Multiscale hierarchical decomposition of images with applications to deblurring, denoising and segmentation

Communications in Mathematical Sciences

Volumn 6, Issue 2, 2008, Pages 281-307

  Tadmor, Eitan ^a   Nezzar, Suzanne ^b   Vese, Luminita ^c  

^a Bldg 142   (United States)

^b RICHARD STOCKTON COLLEGE OF NEW JERSEY   (United States)

^c UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Image deblurring; Image decomposition; Multiscale expansion; Natural images; Segmentation; Total variation

Indexed keywords

EID: 49349102808     PISSN: 15396746     EISSN: None     Source Type: Journal    
DOI: 10.4310/cms.2008.v6.n2.a2     Document Type: Article

References (36)

1. R. Acar and C.R. Vogel, Analysis of bounded variation penalty methods of ill-posed problems, Inverse Problems, 10, 1217-1229, 1994.
2. L. Ambrosio and V. Tortorelli, On the approximation of free discontinuity problems, Bollettino U.M.I., (7)6-B, 105-123, 1992.
3. C. Bennet and R. Sharpley, Interpolation of Operators, Academic Press, 1988.
4. J. Bergh and J. Lofstrom, Interpolation Spaces, Springer Verlag, 1976.
5. F. Andreu-Vaillo, V. Caselles and J.M. Mazón, Parabolic Quasilinear Equations Minimizing Linear Growth Flunctionals, Birkhäuser Verlag, 2004.
6. G. Aubert and L. Vese, A variational method in image recovery, SIAM J. Numer. Anal., 34, 1948-1979, 1997.
7. A.Z. Averbuch, R.R. Coifman, F.G. Meyer and J.O. Strömberg, Multi-layered image representation: application to image compression, in Proceedings of the International Conference on Image Processing (ICIP 1998), Chicago, IL, 1998, IEEE Computer Society Press, Los Alamitos, CA, 2, 292-296, 1998.
8. A. Buades, B. Coll and J.M. Morel, A review of image denoising algorithms, with a new one, Multiscale Model. Simul., 4(2), 490-530, 2005.
9. A. Chambolle and P.L. Lions, Image restoration by constrained total variation minimization and variants, SPIE Electronic Imaging Proceedings, Vol. 2567, 1995.
10. A. Chambolle and P.L. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76, 167-188, 1997.
11. F. Demengel and R. Temam, Convex functions of a measure and applications, Indiana Univ. Mathematics Journal, 33(5), 673-709, 1984.
12. R.A. DeVore and G.G. Lorentz, Constructive Approximation, Springer-Verlag, Grundlehren der mathematischen Wissenschaften, 303, 1993.
13. S. Esedoglu and S. Osher, Decomposition of images by the anisotropic Rudin-Osher-Fatemi model, Comm. Pure Appl. Math., LVII, 1609-1626, 2004.
14. S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Trans. Pattern Anal. Mach. Intell., 6, 721-741, 1984.
15. C.W. Groetsch and O. Scherzer, Inverse scale space theory for inverse problems, Lecture Notes in Computer Science; Proc. Third International Conference on Scale-Space and Morphology in Computer Vision, 2106, 317-325, 2001.
16. J. Jost, Partial Differential Equations, Springer, 2007.
17. N. Kruglyak, The K-functional and Calderón-Zygmund type decompositions, in Interpolation Theory and Applications, L. De Carli and M. Milman, eds, Contemporary Mathematics, Amer. Math. Soc., 2007.
18. D. Levy, D. Paquin, E. Schreibmann and L. Xing, Multiscale image registration, Mathematical Biosciences and Engineering, 3, 389-418, 2006.
19. F.G. Meyer, A.Z. Averbuch and R.R. Coifman, Multilayered image representation: application to image compression, IEEE Trans. Image Process., 11, 1072-1080, 2002.
20. Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, University Lecture Series Volume 22, AMS, 2002.
21. V.A. Morozov, Methods for Solving Incorrectly Posed Problems, Springer Verlag, Heidelberg, 1984.
22. V.A. Morozov, Regularization Methods for Ill-Posed Problems, Florida, CRC Press, 1993.
23. D. Mumford and J. Shah, Optimal Approximations by Piecewise Smooth Functions and Associated Variational Problems, Communications on Pure and Applied Mathematics, Vol. XLVII, 577-685, 1989.
24. 2) Decompositions, Ph.D. Thesis, UCLA, May, 2003.
25. S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An iterative regularization method for total variation-based image restoration, Multiscale Model. Simul., 4(2), 460-489, 2005.
26. L. Rudin and V. Caselles, Image Recovery via Multiscale Total Variation, in Proceedings of the Second European Conference on Image Processing, Palma, Spain, 1995.
27. D. Paquin, D. Levy and L. Xing, Hybrid multiscale landmark and deformable registration, Mathematical Biosciences and Engineering, 4, 711-737, 2007.
28. L. Rudin, P.L. Lions and S. Osher, Multiplicative denoising and deblurring: theory and algorithms, in S. Osher and N. Paragios, Editors, Geometric Level Set Methods in Imaging, Vision and Graphics, Springer, 103-119, 2003.
29. L. Rudin, Images, Numerical Analysis of Singularities and Shock Filters, Ph.D. Thesis, CS Dept., CalTech, Pasadena, 1987.
30. L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60, 259-268, 1992.
31. L. Rudin and S. Osher, Total variation based image restoration with free local constraints, Proc. IEEE ICIP, Austin (Texas) USA, Vol. I, 31-35, 1994.
32. A.J. Smola and B. Scholkop, A tutorial on support vector regression, Stat. and Computing, 14, 199-222, 2004.
33. E. Tadmor, S. Nezzar and L. Vese, A multiscale image representation using hierarchical (BV,L2) decomposition, Multiscale Modeling and Simulation: A SIAM Inter. J., 2(4), 554-579, 2004.
34. A.N. Tikhonov and V.A. Arsenin, Solution of Ill-posed Problems, Winston & Sons, Washington, 1977.
35. V. Vapnik, Statistical Learning Theory, John Wiley and Sons, 1998.
36. L. Vese, A study in the BV space of a denoising-deblurring variational problem, Appl. Math. Optim., 44, 131-161, 2001.