Modeling Multiscale Subbands of Photographic Images with Fields of Gaussian Scale Mixtures

The local statistical properties of photographic images, when represented in a multi-scale basis, have been described using Gaussian scale mixtures. Here, we use this local description as a substrate for constructing a global field of Gaussian scale mixtures (FoGSM). Specifically, we model multi-scale subbands as a product of an exponentiated homogeneous Gaussian Markov random field (hGMRF) and a second independent hGMRF. We show that parameter estimation for this model is feasible, and that samples drawn from a FoGSM model have marginal and joint statistics similar to subband coefficients of photographic images. We develop an algorithm for removing additive Gaussian white noise based on the FoGSM model, and demonstrate denoising performance comparable with state-of-the-art methods.

[1]  Edward H. Adelson,et al.  The Laplacian Pyramid as a Compact Image Code , 1983, IEEE Trans. Commun..

[2]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[3]  Rama Chellappa,et al.  Texture synthesis and compression using Gaussian-Markov random field models , 1985, IEEE Transactions on Systems, Man, and Cybernetics.

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

[5]  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.

[6]  Stuart German,et al.  Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images , 1988 .

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

[8]  William H. Press,et al.  Numerical recipes , 1990 .

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

[10]  John W. Woods,et al.  Compound Gauss-Markov random fields for image estimation , 1991, IEEE Trans. Signal Process..

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

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

[13]  Donald Geman,et al.  Nonlinear image recovery with half-quadratic regularization , 1995, IEEE Trans. Image Process..

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

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

[16]  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).

[17]  José M. N. Leitão,et al.  Unsupervised image restoration and edge location using compound Gauss-Markov random fields and the MDL principle , 1997, IEEE Trans. Image Process..

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

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

[20]  David Mumford,et al.  Statistics of natural images and models , 1999, Proceedings. 1999 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (Cat. No PR00149).

[21]  Lucas C. Parra,et al.  Higher-Order Statistical Properties Arising from the Non-Stationarity of Natural Signals , 2000, NIPS.

[22]  Aapo Hyvärinen,et al.  Topographic ICA as a Model of Natural Image Statistics , 2000, Biologically Motivated Computer Vision.

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

[24]  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).

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

[26]  Geoffrey E. Hinton,et al.  Learning Sparse Topographic Representations with Products of Student-t Distributions , 2002, NIPS.

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

[28]  Gerhard Winkler,et al.  Image Analysis, Random Fields and Markov Chain Monte Carlo Methods: A Mathematical Introduction , 2002 .

[29]  Anuj Srivastava,et al.  Universal Analytical Forms for Modeling Image Probabilities , 2002, IEEE Trans. Pattern Anal. Mach. Intell..

[30]  Zhou Wang,et al.  Local Phase Coherence and the Perception of Blur , 2003, NIPS.

[31]  Aapo Hyvärinen,et al.  Bubbles: a unifying framework for low-level statistical properties of natural image sequences. , 2003, Journal of the Optical Society of America. A, Optics, image science, and vision.

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

[33]  Yee Whye Teh,et al.  Energy-Based Models for Sparse Overcomplete Representations , 2003, J. Mach. Learn. Res..

[34]  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).

[35]  Michael I. Jordan,et al.  An Introduction to Variational Methods for Graphical Models , 1999, Machine Learning.

[36]  William T. Freeman,et al.  Learning Low-Level Vision , 1999, Proceedings of the Seventh IEEE International Conference on Computer Vision.

[37]  Song-Chun Zhu,et al.  Filters, Random Fields and Maximum Entropy (FRAME): Towards a Unified Theory for Texture Modeling , 1998, International Journal of Computer Vision.

[38]  G..  A Theory for Multiresolution Signal Decomposition : The Wavelet Representation , 2004 .

[39]  Michael J. Black,et al.  Fields of Experts: a framework for learning image priors , 2005, 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05).

[40]  L. Held,et al.  Gaussian Markov Random Fields: Theory And Applications (Monographs on Statistics and Applied Probability) , 2005 .

[41]  Michael S. Lewicki,et al.  A Hierarchical Bayesian Model for Learning Nonlinear Statistical Regularities in Nonstationary Natural Signals , 2005, Neural Computation.

[42]  Robert M. Gray,et al.  Toeplitz and Circulant Matrices: A Review , 2005, Found. Trends Commun. Inf. Theory.

[43]  Peter V. Gehler,et al.  Products of Edge-perts , 2005, NIPS.

[44]  Leonhard Held,et al.  Gaussian Markov Random Fields: Theory and Applications , 2005 .

[45]  Robert M. Gray,et al.  Toeplitz And Circulant Matrices: A Review (Foundations and Trends(R) in Communications and Information Theory) , 2006 .

[46]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[47]  Eero P. Simoncelli,et al.  Image Denoising with an Orientation-Adaptive Gaussian Scale Mixture Model , 2006, 2006 International Conference on Image Processing.

[48]  Michael Elad,et al.  Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries , 2006, IEEE Transactions on Image Processing.

[49]  Edward H. Adelson,et al.  Learning Gaussian Conditional Random Fields for Low-Level Vision , 2007, 2007 IEEE Conference on Computer Vision and Pattern Recognition.

[50]  Thomas Hofmann,et al.  Statistical Modeling of Images with Fields of Gaussian Scale Mixtures , 2007 .

[51]  Alessandro Foi,et al.  Image Denoising by Sparse 3-D Transform-Domain Collaborative Filtering , 2007, IEEE Transactions on Image Processing.

[52]  Eero P. Simoncelli,et al.  Optimal Denoising in Redundant Bases , 2007, 2007 IEEE International Conference on Image Processing.

[53]  Javier Portilla,et al.  Image Restoration Using Space-Variant Gaussian Scale Mixtures in Overcomplete Pyramids , 2008, IEEE Transactions on Image Processing.