Image denoising using scale mixtures of Gaussians in the wavelet domain

We describe a method for removing noise from digital images, based on a statistical model of the coefficients of an overcomplete multiscale oriented basis. Neighborhoods of coefficients at adjacent positions and scales are modeled as the product of two independent random variables: a Gaussian vector and a hidden positive scalar multiplier. The latter modulates the local variance of the coefficients in the neighborhood, and is thus able to account for the empirically observed correlation between the coefficient amplitudes. Under this model, the Bayesian least squares estimate of each coefficient reduces to a weighted average of the local linear estimates over all possible values of the hidden multiplier variable. We demonstrate through simulations with images contaminated by additive white Gaussian noise that the performance of this method substantially surpasses that of previously published methods, both visually and in terms of mean squared error.

[1]  H. Robbins.  The Empirical Bayes Approach to Statistical Decision Problems , 1964 .

[2]  G. C. Tiao,et al.  Bayesian inference in statistical analysis , 1973 .

[3]  John P. Rossi,et al.  Digital Techniques for Reducing Television Noise , 1978 .

[4]  Jong-Sen Lee,et al.  Digital Image Enhancement and Noise Filtering by Use of Local Statistics , 1980, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[5]  H. Brehm,et al.  Description and generation of spherically invariant speech-model signals , 1987 .

[6]  M. West.  On scale mixtures of normal distributions , 1987 .

[7]  D J Field,et al.  Relations between the statistics of natural images and the response properties of cortical cells. , 1987, Journal of the Optical Society of America. A, Optics and image science.

[8]  Andrew B. Watson,et al.  The cortex transform: rapid computation of simulated neural images , 1987 .

[9]  Stéphane Mallat,et al.  A Theory for Multiresolution Signal Decomposition: The Wavelet Representation , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[10]  J.G. Daugman,et al.  Entropy reduction and decorrelation in visual coding by oriented neural receptive fields , 1989, IEEE Transactions on Biomedical Engineering.

[11]  Bernhard Wegmann,et al.  Statistical dependence between orientation filter outputs used in a human-vision-based image code , 1990, Other Conferences.

[12]  Edward H. Adelson,et al.  The Design and Use of Steerable Filters , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[13]  Edward H. Adelson,et al.  Shiftable multiscale transforms , 1992, IEEE Trans. Inf. Theory.

[14]  George E. P. Box,et al.  Bayesian Inference in Statistical Analysis: Box/Bayesian , 1992 .

[15]  Jerome M. Shapiro,et al.  Embedded image coding using zerotrees of wavelet coefficients , 1993, IEEE Trans. Signal Process..

[16]  I. Johnstone,et al.  Ideal spatial adaptation by wavelet shrinkage , 1994 .

[17]  D. Ruderman.  The statistics of natural images , 1994 .

[18]  Daniel B. Nelson,et al.  ARCH MODELS a , 1994 .

[19]  Tim Bollerslev,et al.  Chapter 49 Arch models , 1994 .

[20]  D. Donoho,et al.  Translation-Invariant De-Noising , 1995 .

[21]  Peter Kovesi,et al.  Image Features from Phase Congruency , 1995 .

[22]  David J. Field,et al.  Emergence of simple-cell receptive field properties by learning a sparse code for natural images , 1996, Nature.

[23]  Edward H. Adelson,et al.  Noise removal via Bayesian wavelet coring , 1996, Proceedings of 3rd IEEE International Conference on Image Processing.

[24]  Michael T. Orchard,et al.  Image coding based on mixture modeling of wavelet coefficients and a fast estimation-quantization framework , 1997, Proceedings DCC '97. Data Compression Conference.

[25]  Eero P. Simoncelli.  Statistical models for images: compression, restoration and synthesis , 1997, Conference Record of the Thirty-First Asilomar Conference on Signals, Systems and Computers (Cat. No.97CB36136).

[26]  Daniel L. Ruderman,et al.  Origins of scaling in natural images , 1996, Vision Research.

[27]  Dirk Roose,et al.  Wavelet-based image denoising using a Markov random field a priori model , 1997, IEEE Trans. Image Process..

[28]  Terrence J. Sejnowski,et al.  The “independent components” of natural scenes are edge filters , 1997, Vision Research.

[29]  H. Chipman,et al.  Adaptive Bayesian Wavelet Shrinkage , 1997 .

[30]  Antonin Chambolle,et al.  Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage , 1998, IEEE Trans. Image Process..

[31]  B. Vidakovic.  Nonlinear wavelet shrinkage with Bayes rules and Bayes factors , 1998 .

[32]  J. Pesquet,et al.  Multiscale regularization in Besov spaces , 1998, Conference Record of Thirty-Second Asilomar Conference on Signals, Systems and Computers (Cat. No.98CH36284).

[33]  B. Silverman,et al.  Wavelet thresholding via a Bayesian approach , 1998 .

[34]  Robert D. Nowak,et al.  Wavelet-based statistical signal processing using hidden Markov models , 1998, IEEE Trans. Signal Process..

[35]  Martin Vetterli,et al.  Spatially adaptive wavelet thresholding with context modeling for image denoising , 1998, Proceedings 1998 International Conference on Image Processing. ICIP98 (Cat. No.98CB36269).

[36]  Martin J. Wainwright,et al.  Scale Mixtures of Gaussians and the Statistics of Natural Images , 1999, NIPS.

[37]  Eero P. Simoncelli.  Bayesian Denoising of Visual Images in the Wavelet Domain , 1999 .

[38]  Eero P. Simoncelli,et al.  Image compression via joint statistical characterization in the wavelet domain , 1999, IEEE Trans. Image Process..

[39]  Rafael Fonolla Navarro,et al.  Duality of log-polar image representations in the space and spatial-frequency domains , 1999, IEEE Trans. Signal Process..

[40]  Pierre Moulin,et al.  Analysis of Multiresolution Image Denoising Schemes Using Generalized Gaussian and Complexity Priors , 1999, IEEE Trans. Inf. Theory.

[41]  Hyvarinen.  Sparse code shrinkage: denoising of nongaussian data by maximum likelihood estimation , 1999, Neural computation.

[42]  Kannan Ramchandran,et al.  Low-complexity image denoising based on statistical modeling of wavelet coefficients , 1999, IEEE Signal Processing Letters.

[43]  E. Candès,et al.  Ridgelets: a key to higher-dimensional intermittency? , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[44]  Aapo Hyvärinen,et al.  Sparse Code Shrinkage: Denoising of Nongaussian Data by Maximum Likelihood Estimation , 1999, Neural Computation.

[45]  Michael T. Orchard,et al.  Spatially adaptive image denoising under overcomplete expansion , 2000, Proceedings 2000 International Conference on Image Processing (Cat. No.00CH37101).

[46]  Eero P. Simoncelli,et al.  Image denoising using a local Gaussian scale mixture model in the wavelet domain , 2000, SPIE Optics + Photonics.

[47]  Jean-Luc Starck,et al.  Very high quality image restoration by combining wavelets and curvelets , 2001, SPIE Optics + Photonics.

[48]  Eero P. Simoncelli,et al.  Random Cascades on Wavelet Trees and Their Use in Analyzing and Modeling Natural Images , 2001 .

[49]  Martin J. Wainwright,et al.  Adaptive Wiener denoising using a Gaussian scale mixture model in the wavelet domain , 2001, Proceedings 2001 International Conference on Image Processing (Cat. No.01CH37205).

[50]  N. Kingsbury.  Complex Wavelets for Shift Invariant Analysis and Filtering of Signals , 2001 .

[51]  Vasily Strela,et al.  Denoising Via Block Wiener Filtering in Wavelet Domain , 2001 .

[52]  Justin K. Romberg,et al.  Bayesian tree-structured image modeling using wavelet-domain hidden Markov models , 2001, IEEE Trans. Image Process..

[53]  Robert D. Nowak,et al.  Wavelet-based image estimation: an empirical Bayes approach using Jeffrey's noninformative prior , 2001, IEEE Trans. Image Process..

[54]  Emmanuel J. Candès,et al.  The curvelet transform for image denoising , 2002, IEEE Trans. Image Process..

[55]  Levent Sendur,et al.  Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency , 2002, IEEE Trans. Signal Process..

[56]  P. Besbeas,et al.  Empirical Bayes approach to block wavelet function estimation , 2002 .

[57]  Eero P. Simoncelli,et al.  Image Denoising using Gaussian Scale Mixtures in the Wavelet Domain , 2002 .

[58]  Aleksandra Pizurica,et al.  A joint inter- and intrascale statistical model for Bayesian wavelet based image denoising , 2002, IEEE Trans. Image Process..

[59]  I. Selesnick,et al.  Bivariate shrinkage with local variance estimation , 2002, IEEE Signal Processing Letters.

[60]  Eero P. Simoncelli,et al.  Image restoration using Gaussian scale mixtures in the wavelet domain , 2003, Proceedings 2003 International Conference on Image Processing (Cat. No.03CH37429).

[61]  Eero P. Simoncelli,et al.  A Parametric Texture Model Based on Joint Statistics of Complex Wavelet Coefficients , 2000, International Journal of Computer Vision.