P Erez and Heitz : Restriction of a Markov Random Field and Multiresolution Image

| The association of statistical models and mul-tiresolution data analysis in a consistent and tractable mathematical framework remains an intricate theoretical and practical issue. Several consistent approaches have been proposed recently to combine Markov Random Field (MRF) models and multiresolution algorithms in image analysis: renormalization group, subsampling of stochastic processes, MRF's deened on trees or pyramids, etc. For the simulation or a practical use of these models in statistical estimation, an important issue is the preservation of the local Markovian property of the representation at the diierent resolution levels. It is shown in this paper that this key problem may be studied by considering the restriction of a Markov random eld (deened on some simple nite nondirected graph) to a part of its original site set. Several general properties of the restricted eld are derived. The general form of the distribution of the restriction is given. \Locality" of the eld is studied by exhibiting a neighborhood structure with respect to which the restricted eld is a MRF. Suucient conditions for the new neighborhood structure to be \minimal" are derived. Several consequences of these general results related to various \multiresolution" MRF-based modeling approaches in image analysis are presented.