Thresholding estimators for linear inverse problems and deconvolutions

Annals of Statistics

Volumn 31, Issue 1, 2003, Pages 58-109

  Kalifa, Jérôme ^a,c   Mallat, Stéphane ^b,d  

^a ECOLE POLYTECHNIQUE   (France)

^b NEW YORK UNIVERSITY   (United States)

^c CENTRE DE MATHÉMATIQUES APPLIQUÉES   (France)

^d NEW YORK UNIVERSITY   (United States)

Author keywords

Hyperbolic deconvolution; Ill posed inverse problems; Minimax optimality; Wavelet packets

Indexed keywords

EID: 21144431817     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/aos/1046294458     Document Type: Article

References (29)

1. ABRAMOVICH, F. and SILVERMAN, B. W. (1998). Wavelet decomposition approaches to statistical inverse problems. Biometrika 85 115-129.
2. ALVAREZ, L. and MOREL, J.-M. (1994). Formalization and computational aspects of image analysis. Acta Numer. 3 1-59.
3. COIFMAN, R. R. and DONOHO, D. (1995). Translation invariant de-noising. Technical Report 475, Dept. Statistics, Stanford Univ.
4. COIFMAN, R. R., MEYER, Y. and WICKERHAUSER, M. V. (1992). Wavelet analysis and signal processing. In Wavelets and Their Applications (M. B. Ruskai, G. Beylkin, R. Coifman et al., eds.) 153-178. Jones and Bartlett, Boston.
5. DAUBECHIES, I. (1988). Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 41 909-996.
6. DEVORE, R. A., JAWERTH, B. and LUCIER, B. J. (1992). Image compression through wavelet transform coding. IEEE Trans. Inform. Theory 38 719-746.
7. DONOHO, D. (1993). Unconditional bases are optimal bases for data compression and for statistical estimation. Appl. Comput. Harmon. Anal. 1 100-115.
8. DONOHO, D. (1995). Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition. Appl. Comput. Harmon. Anal. 2 101-126.
9. DONOHO, D. and JOHNSTONE, I. (1994). Ideal spatial adaptation via wavelet shrinkage. Biometrika 81 425-455.
10. DONOHO, D., JOHNSTONE, I., KERKYACHARIAN, G. and PICARD, D. (1995). Wavelet shrinkage: Asymptopia? (with discussion). J. Roy. Statist. Soc. Ser. B 57 301-369.
11. DONOHO, D., Liu, R. C. and MACGIBBON, K. B. (1990). Minimax risk over hyperrectangles, and implications. Ann. Statist. 18 1416-1437.
12. DONOHO, D., MALLAT, S. and VON SACHS, R. (1996). Estimating covariances of locally stationary processes: Consistency of best basis methods. In Proc. IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis 337-340. IEEE, New York.
13. IBRAGIMOV, I. A. and KHAS'MINSKII, R. Z. (1982). Bounds for the quality of nonparametric regression estimates. Theory Probab. Appl. 27 81-94.
14. JOHNSTONE, I. M. and SILVERMAN, B. W. (1995). Wavelet threshold estimators for data with correlated noise. Technical report, Dept. Statistics, Stanford Univ.
15. KALIFA, J. (1999). Restauration minimax et déconvolution dans une base d'ondelettes miroirs. Ph. D. thesis, Ecole Polytechnique.
16. KALIFA, J. and MALLAT, S. (1999). Minimax restoration and deconvolution. Bayesian Inference in Wavelet Based Models. Lecture Notes in Statist. 141 115-138. Springer, Berlin.
17. KALIFA, J., MALLAT, S. and ROUGE, B. (2003). Deconvolution by thresholding in mirror wavelet. IEEE Trans. Image Process. To appear.
18. KOLACZYK, E. (1996). A wavelet shrinkage approach to tomographic image reconstruction. J. Amer. Statist. Assoc. 91 1079-1090.
19. LEE, N.-Y. and LUCIER, B. J. (2001). Wavelet methods for inverting the Radon transform with noisy data. IEEE Trans. Image Process. 10 79-94.
20. MALLAT, S. (2000). A Wavelet Tour of Signal Processing, 2nd ed. Academic Press, New York.
21. MEYER, Y. (1992). Wavelets and Operators. Cambridge Univ. Press.
22. O'SULLIVAN, F. (1986). A statistical perspective on ill-posed inverse problems. Statist. Sci. 1 502-527.
23. PENSKY, M. and VIDAKOVIC, B. (1999). Adaptive wavelet estimator for nonparametric density deconvolution. Ann. Statist. 27 2033-2053.
24. ROUGE, B. (1993). Remarks about space-frequency and space-scale representations to clean and restore noisy images in satellite frameworks. In Progress in Wavelet Analysis and Applications (Y. Meyer and S. Roques, eds.) 433-442. Frontières, Paris.
25. ROUGE, B. (1997). Théorie de la chaine image optique et restauration. Ph. D. thesis, Univ. Paris-Dauphine.
26. TSYBAKOV, A. B. and CAVALIER, L. (2001). Sharp adaptation for inverse problems with random noise. Probab. Theory Related Fields 123 323-354.
27. WANG, Y. (1997). Minimax estimation via wavelets for indirect long-memory data. J. Statist. Plann. Inference 64 45-55.
28. WICKERHAUSER, M. V. (1994). Adapted Wavelet Analysis from Theory to Software. Peters, Natick, MA.
29. ZIEMER, W. P. (1989). Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation. Springer, Berlin.