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

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  • 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.
    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.


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  • [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.


    R. A. Adams and J. J. Fournier, Sobolev Spaces, volume 140. Academic Press, 2nd edition, 2003.


    M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, Wiley, New York, 2000.doi: 10.1002/9781118032824.


    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.


    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.


    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.


    W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations, Birkhäuser, 2003.doi: 10.1007/978-3-0348-7605-6.


    E. Bänsch and K. Mikula, A coarsening finite element strategy in images selective smoothing, Computing and Visualization in Science, 1 (1997), 53-61.


    C. Bazan and P. Blomgren, Adaptive finite element method for image processing, In Proceedings of the COMSOL Multiphysics Conference 2005 Boston, 2005.


    E. C. Bingham, Fluidity and Plasticty, International chemical series. McGraw-Hill Book Company, inc., 1922.


    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.


    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.


    F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.doi: 10.1007/978-1-4612-3172-1.


    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.


    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.


    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.


    P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, 1978.


    P. Clément, Approximation by finite element functions using local regularization, RAIRO Analyse Numérique, 9 (1975), 77-84.


    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.


    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.


    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.


    G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976.


    I. Ekeland and R. Témam, Convex Analysis and Variational Problems, Classics Appl. Math. 28, SIAM, 1999.doi: 10.1137/1.9781611971088.


    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.


    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.


    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.


    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.


    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.


    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.


    E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, 1984.doi: 10.1007/978-1-4684-9486-0.


    R. Glowinski, J. L. Lions and R. Trémolières, Analyse Numérique des Inéquations Variationelles, Dunod, Paris, France, 1976.


    E. J. Gumbel, Les valeurs extrêmes des distributions statistiques, Annales de l'institut Henri Poincaré, 5 (1935), 115-158.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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, 223-234.


    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.


    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.


    L. Rudin, MTV-multiscale Total Variation Principle for a Pde-based Solution to Nonsmooth Ill-posed Problem, Technical report, Cognitech, Inc. Talk presented at the Workshop on Mathematical Methods in Computer Vision, University of Minnesota, 1995.


    D. Strong and T. Chan, Spatially and Scale Adaptive Total Variation Based Regularization and Anisotropic Diffusion in Image Processing, Technical report, UCLA, 1996.


    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.


    R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam, 1977.


    R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques, Wiley Teubner, 1996.


    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.


    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.

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