# American Institute of Mathematical Sciences

November  2013, 7(4): 1183-1214. doi: 10.3934/ipi.2013.7.1183

## Image denoising: Learning the noise model via nonsmooth PDE-constrained optimization

 1 Research Center on Mathematical Modelling (MODEMAT), Escuela Politécnica Nacional de Quito, Ladrón de Guevara E11-253, Quito, 170109,, Ecuador 2 Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, United Kingdom

Received  July 2012 Revised  September 2013 Published  November 2013

We propose a nonsmooth PDE-constrained optimization approach for the determination of the correct noise model in total variation (TV) image denoising. An optimization problem for the determination of the weights corresponding to different types of noise distributions is stated and existence of an optimal solution is proved. A tailored regularization approach for the approximation of the optimal parameter values is proposed thereafter and its consistency studied. Additionally, the differentiability of the solution operator is proved and an optimality system characterizing the optimal solutions of each regularized problem is derived. The optimal parameter values are numerically computed by using a quasi-Newton method, together with semismooth Newton type algorithms for the solution of the TV-subproblems.
Citation: Juan Carlos De los Reyes, Carola-Bibiane Schönlieb. Image denoising: Learning the noise model via nonsmooth PDE-constrained optimization. Inverse Problems & Imaging, 2013, 7 (4) : 1183-1214. doi: 10.3934/ipi.2013.7.1183
##### References:
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Novaga and T. Pock, An Introduction to Total Variation for Image Analysis, Theoretical Foundations and Numerical Methods for Sparse Recovery (ed. M. Fornasier,), Radon Series on Computational and Applied Mathematics, De Gruyter Verlag, 263-340, 2010. doi: 10.1515/9783110226157.263.  Google Scholar [17] T. F. Chan and S. Esedoglu, Aspects of total variation regularized $L^1$ function approximation, SIAM J. Appl. Math., 65 (2005), 1817-1837. doi: 10.1137/040604297.  Google Scholar [18] T. F. Chan and J. J. Shen, Image Processing and Analysis - Variational, PDE, Wavelet, and Stochastic Methods, SIAM, (2005). doi: 10.1137/1.9780898717877.  Google Scholar [19] G. Dal Maso, An introduction to Gamma-convergence, Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar [20] J. C. 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Zeevi, Estimation of optimal PDE-based denoising in the SNR sense, Image Processing, IEEE Transactions on, 15 (2006), 2269-2280. doi: 10.1109/TIP.2006.875248.  Google Scholar [34] M. Hintermüller, Y. Dong and M. M. Rincon-Camacho, Automated regularization parameter selection in multi-scale total variation models for image restoration, Journal of Mathematical Imaging and Vision, 40 (2011), 82-104. doi: 10.1007/s10851-010-0248-9.  Google Scholar [35] M. Hintermüller and K. Kunisch, Total bounded variation regularization as a bilaterally constrained optimization problem,, SIAM Journal on Applied Mathematics, 64 (): 1311.  doi: 10.1137/S0036139903422784.  Google Scholar [36] M. Hintermüller and A. Langer, Subspace Correction Methods for a Class of Non-Smooth and Non-Additive Convex Variational Problems in Image Processing, Accepted by SIAM J. Imaging Sciences, 2013, 34 pp. http://math.uni-graz.at/mobis/publications/SFB-Report-2012-021.pdf. Google Scholar [37] M. Hintermüller and G. Stadler, An infeasible primal-dual algorithm for total bounded variation-based inf-convolution-type image restoration, SIAM Journal on Scientific Computing, 28 (2006), 1-23. doi: 10.1137/040613263.  Google Scholar [38] J. M. Chung, M. Chung and D. P. O'Leary, Designing optimal spectral filters for inverse problems, SIAM Journal on Scientific Computing, 33 (2011), 3132-3152. doi: 10.1137/100812938.  Google Scholar [39] T. Le, R. Chartrand and T. J. Asaki, A variational approach to reconstructing images corrupted by Poisson noise, J. Math. Imaging Vision, 27 (2007), 257-263. doi: 10.1007/s10851-007-0652-y.  Google Scholar [40] Risheng Liu, Zhouchen Lin, Wei Zhang and Zhixun Su, Learning PDEs for Image Restoration via Optimal Control, ECCV 2010. doi: 10.1007/978-3-642-15549-9_9.  Google Scholar [41] K. Kunisch and T. Pock, A bilevel optimization approach for parameter learning in variational models, SIAM Journal on Imaging Sciences, 6 (2013), 938-983. doi: 10.1137/120882706.  Google Scholar [42] B. Manz et al., L. F. Gladden and P. B. Warren, Flow and dispersion in porous media: Lattice-Boltzmann and NMR studies, AIChE Journal, 45 (1999), 1845-1854. Google Scholar [43] M. D. Mantle et al., A. J. Sederman and L. F. Gladden, Single- and two-phase flow in fixed-bed reactors: MRI flow visualisation and lattice-Boltzmann simulations, Chemical Engineering Science, 56 (2001), 523-529. Google Scholar [44] V. A. Morozov, Regularization Methods for Ill-posed Problems, CRC Press, Boca Raton, 1993.  Google Scholar [45] M. Nikolova, A variational approach to remove outliers and impulse noise, J. Math. Imaging Vision, 20 (2004), 99-120. doi: 10.1023/B:JMIV.0000011920.58935.9c.  Google Scholar [46] G. Peyré and J. Fadili, Learning analysis sparsity priors, Proc. of Sampta'11, (2011). Google Scholar [47] A. Sawatzky, C. Brune, J. Müller and M. Burger, Total variation processing of images with poisson statistics, Proceedings of the 13th International Conference on Computer Analysis of Images and Patterns, 5702 (2009), 533-540. doi: 10.1007/978-3-642-03767-2_65.  Google Scholar [48] F. R. Schmidt and D. Cremers, A closed-form solution for image sequence segmentation with dynamical shape priors, in Pattern Recognition (Proc. DAGM), 2009. doi: 10.1007/978-3-642-03798-6_4.  Google Scholar [49] D. Strong, J.-F. Aujol and T. Chan, Scale recognition, regularization parameter selection, and Meyers G norm in total variation regularization, SIAM Journal on Multiscale Modeling and Simulation, 5 (2006), 273-303. doi: 10.1137/040621624.  Google Scholar [50] S. P. Sullivan et al., F. M. Sani, M. L. Johns and L. F. Gladden, Simulation of packed bed reactors using lattice Boltzmann methods, Chemical Engineering Science, 60 (2005), 3405-3418. Google Scholar [51] M. F. Tappen, Utilizing variational optimization to learn Markov random fields, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR2007), 2007. doi: 10.1109/CVPR.2007.383037.  Google Scholar [52] M. F. Tappen, C. Liu, E. H. Adelson and W. T. Freeman, Learning Gaussian conditional random fields for low-level vision, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR2007), 2007. doi: 10.1109/CVPR.2007.382979.  Google Scholar [53] I. Tosic, I. Jovanovic, P. Frossard, M. Vetterli and N. Duric, Ultrasound tomography with learned dictionaries, IEEE International Conference on Acoustics, Speech, and Signal Processing, Dallas, Texas, International Conference on Acoustics Speech and Signal Processing ICASSP, 2010. doi: 10.1109/ICASSP.2010.5495211.  Google Scholar [54] L. Vese, A study in the BV space of a denoising-deblurring variational problem, Appl. Math. Optim., 44 (2001), 131-161. doi: 10.1007/s00245-001-0017-7.  Google Scholar [55] C. R. Vogel, Computational Methods for Inverse Problems, SIAM, vol. 10, 2002. doi: 10.1137/1.9780898717570.  Google Scholar [56] C. R. Vogel and M. E. Oman, Iterative methods for total variation denoising, SIAM J. Sci. Comput., 17 (1996), 227-238, Special Issue on Iterative Methods in Numerical Linear Algebra (Breckenridge, CO, 1994). doi: 10.1137/0917016.  Google Scholar [57] A. M. Yip and F. Park, Solution Dynamics, Causality, and Critical Behavior of the Regularization Parameter in Total Variation Denoising Problems, CAM reports 03-59, 2003. Google Scholar

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##### References:
 [1] R. Acar and C. R. Vogel, Analysis of bounded variation penalty methods for ill-posed problems, Inverse Problems, 10 (1994), 1217-1229. doi: 10.1088/0266-5611/10/6/003.  Google Scholar [2] A. Almansa, C. Ballester, V. Caselles and G. Haro, A TV based restoration model with local constraints, J. Sci. Comput., 34 (2008), 209-236. doi: 10.1007/s10915-007-9160-x.  Google Scholar [3] G. Aubert and J.-F. Aujol, A variational approach to remove multiplicative noise, SIAM Journal on Applied Mathematics, 68 (2008), 925-946. doi: 10.1137/060671814.  Google Scholar [4] G. Aubert and L. Vese, A variational method in image recovery, SIAM J. Numer. Anal., 34 (1997), 1948-1979. doi: 10.1137/S003614299529230X.  Google Scholar [5] J.-F. Aujol and G. Gilboa, Constrained and SNR-based solutions for TV-Hilbert space image denoising, Journal of Mathematical Imaging and Vision, 26 (2006), 217-237. doi: 10.1007/s10851-006-7801-6.  Google Scholar [6] F. Baus, M. Nikolova and G. Steidl, Smooth objectives composed of asymptotically affine data-fidelity and regularization: Bounds for the minimizers and parameter choice, Journal of Mathematical Imaging and Vision, Published Online (2013). Google Scholar [7] M. Bertalmio, V. Caselles, B. Rougé and A. Solé, TV based image restoration with local constraints, Journal of Scientific Computing, 19 (2003), 95-122. doi: 10.1023/A:1025391506181.  Google Scholar [8] A. Bovik, Handbook of Image and Video Processing, Academic Press, 2000. Google Scholar [9] G. Bouchitté, A. Braides and G. Buttazzo, Relaxation results for some free discontinuity problems, J. Reine Angew Math., 458 (1995), 1-18. doi: 10.1515/crll.1995.458.1.  Google Scholar [10] G. Bouchitté and G. Buttazzo, New lower semicontinuity results for nonconvex functionals defined on measures, Nonlinear Anal. TMA, 15 (1990), 679-692. doi: 10.1016/0362-546X(90)90007-4.  Google Scholar [11] G. Bouchitté and G. Buttazzo, Relaxation for a class of nonconvex functionals defined on measures, Ann. Inst. H. Poincaré, 10 (1993), 345-361.  Google Scholar [12] J. F. Cai, R. H. Chan and M. Nikolova, Two-phase approach for deblurring images corrupted by impulse plus Gaussian noise, Inverse Problems and Imaging, 2 (2008), 187-204. doi: 10.3934/ipi.2008.2.187.  Google Scholar [13] E. Casas and L. Fernández, Distributed control of systems governed by a general class of quasilinear elliptic equations, J. Differential Equations, 104 (1993), 20-47. doi: 10.1006/jdeq.1993.1062.  Google Scholar [14] A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vision, 20 (2004), 89-97. doi: 10.1023/B:JMIV.0000011321.19549.88.  Google Scholar [15] A. Chambolle and P.-L. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167-188. doi: 10.1007/s002110050258.  Google Scholar [16] A. Chambolle, V. Caselles, D. Cremers, M. Novaga and T. Pock, An Introduction to Total Variation for Image Analysis, Theoretical Foundations and Numerical Methods for Sparse Recovery (ed. M. Fornasier,), Radon Series on Computational and Applied Mathematics, De Gruyter Verlag, 263-340, 2010. doi: 10.1515/9783110226157.263.  Google Scholar [17] T. F. Chan and S. Esedoglu, Aspects of total variation regularized $L^1$ function approximation, SIAM J. Appl. Math., 65 (2005), 1817-1837. doi: 10.1137/040604297.  Google Scholar [18] T. F. Chan and J. J. Shen, Image Processing and Analysis - Variational, PDE, Wavelet, and Stochastic Methods, SIAM, (2005). doi: 10.1137/1.9780898717877.  Google Scholar [19] G. Dal Maso, An introduction to Gamma-convergence, Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar [20] J. C. De los Reyes, Optimal control of a class of variational inequalities of the second kind, SIAM Journal on Control and Optimization, 49 (2011), 1629-1658. doi: 10.1137/090764438.  Google Scholar [21] J. C. De los Reyes, Optimization of mixed variational inequalities arising in flow of viscoplastic materials, Computational Optimization and Applications, 52 (2012), 757-784. doi: 10.1007/s10589-011-9435-x.  Google Scholar [22] F. Demengel and R. Temam, Convex functions of a measure and applications, Indiana Univ. Math. J., 33 (1984), 673-709. doi: 10.1512/iumj.1984.33.33036.  Google Scholar [23] D. C. Dobson and C. R. Vogel, Convergence of an iterative method for total variation denoising, SIAM J. Numer. Anal., 34 (1997), 1779-1791. doi: 10.1137/S003614299528701X.  Google Scholar [24] J. Domke, Generic methods for optimization-based modeling, in International Conference on Artificial Intelligence and Statistics, 318-326, 2012. Google Scholar [25] J. Domke, Learning graphical model parameters with approximate marginal inference, Published Online, PAMI, (2013). doi: 10.1109/TPAMI.2013.31.  Google Scholar [26] V. Duval, J.-F. Aujol and Y. Gousseau, The TVL1 model: A geometric point of view, SIAM Journal on Multiscale Modeling and Simulation, 8 (2009), 154-189. doi: 10.1137/090757083.  Google Scholar [27] M. Fornasier, V. Naumova and S. V. Pereverzyev, Parameter Choice Strategies for Multi-Penalty Regularization, preprint, July 2013. http://www-m15.ma.tum.de/foswiki/pub/M15/Allgemeines/PublicationsEN/parameter_choice_strategies.pdf. doi: 10.1088/0266-5611/29/7/075002.  Google Scholar [28] K. Frick, P. Marnitz and A. Munk, Statistical multiresolution dantzig estimation in imaging: Fundamental concepts and algorithmic framework, Electron. J. Stat., 6 (2012), 231-268. doi: 10.1214/12-EJS671.  Google Scholar [29] K. Frick, P. Marnitz and A. Munk, Shape constrained regularization by statistical multiresolution for inverse problems, Inverse Problems, 28 (2012), 065006. doi: 10.1088/0266-5611/28/6/065006.  Google Scholar [30] K. Frick, P. Marnitz and A. Munk, Statistical multiresolution estimation for variational imaging: With an application in poisson-biophotonics, Journal of Mathematical Imaging and Vision, 46 (2013), 370-387. doi: 10.1007/s10851-012-0368-5.  Google Scholar [31] P. Getreuer, M. Tong and L. A. Vese, A variational model for the restoration of MR images corrupted by blur and Rician noise, in Advances in Visual Computing, Springer Berlin-Heidelberg-Chicago, 686-698, 2012. doi: 10.1007/978-3-642-24028-7_63.  Google Scholar [32] G. Gilboa, N. Sochen and Y. Y. Zeevi, Texture preserving variational denoising using an adaptive fidelity term, in Proc. VLsM, (Vol. 3), 2003. Google Scholar [33] G. Gilboa, N. Sochen and Y. Y. Zeevi, Estimation of optimal PDE-based denoising in the SNR sense, Image Processing, IEEE Transactions on, 15 (2006), 2269-2280. doi: 10.1109/TIP.2006.875248.  Google Scholar [34] M. Hintermüller, Y. Dong and M. M. Rincon-Camacho, Automated regularization parameter selection in multi-scale total variation models for image restoration, Journal of Mathematical Imaging and Vision, 40 (2011), 82-104. doi: 10.1007/s10851-010-0248-9.  Google Scholar [35] M. Hintermüller and K. Kunisch, Total bounded variation regularization as a bilaterally constrained optimization problem,, SIAM Journal on Applied Mathematics, 64 (): 1311.  doi: 10.1137/S0036139903422784.  Google Scholar [36] M. Hintermüller and A. Langer, Subspace Correction Methods for a Class of Non-Smooth and Non-Additive Convex Variational Problems in Image Processing, Accepted by SIAM J. Imaging Sciences, 2013, 34 pp. http://math.uni-graz.at/mobis/publications/SFB-Report-2012-021.pdf. Google Scholar [37] M. Hintermüller and G. Stadler, An infeasible primal-dual algorithm for total bounded variation-based inf-convolution-type image restoration, SIAM Journal on Scientific Computing, 28 (2006), 1-23. doi: 10.1137/040613263.  Google Scholar [38] J. M. Chung, M. Chung and D. P. O'Leary, Designing optimal spectral filters for inverse problems, SIAM Journal on Scientific Computing, 33 (2011), 3132-3152. doi: 10.1137/100812938.  Google Scholar [39] T. Le, R. Chartrand and T. J. Asaki, A variational approach to reconstructing images corrupted by Poisson noise, J. Math. Imaging Vision, 27 (2007), 257-263. doi: 10.1007/s10851-007-0652-y.  Google Scholar [40] Risheng Liu, Zhouchen Lin, Wei Zhang and Zhixun Su, Learning PDEs for Image Restoration via Optimal Control, ECCV 2010. doi: 10.1007/978-3-642-15549-9_9.  Google Scholar [41] K. Kunisch and T. Pock, A bilevel optimization approach for parameter learning in variational models, SIAM Journal on Imaging Sciences, 6 (2013), 938-983. doi: 10.1137/120882706.  Google Scholar [42] B. Manz et al., L. F. Gladden and P. B. Warren, Flow and dispersion in porous media: Lattice-Boltzmann and NMR studies, AIChE Journal, 45 (1999), 1845-1854. Google Scholar [43] M. D. Mantle et al., A. J. Sederman and L. F. Gladden, Single- and two-phase flow in fixed-bed reactors: MRI flow visualisation and lattice-Boltzmann simulations, Chemical Engineering Science, 56 (2001), 523-529. Google Scholar [44] V. A. Morozov, Regularization Methods for Ill-posed Problems, CRC Press, Boca Raton, 1993.  Google Scholar [45] M. Nikolova, A variational approach to remove outliers and impulse noise, J. Math. Imaging Vision, 20 (2004), 99-120. doi: 10.1023/B:JMIV.0000011920.58935.9c.  Google Scholar [46] G. Peyré and J. Fadili, Learning analysis sparsity priors, Proc. of Sampta'11, (2011). Google Scholar [47] A. Sawatzky, C. Brune, J. Müller and M. Burger, Total variation processing of images with poisson statistics, Proceedings of the 13th International Conference on Computer Analysis of Images and Patterns, 5702 (2009), 533-540. doi: 10.1007/978-3-642-03767-2_65.  Google Scholar [48] F. R. Schmidt and D. Cremers, A closed-form solution for image sequence segmentation with dynamical shape priors, in Pattern Recognition (Proc. DAGM), 2009. doi: 10.1007/978-3-642-03798-6_4.  Google Scholar [49] D. Strong, J.-F. Aujol and T. Chan, Scale recognition, regularization parameter selection, and Meyers G norm in total variation regularization, SIAM Journal on Multiscale Modeling and Simulation, 5 (2006), 273-303. doi: 10.1137/040621624.  Google Scholar [50] S. P. Sullivan et al., F. M. Sani, M. L. Johns and L. F. Gladden, Simulation of packed bed reactors using lattice Boltzmann methods, Chemical Engineering Science, 60 (2005), 3405-3418. Google Scholar [51] M. F. 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