Regularization by fractional filter methods and data smoothing

Inverse Problems

Volumn 24, Issue 2, 2008, Pages

  Klann, E ^a   Ramlau, R ^a  

^a JOHANN RADON INSTITUTE FOR COMPUTATIONAL AND APPLIED MATHEMATICS   (Austria)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; MATHEMATICAL OPERATORS; NUMERICAL METHODS; WAVELET TRANSFORMS;

FRACTIONAL FILTERS;

PROBLEM SOLVING;

EID: 42549107286     PISSN: 02665611     EISSN: 13616420     Source Type: Journal    
DOI: 10.1088/0266-5611/24/2/025018     Document Type: Article

References (28)

1. Engl H W, Hanke M and Neubauer A 1996 Regularization of Inverse Problems (Mathematics and its Applications 375) (Dordrecht: Kluwer) 321 p
2. Louis A K 1989 Inverse und schlecht gestellte Probleme (Inverse and Ill-Posed Problems) (Teubner Studienbücher Mathematik) (Stuttgart: B G Teubner) 205 S
3. Morozov V A 1984 Methods for Solving Incorrectly Posed Problems 18 (Berlin: Springer) 257 p
4. Jonas P and Louis A K 2001 A Sobolev space analysis of linear regularization methods for ill-posed problems J. Inverse Ill-posed Probl. 9 59-74
5. Cohen A, Hoffmann M and Reiss M 2004 Adaptive wavelet Galerkin methods for linear inverse problems SIAM J. Numer. Anal. 42 1479-1501
6. Ramlau R and Teschke G 2004 Regularization of Sobolev embedding operators and applications to medical imaging and meteorological data: I. Regularization of Sobolev embedding operators Sampl. Theory Signal Image Process. 3 175-96
7. Klann E 2006 Regularization of linear ill-posed problems in two steps: combination of data smoothing and reconstruction methods PhD Thesis University of Bremen
8. Klann E, Maass P and Ramlau R 2006 Two-step regularization methods for linear inverse problems J. Inverse Ill-Posed Probl. 14 583-607
9. Bertero M and Boccacci P 1998 Introduction to Inverse Problems in Imaging vol 11 (Bristol: Institute of Physics Publishing) 351 p
10. Lamm P K 2001 Some recent developments and open problems in solution methods for mathematical inverse problems Proc. 24 National Congress of Applied and Computational Mathematics (Belo Horizonte, Brazil, 10-13 Sept.) Preprint
11. Rudin L I, Osher S and Fatemi E 1992 Nonlinear total variation based noise removal algorithms Physica D 60 259-68
12. Dobson D C and Santosa F 1996 Recovery of blocky images from noisy and blurred data SIAM J. Appl. Math. 56 1181-98
13. Chambolle A and Lions P-L 1997 Image recovery via total variation minimization and related problems Numer. Math. 76 167-88
14. Vogel C R and Oman M E 1998 Fast, robust total variation-based reconstruction of noisy, blurred images IEEE Trans. Image Process. 7 813-24
15. Vogel Curtis R 2002 Computational methods for inverse problems Front. Appl. Math. 23 (Philadelphia, PA: Society for Industrial and Applied Mathematics)
16. Daubechies I, Defrise M and DeMol C 2004 An iterative thresholding algorithm for linear inverse problems with a sparsity constraint Commun. Pure Appl. Math. 57 1413-57
17. Leonov A S and Yagola A S 1998 Special regularizing methods for ill-posed problems with sourcewise represented solutions Inverse Probl. 14 1539-50
18. Daubechies I 1992 Ten Lectures on Wavelets (CBMS-NSF Regional Conference Series in Applied Mathematics 61) vol 19 (Philadelphia, PA: Society for Industrial and Applied Mathematics) 357 p
19. Louis A K, Maass P and Rieder A 1997 Wavelets: Theory and Applications (Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs and Tracts 17) (Chichester: Wiley) 324 p
20. Mallat S 1998 A Wavelet Tour of Signal Processing vol 24 (San Diego, CA: Academic) 577 p
21. Strang G and Nguyen T 1996 Wavelets and Filter Banks (Wellesley, MA: Wellesley-Cambridge Press)
22. Cohen Albert 2003 Numerical Analysis of Wavelet Methods (Studies in Mathematics and Its Applications 32) vol 18 (Amsterdam: North-Holland) 336 p
23. Cohen A, DeVore R, Kerkyacharian G and Picard D 2001 Maximal spaces with given rate of convergence for thresholding algorithms Appl. Comput. Harmon. Anal. 11 167-91
24. Donoho D L and Johnstone I M 1994 Ideal spatial adaptation by wavelet shrinkage Biometrika 81 425-55
25. Donoho D L and Johnstone I M 1998 Minimax estimation via wavelet shrinkage Ann. Stat. 26 879-921
26. Donoho D L, Johnstone I M, Kerkyacharian G and Picard D 1995 Wavelet shrinkage: asymptopia? J. R. Stat. Soc. B 57 301-69
27. Waugh F V and Abel M E 1967 On fractional powers of a matrix J. Am. Stat. Assoc. 62 1018-21
28. Wan X Q, Wang Y B and Yamamoto M 2006 Detection of irregular points by regularization in numerical differentiation and application to edge detection Inverse Probl. 22 1089-103