Some Results on the Denoising Problem in the Elliptically Distributed Context

The purpose of this paper is to revisit the denoising problem of a multivariate elliptically symmetric random vector corrupted by a multivariate elliptically symmetric noise. The investigation begins with a review of multivariate elliptically distributed random vectors and of their basic but important properties. A focus is made on the Gaussian scale mixtures, a subclass of the elliptical distributions. In the second part, the denoising problem is revisited in this elliptical context following two classical directions: the minimum mean square error and the maximum a posteriori estimation approaches. Although subject to some restrictions, this investigation extends the two recent studies by Alecu and Selesnick. The practical use of the proposed estimators and some possible special behaviors are then discussed through various illustrative examples.

[1]  Te-Won Lee,et al.  On the multivariate Laplace distribution , 2006, IEEE Signal Processing Letters.

[2]  Anatoly M. Vershik,et al.  Some Characteristic Properties of Gaussian Stochastic Processes , 1964 .

[3]  D. Widder,et al.  The Laplace Transform , 1943, The Mathematical Gazette.

[4]  H. Teicher.  On the Mixture of Distributions , 1960 .

[5]  M. Kendall.  Probability and Statistical Inference , 1956, Nature.

[6]  Thierry Pun,et al.  The Gaussian Transform of Distributions: Definition, Computation and Application , 2006, IEEE Transactions on Signal Processing.

[7]  John G. Proakis,et al.  Probability, random variables and stochastic processes , 1985, IEEE Trans. Acoust. Speech Signal Process..

[8]  S. Kay.  Fundamentals of statistical signal processing: estimation theory , 1993 .

[9]  竹中 茂夫.  G.Samorodnitsky,M.S.Taqqu:Stable non-Gaussian Random Processes--Stochastic Models with Infinite Variance , 1996 .

[10]  Guy P. Nason,et al.  On the sum of t and Gaussian random variables , 2006 .

[11]  D. Owen.  Handbook of Mathematical Functions with Formulas , 1965 .

[12]  P. J. Green,et al.  Probability and Statistical Inference , 1978 .

[13]  D. Applebaum.  Stable non-Gaussian random processes , 1995, The Mathematical Gazette.

[14]  Kung Yao,et al.  A representation theorem and its applications to spherically-invariant random processes , 1973, IEEE Trans. Inf. Theory.

[15]  Samuel Kotz,et al.  The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance , 2001 .

[16]  M. Bilodeau,et al.  Theory of multivariate statistics , 1999 .

[17]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[18]  Douglas Kelker,et al.  Infinite Divisibility and Variance Mixtures of the Normal Distribution , 1971 .

[19]  K. Chu.  Estimation and decision for linear systems with elliptical random processes , 1972, CDC 1972.

[20]  Y. Kano.  Consistency Property of Elliptic Probability Density Functions , 1994 .

[21]  Ivan W. Selesnick,et al.  The Estimation of Laplace Random Vectors in Additive White Gaussian Noise , 2008, IEEE Transactions on Signal Processing.

[22]  D. F. Andrews,et al.  Scale Mixtures of Normal Distributions , 1974 .

[23]  Morris L. Eaton.  On the Projections of Isotropic Distributions , 1981 .

[24]  R. Lord,et al.  THE USE OF THE HANKEL TRANSFORM IN STATISTICS: I. GENERAL THEORY AND EXAMPLES , 1954 .

[25]  Bernard C. Picinbono.  Spherically invariant and compound Gaussian stochastic processes (Corresp.) , 1970, IEEE Trans. Inf. Theory.

[26]  I. J. Schoenberg.  Metric spaces and completely monotone functions , 1938 .

[27]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[28]  Martin J. Wainwright,et al.  Image denoising using scale mixtures of Gaussians in the wavelet domain , 2003, IEEE Trans. Image Process..

[29]  D. Freedman,et al.  A dozen de Finetti-style results in search of a theory , 1987 .

[30]  Lionel Bombrun,et al.  Fisher Distribution for Texture Modeling of Polarimetric SAR Data , 2008, IEEE Geoscience and Remote Sensing Letters.

[31]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[32]  M. Taqqu,et al.  Stable Non-Gaussian Random Processes : Stochastic Models with Infinite Variance , 1995 .

[33]  Joseph Tabrikian,et al.  Blind Separation of Independent Sources Using Gaussian Mixture Model , 2007, IEEE Transactions on Signal Processing.

[34]  M. S. Bartlett,et al.  The vector representation of a sample , 1934, Mathematical Proceedings of the Cambridge Philosophical Society.

[35]  Steeve Zozor,et al.  Revisiting the denoising problem in the context of elliptical distributions , 2008, 2008 16th European Signal Processing Conference.

[36]  Aarnout Brombacher,et al.  Probability... , 2009, Qual. Reliab. Eng. Int..

[37]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[38]  W. PEDDIE,et al.  The Scientific Papers of James Clerk Maxwell , 1927, Nature.

[39]  Murray Aitkin,et al.  Statistical modelling of artificial neural networks using the multi-layer perceptron , 2003, Stat. Comput..

[40]  Center-similar distributions with applications in multivariate analysis , 2003 .

[41]  G. Box.  MULTI-FACTOR DESIGNS OF FIRST ORDER , 1952 .

[42]  A. Öztürk,et al.  Non-Gaussian random vector identification using spherically invariant random processes , 1993 .

[43]  Samuel Kotz,et al.  Some extremal type elliptical distributions , 2001 .

[44]  Ivan W. Selesnick,et al.  An elliptically contoured exponential mixture model for wavelet based image denoising , 2007 .

[45]  Joel Goldman,et al.  Detection in the presence of spherically symmetric random vectors , 1976, IEEE Trans. Inf. Theory.

[46]  G. Simons,et al.  On the theory of elliptically contoured distributions , 1981 .

[47]  Applications of fractional calculus to spherical (radial) probability models and generalizations , 1975 .

[48]  N. L. Johnson,et al.  Continuous Univariate Distributions. , 1995 .

[49]  R. Williamson.  Multiply monotone functions and their Laplace transforms , 1956 .

[50]  Joseph Lipka,et al.  A Table of Integrals , 2010 .

[51]  Gabriela Koreisová,et al.  Scientific Papers , 1911, Nature.

[52]  N. L. Johnson,et al.  Continuous Multivariate Distributions: Models and Applications , 2005 .

[53]  S. Kotz,et al.  Symmetric Multivariate and Related Distributions , 1989 .

[54]  Henri Maître,et al.  A new statistical model for Markovian classification of urban areas in high-resolution SAR images , 2004, IEEE Transactions on Geoscience and Remote Sensing.

[55]  N. L. Johnson,et al.  Continuous Multivariate Distributions, Volume 1: Models and Applications , 2019 .

[56]  K. Shimizu,et al.  The generalized symmetric Laplace distribution on the sphere , 2008 .

[57]  M. A. Chmielewski,et al.  Elliptically Symmetric Distributions: A Review and Bibliography , 1981 .

[58]  J. Keilson,et al.  MIXTURES OF DISTRIBUTIONS, MOMENT INEQUALITIES AND MEASURES OF EXPONENTIALITY AND NORMALITY' , 1974 .

[59]  Yacine Chitour,et al.  Exact Maximum Likelihood Estimates for SIRV Covariance Matrix: Existence and Algorithm Analysis , 2008, IEEE Transactions on Signal Processing.

[60]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[61]  Saralees Nadarajah,et al.  The Kotz-type distribution with applications , 2003 .

[62]  Carl E. Pearson,et al.  Functions of a complex variable - theory and technique , 2005 .

[63]  O. Arslan.  A new class of multivariate distributions: Scale mixture of Kotz-type distributions , 2005 .