{"title": "Scale Mixtures of Gaussians and the Statistics of Natural Images", "book": "Advances in Neural Information Processing Systems", "page_first": 855, "page_last": 861, "abstract": null, "full_text": "Scale Mixtures of Gaussians and the \n\nStatistics of Natural Images \n\nMartin J. Wainwright \nStochastic Systems Group \nElectrical Engineering & CS \n\nMIT, Building 35-425 \nCambridge, MA 02139 \n\nmjwain@mit.edu \n\nEero P. Simoncelli \n\nCtr. for Neural Science, and \n\nCourant Inst. of Mathematical Sciences \n\nNew York University \nNew York, NY 10012 \neero. simoncelli@nyu.edu \n\nAbstract \n\nThe statistics of photographic images, when represented using \nmultiscale (wavelet) bases, exhibit two striking types of non(cid:173)\nGaussian behavior. First, the marginal densities of the coefficients \nhave extended heavy tails. Second, the joint densities exhibit vari(cid:173)\nance dependencies not captured by second-order models. We ex(cid:173)\namine properties of the class of Gaussian scale mixtures, and show \nthat these densities can accurately characterize both the marginal \nand joint distributions of natural image wavelet coefficients. This \nclass of model suggests a Markov structure, in which wavelet coeffi(cid:173)\ncients are linked by hidden scaling variables corresponding to local \nimage structure. We derive an estimator for these hidden variables, \nand show that a nonlinear \"normalization\" procedure can be used \nto Gaussianize the coefficients. \n\nRecent years have witnessed a surge of interest in modeling the statistics of natural \nimages. Such models are important for applications in image processing and com(cid:173)\nputer vision, where many techniques rely (either implicitly or explicitly) on a prior \ndensity. A number of empirical studies have demonstrated that the power spectra \nof natural images follow a 1/ f'Y law in radial frequency, where the exponent \"f is \ntypically close to two [e.g., 1]. Such second-order characterization is inadequate, \nhowever, because images usually exhibit highly non-Gaussian behavior. For in(cid:173)\nstance, the marginals of wavelet coefficients typically have much heavier tails than \na Gaussian [2]. Furthermore, despite being approximately decorrelated (as sug(cid:173)\ngested by theoretical analysis of 1/ f processes [3]), orthonormal wavelet coefficients \nexhibit striking forms of statistical dependency [4, 5]. In particular, the standard \ndeviation of a wavelet coefficient typically scales with the absolute values of its \nneighbors [5]. \n\nA number of researchers have modeled the marginal distributions of wavelet coef(cid:173)\nficients with generalized Laplacians, py(y) ex exp( -Iy/ AlP) [e.g. 6, 7, 8]. Special \ncases include the Gaussian (p = 2) and the Laplacian (p = 1), but appropriate ex-\nResearch supported by NSERC 1969 fellowship 160833 to MJW, and NSF CAREER grant \nMIP-9796040 to EPS. \n\n\f856 \n\nM J Wainwright and E. P. Simoncelli \n\nMixing density \n\nGSM density \n\nJZ(t) \n\nl/JZ({3-~) \n\nsymmetrized Gamma \nStudent: \n[1/(>,2 + y2)]t3, {3>~ \n\nGSM char. function \nl+w -, ,),>0 \n( \n\nt'l ) \n\n'Y \n\nNo explicit form \n\nPositive, J~ - stable \n\na-stable \n\nexp (-IAW~), a E (0,2] \n\nNo explicit form \n\ngeneralized Laplacian: \nexp (-Iy / AlP), p E (0,2] \n\nNo explicit form \n\nTable 1. Example densities from the class of Gaussian scale mixtures. Zh) de(cid:173)\nnotes a positive gamma variable, with density p(z) = [l/rh)] z\"Y- 1 exp (-z). \nThe characteristic \nas \n\u00a2\",(t) ~ J~oo p(x) exp (jxt) dx . \n\nrandom variable \n\nfunction of a \n\nis defined \n\nx \n\nponents for natural images are typically less than one. Simoncelli [5, 9] has modeled \nthe variance dependencies of pairs of wavelet coefficients. Romberg et al. [10] have \nmodeled wavelet densities using two-component mixtures of Gaussians. Huang and \nMumford [11] have modeled marginal densities and cross-sections of joint densities \nwith multi-dimensional generalized Laplacians. \n\nIn the following sections, we explore the semi-parametric class of Gaussian scale \nmixtures. We show that members of this class satisfy the dual requirements of \nbeing heavy-tailed, and exhibiting multiplicative scaling between coefficients. We \nalso show that a particular member of this class, in which the multiplier variables \nare distributed according to a gamma density, captures the range of joint statistical \nbehaviors seen in wavelet coefficients of natural images. We derive an estimator for \nthe multipliers, and show that a nonlinear \"normalization\" procedure can be used \nto Gaussianize the wavelet coefficients. Lastly, we form random cascades by linking \nthe multipliers on a multiresolution tree. \n\n1 Scale Mixtures of Gaussians \n\nequality in distribution; z 2:: \u00b0 is a scalar random variable; U f'V N(O, Q) is a \nA random vector Y is a Gaussian scale mixture (GSM) if Y 4 zU, where 4 denotes \n\nGaussian random vector; and z and U are independent. \nAs a consequence, any GSM variable has a density given by an integral: \n\n100 \n\npy(Y) = -00 [21r]~ Iz2Q1 1/ 2 exp \n\n-\n\n2z2 \n\n