The infinite Hidden Markov random field model

Dirichlet process (DP) mixture models have recently emerged in the cornerstone of nonparametric Bayesian statistics as promising candidates for clustering applications where the number of clusters is unknown a priori. Hidden Markov random field (HMRF) models are parametric statistical models widely used for image segmentation, as they appear naturally in problems where a spatially-constrained clustering scheme is asked for. A major limitation of HMRF models concerns the automatic selection of the proper number of their states, i.e. the number of segments derived by the image segmentation procedure. Typically, for this purpose, various likelihood based criteria are employed. Nevertheless, such methods often fail to yield satisfactory results, exhibiting significant overfitting proneness. Recently, higher order conditional random field models using potentials defined on superpixels have been considered as alternatives tackling these issues. Still, these models are in general computationally inefficient, a fact that limits their widespread adoption in practical applications. To resolve these issues, in this paper we introduce a novel, nonparametric Bayesian formulation for the HMRF model, the infinite HMRF model. We describe an efficient variational Bayesian inference algorithm for the proposed model, and we apply it to a series of image segmentation problems, demonstrating its advantages over existing methodologies.