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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 |
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. |
[2] |
R. A. Adams and J. J. Fournier, Sobolev Spaces, volume 140. Academic Press, 2nd edition, 2003. |
[3] |
M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, Wiley, New York, 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), 117-137.
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), 209-236.
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), 034011, 22 pp.
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), 53-61. |
[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, inc., 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), 13-38.
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), 2431-2444.
doi: 10.1137/S0036142994264079. |
[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. |
[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. |
[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. |
[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. |
[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), 77-84. |
[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. |
[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. |
[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. |
[22] |
G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976. |
[23] |
I. Ekeland and R. Témam, Convex Analysis and Variational Problems, Classics Appl. Math. 28, SIAM, 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), 533-556.
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), 231-268.
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), 370-387.
doi: 10.1007/s10851-012-0368-5. |
[27] |
M. Fried, Multichannel image segmentation using adaptive finite elements, Computing and Visualization in Science, 12 (2009), 125-135.
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), 237-266.
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), 1950-1976.
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, Paris, France, 1976. |
[32] |
E. J. Gumbel, Les valeurs extrêmes des distributions statistiques, Annales de l'institut Henri Poincaré, 5 (1935), 115-158. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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), 3317-3341.
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), 259-268.
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, Cognitech, Inc. Talk presented at the Workshop on Mathematical Methods in Computer Vision, University of Minnesota, 1995. |
[49] |
D. Strong and T. Chan, Spatially and Scale Adaptive Total Variation Based Regularization and Anisotropic Diffusion in Image Processing, Technical report, UCLA, 1996. |
[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. |
[51] |
R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam, 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), 1217-1229.
doi: 10.1088/0266-5611/10/6/003. |
[2] |
R. A. Adams and J. J. Fournier, Sobolev Spaces, volume 140. Academic Press, 2nd edition, 2003. |
[3] |
M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, Wiley, New York, 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), 117-137.
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), 209-236.
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), 034011, 22 pp.
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), 53-61. |
[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, inc., 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), 13-38.
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), 2431-2444.
doi: 10.1137/S0036142994264079. |
[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. |
[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. |
[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. |
[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. |
[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), 77-84. |
[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. |
[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. |
[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. |
[22] |
G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976. |
[23] |
I. Ekeland and R. Témam, Convex Analysis and Variational Problems, Classics Appl. Math. 28, SIAM, 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), 533-556.
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), 231-268.
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), 370-387.
doi: 10.1007/s10851-012-0368-5. |
[27] |
M. Fried, Multichannel image segmentation using adaptive finite elements, Computing and Visualization in Science, 12 (2009), 125-135.
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), 237-266.
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), 1950-1976.
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, Paris, France, 1976. |
[32] |
E. J. Gumbel, Les valeurs extrêmes des distributions statistiques, Annales de l'institut Henri Poincaré, 5 (1935), 115-158. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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), 3317-3341.
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), 259-268.
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, Cognitech, Inc. Talk presented at the Workshop on Mathematical Methods in Computer Vision, University of Minnesota, 1995. |
[49] |
D. Strong and T. Chan, Spatially and Scale Adaptive Total Variation Based Regularization and Anisotropic Diffusion in Image Processing, Technical report, UCLA, 1996. |
[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. |
[51] |
R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam, 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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