A Class of Heavy-tailed Multivariate Non-Gaussian Probability Models for Wavelet Coefficients

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

It is well documented that the statistical distribution of wavelet coefficients for natural images is non-Gaussian and that neighboring coefficients are highly dependent. In this paper, we propose a new multivariate non-Gaussian probability model to capture the dependencies among neighboring wavelet coefficients in the same scale. The model can be expressed as K exp(-||w||) where w is an N-element vector of wavelet coefficients and ||w|| is a convex combination of l 2 norms over subspaces of R N. This model includes the commonly used independent Laplacian model as a special case but it has many more degrees of freedom. Based on this model, the corresponding non-linear threshold (shrinkage) function for denoising is derived using Bayesian estimation theory. Although this function does not have a closed-form solution, a successive substitution method can be used to numerically compute it.

Original languageEnglish (US)
Title of host publicationProceedings of SPIE - The International Society for Optical Engineering
EditorsM.A. Unser, A. Aldroubi, A.F. Laine
Pages1-8
Number of pages8
Volume5207
Edition1
StatePublished - 2003
EventWavelets: Applications in Signal and Image Processing X - San Diego, CA, United States
Duration: Aug 4 2003Aug 8 2003

Other

OtherWavelets: Applications in Signal and Image Processing X
CountryUnited States
CitySan Diego, CA
Period8/4/038/8/03

Fingerprint

coefficients
statistical distributions
shrinkage
norms
Substitution reactions
degrees of freedom
substitutes
thresholds

Keywords

  • Denoising
  • Non-Gaussian
  • Statistical model
  • Wavelet

ASJC Scopus subject areas

  • Electrical and Electronic Engineering
  • Condensed Matter Physics

Cite this

Shi, F., & Selesnick, I. (2003). A Class of Heavy-tailed Multivariate Non-Gaussian Probability Models for Wavelet Coefficients. In M. A. Unser, A. Aldroubi, & A. F. Laine (Eds.), Proceedings of SPIE - The International Society for Optical Engineering (1 ed., Vol. 5207, pp. 1-8)

A Class of Heavy-tailed Multivariate Non-Gaussian Probability Models for Wavelet Coefficients. / Shi, Fei; Selesnick, Ivan.

Proceedings of SPIE - The International Society for Optical Engineering. ed. / M.A. Unser; A. Aldroubi; A.F. Laine. Vol. 5207 1. ed. 2003. p. 1-8.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Shi, F & Selesnick, I 2003, A Class of Heavy-tailed Multivariate Non-Gaussian Probability Models for Wavelet Coefficients. in MA Unser, A Aldroubi & AF Laine (eds), Proceedings of SPIE - The International Society for Optical Engineering. 1 edn, vol. 5207, pp. 1-8, Wavelets: Applications in Signal and Image Processing X, San Diego, CA, United States, 8/4/03.
Shi F, Selesnick I. A Class of Heavy-tailed Multivariate Non-Gaussian Probability Models for Wavelet Coefficients. In Unser MA, Aldroubi A, Laine AF, editors, Proceedings of SPIE - The International Society for Optical Engineering. 1 ed. Vol. 5207. 2003. p. 1-8
Shi, Fei ; Selesnick, Ivan. / A Class of Heavy-tailed Multivariate Non-Gaussian Probability Models for Wavelet Coefficients. Proceedings of SPIE - The International Society for Optical Engineering. editor / M.A. Unser ; A. Aldroubi ; A.F. Laine. Vol. 5207 1. ed. 2003. pp. 1-8
@inproceedings{2d1128d62173458188e89ccb5d1fdd61,
title = "A Class of Heavy-tailed Multivariate Non-Gaussian Probability Models for Wavelet Coefficients",
abstract = "It is well documented that the statistical distribution of wavelet coefficients for natural images is non-Gaussian and that neighboring coefficients are highly dependent. In this paper, we propose a new multivariate non-Gaussian probability model to capture the dependencies among neighboring wavelet coefficients in the same scale. The model can be expressed as K exp(-||w||) where w is an N-element vector of wavelet coefficients and ||w|| is a convex combination of l 2 norms over subspaces of R N. This model includes the commonly used independent Laplacian model as a special case but it has many more degrees of freedom. Based on this model, the corresponding non-linear threshold (shrinkage) function for denoising is derived using Bayesian estimation theory. Although this function does not have a closed-form solution, a successive substitution method can be used to numerically compute it.",
keywords = "Denoising, Non-Gaussian, Statistical model, Wavelet",
author = "Fei Shi and Ivan Selesnick",
year = "2003",
language = "English (US)",
volume = "5207",
pages = "1--8",
editor = "M.A. Unser and A. Aldroubi and A.F. Laine",
booktitle = "Proceedings of SPIE - The International Society for Optical Engineering",
edition = "1",

}

TY - GEN

T1 - A Class of Heavy-tailed Multivariate Non-Gaussian Probability Models for Wavelet Coefficients

AU - Shi, Fei

AU - Selesnick, Ivan

PY - 2003

Y1 - 2003

N2 - It is well documented that the statistical distribution of wavelet coefficients for natural images is non-Gaussian and that neighboring coefficients are highly dependent. In this paper, we propose a new multivariate non-Gaussian probability model to capture the dependencies among neighboring wavelet coefficients in the same scale. The model can be expressed as K exp(-||w||) where w is an N-element vector of wavelet coefficients and ||w|| is a convex combination of l 2 norms over subspaces of R N. This model includes the commonly used independent Laplacian model as a special case but it has many more degrees of freedom. Based on this model, the corresponding non-linear threshold (shrinkage) function for denoising is derived using Bayesian estimation theory. Although this function does not have a closed-form solution, a successive substitution method can be used to numerically compute it.

AB - It is well documented that the statistical distribution of wavelet coefficients for natural images is non-Gaussian and that neighboring coefficients are highly dependent. In this paper, we propose a new multivariate non-Gaussian probability model to capture the dependencies among neighboring wavelet coefficients in the same scale. The model can be expressed as K exp(-||w||) where w is an N-element vector of wavelet coefficients and ||w|| is a convex combination of l 2 norms over subspaces of R N. This model includes the commonly used independent Laplacian model as a special case but it has many more degrees of freedom. Based on this model, the corresponding non-linear threshold (shrinkage) function for denoising is derived using Bayesian estimation theory. Although this function does not have a closed-form solution, a successive substitution method can be used to numerically compute it.

KW - Denoising

KW - Non-Gaussian

KW - Statistical model

KW - Wavelet

UR - http://www.scopus.com/inward/record.url?scp=1242330935&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=1242330935&partnerID=8YFLogxK

M3 - Conference contribution

VL - 5207

SP - 1

EP - 8

BT - Proceedings of SPIE - The International Society for Optical Engineering

A2 - Unser, M.A.

A2 - Aldroubi, A.

A2 - Laine, A.F.

ER -