The Annals of Statistics

Thresholding estimators for linear inverse problems and deconvolutions

Jérôme Kalifa and Stéphane Mallat

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Abstract

Thresholding algorithms in an orthonormal basis are studied to estimate noisy discrete signals degraded by a linear operator whose inverse is not bounded. For signals in a set $\Theta$, sufficient conditions are established on the basis to obtain a maximum risk with minimax rates of convergence. Deconvolutions with kernels having a Fourier transform which vanishes at high frequencies are examples of unstable inverse problems, where a thresholding in a wavelet basis is a suboptimal estimator. A new "mirror wavelet" basis is constructed to obtain a deconvolution risk which is proved to be asymptotically equivalent to the minimax risk over bounded variation signals. This thresholding estimator is used to restore blurred satellite images.

Article information

Source
Ann. Statist., Volume 31, Number 1 (2003), 58-109.

Dates
First available in Project Euclid: 26 February 2003

Permanent link to this document
https://projecteuclid.org/euclid.aos/1046294458

Digital Object Identifier
doi:10.1214/aos/1046294458

Mathematical Reviews number (MathSciNet)
MR1962500

Zentralblatt MATH identifier
1102.62318

Subjects
Primary: 49K35: Minimax problems 34A55: Inverse problems
Secondary: 42C40: Wavelets and other special systems

Keywords
Ill-posed inversed problems minimax optimality wavelet packets hyperbolic deconvolution

Citation

Kalifa, Jérôme; Mallat, Stéphane. Thresholding estimators for linear inverse problems and deconvolutions. Ann. Statist. 31 (2003), no. 1, 58--109. doi:10.1214/aos/1046294458. https://projecteuclid.org/euclid.aos/1046294458


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References

  • [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., MEy ER, Y. and WICKERHAUSER, M. V. (1992). Wavelet analysis and signal processing. In Wavelets and Their Applications (M. B. Ruskai, G. Bey lkin, 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., KERKy ACHARIAN, G. and PICARD, D. (1995). Wavelet shrinkage: Asy mptopia? (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 hy perrectangles, 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 Sy mposium on Time-Frequency and Time-Scale Analy sis 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 Poly technique.
  • [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 ROUGÉ, B. (2003). Deconvolution by thresholding in mirror wavelet. IEEE Trans. Image Process. To appear.
  • [18] KOLACZy K, 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] MEy ER, 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] ROUGÉ, B. (1993). Remarks about space-frequency and space-scale representations to clean and restore noisy images in satellite frameworks. In Progress in Wavelet Analy sis and Applications (Y. Meyer and S. Roques, eds.) 433-442. Frontières, Paris.
  • [25] ROUGÉ, B. (1997). Théorie de la chaine image optique et restauration. Ph. D. thesis, Univ. Paris-Dauphine.
  • [26] TSy BAKOV, 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 Analy sis from Theory to Software. Peters, Natick, MA.
  • [29] ZIEMER, W. P. (1989). Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation. Springer, Berlin.
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