# American Institute of Mathematical Sciences

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:
 [1] R. Acar and C. R. Vogel, Analysis of bounded variation penalty methods for ill-posed problems,, Inverse Prolems, 10 (1994), 1217. doi: 10.1088/0266-5611/10/6/003. [2] R. A. Adams and J. J. Fournier, Sobolev Spaces, volume 140., Academic Press, (2003). [3] M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis,, Wiley, (2000). doi: 10.1002/9781118032824. [4] J. Alberty, C. Carstensen and S. Funken, Remarks around 50 lines of matlab: Short finite element implementation,, Numerical Algorithms, 20 (1999), 117. doi: 10.1023/A:1019155918070. [5] A. Almansa, C. Ballester, V. Caselles and G. Haro, A TV based restoration model with local constraints,, Journal Scientific Computing, 34 (2008), 209. doi: 10.1007/s10915-007-9160-x. [6] W. Bangerth and A. Joshi, Adaptive finite element methods for the solution of inverse problems in optical tomography,, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/3/034011. [7] W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations,, Birkhäuser, (2003). doi: 10.1007/978-3-0348-7605-6. [8] E. Bänsch and K. Mikula, A coarsening finite element strategy in images selective smoothing,, Computing and Visualization in Science, 1 (1997), 53. [9] C. Bazan and P. Blomgren, Adaptive finite element method for image processing,, In Proceedings of the COMSOL Multiphysics Conference 2005 Boston, (2005). [10] E. C. Bingham, Fluidity and Plasticty,, International chemical series. McGraw-Hill Book Company, (1922). [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. doi: 10.1016/j.apnum.2004.06.012. [12] D. Braess and R. Verfürth, A posteriori error estimators for the Raviart-Thomas element,, SIAM J. Numer. Anal., 33 (1996), 2431. doi: 10.1137/S0036142994264079. [13] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods,, Springer-Verlag, (1991). doi: 10.1007/978-1-4612-3172-1. [14] A. Chambolle, An algorithm for total variation minimization and application,, Journal of Mathematical Imaging and Vision, 20 (2004), 89. doi: 10.1023/B:JMIV.0000011321.19549.88. [15] A. Chambolle and P-L. Lions., Image recovery via total variation minimization and related problems,, Numerische Mathematik, 76 (1997), 167. doi: 10.1007/s002110050258. [16] Q. Chang and I-L. Chern, Acceleration methods for total variation-based image denoising,, SIAM J. Applied Mathematics, 25 (2003), 982. doi: 10.1137/S106482750241534X. [17] P. G. Ciarlet, The Finite Element Method for Elliptic Problems,, North Holland, (1978). [18] P. Clément, Approximation by finite element functions using local regularization,, RAIRO Analyse Numérique, 9 (1975), 77. [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. doi: 10.1016/j.jnnfm.2006.09.002. [20] D. C. Dobson and C. R. Vogel, Convergence of an iterative method for total variation denoising,, SIAM J. Numer. Anal., 34 (1997), 1779. doi: 10.1137/S003614299528701X. [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. doi: 10.1007/s10851-010-0248-9. [22] G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics,, Springer-Verlag, (1976). [23] I. Ekeland and R. Témam, Convex Analysis and Variational Problems,, Classics Appl. Math. 28, (1999). doi: 10.1137/1.9781611971088. [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. doi: 10.1051/m2an:2003041. [25] K. Frick and P. Marnitz, Statistical multiresolution dantzig estimation in imaging: Fundamental concepts and algorithmic framework,, Electronic Journal of Statistics, 6 (2012), 231. doi: 10.1214/12-EJS671. [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. doi: 10.1007/s10851-012-0368-5. [27] M. Fried, Multichannel image segmentation using adaptive finite elements,, Computing and Visualization in Science, 12 (2009), 125. doi: 10.1007/s00791-007-0082-9. [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. doi: 10.1093/imamat/61.3.237. [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. doi: 10.1137/S0036139998335165. [30] E. Giusti, Minimal Surfaces and Functions of Bounded Variation,, Birkhäuser, (1984). doi: 10.1007/978-1-4684-9486-0. [31] R. Glowinski, J. L. Lions and R. Trémolières, Analyse Numérique des Inéquations Variationelles,, Dunod, (1976). [32] E. J. Gumbel, Les valeurs extrêmes des distributions statistiques,, Annales de l'institut Henri Poincaré, 5 (1935), 115. [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. doi: 10.1088/0266-5611/23/3/017. [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. [35] E. Haber, S. Heldmann and U. Ascher, Adaptive finite volume method for distributed non-smooth parameter identification,, Inverse Problems, 23 (2007), 1659. doi: 10.1088/0266-5611/23/4/017. [36] E. Haber, S. Heldmann and J. Modersitzki, An octree method for parametric image registration,, SIAM J. on Scientific Computing, 29 (2007), 2008. doi: 10.1137/060662605. [37] E. Haber, S. Heldmann and J. Modersitzki, Adaptive mesh refinement for non parametric image registration,, SIAM J. on Scientific Computing, 30 (2008), 3012. doi: 10.1137/070687724. [38] E. Hashrova, Z. Kabluchko and A. Wübker, Extremes of independent chi-square random vectors,, Extremes, 15 (2012), 35. doi: 10.1007/s10687-010-0125-3. [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. doi: 10.1051/cocv:2007057. [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. doi: 10.1137/S1052623401383558. [41] M. Hintermüller and K. Kunisch, Total bounded variation regularization as bilaterally constrained optimization problem,, SIAM J. Appl. Math., 64 (2004), 1311. doi: 10.1137/S0036139903422784. [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. doi: 10.1137/040613263. [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. doi: 10.1016/j.csda.2011.08.018. [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. doi: 10.1137/040605412. [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. [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. doi: 10.1016/S0045-7825(03)00262-7. [47] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D, 60 (1992), 259. doi: 10.1016/0167-2789(92)90242-F. [48] L. Rudin, MTV-multiscale Total Variation Principle for a Pde-based Solution to Nonsmooth Ill-posed Problem,, Technical report, (1995). [49] D. Strong and T. Chan, Spatially and Scale Adaptive Total Variation Based Regularization and Anisotropic Diffusion in Image Processing,, Technical report, (1996). [50] D. Strong and T. Chan, Edge-preserving and scale-dependent properties of total variation regularization,, Inverse Problems, 19 (2003). doi: 10.1088/0266-5611/19/6/059. [51] R. Temam, Navier-Stokes Equations,, North-Holland, (1977). [52] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques,, Wiley Teubner, (1996). [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. [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.

show all references

##### References:
 [1] R. Acar and C. R. Vogel, Analysis of bounded variation penalty methods for ill-posed problems,, Inverse Prolems, 10 (1994), 1217. doi: 10.1088/0266-5611/10/6/003. [2] R. A. Adams and J. J. Fournier, Sobolev Spaces, volume 140., Academic Press, (2003). [3] M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis,, Wiley, (2000). doi: 10.1002/9781118032824. [4] J. Alberty, C. Carstensen and S. Funken, Remarks around 50 lines of matlab: Short finite element implementation,, Numerical Algorithms, 20 (1999), 117. doi: 10.1023/A:1019155918070. [5] A. Almansa, C. Ballester, V. Caselles and G. Haro, A TV based restoration model with local constraints,, Journal Scientific Computing, 34 (2008), 209. doi: 10.1007/s10915-007-9160-x. [6] W. Bangerth and A. Joshi, Adaptive finite element methods for the solution of inverse problems in optical tomography,, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/3/034011. [7] W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations,, Birkhäuser, (2003). doi: 10.1007/978-3-0348-7605-6. [8] E. Bänsch and K. Mikula, A coarsening finite element strategy in images selective smoothing,, Computing and Visualization in Science, 1 (1997), 53. [9] C. Bazan and P. Blomgren, Adaptive finite element method for image processing,, In Proceedings of the COMSOL Multiphysics Conference 2005 Boston, (2005). [10] E. C. Bingham, Fluidity and Plasticty,, International chemical series. McGraw-Hill Book Company, (1922). [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. doi: 10.1016/j.apnum.2004.06.012. [12] D. Braess and R. Verfürth, A posteriori error estimators for the Raviart-Thomas element,, SIAM J. Numer. Anal., 33 (1996), 2431. doi: 10.1137/S0036142994264079. [13] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods,, Springer-Verlag, (1991). doi: 10.1007/978-1-4612-3172-1. [14] A. Chambolle, An algorithm for total variation minimization and application,, Journal of Mathematical Imaging and Vision, 20 (2004), 89. doi: 10.1023/B:JMIV.0000011321.19549.88. [15] A. Chambolle and P-L. Lions., Image recovery via total variation minimization and related problems,, Numerische Mathematik, 76 (1997), 167. doi: 10.1007/s002110050258. [16] Q. Chang and I-L. Chern, Acceleration methods for total variation-based image denoising,, SIAM J. Applied Mathematics, 25 (2003), 982. doi: 10.1137/S106482750241534X. [17] P. G. Ciarlet, The Finite Element Method for Elliptic Problems,, North Holland, (1978). [18] P. Clément, Approximation by finite element functions using local regularization,, RAIRO Analyse Numérique, 9 (1975), 77. [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. doi: 10.1016/j.jnnfm.2006.09.002. [20] D. C. Dobson and C. R. Vogel, Convergence of an iterative method for total variation denoising,, SIAM J. Numer. Anal., 34 (1997), 1779. doi: 10.1137/S003614299528701X. [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. doi: 10.1007/s10851-010-0248-9. [22] G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics,, Springer-Verlag, (1976). [23] I. Ekeland and R. Témam, Convex Analysis and Variational Problems,, Classics Appl. Math. 28, (1999). doi: 10.1137/1.9781611971088. [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. doi: 10.1051/m2an:2003041. [25] K. Frick and P. Marnitz, Statistical multiresolution dantzig estimation in imaging: Fundamental concepts and algorithmic framework,, Electronic Journal of Statistics, 6 (2012), 231. doi: 10.1214/12-EJS671. [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. doi: 10.1007/s10851-012-0368-5. [27] M. Fried, Multichannel image segmentation using adaptive finite elements,, Computing and Visualization in Science, 12 (2009), 125. doi: 10.1007/s00791-007-0082-9. [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. doi: 10.1093/imamat/61.3.237. [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. doi: 10.1137/S0036139998335165. [30] E. Giusti, Minimal Surfaces and Functions of Bounded Variation,, Birkhäuser, (1984). doi: 10.1007/978-1-4684-9486-0. [31] R. Glowinski, J. L. Lions and R. Trémolières, Analyse Numérique des Inéquations Variationelles,, Dunod, (1976). [32] E. J. Gumbel, Les valeurs extrêmes des distributions statistiques,, Annales de l'institut Henri Poincaré, 5 (1935), 115. [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. doi: 10.1088/0266-5611/23/3/017. [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. [35] E. Haber, S. Heldmann and U. Ascher, Adaptive finite volume method for distributed non-smooth parameter identification,, Inverse Problems, 23 (2007), 1659. doi: 10.1088/0266-5611/23/4/017. [36] E. Haber, S. Heldmann and J. Modersitzki, An octree method for parametric image registration,, SIAM J. on Scientific Computing, 29 (2007), 2008. doi: 10.1137/060662605. [37] E. Haber, S. Heldmann and J. Modersitzki, Adaptive mesh refinement for non parametric image registration,, SIAM J. on Scientific Computing, 30 (2008), 3012. doi: 10.1137/070687724. [38] E. Hashrova, Z. Kabluchko and A. Wübker, Extremes of independent chi-square random vectors,, Extremes, 15 (2012), 35. doi: 10.1007/s10687-010-0125-3. [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. doi: 10.1051/cocv:2007057. [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. doi: 10.1137/S1052623401383558. [41] M. Hintermüller and K. Kunisch, Total bounded variation regularization as bilaterally constrained optimization problem,, SIAM J. Appl. Math., 64 (2004), 1311. doi: 10.1137/S0036139903422784. [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. doi: 10.1137/040613263. [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. doi: 10.1016/j.csda.2011.08.018. [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. doi: 10.1137/040605412. [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. [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. doi: 10.1016/S0045-7825(03)00262-7. [47] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D, 60 (1992), 259. doi: 10.1016/0167-2789(92)90242-F. [48] L. Rudin, MTV-multiscale Total Variation Principle for a Pde-based Solution to Nonsmooth Ill-posed Problem,, Technical report, (1995). [49] D. Strong and T. Chan, Spatially and Scale Adaptive Total Variation Based Regularization and Anisotropic Diffusion in Image Processing,, Technical report, (1996). [50] D. Strong and T. Chan, Edge-preserving and scale-dependent properties of total variation regularization,, Inverse Problems, 19 (2003). doi: 10.1088/0266-5611/19/6/059. [51] R. Temam, Navier-Stokes Equations,, North-Holland, (1977). [52] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques,, Wiley Teubner, (1996). [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. [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.
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