Sur un problème d'estimation pour des processus de Poisson composés et filtrés - On a problem of estimation for composed and filtered Poisson processes

Sur un problème d'estimation pour des processus de Poisson composés et filtrés

On a problem of estimation for composed and filtered Poisson processes

Alfred O. Hero III

Dept. of Electrical Engineering and Computer Science University of Michigan, Ann Arbor, MI 48109-2122, USA

Corresponding Author Email: 
hero©eecs.umich.edu
Page: 
493-502
|
Received: 
16 February 1998
| |
Accepted: 
N/A
| | Citation

OPEN ACCESS

Abstract: 

Compound and filtered Poison processes are useful models for many applications in signal processing, image processing, and communication. One of the earliest imaging applications of these models was proposed by Bernard Picinbono in a 1955 paper on silver dye photographs . In this paper we treat a generalized model with the primiary objective being to estimate parameters of the filtered Poisson process in the presence of spatial smoothing and additive Gaussian noise. By imbedding the estimation problem into the context of information theory we decompose the model into the cascade of a discrete event Poisson process channel and a continuous Gaussian waveform channel. This naturally leads to a expectation-maximization (EM) type estimation algorithm and a distortion-rate lower bound on estimation error.

Résumé

Les processus de Poisson composés et filtrés forment une classe de modèles très utile pour certaines applications en traitement du signal, traitement de l'image, et télécommunications. Une des premières applications de ce type de modèle en traitement de l'image a été proposée par Bernard Picinbono en 1955 pour la distribution des grains d'argent dans un film photographique. Ici on introduit un modeèle de Picinbono généralisé dont l'objectif est d'estimer les paramètres du processus de Poisson filtré en présence de lissage spatial et de bruit additif Gaussien. En posant le problème de l'estimation dans le contexte de la théorie de l'information, on est conduit à une représentation du modèle par la composition d'un canal Poissonnien et d'un canal Gaussien. Cette composition mène naturellement à un estimateur paramétrique du type «expectation-maximization» (EM) et à une borne du type «distortion-rate» sur l'erreur d'estimation.

Keywords: 

Boolean images, granulometry, EM algorithm, Shannon bound, coverage processes

Mots clés

Images Booléennes, granulométrie, algorithme EM, borne de Shannon, processus de recouvrement

1. Introduction
2. Un Modèle Général
3. Représentation Statistique Par Un Canal D'information
4. Un Estimateur MAP Du Type EM
5. Une Borne Inférieure Sur L'erreur D'estimation
6. Application Au Processus De Poisson Filtré
7. Conclusion
8. Remerciements
  References

[1] J . Amoss and F. Davidson, «Detection of weak optical images with photon counting techniques», Applied Optics, vol . 11, pp . 1793-1799, 1972.

[2] C . Andrieu and P. Duvaut, «Bayesian estimation and detection of shot nois processes using reversible jumps», in Proc. IEEE Int. Conf. Acoust., Speech, and Sig . Proc ., pp . 3681-3684, Munich, Germany, 1997 .

[3] N. Antoniadis and A . O . Hero, «Time delay estimation for filtered Poisson processes using an EM-type algorithm», IEEE Trans . on Signal Processing , vol . 42, no . 8, pp . 2112-2123, 1994 .

[4] I. Bar-David, «Information in the time of arrival of a photon packet : capacity of PPM channels», J. Opt. Soc. Am., vol . 63, No. 2, pp. 166-170, Feb. 1973.

[5] I. Bar-David, «Minimum-mean-square-error estimation of photon pulse delay», IEEE Trans. on Inform . Theory, pp. 326-330, May 1975 .

[6] T. Berger, Rate Distortion Theory : A Mathematical Basis for Data Compression, Prentice-Hall, Englewood Cliffs NJ, 1971 .

[7] A. M. Bruckstein, T. J. Shan, and T. Kailath, «The resolution of overlapping echos» , IEEE Trans. Acoust., Speech, and Sig. Proc., vol . ASSP-33, pp. 1357-1368, Dec. 1985.

[8] N . H . Clinthorne, W. L . Rogers, L . Shao, and K . Korat, «A hybrid maximum likelihood position computer for scintillation cameras», IEEE Trans. Nuclear Science, vol . NS-37, no. 2, pp . 658-663, 1990 .

[9] A . P. Dempster, N . M . Laird, and D . B . Rubin, «Maximum likelihood from incomplete data via the EM algorithm», J. Royal Statistical Society, Sen B , vol. 39, no . 1, pp . 1-38, 1977 .

[10] P. J . Diggle, «Binary mosaics and the spatial pattern of heather», Biometrics , vol . 37, pp . 531-539, 1981 .

[11] L . J . Dorfman, «The distribution of conduction velocities (DCV) in peripheral nerves : A review», Muscle Nerve, vol . 7, pp . 2-11, 1984 .

[12] A . R. Eckler, «A survey of coverage problems associated with point and area targets», Technometrics, vol . 11, pp. 561-589, 1969 .

[13] H. Elias, Stereology, Springer, Berlin, 1967 .

[14] P. Faure, «Theoretical models of reverberation noise», J. Acoust. Soc. Am. , vol. 36, pp . 259-268, 1964 .

[15] R . M . Gagliardi and S . Karp, Optical Communications, Wiley, New York , 1976 .

[16] R. G . Gallager, Information Theory and Reliable Communication, Wiley, 1968 .

[17] F. Gatti and V. Svelto, «Review of theories and experiments of resolving time with scintillation counters», Nuclear Instruments and Methods, vol . 43, pp. 248-268, 1966 .

[18] J . A . Gubner, «Computation of shot-noise probability distributions and densities», SIAM J. Sci. Comput., vol . 17, pp . 750-761, May 1996 .

[19] P. Hall, Introduction to the theory of coverage processes, Wiley, New York , 1988 .

[20] A . O . Hero, «Timing estimation for a filtered Poisson process in Gaussian noise», IEEE Trans. on Inform. Theory, vol . 37, no. 1, pp . 92-106, Jan. 1991 .

[21] A . O. Hero, «Lower bounds on estimator performance for energy invariant parameters of multi-dimensional Poisson processes», IEEE Trans. on Inform . Theory, vol . 35, pp . 843-858, July 1989.

[22] A. O. Hero, «Recovering photon-intensity information from continuous photo-detector measurements», in Proceedings of the 25-th Conference on Information Sciences and Systems, pp. 643-648, Johns Hopkins University, Baltimore, MD, Mar. 1991.

[23] A. O . Hero, «Theoretical limits for optical position estimation using imaging arrays», in Actes du Colloque GRETSI, pp. 793-796, Juan-les-Pins, France , Sept. 1991 .

[24] A . O . Hero, N . Antoniadis, N . H . Clinthorne, and W. L . Rogers, «Optimal and sub-optimal post-detection timing estimators for PET», Proc. of IEEE Nuclear Science Symposium, vol . NS-37, no . 2, pp . 725-729, April 1990 .

[25] A. O. Hero, N . Antoniadis, N . H . Clinthorne, W. L. Rogers, and G . D . Hutchins, «Optimal and sub-optimal post-detection timing estimators for PET», IEEE Trans. Nuclear Science, vol . NS-37, no . 2, pp . 725-729, 1990 .

[26] A. O . Hero and J. A . Fessier, «A recursive algorithm for computing CR-type bounds on estimator covariance», IEEE Trans . on Inform. Theory, vol. 40 , pp . 1205-1210, July 1994 .

[27] T. T. Kadota, «Approximately optimum detection of deterministic signals in Gaussian and compound Poisson noise», IEEE Trans . on Inform. Theory , vol . 34, pp . 1517-1527, Nov . 1988 .

[28] L. Kazovsky, «Beam position estimation by means of detector arrays», Opt. Quantum Electron., vol. 13, pp. 201-208, 1981.

[29] J. M . Mendel, «White noise estimators for seismic data processing in oil exploration», IEEETrans. Automatic Control, vol . AC-22, no . 5, pp. 694-706, Oct. 1977 .

[30] P. N . Misra and H. W. Sorenson, «Parameter estimation in Poisson processes», IEEE Trans . on Inform. Theory, pp. 87-90, Jan . 1975 .

[31] J . J . O'Reilly, «Generating functions and bounds in optical communications», in Problems of randomness in communications engineering, K . Cattermole and J . O'Reilly, editors, chapter 7, pp . 119-133, Wiley, New York, Nov . 1987 .

[32] B . Picinbono, «Modèle statistique suggéré par la distribution de grains d'argent dans les films photographiques», Comptes Rendus de l'Académie des Sciences, vol . Séance du 6 Juin, pp . 2206-2208, 1955 .

[33] B . D . Ripley, Spatial statistics, Wiley, New York, 1981 .

[34] D. J. Sakrison and V. R. Algazi, «Comparison of line-by-line and two - dimensional encoding of random images», IEEE Trans. on Inform. Theory, vol. IT-17, no. 4, pp . 386-398, July 1971 .

[35] P. Salomon and T. Glavich, «Image signal processing in sub-pixel accuracy star trackers», in Proc. Soc. Phot-Opt. Instrum. Eng., pp . 64–74, 1980 .

[36] R. Schoonhoven, D. F. Stegeman, and J. P. C. de Weerd, «The forward problem in electroneurography – I: A generalized volume conductor model», IEEE Trans. on Biomed. Eng., vol. BME-33, pp. 327–334, 1986.

[37] J . Serra, Image analysis and mathematical morphology, Academic Press, New York, 1982 .

[38] D. L . Snyder and M . I . Miller, Random Point Processes in Time and Space , Springer-Verlag, New York, 1991 .

[39] M . N. Wernick and G. M . Morris, «Image classification at low light levels» , J. Opt. Soc . Am ., vol. 3, pp . 2179–2187, 1986 .