Discrete Markov image modeling and inference on the quadtree

Noncasual Markov (or energy-based) models are widely used in early vision applications for the representation of images in high-dimensional inverse problems. Due to their noncausal nature, these models generally lead to iterative inference algorithms that are computationally demanding. In this paper, we consider a special class of nonlinear Markov models which allow one to circumvent this drawback. These models are defined as discrete Markov random fields (MRF) attached to the nodes of a quadtree. The quadtree induces causality properties which enable the design of exact, noniterative inference algorithms, similar to those used in the context of Markov chain models. We first introduce an extension of the Viterbi algorithm which enables exact maximum a posteriori (MAP) estimation on the quadtree. Two other algorithms, related to the MPM criterion and to Bouman and Shapiro's (1994) sequential-MAP (SMAP) estimator are derived on the same hierarchical structure. The estimation of the model hyper parameters is also addressed. Two expectation-maximization (EM)-type algorithms, allowing unsupervised inference with these models are defined. The practical relevance of the different models and inference algorithms is investigated in the context of image classification problem, on both synthetic and natural images.