Electronic Journal of Statistics

Blockwise SVD with error in the operator and application to blind deconvolution

S. Delattre, M. Hoffmann, D. Picard, and T. Vareschi

Full-text: Open access

Abstract

We consider linear inverse problems in a nonparametric statistical framework. Both the signal and the operator are unknown and subject to error measurements. We establish minimax rates of convergence under squared error loss when the operator admits a blockwise singular value decomposition (blockwise SVD) and the smoothness of the signal is measured in a Sobolev sense. We construct a nonlinear procedure adapting simultaneously to the unknown smoothness of both the signal and the operator and achieving the optimal rate of convergence to within logarithmic terms. When the noise level in the operator is dominant, by taking full advantage of the blockwise SVD property, we demonstrate that the block SVD procedure outperforms classical methods based on Galerkin projection [14] or nonlinear wavelet thresholding [18]. We subsequently apply our abstract framework to the specific case of blind deconvolution on the torus and on the sphere.

Article information

Source
Electron. J. Statist. Volume 6 (2012), 2274-2308.

Dates
First available in Project Euclid: 30 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1354284420

Digital Object Identifier
doi:10.1214/12-EJS745

Mathematical Reviews number (MathSciNet)
MR3020263

Zentralblatt MATH identifier
1295.62030

Subjects
Primary: 62G05: Estimation 62G99: None of the above, but in this section
Secondary: 65J20: Improperly posed problems; regularization 65J22: Inverse problems

Keywords
Blind deconvolution blockwise SVD circular and spherical deconvolution nonparametric adaptive estimation linear inverse problems error in the operator

Citation

Delattre, S.; Hoffmann, M.; Picard, D.; Vareschi, T. Blockwise SVD with error in the operator and application to blind deconvolution. Electron. J. Statist. 6 (2012), 2274--2308. doi:10.1214/12-EJS745. https://projecteuclid.org/euclid.ejs/1354284420


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References

  • [1] F. Abramovich and B.W. Silverman. Wavelet decomposition approaches to statistical inverse problems., Biometrika, 85:115–129, 1998.
  • [2] O. Bousquet. A Bennett concentration inequality and its application to suprema of empirical processes., C.R. Math. Acad. Sci. Paris, 334:495–500, 2002.
  • [3] T. Cai. Adaptive wavelet estimation: A block thresholding and oracle inequality approach., Ann. Statist., 27:898–924, 1999.
  • [4] T. Cai and H. Zou. A data-driven block thresholding approach to wavelet estimation., Ann. Statist., 37:569–595, 2009.
  • [5] L. Cavalier and N.W. Hengartner. Adaptive estimation for inverse problems with noisy operators., Inverse Problems, 21 :1345–1361, 2005.
  • [6] L. Cavalier and M. Raimondo. Wavelet deconvolution with noisy eigenvalues., IEEE Transactions on signal processing, 55 :2414–2424, 2007.
  • [7] L. Cavalier and M. Raimondo. Wavelet deconvolution with noisy eigenvalues., IEEE Trans. SIgnal Processing, 55 :2414–2424, 2007.
  • [8] L. Cavalier and A.B. Tsybakov. Sharp adaptation for inverse problems with random noise., Probab. Theory Relat. Fields, 123:323–254, 2002.
  • [9] F. Comte and C. Lacour. Data driven density estimation in presence of unknown convolution operator., J. Royal Stat. Soc., Ser B., 73:601–627, 2011.
  • [10] J.G. McNally, C. Preza, J.A. Conchello and L.J. Thomas. Artifacts in computational optical-sectioning microscopy., J. Opt. Soc. Am. A, 11 :1056–1067, 1994.
  • [11] K.R. Davidson and S.J. Szarek., Local operator theory, random matrices and Banach spaces. In Handbook on the Geometry of Banach Spaces 1. North-Holland, Amsterdam, (w. b. johnson and j. lindenstrauss, eds.) edition, 2001.
  • [12] D. Donoho. Nonlinear solution of linear inverse problems by wavelet–vaguelette decomposition., Appl. Comput. Harmon. Anal., 2:101–126, 1995.
  • [13] D.L. Donoho and I.M. Johnstone. Adapting to unknown smoothness via wavelet shrinkage., J. Amer. Statist. Assoc., 90 :1200–1224, 1995.
  • [14] S. Efromovich and V. Kolchinskii On inverse problems with unknown operators., IEEE Transf. Inf. Theory, 47 :2876–2894, 2001.
  • [15] S.F. Gibson and F. Lanni. Diffraction by a circular aperture as a model for three-dimensional optical microscopy., J. Opt. Soc. Am. A, 6, 1357–1367.
  • [16] D.M. Healy, H. Hendriks and P.T. Kim. Spherical deconvolution., J. Multivariate Anal., 67, 1–22.
  • [17] A. Cohen, M. Hoffmann and M. Reiß. Adaptive wavelet Galerkin methods for linear inverse problems., SIAM J. Numer. Anal., 42 :1479–1501, 2004.
  • [18] M. Hoffmann and M. Reiß. Nonlinear estimation for linear inverse problems with error in the operator., Ann. Statist., 36:310–336, 2008.
  • [19] I.A. Ibragimov and R.Z Hasminskii., Statistical Estimation. Asymptotic Theory. Springer-Verlag, 1981.
  • [20] J. Johannes. Deconvolution with unknown error distribution., Ann. Statist., 37 :2301–2323, 2009.
  • [21] J. Johannes and M. Schwarz. Adaptive circular deconvolution by model selection under unknown error distribution. arXiv :0912.1207, 2012.
  • [22] J. Johannes and M. Schwarz. Adaptive gaussian inverse regression with partially unknown operator. arXiv :1204.1226v1 [math.ST], 2012.
  • [23] H.E. Keller., Handbook of Biological Confocal Microscopy. Plenum Press, New York, 2nd edition edition, 1995. chapter Objective lenses for confocal microscopy.
  • [24] P. Hall, G. Kerkyacharian and D. Picard. On the minimax optimality of block thresh- olded wavelet estimators., Statist. Sinica, 9:33–50, 1999.
  • [25] Koo Kim and Luo. Weyl eigenvalue asymptotics and sharp adaptation on vector bundles., Journal of Multivariate Analysis, page 1962–1978, 2009.
  • [26] P.T. Kim and Y.Y. Koo Optimal spherical deconvolution., J. Multivariate Anal., 80:21–42, 2002.
  • [27] P. Massart., Concentration inequalities and model selection. Ecole d’Eté de Probabilités de Saint-Flour XXXIII (Jean Picard ed.), Lecture Notes in Mathematics 1986. Springer, 2007.
  • [28] M.H. Neumann. On the effect of estimating the error density in nonparametric deconvolution., J. Nonparametr. Statist., 7:307–330, 1997.
  • [29] G. Kerkyacharian, T.M. Pham Ngoc and D. Picard. Localized deconvolution on the sphere., Ann. Statist., 39 :1042–1068, 2011.
  • [30] M. Nussbaum and S.V. Pereverzev The degree of ill-posedness in stochastic and deterministic models., Preprint No. 509, Weierstrass Institute (WIAS), Berlin, 1999.
  • [31] P. Pankajakshan, L. Blanc-Féraud, B. Zhang, Z. Kam, J-C. Olivo-Marin and J. Zerubia. Parametric blind deconvolution for confocal laser scanning microscopy (clsm)-proof of concept., Projet INRIA Ariana, research report 6493, 2008.
  • [32] G. Kerkyacharian, G. Kyriazis, E. Le Pennec, P. Petrushev and D. Picard. Inversion of noisy radon transform by svd based needlets., Appl. Comput. Harmonic Anal., 28:24–45, 2010.
  • [33] I. Johnstone, G. Kerkyacharian, D. Picard and M. Raimondo. Wavelet deconvolution in a periodic setting., J. R. Stat. Soc. Ser. B Stat. Methodol., 66:547–573, 2004.
  • [34] G. Reiner, C. Cremer, S. Hell and E.H.K. Stelzer. Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index., J. Microscopy, 169:391–405, 1993.
  • [35] P.A. Stokseth. Properties of a defocused optical system., J. Opt. Soc. Am. A, 59 :1314–1321, 1969.
  • [36] A.B. Tsybakov. On the best rate of adaptive estimation in some inverse problems., C.R. Acad. Sci. Paris Sér. I Math., 330:835–840, 2000.
  • [37] A.C.M. van Rooij and F.H. Ruymgaart. Regularized deconvolution on the circle and the sphere., In Nonparametric Functional Estimation and Related Topics (Spetses, 1990). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 335:679–690, 1990.
  • [38] N.J. Vilenkin., Fonctions spéciales et théorie de la représentation des groupes. Monographies Universitaires de Mathématiques, 33. Dunod, Paris, 1969.