Multiscale and critical Markov random fields

摘要

A scale-based approach to analyze Markov random fields is described. The work presented relies on the idea that the behavior of these random fields can be better understood when examined at different scales. The identification of a nontrivial scale invariance for Markov random fields (MRFs) is used to subdivide the parameter space into three distinct regions of operations. This subdivision is based on the asymptotic behavior of the MRF's parameters under generalized nonlinear scale transformations.;In constructing these scale transformations of MRF models, we use the renormalization group theory, whose main criterion is the invariance of the partition function of the MRF across scale. We distinguish two forms of partition function scale invariance: functional invariance and point-wise invariance. While we require the latter form of invariance in constructing multiscale models, the former is a stronger property and defines a class of self-similar MRF models.;One of the main contributions of our work lies in identifying the class of self-similar MRFs, which we call critical MRFs. The concept of phase transition and its usefulness in characterizing these models are presented. We show that the statistical properties of critical MRFs are given by homogeneous functions, making them potential models for self-similar phenomena.;This observation provides a way to enlarge the class of probabilistic fractal models from isotropic Gaussian fractional models to anisotropic non-Gaussian MRF-based fractals. We illustrate how MRFs-based patterns can be efficiently and systematically generated in a coarse-to-fine manner, once the set of parameters at which the MRF is critical are determined.