Some results on the denoising problem in the elliptically distributed context

IEEE Transactions on Signal Processing

Volumn 58, Issue 1, 2010, Pages 134-150

  Zozor, Steeve ^a   Vignat, Christophe ^b,c,d  

^a Grenoble Informatics Laboratory   (France)

^b Université de Marne la Vallée   (France)

^c L S S   (France)

^d UNIVERSITÉ PARIS SACLAY   (France)

Author keywords

Denoising; Elliptically symmetric distributions; Estimation; Maximum a posteriori; Minimum mean square error

Indexed keywords

DE-NOISING; ELLIPTICALLY SYMMETRIC DISTRIBUTIONS; MAXIMUM A POSTERIORI; MINIMUM MEAN SQUARE ERRORS; SYMMETRIC DISTRIBUTIONS;

BLOCK CODES; MEAN SQUARE ERROR;

ESTIMATION;

EID: 72949090875     PISSN: 1053587X     EISSN: None     Source Type: Journal    
DOI: 10.1109/TSP.2009.2030840     Document Type: Article

References (56)

1. T. I. Alecu, S. Voloshynovskiy, and T. Pun, "The Gaussian transform of distributions: Definition, computation and application," IEEE Trans. Signal Process., vol.54, pp. 2976-2985, Aug. 2006.
2. F. Shi and I. W. Selesnick, "An elliptically contoured exponential mixture model for wavelet based image denoising," Appl. Comput. Harmon. Anal., vol.23, no.1, pp. 131-151, Jul. 2007.
3. I. W. Selesnick, "The estimation of Laplace random vectors in additive white Gaussian noise," IEEE Trans. Signal Process., vol.56, pp. 3482-3496, Aug. 2008.
4. Y. Chitour and F. Pascal, "Exact maximum likelihood estimates for SIRV covariance matrix: Existence and algorithm analysis," IEEE Trans. Signal Process., vol.56, pp. 4563-4573, Oct. 2008.
5. J. C. Maxwell, The Scientific Papers of James Clerk Maxwell, W. D. Nieven, Ed. New York: Dover, 1952, vol.1.
6. M. S. Bartlett, "The vector representation of a sample," Proc. Cambridge Phil. Soc., vol.30, no.3, pp. 327-340, Jul. 1934.
7. M. A. Chmielewski, "Elliptically symmetric distributions: A review and bibliography," Int. Stat. Rev., vol.49, no.1, pp. 67-74, Apr. 1981.
8. K.-C. Chu, "Estimation and decision for linear systems with elliptical random processes," IEEE Trans. Autom. Control, vol.AC-18, pp. 499-505, Oct. 1973.
9. K. Yao, "A representation theorem and its applications to sphericallyinvariant random processes," IEEE Trans. Inf. Theory, vol.IT-19, pp. 600-608, Sep. 1973.
10. J. Goldman, "Detection in the presence of spherically symmetric random vectors," IEEE Trans. Inf. Theory, vol.IT-22, pp. 52-59, Jan. 1976.
11. G. E. P. Box, "Multi-factor designs of first order," Biometrika, vol. 39, no. 1/2, pp. 49-57, May 1952.
12. K. T. Fang, S. Kotz, and K. W. Ng, Symmetric Multivariate and Related Distributions, ser. Monographs on Statistics and Probability. London, U.K.: Chapman & Hall, 1990.
13. M. Rangaswamy, D. Weiner, and A. Öztürk, "Non-Gaussian random vector identification using spherically invariant random processes," IEEE Trans. Aerosp. Electron. Syst., vol.26, pp. 111-124, Jan. 1993.
14. J. Portilla, V. Strela, J. Wainwright, and R. P. Simoncelli, "Image denoising using scale mixtures of Gaussians in the wavelet domain," IEEE Trans. Image Process., vol.12, pp. 1338-1351, Nov. 2003.
15. K. Todros and J. Tabrikian, "Blind separation of independent sources using Gaussian mixture model," IEEE Trans. Signal Process., vol.55, p. 3645, Jul. 2007.
16. W. Feller, An Introduction to Probability Theory and Its Applications. New York: Wiley, 1971, vol.2.
17. D. F. Andrews and C. L. Mallows, "Scale mixtures of normal distributions,"J. Royal Stat. Soc. B, vol.36, no.1, pp. 99-102, 1974.
18. B. Picinbono, "Spherically invariant and compound Gaussian stochastic processes," IEEE Trans. Inf. Theory, vol.IT-17, pp. 77-79, Jan. 1970.
19. R. Lord, "The use of the Hankel transform in statistics I. General theory and examples," Biometrika, vol. 41, no. 1/2, pp. 44-55, Jun. 1954.
20. M. Bilodeau and D. Brenner, Theory of Multivariate Statistics. New York: Springer, 1999.
21. M. L. Eaton, "On the projections of isotropic distributions," Ann. Stat., vol.9, no.2, pp. 391-400, Mar. 1981.
22. R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1985.
23. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. New York: McGraw-Hill, 1991.
24. S. Cambanis, S. Huang, and G. Simons, "On the theory of elliptically contoured distributions," J. Multivar. Anal., vol.11, no.3, pp. 368-385, Sep. 1981.
25. A. G. Laurent, "Applications of fractional calculus to spherical (radial) probability models and generalizations," in Fractional Calculus and Its Applications, ser. Lecture Notes in Mathematics, A. Dold and B. Eckmann, Eds. Berlin, Germany: Springer-Verlag, 1975, vol.457, pp. 256-266.
26. Z. Yang and S. Kotz, "Center-similar distribution with applications in multivariate analysis," Stat. Probab. Lett., vol.64, no.4, pp. 335-345, 2003.
27. J. Keilson and F. W. Steutel, "Mixture of distributions, moment inequalities and measures of exponentiality and normality," Ann. Probab., vol.2, no.1, pp. 112-130, Feb. 1974.
28. H. Teicher, "On the mixture of distributions," Ann. Math. Stat., vol.31, no.1, pp. 55-73, Mar. 1960.
29. R. E. Williamson, "Multiply monotone functions and their Laplace transforms," Duke Math. J., vol.23, no.2, pp. 189-207, Jun. 1956.
30. S. Kotz, N. Balakrishnan, and N. L. Johnson, Continuous Multivariate Distributions, Models and Applications, 2 ed. New York: Wiley, 2000, vol.1.
31. M.West, "On scale mixtures of normal distributions," Biometrika, vol.74, no.3, pp. 646-648, Sep. 1987.
32. A. M. Vershik, "Some characteristic properties of Gaussian stochastic processes," Theory Probab. Appl., vol.9, no.2, pp. 353-356, 1964.
33. D. Kelker, "Infinite divisibility and variance mixture of normal distributions," Ann. Math. Stat., vol.42, no.2, pp. 802-808, Apr. 1971.
34. P. Diaconis and D. Freedman, "A dozen de Finetti-style results in search of a theory," Ann. l'Inst. Henri Poincaré (B) Prob. Stat., vol.23, no.2, pp. 397-423, 1987.
35. D. V. Widder, The Laplace Transform, ser. Princeton Mathematical Series. Princeton, NJ: Princeton Univ. Press, 1946.
36. I. J. Schoenberg, "Metric spaces and completely monotone functions," Ann. Math., vol.39, no.4, pp. 811-841, Oct. 1938.
37. S. Nadarajah, "TheKotz-type distribution with applications," Statistics, vol.37, no.4, pp. 341-358, Jul. 2003.
38. S.Kotz and S. Nadarajah, "Some extremal type elliptical distributions," Stat. Proba.. Lett., vol.54, pp. 171-182, 2001.
39. O. Arslan, "A new class of multivariate distributions: Scale mixture of Kotz-type distributions," Stat. Probab. Lett., vol.75, no.1, pp. 18-28, Nov. 2005.
40. Y. Kano, "Consistency property of elliptic probability density functions," J. Multivar. Anal., vol.51, no.1, pp. 139-147, Oct. 1994.
41. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. San Diego, CA: Academic, 1980.
42. G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes. Stochastic Models With Infinite Variance. New York: Chapman & Hall, 1994.
43. H.-Y. Siew and K. Shimizu, "The generalized symmetric Laplace distribution on the sphere," Stat. Methodol., vol.5, no.6, pp. 487-501, Nov. 2008.
44. N. Mukhopadhyay, Probability and Statistical Inference, ser. Statistics: Textbooks and Monographs, 5 ed. New York: Marcel Dekker, 2000, vol.162.
45. S. Kotz, T. J. Kozubowski, and K. Podgórski, The Laplace Distribution and Generalizations: A Revisit With Applications to Communications, Economics, Engineering, and Finance. Berlin, Germany: Birkhäuser, 2001.
46. T. Eltoft, T. Kim, and T.-W. Lee, "On the multivariate Laplace distribution," IEEE Signal Process. Lett., vol.13, pp. 300-303, May 2006.
47. L. Bombrun and J.-M. Beaulieu, "Fisher distribution for texture modeling of polarimetric SAR data," IEEE Geosci. Remote Sens. Lett., vol.5, pp. 512-516, Jul. 2008.
48. C. Tison, J.-M. Nicolas, and F. T. Maître, "A new statistical model for Markovian classification of urban areas in high-resolution SAR images," IEEE Trans. Geosci. Remote Sens., vol.42, pp. 2046-2057, Oct. 2004.
49. T. J. Kozubowski, "On the vertical density of the multivariate exponential power distribution," Statistics, vol.36, no.3, pp. 219-221, 2002.
50. M. D. Ernst, "A multivariate generalized Laplace distribution," Comp. Stat., vol.13, no.2, pp. 227-232, 1998.
51. S. Zozor and C. Vignat, "Revisiting the denoising problem in the context of elliptical distributions," in Proc. Eur. Signal Process. Conf. (EUSIPCO), Lausanne, Switzerland, Aug. 2008, pp. 25-29.
52. G. F. Carrier, M. Krook, and C. E. Pearson, Function of a Complex Variable: Theory and Technique. Philadelphia, PA: SIAM, 2005.
53. N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, 2nd ed. New York: Wiley, 1995, vol.2.
54. S. M. Kay, Fundamentals for Statistical Signal Processing: Estimation Theory, ser. Upper Saddle River, NJ: Prentice-Hall, 1993, vol.1.
55. G. P. Nason, "On the sum of t and Gaussian random variables," Stat. Probab. Lett., vol.76, no.12, Jul. 2006.
56. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1970.