, Image processing with nonlocal spectral bases,, SIAM Multiscale Modeling and Simulation, 7 (2008), 703. doi: 10.1137/07068881X. Google Scholar [40] G. Peyré, S. Bougleux and L. Cohen, Nonlocal regularization of inverse problems,, in ECCV 8: European Conference on Computer Vision, (2008). Google Scholar [41] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Phys. D, 60 (1992), 259. doi: 10.1016/0167-2789(92)90242-F. Google Scholar [42] G. Sapiro, Geometric Partial Differential Equations and Image Analysis,, Cambridge University Press, (2009). doi: 10.1017/CBO9780511626319. Google Scholar [43] O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging,, Appl. Math. Sci., (2009). Google Scholar [44] A. Tikhonov and V. Arsenin, Solution of Ill-Posed Problems,, Wiley, (1977). Google Scholar [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. doi: 10.1109/ICCV.1998.710815. Google Scholar [46] L. Vese and S. Osher, Modeling textures with total variation minimization and oscillating patterns in image processing,, J. Sci. Comput., 19 (2003), 553. doi: 10.1023/A:1025384832106. Google Scholar [47] J. Weickert, Anisotropic Diffusion in Image Processing,, Teubner, (1998). Google Scholar [48] L. P. Yaroslavsky, Digital Picture Processing. An Introduction,, Springer Seriesin Information Sciences, (1985). doi: 10.1007/978-3-642-81929-2. Google Scholar [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. Google Scholar [50] J. Yuan, C. Schnörr and G. Steidl, Convex Hodge decomposition and regularization of image flows,, J. Math. Imaging Vis., 33 (2009), 169. doi: 10.1007/s10851-008-0122-1. Google Scholar [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. doi: 10.1137/090746379. Google Scholar [52] X. Zhang and T. Chan, Wavelet inpainting by nonlocal total variation,, Inverse Problems and Imaging, 4 (2010), 191. doi: 10.3934/ipi.2010.4.191. Google Scholar [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). Google Scholar [54] D. Zhou and B. Schölkopf, Regularization on discrete spaces,, in Pattern Recognition, (2005), 361. Google Scholar

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. doi: 10.1007/s00245-004-0812-z. Google Scholar [2] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing. Partial Differential Equations and the Calculus of Variations,, Appl. Math. Sci., (2006). Google Scholar [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. doi: 10.1007/s10851-005-4783-8. Google Scholar [4] J. F. Aujol and A. Chambolle, Dual norms and image decomposition models,, International Journal of Computer Vision, 63 (2005), 85. Google Scholar [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. doi: 10.1007/s10851-006-7801-6. Google Scholar [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. Google Scholar [7] L. Bartholdi, T. Schick, N. Smale and S. Smale, Hodge theory on metric spaces,, , (2009). Google Scholar [8] X. Bresson and T. F. Chan, Nonlocal Unsupervised Variational Image Segmentation Models,, UCLA C.A.M. Report 08-67, (2008), 08. Google Scholar [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. doi: 10.1137/040616024. Google Scholar [10] A. Buades, B. Coll and J.-M. Morel, Image Enhancement by Nonlocal Reverse Heat Equation,, Technical Report 22, (2006). Google Scholar [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. Google Scholar [12] A. Chambolle, An algorithm for total variation minimization and applications,, J. Math. Imaging Vis., 20 (2004), 89. doi: 10.1023/B:JMIV.0000011321.19549.88. Google Scholar [13] A. Chambolle and P. L. Lions, Image recovery via total variational minimization and related problems,, Numer. Math., 76 (1997), 167. doi: 10.1007/s002110050258. Google Scholar [14] T. F. Chan and S. Esedoglu, Aspects of total variation regularized L1 function approximation,, SIAM J. Appl. Math., 65 (2005), 1817. doi: 10.1137/040604297. Google Scholar [15] T. F. Chan, A. Marquina and P. Mulet, High-order total variation based image restoration,, SIAM J. Sci. Comput., 22 (2000), 503. doi: 10.1137/S1064827598344169. Google Scholar [16] T. F. Chan and J. Shen, Image Processing and Analysis. Variational, PDE, Wavelet, and Stochastic Methods,, Society for Industrial and Applied Mathematics (SIAM), (2005). doi: 10.1137/1.9780898717877. Google Scholar [17] F. R. K. Chung, Spectral Graph Theory,, CBMS Regional Conference Series in Mathematics, (1997). Google Scholar [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. Google Scholar [19] G. Gilboa, J. Darbon, S. Osher and T. Chan, Nonlocal Convex Functionals for Image Regularization,, Technical Report 06-57, (2006), 06. Google Scholar [20] G. Gilboa and S. Osher, Nonlocal linear image regularization and supervised segmentation,, Multiscale Model. Simul., 6 (2007), 595. doi: 10.1137/060669358. Google Scholar [21] G. Gilboa and S. Osher, Nonlocal operators with applications to image processing,, Multiscale Model. Simul., 7 (2008), 1005. doi: 10.1137/070698592. Google Scholar [22] G. Gilboa and S. Osher, Nonlocal evolutions for image regularization,, Proceedings of SPIE, (6498). doi: 10.1117/12.714701. Google Scholar [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. doi: 10.1109/TIP.2006.875248. Google Scholar [24] Y. Jin, J. Jost and G. Wang, Nonlocal version of Osher-Solé-Vese model,, J. Math. Imaging Vision, 44 (2012), 99. doi: 10.1007/s10851-011-0313-z. Google Scholar [25] Y. Jin, J. Jost and G. Wang, A new nonlocal $H^1$ model for image denoising,, J. Math. Imaging Vision, 48 (2014), 93. doi: 10.1007/s10851-012-0395-2. Google Scholar [26] J. Jost, Equilibrium maps between metric spaces,, Calc. Var., 2 (1994), 173. doi: 10.1007/BF01191341. Google Scholar [27] J. Jost, Riemannian Geometry and Geometric Analysis,, Sixth edition, (2011). doi: 10.1007/978-3-642-21298-7. Google Scholar [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. doi: 10.1007/978-3-642-02256-2_34. Google Scholar [29] S. Kindermann, S. Osher and P. W. Jones, Deblurring and denoising of images by nonlocal functionals,, Multiscale Model. Simul., 4 (2005), 1091. doi: 10.1137/050622249. Google Scholar [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. Google Scholar [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. doi: 10.5201/ipol.2012.llm-ksvd. Google Scholar [32] Y. Lou, X. Zhang, S. Osher and A. Bertozzi, Image recovery via nonlocal operators,, Journal of Scientific Computing, 42 (2010), 185. doi: 10.1007/s10915-009-9320-2. Google Scholar [33] Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations,, University Lecture Series, (2001). Google Scholar [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., (). Google Scholar [35] S. J. Osher and S. Esedoglu, Decomposition of images by the anisotropic Rudin-Osher-Fatemi model,, Comm. Pure Appl. Math., 57 (2004), 1609. doi: 10.1002/cpa.20045. Google Scholar [36] S. Osher and R. P. Fedkiw, Level set methods and dynamic implicit surfaces,, Appl. Mech. Rev., 57 (2004). doi: 10.1115/1.1760520. Google Scholar [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. doi: 10.1137/S1540345902416247. Google Scholar [38] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion,, PAMI, 12 (1990), 629. Google Scholar [39] G. Peyré, Image processing with nonlocal spectral bases,, SIAM Multiscale Modeling and Simulation, 7 (2008), 703. doi: 10.1137/07068881X. Google Scholar [40] G. Peyré, S. Bougleux and L. Cohen, Nonlocal regularization of inverse problems,, in ECCV 8: European Conference on Computer Vision, (2008). Google Scholar [41] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Phys. D, 60 (1992), 259. doi: 10.1016/0167-2789(92)90242-F. Google Scholar [42] G. Sapiro, Geometric Partial Differential Equations and Image Analysis,, Cambridge University Press, (2009). doi: 10.1017/CBO9780511626319. Google Scholar [43] O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging,, Appl. Math. Sci., (2009). Google Scholar [44] A. Tikhonov and V. Arsenin, Solution of Ill-Posed Problems,, Wiley, (1977). Google Scholar [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. doi: 10.1109/ICCV.1998.710815. Google Scholar [46] L. Vese and S. Osher, Modeling textures with total variation minimization and oscillating patterns in image processing,, J. Sci. Comput., 19 (2003), 553. doi: 10.1023/A:1025384832106. Google Scholar [47] J. Weickert, Anisotropic Diffusion in Image Processing,, Teubner, (1998). Google Scholar [48] L. P. Yaroslavsky, Digital Picture Processing. An Introduction,, Springer Seriesin Information Sciences, (1985). doi: 10.1007/978-3-642-81929-2. Google Scholar [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. Google Scholar [50] J. Yuan, C. Schnörr and G. Steidl, Convex Hodge decomposition and regularization of image flows,, J. Math. Imaging Vis., 33 (2009), 169. doi: 10.1007/s10851-008-0122-1. Google Scholar [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. doi: 10.1137/090746379. Google Scholar [52] X. Zhang and T. Chan, Wavelet inpainting by nonlocal total variation,, Inverse Problems and Imaging, 4 (2010), 191. doi: 10.3934/ipi.2010.4.191. Google Scholar [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). Google Scholar [54] D. Zhou and B. Schölkopf, Regularization on discrete spaces,, in Pattern Recognition, (2005), 361. Google Scholar
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