CSDL Home IEEE Transactions on Pattern Analysis & Machine Intelligence 2009 vol.31 Issue No.04 - April

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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.2008.107REFERENCES

- [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.- [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.- [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.- [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.- [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.- [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.- [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.- [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.- [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.- [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.- [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.- [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. |