Statistical Modeling of Images with Fields of Gaussian Scale Mixtures
Abstract
The local statistical properties of photographic images, when represented in a multi-scale basis, have been described using Gaussian scale mixtures (GSMs). Here, we use this local description to construct a global field of Gaussian scale mixtures (FoGSM). Specifically, we model subbands of wavelet coefficients as a product of an exponentiated homogeneous Gaussian Markov random field (hGMRF) and a second independent hGMRF. We show that parameter estimation for FoGSM is feasible, and that samples drawn from an estimated FoGSM model have marginal and joint statistics similar to wavelet coefficients of photographic images. We develop an algorithm for image denoising based on the FoGSM model, and demonstrate substantial improvements over current state-ofthe-art denoising method based on the local GSM model. Many successful methods in image processing and computer vision rely on statistical models for images, and it is thus of continuing interest to develop improved models, both in terms of their ability to precisely capture image structures, and in terms of their tractability when used in applications. Constructing such a model is difficult, primarily because of the intrinsic high dimensionality of the space of images. Two simplifying assumptions are usually made to reduce model complexity. The first is Markovianity: the density of a pixel conditioned on a small neighborhood, is assumed to be independent from the rest of the image. The second assumption is homogeneity: the local density is assumed to be independent of its absolute position within the image. The set of models satisfying both of these assumptions constitute the class of homogeneous Markov random fields (hMRFs). Over the past two decades, studies of photographic images represented with multi-scale multiorientation image decompositions (loosely referred to as "wavelets") have revealed striking nonGaussian regularities and inter and intra-subband dependencies. For instance, wavelet coefficients generally have