Poisson inverse problems

Annals of Statistics

Volumn 34, Issue 5, 2006, Pages 2132-2158

  Antoniadis, Anestis ^a   Bigot, Jéremie ^b  

^a UNIV GRENOBLE ALPES   (France)

^b UNIVERSITÉ PAUL SABATIER   (France)

Author keywords

Adaptive estimation; Besov spaces; Galerkin inversion; Integral equation; Intensity function; Poisson process; Wavelet thresholding; Wavelets

Indexed keywords

EID: 33847378070     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/009053606000000687     Document Type: Article

References (27)

1. ABRAMOVITCH, F. and SILVERMAN, B. W. (1998). Wavelet decomposition approaches to statistical inverse problems. Biometrika 85 115-129. MR1627226
2. BARRON, A. R. and SHEU, C.-H. (1991). Approximation of density functions by sequences of exponential families. Ann. Statist. 19 1347-1369. MR1126328
3. CAVALIER, L. and KOO, J.-Y. (2002). Poisson intensity estimation for tomographic data using a wavelet shrinkage approach. IEEE Trans. Inform. Theory 48 2794-2802. MR1930348
4. COHEN, A., DEVORE, R., KERKYACHARIAN, G. and PICARD, D. (2001). Maximal spaces with given rate of convergence for thresholding algorithms. Appl. Comput. Harmon. Anal. 11167-191. MR1848302
5. COHEN, A., HOFFMANN, M. and REISS, M. (2004). Adaptive wavelet-Galerkin methods for inverse problems. SIAMJ. Numer. Anal. 42 1479-1501. MR2114287
6. CSISZÁR, I. (1975). I-divergence geometry of probability distributions and minimization problems. Ann. Probab. 3 146-158. MR0365798
7. DONOHO, D. L. (1995). Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition. Appl. Comput. Harmon. Anal. 2 101-126. MR1325535
8. DONOHO, D. L., JOHNSTONE, I. M., KERKYACHARIAN, G. and PICARD, D. (1996). Density estimation by wavelet thresholding. Ann. Statist. 24 508-539. MR1394974
9. FAN, J. and KOO, J.-Y. (2002). Wavelet deconvolution. IEEE Trans. Inform. Theory 48 734-747. MR1889978
10. FIGUEIREDO, M. and NOWAK, R. (2003). An EM algorithm for wavelet-based image restoration. IEEE Trans. Image Process. 12 906-916. MR2008658
11. JOHNSTONE, I. M. and SILVERMAN, B. W. (1990). Speed of estimation in positron emission tomography and related inverse problems. Ann. Statist. 18 251-280. MR1041393
12. HALL, P. and SMITH, R. L. (1988). The kernel method for unfolding sphere size distributions. J. Comput. Phys. 74 409-421. MR0930210
13. KIM, W.-C. and KOO, J.-Y. (2002). Inhomogeneous Poisson intensity estimation via information projections onto wavelet subspaces. J. Korean Statist. Soc. 31 343-357. MR2001776
14. KOLACZYK, E. D. (1999). Bayesian multiscale models for Poisson processes. J. Amer. Statist. Assoc. 94 920-933. MR1723303
15. KOLACZYK, E. D. (1999). Wavelet shrinkage estimation of certain Poisson intensity signals using corrected thresholds. Statist. Sinica 9 119-135. MR1678884
16. KOLACZYK, E. D. and NOWAK, R. D. (2004). Multiscale likelihood analysis and complexity penalized estimation. Ann. Statist. 32 500-527. MR2060167
17. KOO, J.-Y. and KIM, W.-C. (1996). Wavelet density estimation by approximation of logdensities. Statist. Probab. Lett. 26 271-278. MR1394903
18. KOVAC, A. and SILVERMAN, B. W. (2000). Extending the scope of wavelet regression methods by coefficient-dependent thresholding. J. Amer. Statist. Assoc. 95 172-183.
19. MASE, S. (1995). Stereological estimation of particle size distributions. Adv. in Appl. Probab. 27 350-366 MR1334818
20. NOWAK, R. D. and KOLACZYK, E. D. (2000). A statistical multiscale framework for Poisson inverse problems. Information-theoretic imaging. IEEE Trans. Inform. Theory 46 1811-1825. MR1790322
21. NYCHKA, D. and COX, D. D. (1989). Convergence rates for regularized solutions of integral equations from discrete noisy data. Ann. Statist. 17 556-572. MR0994250
22. NYCHKA, D., WAHBA, G., GOLDFARB, S. and PUGH, T. (1984). Cross-validated spline methods for the estimation of three-dimensional tumor size distributions from observations on two-dimensional cross sections. J. Amer. Statist. Assoc. 79 832-846. MR0770276
23. O' SULLIVAN, F. (1986). A statistical perspective on ill-posed problems (with discussion). Statist. Sci. 1 502-527. MR0874480
24. REYNAUD-BOURET, P. (2003). Adaptive estimation of the intensity of inhomogeneous Poisson processes via concentration inequalities. Probab. Theory Related Fields 126 103-153. MR1981635
25. SILVERMAN, B. W., JONES, M. C., NYCHKA, D. and WILSON, J. D. (1990). A smoothed EM approach to indirect estimation problems, with particular reference to stereology and emission tomography (with discussion). J. Ray. Statist. Soc. Ser. B 52 271-324. MR1064419
26. SZKUTNIK, Z. (2000). Unfolding intensity function of a Poisson process in models with approximately specified folding operator. Metrika 52 1-26. MR1791921
27. VANNUCCI, M. and CORRADI, F. (1999). Covariance structure of wavelet coefficients: Theory and models in a Bayesian perspective. J. R. Stat. Soc. Ser. B Stat. Methodol. 61 971-986. MR1722251