Multiscale representations of Markov random fields

Recently, a framework for multiscale stochastic modeling was introduced based on coarse-to-fine scale-recursive dynamics defined on trees. This model class has some attractive characteristics which lead to extremely efficient, statistically optimal signal and image processing algorithms. The authors show that this model class is also quite rich. In particular, they describe how 1-D Markov processes and 2-D Markov random fields (MRFs) can be represented within this framework. The recursive structure of 1-D Markov processes makes them simple to analyze, and generally leads to computationally efficient algorithms for statistical inference. On the other hand, 2-D MRFs are well known to be very difficult to analyze due to their noncausal structure, and thus their use typically leads to computationally intensive algorithms for smoothing and parameter identification. In contrast, their multiscale representations are based on scale-recursive models and thus lead naturally to scale-recursive algorithms, which can be substantially more efficient computationally than those associated with MRF models. In 1-D, the multiscale representation is a generalization of the midpoint deflection construction of Brownian motion. The representation of 2-D MRFs is based on a further generalization to a “midline” deflection construction. The exact representations of 2-D MRFs are used to motivate a class of multiscale approximate MRF models based on one-dimensional wavelet transforms. They demonstrate the use of these latter models in the context of texture representation and, in particular, they show how they can be used as approximations for or alternatives to well-known MRF texture models