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Computationally efficient estimators for sequential and resolution-limited inverse problems. (English) Zbl 1349.62208
Summary: A common problem in the sciences is that a signal of interest is observed only indirectly, through smooth functionals of the signal whose values are then obscured by noise. In such inverse problems, the functionals dampen or entirely eliminate some of the signal’s interesting features. This makes it difficult or even impossible to fully reconstruct the signal, even without noise. In this paper, we develop methods for handling sequences of related inverse problems, with the problems varying either systematically or randomly over time. Such sequences often arise with automated data collection systems, like the data pipelines of large astronomical instruments such as the Large Synoptic Survey Telescope (LSST). The LSST will observe each patch of the sky many times over its lifetime under varying conditions. A possible additional complication in these problems is that the observational resolution is limited by the instrument, so that even with many repeated observations, only an approximation of the underlying signal can be reconstructed. We propose an efficient estimator for reconstructing a signal of interest given a sequence of related, resolution-limited inverse problems. We demonstrate our method’s effectiveness in some representative examples and provide theoretical support for its adoption.
MSC:
62H12 Estimation in multivariate analysis
62H35 Image analysis in multivariate analysis
65F22 Ill-posedness and regularization problems in numerical linear algebra
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
62P35 Applications of statistics to physics
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[1] Backus, G. and Gilbert, F. (1968). The resolving power of gross Earth data. Geophysical Journal of the Royal Astronomical Society 16 169-205. · Zbl 0177.54102
[2] Bahadur, R. R. (1954). Sufficiency and statistical decision functions. The Annals of Mathematical Statistics 25 423-462. · Zbl 0057.35604
[3] Beran, R. (2000). Scatterplot smoothers: superefficiency through basis economy. Journal of the American Statistical Association 95 155-171. · Zbl 1013.62073
[4] Berenstein, C. and Patrick, E. V. (1990). Exact deconvolution for multiple convolution operators - an overview, plus performance characterizations for imaging sensors. Proceedings of the IEEE 78 723-734.
[5] Bertero, M. and Boccacci, P. (1998). Introduction to Inverse Problems in Imaging . IOP Publishing, Bristol. · Zbl 0914.65060
[6] Bertero, M. and Boccacci, P. (2000a). Image restoration methods for the Large Binocular Telescope. Astronomy and Astrophysics Supplement Series 147 323-333.
[7] Bertero, M. and Boccacci, P. (2000b). Application of the OS-EM method to the restoration of Large Binocular Telescope images. Astronomy and Astrophysics Supplement Series 144 181-186.
[8] Brown, L. D. (1975). Estimation with incompletely specified loss functions (the case of several location parameters. Journal of the American Statistical Association 70 417-427. · Zbl 0333.62013
[9] Brown, L. D., Nie, H. and Xie, X. (2012). Ensemble minimax estimation for multivariate normal means. The Annals of Statistics .
[10] Candés, E. J. and Donoho, D. L. (2002). Recovering edges in ill-posed inverse problems: optimality of curvelet frames. Annals of Statistics 30 784-842. · Zbl 1101.62335
[11] Casey, S. and Walnut, D. (1994). Systems of convolution equations, deconvolutions, shannon sampling, and the wavelet and Gabor transforms. SIAM Review 36 537-577. · Zbl 0814.45001
[12] Cavalier, L. (2008). Nonparametric statistical inverse problems. Inverse Problems 24 . · Zbl 1137.62323
[13] Cavalier, L. and Tsybakov, A. B. (2002). Sharp adaptation for inverse problems with random noise. Probability Theory and Related Fields 123 323-354. · Zbl 1039.62031
[14] Cavalier, L., Golubev, G. K., Picard, D. and Tsybakov, A. B. (2002). Oracle inequlities for inverse problems. Annals of Statistics 30 843-874. · Zbl 1029.62032
[15] Correia, S., Carbillet, M., Boccacci, P., Bertero, M. and Fini, L. (2002). Restoration of interferometric images. Astronomy & Astropysics 387 733-743.
[16] Donoho, D. L. (1995). Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition. Applied and Computational Harmonic Analysis 101-126. · Zbl 0826.65117
[17] Donoho, D. L. and Johnstone, I. M. (1995). Adapting to unknown smoothness via wavelet shrinkage. Journal of the American Statistical Association 90 1200-1224. · Zbl 0869.62024
[18] Gallager, R. G. (2008). Principals of Digital Communication . Cambridge. · Zbl 1183.94001
[19] Goodman, N. R. (1963). Statistical analysis based on a certain multivariate complex Gaussian distribution. The Annals of Mathematical Statistics 34 152-177. · Zbl 0122.36903
[20] Gray, R. M. (2001). Toeplitz and circulant matrices: a review .
[21] Halko, N., Martinsson, P. G. and Tropp, J. A. (2009). Finding structure with randomness: Stochastic algorithms for constructing approximate matrix decompositions. California Inst. Tech., Sep. 2009 ACM Report 2009-05 . · Zbl 1269.65043
[22] Horn, R. A. and Johnson, C. R. (1985). Matrix Analysis . Cambridge University Press. · Zbl 0576.15001
[23] Ólafsson, G. and Quinto, E. T. (2005). The Radon Transform, Inverse Problems, and Tomography: Short Course . American Mathematical Society, Atlanta Georgia. · Zbl 1085.44001
[24] O’Sullivan, F. (1986). A statistical perspective on ill-posed inverse problems. Statistical Science 1 502-527. · Zbl 0625.62110
[25] Piana, M. and Bertero, M. (1996). Regularized deconvolution of multiple images of the same object. J. Opt. Soc. Am. A 13 1516-1523.
[26] Robertson, T., Wright, F. T. and Dykstra, R. L. (1988). Order Restricted Statistical Inference . John Wiley and Sons, Great Britain. · Zbl 0645.62028
[27] Schreier, P. J. and Scharf, L. L. (2003). Second-order analysis of improper complex random vectors and processes. Signal Processing, IEEE Transactions on 51 714-725. · Zbl 1369.94398
[28] Schreier, P. J., Scharf, L. L. and Mullis, C. T. (2005). Detection and estimation of improper complex random signals. Information Theory, IEEE Transactions on 51 306-312. · Zbl 1303.94013
[29] Seber, G. A. F. and Lee, A. J. (2003). Linear Regression Analysis . John Wiley and Sons, Great Britain. · Zbl 1029.62059
[30] Starck, J. L., Pantin, E. and Murtagh, F. (2002). Deconvolution in Astronomy: a review. Publications of the Astronomy Society of the Pacific 114 1051-1069.
[31] Stein, C. M. (1981). Estimation of the mean of a multivariate normal distribution. The Annals of Statistics 90 1247-1256.
[32] Tenorio, L. (2001). Statistical regularization of inverse problems. SIAM Review 43 347-366. · Zbl 0976.65114
[33] van Dyk, D., Connors, A., Esch, D. N., Freeman, P., Kang, H., Karovska, M., Kashyap, V., Siemiginowska, A. and Zezas, A. (2006). Deconvolution in high-energy Astrophysics: science, instrumentation, and methods. Bayesian Analysis 1 189-236. · Zbl 1331.85008
[34] Wahba, G. (1990). Spline Models for Observational Data . SIAM, Philadelphia, PA. · Zbl 0813.62001
[35] Wooding, R. (1956). The multivariate distribution of complex normal variables. Biometrika 43 212-215. · Zbl 0070.36204
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