American Institute of Mathematical Sciences

August  2014, 8(3): 685-711. doi: 10.3934/ipi.2014.8.685

An adaptive finite element method in $L^2$-TV-based image denoising

 1 Department of Mathematics, Humboldt-University of Berlin, Unter den Linden 6, 10099 Berlin, Germany 2 START-Project "Interfaces and Free Boundaries" and SFB "Mathematical Optimization and Applications in Biomedical Science", Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, A-8010 Graz, Austria

Received  October 2012 Revised  June 2013 Published  August 2014

The first order optimality system of a total variation regularization based variational model with $L^2$-data-fitting in image denoising ($L^2$-TV problem) can be expressed as an elliptic variational inequality of the second kind. For a finite element discretization of the variational inequality problem, an a posteriori error residual based error estimator is derived and its reliability and (partial) efficiency are established. The results are applied to solve the $L^2$-TV problem by means of the adaptive finite element method. The adaptive mesh refinement relies on the newly derived a posteriori error estimator and on an additional heuristic providing a local variance estimator to cope with noisy data. The numerical solution of the discrete problem on each level of refinement is obtained by a superlinearly convergent algorithm based on Fenchel-duality and inexact semismooth Newton techniques and which is stable with respect to noise in the data. Numerical results justifying the advantage of adaptive finite elements solutions are presented.
Citation: Michael Hintermüller, Monserrat Rincon-Camacho. An adaptive finite element method in $L^2$-TV-based image denoising. Inverse Problems & Imaging, 2014, 8 (3) : 685-711. doi: 10.3934/ipi.2014.8.685
References:
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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 [22] G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976.  Google Scholar [23] I. Ekeland and R. Témam, Convex Analysis and Variational Problems, Classics Appl. Math. 28, SIAM, 1999. doi: 10.1137/1.9781611971088.  Google Scholar [24] X. Feng and A. Prohl, Analysis of total variation flow and its finite element approximations, ESAIM: Mathematical Modelling and Numerical Analysis, 37 (2003), 533-556. doi: 10.1051/m2an:2003041.  Google Scholar [25] K. Frick and P. Marnitz, Statistical multiresolution dantzig estimation in imaging: Fundamental concepts and algorithmic framework, Electronic Journal of Statistics, 6 (2012), 231-268. doi: 10.1214/12-EJS671.  Google Scholar [26] K. Frick, P. Marnitz and A. 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Lions and R. Trémolières, Analyse Numérique des Inéquations Variationelles, Dunod, Paris, France, 1976. Google Scholar [32] E. J. Gumbel, Les valeurs extrêmes des distributions statistiques, Annales de l'institut Henri Poincaré, 5 (1935), 115-158.  Google Scholar [33] M. Guven et al, Effect of discretization error analysis and adaptive mesh generation in diffuse optical absorption imaging: I, Inverse Problems, 23 (2007), 1115-1133. doi: 10.1088/0266-5611/23/3/017.  Google Scholar [34] M. Guven et al, Discretization error analysis and adaptive meshing algorithms for fluorescence diffuse optical tomography: Part I, IEEE Trans. Med. Imag., 29 (2010), 217-229. Google Scholar [35] E. Haber, S. Heldmann and U. Ascher, Adaptive finite volume method for distributed non-smooth parameter identification, Inverse Problems, 23 (2007), 1659-1676. doi: 10.1088/0266-5611/23/4/017.  Google Scholar [36] E. Haber, S. Heldmann and J. 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Kunisch, The primal-dual active set strategy as a semismooth newton method, SIAM Journal on Optimization, 13 (2002), 865-888. doi: 10.1137/S1052623401383558.  Google Scholar [41] M. Hintermüller and K. Kunisch, Total bounded variation regularization as bilaterally constrained optimization problem, SIAM J. Appl. Math., 64 (2004), 1311-1333. doi: 10.1137/S0036139903422784.  Google Scholar [42] 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 [43] T. Hotz, P. Marnitz, R. Stichtenoth, L. Davies, Z. Kabluchko and A. Munk, Locally adaptive image denoising by a statistical multiresolution criterion, Comput. Stat. Data Anal., 56 (2012), 543-558. doi: 10.1016/j.csda.2011.08.018.  Google Scholar [44] S. Osher, M. Burger, D. Goldfarb, J. Xu and W. 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Talk presented at the Workshop on Mathematical Methods in Computer Vision, University of Minnesota, 1995. Google Scholar [49] D. Strong and T. Chan, Spatially and Scale Adaptive Total Variation Based Regularization and Anisotropic Diffusion in Image Processing, Technical report, UCLA, 1996. Google Scholar [50] D. Strong and T. Chan, Edge-preserving and scale-dependent properties of total variation regularization, Inverse Problems, 19 (2003), S165-S187. doi: 10.1088/0266-5611/19/6/059.  Google Scholar [51] R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam, 1977.  Google Scholar [52] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques, Wiley Teubner, 1996. Google Scholar [53] C. R. Vogel, Computational Methods for Inverse Problems, volume 23 of Frontiers Appl. Math. SIAM-Society of Industrial and Applied Mathematics, 2002. doi: 10.1137/1.9780898717570.  Google Scholar [54] G. Winkler, Image Analysis, Random Fields And Markov Chain Monte Carlo Methods: A Mathematical Introduction, Applications of mathematics. Springer, 2003. doi: 10.1007/978-3-642-55760-6.  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 Prolems, 10 (1994), 1217-1229. doi: 10.1088/0266-5611/10/6/003.  Google Scholar [2] R. A. Adams and J. J. Fournier, Sobolev Spaces, volume 140. Academic Press, 2nd edition, 2003.  Google Scholar [3] M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, Wiley, New York, 2000. doi: 10.1002/9781118032824.  Google Scholar [4] J. Alberty, C. Carstensen and S. Funken, Remarks around 50 lines of matlab: Short finite element implementation, Numerical Algorithms, 20 (1999), 117-137. doi: 10.1023/A:1019155918070.  Google Scholar [5] A. Almansa, C. Ballester, V. Caselles and G. Haro, A TV based restoration model with local constraints, Journal Scientific Computing, 34 (2008), 209-236. doi: 10.1007/s10915-007-9160-x.  Google Scholar [6] W. Bangerth and A. Joshi, Adaptive finite element methods for the solution of inverse problems in optical tomography, Inverse Problems, 24 (2008), 034011, 22 pp. doi: 10.1088/0266-5611/24/3/034011.  Google Scholar [7] W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations, Birkhäuser, 2003. doi: 10.1007/978-3-0348-7605-6.  Google Scholar [8] E. Bänsch and K. Mikula, A coarsening finite element strategy in images selective smoothing, Computing and Visualization in Science, 1 (1997), 53-61. Google Scholar [9] C. Bazan and P. Blomgren, Adaptive finite element method for image processing, In Proceedings of the COMSOL Multiphysics Conference 2005 Boston, 2005. Google Scholar [10] E. C. Bingham, Fluidity and Plasticty, International chemical series. McGraw-Hill Book Company, inc., 1922. Google Scholar [11] V. Bostan, W. Han and B. D. Reddy, A posteriori error estimation and adaptive solution of elliptic variational inequalities of the second kind, Applied Numerical Mathematics, 52 (2005), 13-38. doi: 10.1016/j.apnum.2004.06.012.  Google Scholar [12] D. Braess and R. Verfürth, A posteriori error estimators for the Raviart-Thomas element, SIAM J. Numer. Anal., 33 (1996), 2431-2444. doi: 10.1137/S0036142994264079.  Google Scholar [13] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3172-1.  Google Scholar [14] A. Chambolle, An algorithm for total variation minimization and application, Journal of Mathematical Imaging and 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, Numerische Mathematik, 76 (1997), 167-188. doi: 10.1007/s002110050258.  Google Scholar [16] Q. Chang and I-L. Chern, Acceleration methods for total variation-based image denoising, SIAM J. Applied Mathematics, 25 (2003), 982-994. doi: 10.1137/S106482750241534X.  Google Scholar [17] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, 1978.  Google Scholar [18] P. Clément, Approximation by finite element functions using local regularization, RAIRO Analyse Numérique, 9 (1975), 77-84.  Google Scholar [19] E. J. Dean, R. Glowinski and G. Guidoboni, On the numerical simulation of Bingham visco-plastic flow: Old and new results, Journal of Non-Newtonian Fluid Mechanics, 142 (2007), 36-62. doi: 10.1016/j.jnnfm.2006.09.002.  Google Scholar [20] 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 [21] Y. Dong, M. Hintermüller 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 [22] G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976.  Google Scholar [23] I. Ekeland and R. Témam, Convex Analysis and Variational Problems, Classics Appl. Math. 28, SIAM, 1999. doi: 10.1137/1.9781611971088.  Google Scholar [24] X. Feng and A. Prohl, Analysis of total variation flow and its finite element approximations, ESAIM: Mathematical Modelling and Numerical Analysis, 37 (2003), 533-556. doi: 10.1051/m2an:2003041.  Google Scholar [25] K. Frick and P. Marnitz, Statistical multiresolution dantzig estimation in imaging: Fundamental concepts and algorithmic framework, Electronic Journal of Statistics, 6 (2012), 231-268. doi: 10.1214/12-EJS671.  Google Scholar [26] K. Frick, P. Marnitz and A. Munk, Statistical multiresolution estimation for variational imaging: with an application in Poisson-biophotonics, J. Math. Imaging Vision, 46 (2013), 370-387. doi: 10.1007/s10851-012-0368-5.  Google Scholar [27] M. Fried, Multichannel image segmentation using adaptive finite elements, Computing and Visualization in Science, 12 (2009), 125-135. doi: 10.1007/s00791-007-0082-9.  Google Scholar [28] I. A. Frigaard and O. Scherzer, Uniaxial exchange flows of two Bingham fluids in a cylindrical duct, IMA journal of applied mathematics, 61 (1998), 237-266. doi: 10.1093/imamat/61.3.237.  Google Scholar [29] I. A. Frigaard and O. Scherzer, The effects of yield stress variation on uniaxial exchange flows of two Bingham fluids in a pipe, SIAM J. Appl. Math., 60 (2000), 1950-1976. doi: 10.1137/S0036139998335165.  Google Scholar [30] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, 1984. doi: 10.1007/978-1-4684-9486-0.  Google Scholar [31] R. Glowinski, J. L. Lions and R. Trémolières, Analyse Numérique des Inéquations Variationelles, Dunod, Paris, France, 1976. Google Scholar [32] E. J. Gumbel, Les valeurs extrêmes des distributions statistiques, Annales de l'institut Henri Poincaré, 5 (1935), 115-158.  Google Scholar [33] M. Guven et al, Effect of discretization error analysis and adaptive mesh generation in diffuse optical absorption imaging: I, Inverse Problems, 23 (2007), 1115-1133. doi: 10.1088/0266-5611/23/3/017.  Google Scholar [34] M. Guven et al, Discretization error analysis and adaptive meshing algorithms for fluorescence diffuse optical tomography: Part I, IEEE Trans. Med. Imag., 29 (2010), 217-229. Google Scholar [35] E. Haber, S. Heldmann and U. Ascher, Adaptive finite volume method for distributed non-smooth parameter identification, Inverse Problems, 23 (2007), 1659-1676. doi: 10.1088/0266-5611/23/4/017.  Google Scholar [36] E. Haber, S. Heldmann and J. Modersitzki, An octree method for parametric image registration, SIAM J. on Scientific Computing, 29 (2007), 2008-2023. doi: 10.1137/060662605.  Google Scholar [37] E. Haber, S. Heldmann and J. Modersitzki, Adaptive mesh refinement for non parametric image registration, SIAM J. on Scientific Computing, 30 (2008), 3012-3027. doi: 10.1137/070687724.  Google Scholar [38] E. Hashrova, Z. Kabluchko and A. Wübker, Extremes of independent chi-square random vectors, Extremes, 15 (2012), 35-42. doi: 10.1007/s10687-010-0125-3.  Google Scholar [39] M. Hintermüller, R. H. W. Hoppe, Y. Iliash and M. Kieweg, An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints, ESAIM: COCV, 14 (2008), 540-560. doi: 10.1051/cocv:2007057.  Google Scholar [40] M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as a semismooth newton method, SIAM Journal on Optimization, 13 (2002), 865-888. doi: 10.1137/S1052623401383558.  Google Scholar [41] M. Hintermüller and K. Kunisch, Total bounded variation regularization as bilaterally constrained optimization problem, SIAM J. Appl. Math., 64 (2004), 1311-1333. doi: 10.1137/S0036139903422784.  Google Scholar [42] 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 [43] T. Hotz, P. Marnitz, R. Stichtenoth, L. Davies, Z. Kabluchko and A. Munk, Locally adaptive image denoising by a statistical multiresolution criterion, Comput. Stat. Data Anal., 56 (2012), 543-558. doi: 10.1016/j.csda.2011.08.018.  Google Scholar [44] S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An iterative regularization method for total variation-based image restoration, SIAM Multiscale Model. and Simu., 4 (2005), 460-489. doi: 10.1137/040605412.  Google Scholar [45] T. Preusser and M. Rumpf, An adaptive finite element method for large scale image processing,, In Proceedings of the Second International Conference on Scale-Space Theories in Computer Vision, (): 223.   Google Scholar [46] N. Roquet and P. Saramito, An adaptive finite element method for bingham fluid flows around a cylinder, Computer Methods in Applied Mechanics and Engineering, 192 (2003), 3317-3341. doi: 10.1016/S0045-7825(03)00262-7.  Google Scholar [47] L. I. 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.  Google Scholar [48] L. Rudin, MTV-multiscale Total Variation Principle for a Pde-based Solution to Nonsmooth Ill-posed Problem, Technical report, Cognitech, Inc. 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