Restriction of a Markov Random Field on a Graph and Multiresolution Image Analysis

The association of statistical models and multiresolution data analysis in a consistent and tractable mathematical framework remains and 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, MRFs 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 Field (deened on a nite arbitrary 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 and applications of these general results to various \multiresolution" MRF-based modeling approaches are presented. Restriction d'un champ markovien sur un graphe et analyse d'image multir esolution R esum e : La d eenition de mod eles statistiques multir esolutions se pose comme un pro-bl eme th eorique (et pratique) diicile. Dii erentes approches ont et e propos ees r ecemment pour associer champs markoviens et techniques d'analyse multir esolution de l'image : groupe de renormalisation, sous-echantillonnage de champs al eatoires, d eenition de mod eles marko-viens sur des arbres ou sur des graphes pyramidaux, etc. Pour une simulation eecace de ces mod eles ou leur utilisation dans le contexte de l'estimation bayesienne, il est primordial que des propri et es markoviennes (locales) soient pr eserv ees pour le champ multir esolution. Nous montrons dans cet article que ce probl eme peut ^ etre etudi e en consid erant la restriction d'un champ markovien (d eeni sur un graphe ni non-orient e) a une partie de son support. La distribution ainsi que les propri et es markoviennes du champ restreint sont etablies. Le carac-t ere local de la structure de voisinage associ ee a ce nouveau mod ele markovien est etudi e de faa con d etaill ee. Ces r esultats g en eraux sont appliqu es a dii erentes approches …