Detection in the Presence of Speckle Using Multivariate Mixed Poisson Distributions. Lois de Poisson Mélangées Multivariées : Application À la Détection en Présence de Speckles

Detection in the Presence of Speckle Using Multivariate Mixed Poisson Distributions

Lois de Poisson Mélangées Multivariées : Application À la Détection en Présence de Speckles

Florent Chatelain André Ferrari  Jean-Yves Tourneret  

Gipsa-lab, Grenoble INP, 961 rue de la Houille Blanche, BP 46, 38042 Saint Martin d’Hères

Lab. Fizeau, UNS/OCA/CNRS, Parc Valrose, 06108 Nice cedex 2

IRIT/ENSEEIHT/TéSA, 2 rue Charles Camichel, BP 7122, 31071 Toulouse cedex 7

Page: 
227-238
|
Received: 
18 June 2008
|
Accepted: 
N/A
|
Published: 
30 June 2009
| Citation

OPEN ACCESS

Abstract: 

This paper studied a family of multivariate mixed Poisson distributions that plays a central role in the statistic model associated to photocount vectors. Several estimation methods were studied for the estimation of the unknown parameters associated to these multivariate mixed Poisson distributions. An estimator was derived by considering spatial correlation between adjacent pixels of the image. This estimator performed very similarly when compared to another estimator neglecting the correlation between adjacent pixels of the image. Future investigations include the consideration of stronger prior information for the correlation parameters. This prior information might be obtained from specific data, e.g., coming from the wavefront sensor.

Résumé

L’article étudie l’estimation des paramètres de l’amplitude complexe du front d’onde à partir de mesures d’intensités faible-flux. La corrélation spatiale de l’amplitude complexe du front d’onde est modélisée au sein d’une clique de l’image observée. Le cas général où cette amplitude complexe est de moyenne non nulle est également considéré. Un estimateur classique des moments, ainsi qu’un estimateur des moindres carrés non-linéaires prenant en compte les corrélations spatiales sont proposés. Dans le cadre de la détection d’exo-planètes par imagerie directe, les performances de ces estimateurs et des détecteurs associés sont étudiées et comparées. Les résultats obtenus mettent en évidence la nécessité de disposer d’un a priori plus fort sur la corrélation spatiale du front d’onde afin d’améliorer les performances d’estimation et de détection.

Keywords: 

Speckle, spatial correlation, multivariate mixed Poisson distribution, composite likelihood, method of moments, nonlinear least squares, exoplanet

Mots clés

Speckle, corrélation spatiale, lois de Poisson mélangées multivariées, vraisemblance composite, méthode de moments, moindres carrés non linéaires, détection d’exo-planètes.

1. Introduction
2. Modèle Statistique
3. Estimation des Paramètres et Détection
4. Application à la Détection d’Exoplanètes
5. Conclusion
  References

[1] C. AIME. Analysis of the technique of dark speckles for detection of exo-planets. Journal of Optics A : Pure and Applied Optics, pages 411-421, 2000.

[2] C. AIME and R. SOUMMER. Influence of speckle and Poisson noise on exoplanet detection with a coronograph. In EUSIPCO-04, pages 509-512, Vienna, Austria, Sept. 2004. Elsevier.

[3] C. AIME and F. VAKILI, editors. Direct Imaging of Exoplanets : Science & Techniques. International Astronomical Union Colloquium 200, Cambridge University Press, 2006.

[4] S. K. BAR-LEV, D. BSHOUTY, P. ENIS, G. LETAC, I-Li LU, and D. RICHARDS. The diagonal multivariate natural exponential families and their classification. J. of Theoret. Probab., 7(4):883-929, Oct. 1994.

[5] P. BERNARDO. Which negative multinomial distributions are infinitely divisible ? Bernoulli, 9(6), 2003.

[6] P. BERNARDO. Which multivariate Gamma distributions are infinitely divisible ? Bernoulli, 12(1) :169-189, 2006.

[7] V.F. CANALES and M.P. CAGIGAL. Photon statistics in partially compensated wave fronts. Journal of the Optical Society of America, 16:2550-2554, oct 1999.

[8] F. CHATELAIN, A. FERRARI, and J.-Y. TOURNERET. Parameter estimation for multivariate mixed poisson distributions. In IEEE ICASSP, volume 3, pages 684-687, Toulouse, May. 2006.

[9] F. CHATELAIN and J.-Y. TOURNERET. Composite likelihood estimation for multivariate mixed Poisson distributions. In Proc. IEEESP Workshop Stat. Signal Processing, Bordeaux, France, July 2005.

[10] Florent CHATELAIN, Sophie LAMBERT-LACROIX, and Jean- Yves TOURNERET. Pairwise likelihood estimation for multivariate mixed Poisson models generated by gamma intensities. Statistics and Computing. To appear (published online in Sept. 2008).

[11] A. FERRARI and C. AIME. Étude statistique de la détection d’exoplanètes en imagerie courte pose. In Actes du 18ème colloque GRETSI, pp. 669-672, Toulouse, 2001. http://hdl.handle.net/2042/13384.

[12] A. FERRARI, G. LETAC, and J.-Y. TOURNERET. Multivariate mixed Poisson distributions. In EUSIPCO, Vienna, Austria, Sept. 2004. Elsevier.

[13] J. GOODMAN. Statistical Optics. Wiley, New York, 1985.

[14] J.W. GOODMAN. Statistical properties of laser speckle pattern. In J.C. Dainty, editor, Laser Speckle and Related Phenomena, pages 9-75. Springer-Verlag, Berlin, 1975.

[15] J. GRANDELL. Mixed Poisson Processes. Chapman and Hall, 1997.

[16] N. L. JOHNSON, S. KOTZ, and N. BALAKRISHNAN. Continuous Univariate Distributions, volume 2. John Wiley, New York, 2nd edition, 1995.

[17] N. L. JOHNSON, S. KOTZ, and A. W. KEMP. Univariate discrete distributions. John Wiley, New York, 2nd edition, 1992.

[18] A. LABEYRIE. Images of exo-planets obtainable from dark speckles in adaptive telescopes. Astronomy and Astrophysics, 298:544-548, 1995.

[19] P MCCULLAGH. Tensor methods in statistics. Monographs on statistics and applied probability. Chapman and Hall, London, 1987.

[20] Robb J MUIRHEAD. Aspects of multivariate statistical theory. Wiley series in probability and mathematical statistics. Wiley, New York, 1982.

[21] Bernard PICINBONO. Théorie des signaux et des systèmes avec problèmes résolus. Dunod Université, Paris, 1989.

[22] B. PORAT and B. FRIEDLANDER. Performance analysis of parameter estimation algorithms based on high-order moments. International Journal of adaptive control and signal processing, 3:191-229, 1989.

[23] J.-Y. TOURNERET, A. FERRARI, and G. LETAC. The noncentral Wishart distribution : Properties and application to speckle imaging. In Proc. IEEE-SP Workshop Stat. Signal Processing, Bordeaux, France, jul 2005.

[24] C. VARIN. On composite marginal likelihoods. Advances in Statistical Analysis, 92(1):1-28, fev. 2008.