Nonlinear multiscale representations of Markov random fields

We develop a framework for multiscale representations of Markov random fields (MRFs) using the renormalization group theory. This representation is a nonlinear transformation of the MRFs coupling parameters at successive scale transformations. The marginally stable fixed points of the nonlinear transformation define an important class of self-similar non-Gaussian Markov random fields that we call critical MRFs (CMRFs). The main advantage of this multi-scale representation framework guarantees that all order statistics of the MRFs at different resolutions are preserved. We show that since the partition function in a Gibbs distribution of a CMRF is necessarily scale invariant, all order statistics are generalized homogenous functions. This leads us to closely examine self-similarity in a class of MRFs