Subscribe
Issue No.04 - April (2009 vol.31)
pp: 693-706
Siwei Lyu , University at Albany, State University of New York, Albany
Eero P. Simoncelli , Howard Hughes Medical Institute and New York University, New York
ABSTRACT
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 (FoGSMs). 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.
INDEX TERMS
Image Representation, Statistical, Enhancement, Restoration
CITATION
Siwei Lyu, Eero P. Simoncelli, "Modeling Multiscale Subbands of Photographic Images with Fields of Gaussian Scale Mixtures", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.31, no. 4, pp. 693-706, April 2009, doi:10.1109/TPAMI.2008.107
REFERENCES
 [1] P. Burt and E. Adelson, “The Laplacian Pyramid as a Compact Image Code,” IEEE Trans. Comm., vol. 31, no. 4, pp. 532-540, 1981. [2] D.J. Field, “Relations between the Statistics of Natural Images and the Response Properties of Cortical Cells,” J. Optical Soc. Am., vol. 4, no. 12, pp. 2379-2394, 1987. [3] S.G. Mallat, “A Theory for Multiresolution Signal Decomposition,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, pp. 674-697, 1989. [4] J. Shapiro, “Embedded Image Coding Using Zerotrees of Wavelet Coefficients,” IEEE Trans. Signal Processing, vol. 41, no. 12, pp.3445-3462, Dec. 1993. [5] E.P. Simoncelli, “Statistical Models for Images: Compression, Restoration and Synthesis,” Proc. 31st Asilomar Conf. Signals, Systems and Computers, vol. 1, pp. 673-678, Nov. 1997. [6] R.W. Buccigrossi and E.P. Simoncelli, “Image Compression via Joint Statistical Characterization in the Wavelet Domain,” IEEE Trans. Image Processing, vol. 8, no. 12, pp. 1688-1701, Dec. 1999. [7] E.P. Simoncelli and E.H. Adelson, “Noise Removal via Bayesian Wavelet Coring,” Proc. Third IEEE Int'l Conf. Image Processing, vol. 1, pp. 379-382, Sept. 1996. [8] A. Hyvärinen, P.O. Hoyer, and M. Inki, “Topographic ICA as a Model of Natural Image Statistics,” Proc. First IEEE Int'l Workshop Biologically Motivated Computer Vision, 2000. [9] J. Huang and D. Mumford, “Statistics of Natural Images and Models,” Proc. IEEE Int'l Conf. Computer Vision and Pattern Recognition, 1999. [10] P. Gehler and M. Welling, “Products of “Edge-Perts”,” Proc. Advances in Neural Information Processing Systems, Y. Weiss, B.Schölkopf, and J. Platt, eds., pp. 419-426, 2006. [11] A. Srivastava, X. Liu, and U. Grenander, “Universal Analytical Forms for Modeling Image Probability,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 28, no. 9, pp. 217-232, Sept. 2002. [12] Y. Teh, M. Welling, and S. Osindero, “Energy-Based Models for Sparse Overcomplete Representations,” J. Machine Learning Research, vol. 4, pp. 1235-1260, , 2003. [13] L. Parra, C. Spence, and P. Sajda, “Higher-Order Statistical Properties Arising from the Non-Stationarity of Natural Signals,” Proc. Advances in Neural Information Processing Systems, vol. 13, 2000. [14] L. Sendur and I.W. Selesnick, “Bivariate Shrinkage Functions for Wavelet-Based Denoising Exploiting Interscale Dependency,” IEEE Trans. Signal Processing, vol. 50, no. 11, pp. 2744-2756, 2002. [15] D.F. Andrews and C.L. Mallows, “Scale Mixtures of Normal Distributions,” J. Royal Statistical Soc., Series B, vol. 36, no. 1, pp.99-102, 1974. [16] M.J. Wainwright and E.P. Simoncelli, “Scale Mixtures of Gaussians and the Statistics of Natural Images,” Proc. Advances in Neural Information Processing Systems, S.A. Solla, T.K. Leen, and K.-R. Müller, eds., vol. 12, pp. 855-861, May 2000. [17] J. Portilla, V. Strela, M.J. Wainwright, and E.P. Simoncelli, “Image Denoising Using a Scale Mixture of Gaussians in the Wavelet Domain,” IEEE Trans. Image Processing, vol. 12, no. 11, pp. 1338-1351, Nov. 2003. [18] J. Guerrero-Colon, L. Mancera, and J. Portilla, “Image Restoration Using Space-Variant Gaussian Scale Mixtures in Overcomplete Pyramids,” IEEE Trans. Image Processing, vol. 17, no. 1, pp. 27-41, Jan. 2008. [19] M.J. Wainwright, E.P. Simoncelli, and A.S. Willsky, “Random Cascades on Wavelet Trees and Their Use in Modeling and Analyzing Natural Imagery,” Applied and Computational Harmonic Analysis, vol. 11, no. 1, pp. 89-123, July 2001. [20] J. Romberg, H. Choi, and R. Baraniuk, “Bayesian Tree-Structured Image Modeling Using Wavelet-Domain Hidden Markov Models,” IEEE Trans. Image Processing, vol. 10, no. 7, July 2001. [21] S. Geman and D. Geman, “Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 6, pp. 721-741, 1984. [22] F. Jeng and J. Woods, “Compound Gauss-Markov Random Fields for Image Estimation,” IEEE Trans. Signal Processing, vol. 39, no. 3, pp. 683-697, 1991. [23] S.C. Zhu, Y. Wu, and D. Mumford, “Filters, Random Fields and Maximum Entropy (FRAME): Towards a Unified Theory for Texture Modeling,” Int'l J. Computer Vision, vol. 27, no. 2, pp. 107-126, 1998. [24] W.T. Freeman, E.C. Pasztor, and O.T. Carmichael, “Learning Low-Level Vision,” Int'l J. Computer Vision, vol. 40, no. 1, pp. 25-47, Oct. 2000. [25] M. Tappen, C. Liu, E. Adelson, and W. Freeman, “Learning Gaussian Conditional Random Fields for Low-Level Vision,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 1-8, 2007. [26] P. Winkler, Image Analysis, Random Fields and Markov Chain Monte Carlo Methods, second ed. Springer, 2003. [27] S. Roth and M. Black, “Fields of Experts: A Framework for Learning Image Priors,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, vol. 2, pp. 860-867, , 2005. [28] S. Lyu and E.P. Simoncelli, “Statistical Modeling of Images with Fields of Gaussian Scale Mixtures,” Proc. Advances in Neural Information Processing Systems, B. Schölkopf, J. Platt, and T.Hofmann, eds., vol. 19, May 2007. [29] E.P. Simoncelli, W.T. Freeman, E.H. Adelson, and D.J. Heeger, “Shiftable Multi-Scale Transforms,” IEEE Trans. Information Theory, vol. 38, no. 2, pp. 587-607, special issue on wavelets, Mar. 1992. [30] B. Wegmann and C. Zetzsche, “Statistical Dependencies between Orientation Filter Outputs Used in Human Vision Based Image Code,” Proc. Visual Comm. and Image Processing, vol. 1360, pp. 909-922, 1990. [31] M. Welling, G.E. Hinton, and S. Osindero, “Learning Sparse Topographic Representations with Products of Student $t\hbox{-}{\rm Distributions}$ ,” Proc. Advances in Neural Information Processing Systems, pp. 1359-1366, 2002. [32] H. Rue and L. Held, Gaussian Markov Random Fields: Theory and Applications. Chapman and Hall/CRC, 2005. [33] J. Portilla, V. Strela, M.J. Wainwright, and E.P. Simoncelli, “Adaptive Wiener Denoising Using a Gaussian Scale Mixture Model in the Wavelet Domain,” Proc. Eighth IEEE Int'l Conf. Image Processing, vol. 2, pp. 37-40, Oct. 2001. [34] W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes, second ed. Cambridge, 2002. [35] M.I. Jordan, Z. Ghahramani, T. Jaakkola, and L.K. Saul, “An Introduction to Variational Methods for Graphical Models,” Machine Learning, vol. 37, no. 2, pp. 183-233, citeseer.ist.psu.edu/teh03energy based.htmlciteseer.ist.psu.edu/ 729276.htmlciteseer.ist.psu.edu jordan98introduction.html , 1999. [36] M. Figueiredo and J. Leitäo, “Unsupervised Image Restoration and Edge Location Using Compound Gauss-Markov Random Fields and MDL Principle,” IEEE Trans. Image Processing, vol. 6, no. 8, pp. 1089-1122, 1997. [37] K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image Denoising by Sparse 3D Transform-Domain Collaborative Filtering,” IEEE Trans. Image Processing, vol. 16, no. 6, pp. 1064-1083, 2007. [38] J. Portilla and E.P. Simoncelli, “Image Restoration Using Gaussian Scale Mixtures in the Wavelet Domain,” Proc. 10th IEEE Int'l Conf. Image Processing, vol. 2, pp. 965-968, Sept. 2003. [39] M. Elad and M. Aharon, “Image Denoising via Sparse and Redundant Representations over Learned Dictionaries,” IEEE Trans. Image Processing, vol. 15, no. 12, pp. 3736-3745, Dec. 2006. [40] E.P. Simoncelli and E.H. Adelson, “Subband Transforms,” Subband Image Coding, J.W. Woods, ed., chapter 4, pp. 143-192, 1990. [41] R.R. Coifman and D.L. Donoho, “Translation-Invariant De-Noising,” Wavelets and Statistics, A. Antoniadis and G. Oppenheim, eds., Springer-Verlag, 1995. [42] M. Raphan and E.P. Simoncelli, “Optimal Denoising in Redundant Bases,” Proc. 14th IEEE Int'l Conf. Image Processing, Sept. 2007. [43] Y. Karklin and M.S. Lewicki, “A Hierarchical Bayesian Model for Learning Non-Linear Statistical Regularities in Non-Stationary Natural Signals,” Neural Computation, vol. 17, no. 2, pp. 397-423, 2005. [44] A. Hyvärinen, J. Hurri, and J. Väyrynen, “Bubbles: A Unifying Framework for Low-Level Statistical Properties of Natural Image Sequences,” J. Optical Soc. Am. A, vol. 20, no. 7, pp. 1237-1252, 2003. [45] D.K. Hammond and E.P. Simoncelli, “Image Denoising with an Orientation-Adaptive Gaussian Scale Mixture Model,” Proc. 13th IEEE Int'l Conf. Image Processing, pp. 1433-1436, Oct. 2006. [46] Z. Wang and E.P. Simoncelli, “Local Phase Coherence and the Perception of Blur,” Proc. Advances in Neural Information Processing Systems, vol. 16, 2003. [47] R.M. Gray, “Toeplitz and Circulant Matrices: A Review,” Foundations and Trends in Comm. and Information Theory, vol. 2, no. 3, pp. 155-239, 2006. [48] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge Univ. Press, 2005. [49] R. Chellappa, S. Chatterjee, and R. Bagdazian, “Texture Synthesis and Compression Using Gaussian-Markov Random Field Models,” IEEE Trans. Systems, Man, and Cybernetics, vol. 15, no. 3, pp.298-303, Mar. 1985. [50] D. Geman and C. Yang, “Nonlinear Image Recovery with Half-Quadratic Regularization,” IEEE Trans. Image Processing, vol. 4, no. 7, pp. 932-946, 1995.