MULTISCALE REPRESENTATIONS OF MARKOV RANDOM-FIELDS

Recently, a framework for multiscale stochastic modeling was introduce d based on coarse-to-fine scale-recursive dynamics defined on trees. T his model class has some attractive characteristics which lead to extr emely efficient, statistically optimal signal and image processing alg orithms. In this paper, we show that this model class is also quite ri ch. In particular, we describe how 1-D Markov processes and 2-D Markov random fields (MRF's) can be represented within this framework. The r ecursive structure of 1-D Markov processes makes them simple to analyz e, and generally leads to computationally efficient algorithms for sta tistical inference. On the other hand, 2-D MRF's are well known to be very difficult to analyze due to their noncausal structure, and thus t heir use typically leads to computationally intensive algorithms for s moothing and parameter identification. In contrast, our multiscale rep resentations are based on scale-recursive models and thus lead natural ly to scale-recursive algorithms, which can be substantially more effi cient computationally than those associated with MRF models. In 1-D, t he multiscale representation is a generalization of the midpoint defle ction construction of Brownian motion. The representation of 2-D MRF's is based on a further generalization to a ''midline'' deflection cons truction. The exact representations of 2-D MRF's are used to motivate a class of multiscale approximate MRF models based on one-dimensional wavelet transforms. We demonstrate the use of these latter models in t he context of texture representation and, in particular, we show how t hey can be used as approximations for or alternatives to well-known MR F texture models.