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May  2016, 10(2): 305-325. doi: 10.3934/ipi.2016002

## Forward and backward filtering based on backward stochastic differential equations

 1 Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

Received  March 2013 Revised  January 2015 Published  May 2016

In this paper we explore the problem of reconstruction of blurred and noisy images. The idea presented here provides a new methodology based on advanced tools of stochastic analysis which can be successfully used to solve the inverse problem. In order to solve this problem we use backward stochastic differential equations. The reconstructed image is characterized by smoothing noisy pixels and at the same time enhancing and sharpening edges. Our experiments show that the new approach gives very good results and compares favourably with deterministic partial differential equation methods.
Citation: Dariusz Borkowski. Forward and backward filtering based on backward stochastic differential equations. Inverse Problems and Imaging, 2016, 10 (2) : 305-325. doi: 10.3934/ipi.2016002
##### References:
 [1] M. Aharon, M. Elad and A. Bruckstein, K-SVD: an algorithm for designing overcomplete dictionaries for sparse representation, IEEE Trans. Signal Process., 54 (2006), 4311-4322. doi: 10.1109/TSP.2006.881199. [2] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, Applied Mathematical Sciences, 147, Springer, New York, 2006. doi: 10.1007/978-0-387-44588-5. [3] D. Borkowski, Chromaticity denoising using solution to the Skorokhod problem, in Image Processing Based on Partial Differential Equations, Mathematics and Visualization, 2007, 149-161. doi: 10.1007/978-3-540-33267-1_9. [4] D. Borkowski, Smoothing, enhancing filters in terms of backward stochastic differential equations, in Computer Vision and Graphics, Lect. Notes Comput. Sci., 6374 (2010), 233-240. doi: 10.1007/978-3-642-15910-7_26. [5] D. Borkowski, Euler's approximations to image reconstruction, in Computer Vision and Graphics, Lect. Notes Comput. Sci., 7594 (2012), 30-37. doi: 10.1007/978-3-642-33564-8_4. [6] D. Borkowski and K. Jańczak-Borkowska, Application of backward stochastic differential equations to reconstruction of vector-valued images, in Computer Vision and Graphics, Lect. Notes Comput. Sci., 7594 (2012), 38-47. doi: 10.1007/978-3-642-33564-8_5. [7] A. Buades, B. Coll and J. M. Morel, A non local algorithm for image denoising, in Computer Vision and Pattern Recognition, 2005. CVPR 2005. IEEE Computer Society Conference on, 2 (2005), 60-65. doi: 10.1109/CVPR.2005.38. [8] 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. [9] 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. [10] F. Catte, P. L. Lions, J. M. Morel and T. Coll, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal., 29 (1992), 182-193. doi: 10.1137/0729012. [11] T. F. Chan and J. J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods, SIAM, 2005. doi: 10.1137/1.9780898717877. [12] K. Dabov, A. Foi, V. Katkovnik and K. Egiazarian, Image denoising by sparse 3D transform-domain collaborative filtering, IEEE Trans. Image Process., 16 (2007), 2080-2095. doi: 10.1109/TIP.2007.901238. [13] A. Danielyan, V. Katkovnik and K. Egiazarian, Bm3d frames and variational image deblurring, IEEE Trans. Image Process., 21 (2012), 1715-1728. doi: 10.1109/TIP.2011.2176954. [14] D. L. Donoho and I. M. Johnstone, Ideal spatial adaptation via wavelet shrinkage, Biometrika, 81 (1994), 425-455. doi: 10.1093/biomet/81.3.425. [15] D. Duffie and L. Epstein, Asset pricing with stochastic differential utility, Review of Financial Studies, 5 (1992), 411-436. doi: 10.1093/rfs/5.3.411. [16] D. Duffie and L. Epstein, Stochastic differential utility, Econometrica, 60 (1992), 353-394. doi: 10.2307/2951600. [17] A. Efros and T. Leung, Texture synthesis by non parametric sampling, in Computer Vision, 1999. The Proceedings of the Seventh IEEE International Conference on, 2 (1999), 1033-1038. doi: 10.1109/ICCV.1999.790383. [18] D. Fang, Z. Nanning and X. Jianru, Image smoothing and sharpening based on nonlinear diffusion equation, Signal Process., 88 (2008), 2850-2855. doi: 10.1016/j.sigpro.2008.05.008. [19] S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images, IEEE Trans. Pattern Anal. Mach. Intell., 6 (1984), 721-741. doi: 10.1109/TPAMI.1984.4767596. [20] P. Getreuer, Rudin-Osher-Fatemi total variation denoising using split Bregman, Image Processing On Line, 2 (2012), 74-95. doi: 10.5201/ipol.2012.g-tvd. [21] G. Gilboa, N. Sochen and Y. Y. Zeevi, Forward-and-backward diffusion processes for adaptive image enhancement and denoising, IEEE Trans. Image Process., 11 (2002), 689-703. doi: 10.1109/TIP.2002.800883. [22] T. Goldstein and S. Osher, The split Bregman method for L1 regularized problems, SIAM J. Imag. Sci., 2 (2009), 323-343. doi: 10.1137/080725891. [23] S. Hamadene and J. P. Lepeltie, Zero-sum stochastic differential games and backward equations, Syst. Control Lett., 24 (1995), 259-263. doi: 10.1016/0167-6911(94)00011-J. [24] N. El Karoui, S. Peng ang M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71. doi: 10.1111/1467-9965.00022. [25] V. Katkovnik, A. Danielyan and K. Egiazarian, Decoupled inverse and denoising for image deblurring: variational BM3D-frame technique, in Image Processing (ICIP), 2011 18th IEEE International Conference on, 2011, 3453-3456. doi: 10.1109/ICIP.2011.6116455. [26] M. Lebrun, A. Buades and J. M. Morel, Implementation of the non-local Bayes image denoising, Image Processing On Line, 3 (2013), 1-42. doi: 10.5201/ipol.2013.16. [27] J. Ma, P. Protter, J. San Martín and S. Torres, Numerical method for backward stochastic differential equations, Ann. Appl. Probab., 12 (2002), 302-316. doi: 10.1214/aoap/1015961165. [28] J. Mairal, M. Elad and G. Sapiro, Sparse representation for color image restoration, IEEE Trans. Image Process., 17 (2008), 53-69. doi: 10.1109/TIP.2007.911828. [29] É. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14 (1990), 55-61. doi: 10.1016/0167-6911(90)90082-6. [30] É. Pardoux, Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order, in: Stochastic Analysis and Related Topics VI, Progr. Probab. 42 (1998), 79-127. doi: 10.1007/978-1-4612-2022-0_2. [31] É. Pardoux and S. G. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, Lecture Notes in Control and Inform. Sci., 176 (1992), 200-217. doi: 10.1007/BFb0007334. [32] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 629-639. doi: 10.1109/34.56205. [33] R. Pettersson, Approximations for stochastic differential equations with reflecting convex boundaries, Stochastic Process. Appl., 59 (1995), 295-308. doi: 10.1016/0304-4149(95)00040-E. [34] W. H. Richardson, Bayesian-based iterative method of image restoration, J. Opt. Soc. Am., 62 (1972), 55-59. doi: 10.1364/JOSA.62.000055. [35] 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. [36] B. Smolka and K. N. Plataniotis, On the coupled forward and backward anisotropic diffusion scheme for color image enhancement, in Image and Video Retrieval, Lect. Notes Comput. Sci., 2383 (2002), 70-80. doi: 10.1007/3-540-45479-9_8. [37] L. Słomiński, Euler's approximations of solutions of SDEs with reflecting boundary, Stochastic Process. Appl., 94 (2001), 317-337. doi: 10.1016/S0304-4149(01)00087-4. [38] H. Tanaka, Stochastic differential equations with reflecting boundary condition in convex regions, Hiroshima Math. J., 9 (1979), 163-177. Available from: http://projecteuclid.org/euclid.hmj/1206135203. [39] J. Weickert, Theoretical foundations of anisotropic diffusion in image processing, in Theoretical Foundations of Computer Vision, Computing Supplement, 11 (1996), 221-236. doi: 10.1007/978-3-7091-6586-7_13. [40] J. Weickert, Coherence-enhancing diffusion filtering, Int. J. Comput. Vision, 31 (1999), 111-127. doi: 10.1023/A:1008009714131. [41] M. Welk, G. Gilboa and J. Weickert, Theoretical foundations for discrete forward-and-backward diffusion filtering, in Scale Space and Variational Methods in Computer Vision, Lect. Notes Comput. Sci., 5567 (2009), 527-538. doi: 10.1007/978-3-642-02256-2_44. [42] L. P. Yaroslavsky, Local adaptive image restoration and enhancement with the use of DFT and DCT in a running window, in Wavelet Applications in Signal and Image Processing IV, 2 (October 23, 1996), Proc. SPIE, 2825 (1996), 2-13. doi: 10.1117/12.255218. [43] L. P. Yaroslavsky, K. O. Egiazarian and J. T. Astola, Transform domain image restoration methods: review, comparison, and interpretation, in Nonlinear Image Processing and Pattern Analysis XII, 155 (May 8, 2001), Proc. SPIE, 4304 (2001), 155-169. doi: 10.1117/12.424970.

show all references

##### References:
 [1] M. Aharon, M. Elad and A. Bruckstein, K-SVD: an algorithm for designing overcomplete dictionaries for sparse representation, IEEE Trans. Signal Process., 54 (2006), 4311-4322. doi: 10.1109/TSP.2006.881199. [2] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, Applied Mathematical Sciences, 147, Springer, New York, 2006. doi: 10.1007/978-0-387-44588-5. [3] D. Borkowski, Chromaticity denoising using solution to the Skorokhod problem, in Image Processing Based on Partial Differential Equations, Mathematics and Visualization, 2007, 149-161. doi: 10.1007/978-3-540-33267-1_9. [4] D. Borkowski, Smoothing, enhancing filters in terms of backward stochastic differential equations, in Computer Vision and Graphics, Lect. Notes Comput. Sci., 6374 (2010), 233-240. doi: 10.1007/978-3-642-15910-7_26. [5] D. Borkowski, Euler's approximations to image reconstruction, in Computer Vision and Graphics, Lect. Notes Comput. Sci., 7594 (2012), 30-37. doi: 10.1007/978-3-642-33564-8_4. [6] D. Borkowski and K. Jańczak-Borkowska, Application of backward stochastic differential equations to reconstruction of vector-valued images, in Computer Vision and Graphics, Lect. Notes Comput. Sci., 7594 (2012), 38-47. doi: 10.1007/978-3-642-33564-8_5. [7] A. Buades, B. Coll and J. M. Morel, A non local algorithm for image denoising, in Computer Vision and Pattern Recognition, 2005. CVPR 2005. IEEE Computer Society Conference on, 2 (2005), 60-65. doi: 10.1109/CVPR.2005.38. [8] 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. [9] 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. [10] F. Catte, P. L. Lions, J. M. Morel and T. Coll, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal., 29 (1992), 182-193. doi: 10.1137/0729012. [11] T. F. Chan and J. J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods, SIAM, 2005. doi: 10.1137/1.9780898717877. [12] K. Dabov, A. Foi, V. Katkovnik and K. Egiazarian, Image denoising by sparse 3D transform-domain collaborative filtering, IEEE Trans. Image Process., 16 (2007), 2080-2095. doi: 10.1109/TIP.2007.901238. [13] A. Danielyan, V. Katkovnik and K. Egiazarian, Bm3d frames and variational image deblurring, IEEE Trans. Image Process., 21 (2012), 1715-1728. doi: 10.1109/TIP.2011.2176954. [14] D. L. Donoho and I. M. Johnstone, Ideal spatial adaptation via wavelet shrinkage, Biometrika, 81 (1994), 425-455. doi: 10.1093/biomet/81.3.425. [15] D. Duffie and L. Epstein, Asset pricing with stochastic differential utility, Review of Financial Studies, 5 (1992), 411-436. doi: 10.1093/rfs/5.3.411. [16] D. Duffie and L. Epstein, Stochastic differential utility, Econometrica, 60 (1992), 353-394. doi: 10.2307/2951600. [17] A. Efros and T. Leung, Texture synthesis by non parametric sampling, in Computer Vision, 1999. The Proceedings of the Seventh IEEE International Conference on, 2 (1999), 1033-1038. doi: 10.1109/ICCV.1999.790383. [18] D. Fang, Z. Nanning and X. Jianru, Image smoothing and sharpening based on nonlinear diffusion equation, Signal Process., 88 (2008), 2850-2855. doi: 10.1016/j.sigpro.2008.05.008. [19] S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images, IEEE Trans. Pattern Anal. Mach. Intell., 6 (1984), 721-741. doi: 10.1109/TPAMI.1984.4767596. [20] P. Getreuer, Rudin-Osher-Fatemi total variation denoising using split Bregman, Image Processing On Line, 2 (2012), 74-95. doi: 10.5201/ipol.2012.g-tvd. [21] G. Gilboa, N. Sochen and Y. Y. Zeevi, Forward-and-backward diffusion processes for adaptive image enhancement and denoising, IEEE Trans. Image Process., 11 (2002), 689-703. doi: 10.1109/TIP.2002.800883. [22] T. Goldstein and S. Osher, The split Bregman method for L1 regularized problems, SIAM J. Imag. Sci., 2 (2009), 323-343. doi: 10.1137/080725891. [23] S. Hamadene and J. P. Lepeltie, Zero-sum stochastic differential games and backward equations, Syst. Control Lett., 24 (1995), 259-263. doi: 10.1016/0167-6911(94)00011-J. [24] N. El Karoui, S. Peng ang M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71. doi: 10.1111/1467-9965.00022. [25] V. Katkovnik, A. Danielyan and K. Egiazarian, Decoupled inverse and denoising for image deblurring: variational BM3D-frame technique, in Image Processing (ICIP), 2011 18th IEEE International Conference on, 2011, 3453-3456. doi: 10.1109/ICIP.2011.6116455. [26] M. Lebrun, A. Buades and J. M. Morel, Implementation of the non-local Bayes image denoising, Image Processing On Line, 3 (2013), 1-42. doi: 10.5201/ipol.2013.16. [27] J. Ma, P. Protter, J. San Martín and S. Torres, Numerical method for backward stochastic differential equations, Ann. Appl. Probab., 12 (2002), 302-316. doi: 10.1214/aoap/1015961165. [28] J. Mairal, M. Elad and G. Sapiro, Sparse representation for color image restoration, IEEE Trans. Image Process., 17 (2008), 53-69. doi: 10.1109/TIP.2007.911828. [29] É. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14 (1990), 55-61. doi: 10.1016/0167-6911(90)90082-6. [30] É. Pardoux, Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order, in: Stochastic Analysis and Related Topics VI, Progr. Probab. 42 (1998), 79-127. doi: 10.1007/978-1-4612-2022-0_2. [31] É. Pardoux and S. G. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, Lecture Notes in Control and Inform. Sci., 176 (1992), 200-217. doi: 10.1007/BFb0007334. [32] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 629-639. doi: 10.1109/34.56205. [33] R. Pettersson, Approximations for stochastic differential equations with reflecting convex boundaries, Stochastic Process. Appl., 59 (1995), 295-308. doi: 10.1016/0304-4149(95)00040-E. [34] W. H. Richardson, Bayesian-based iterative method of image restoration, J. Opt. Soc. Am., 62 (1972), 55-59. doi: 10.1364/JOSA.62.000055. [35] 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. [36] B. Smolka and K. N. Plataniotis, On the coupled forward and backward anisotropic diffusion scheme for color image enhancement, in Image and Video Retrieval, Lect. Notes Comput. Sci., 2383 (2002), 70-80. doi: 10.1007/3-540-45479-9_8. [37] L. Słomiński, Euler's approximations of solutions of SDEs with reflecting boundary, Stochastic Process. Appl., 94 (2001), 317-337. doi: 10.1016/S0304-4149(01)00087-4. [38] H. Tanaka, Stochastic differential equations with reflecting boundary condition in convex regions, Hiroshima Math. J., 9 (1979), 163-177. Available from: http://projecteuclid.org/euclid.hmj/1206135203. [39] J. Weickert, Theoretical foundations of anisotropic diffusion in image processing, in Theoretical Foundations of Computer Vision, Computing Supplement, 11 (1996), 221-236. doi: 10.1007/978-3-7091-6586-7_13. [40] J. Weickert, Coherence-enhancing diffusion filtering, Int. J. Comput. Vision, 31 (1999), 111-127. doi: 10.1023/A:1008009714131. [41] M. Welk, G. Gilboa and J. Weickert, Theoretical foundations for discrete forward-and-backward diffusion filtering, in Scale Space and Variational Methods in Computer Vision, Lect. Notes Comput. Sci., 5567 (2009), 527-538. doi: 10.1007/978-3-642-02256-2_44. [42] L. P. Yaroslavsky, Local adaptive image restoration and enhancement with the use of DFT and DCT in a running window, in Wavelet Applications in Signal and Image Processing IV, 2 (October 23, 1996), Proc. SPIE, 2825 (1996), 2-13. doi: 10.1117/12.255218. [43] L. P. Yaroslavsky, K. O. Egiazarian and J. T. Astola, Transform domain image restoration methods: review, comparison, and interpretation, in Nonlinear Image Processing and Pattern Analysis XII, 155 (May 8, 2001), Proc. SPIE, 4304 (2001), 155-169. doi: 10.1117/12.424970.
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