Scale Mixtures of Gaussians and the Statistics of Natural Images

Part of Advances in Neural Information Processing Systems 12 (NIPS 1999)

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Martin J. Wainwright, Eero Simoncelli


The statistics of photographic images, when represented using multiscale (wavelet) bases, exhibit two striking types of non(cid:173) Gaussian behavior. First, the marginal densities of the coefficients have extended heavy tails. Second, the joint densities exhibit vari(cid:173) ance dependencies not captured by second-order models. We ex(cid:173) amine properties of the class of Gaussian scale mixtures, and show that these densities can accurately characterize both the marginal and joint distributions of natural image wavelet coefficients. This class of model suggests a Markov structure, in which wavelet coeffi(cid:173) cients are linked by hidden scaling variables corresponding to local image structure. We derive an estimator for these hidden variables, and show that a nonlinear "normalization" procedure can be used to Gaussianize the coefficients.

Recent years have witnessed a surge of interest in modeling the statistics of natural images. Such models are important for applications in image processing and com(cid:173) puter vision, where many techniques rely (either implicitly or explicitly) on a prior density. A number of empirical studies have demonstrated that the power spectra of natural images follow a 1/ f'Y law in radial frequency, where the exponent "f is typically close to two [e.g., 1]. Such second-order characterization is inadequate, however, because images usually exhibit highly non-Gaussian behavior. For in(cid:173) stance, the marginals of wavelet coefficients typically have much heavier tails than a Gaussian [2]. Furthermore, despite being approximately decorrelated (as sug(cid:173) gested by theoretical analysis of 1/ f processes [3]), orthonormal wavelet coefficients exhibit striking forms of statistical dependency [4, 5]. In particular, the standard deviation of a wavelet coefficient typically scales with the absolute values of its neighbors [5].

A number of researchers have modeled the marginal distributions of wavelet coef(cid:173) ficients with generalized Laplacians, py(y) ex exp( -Iy/ AlP) [e.g. 6, 7, 8]. Special cases include the Gaussian (p = 2) and the Laplacian (p = 1), but appropriate ex- Research supported by NSERC 1969 fellowship 160833 to MJW, and NSF CAREER grant MIP-9796040 to EPS.


M J Wainwright and E. P. Simoncelli