Alternating direction optimization for image segmentation using hidden Markov measure field models

Image segmentation is fundamentally a discrete problem. It consists of finding a partition of the image domain such that the pixels in each element of the partition exhibit some kind of similarity. The solution is often obtained by minimizing an objective function containing terms measuring the consistency of the candidate partition with respect to the observed image, and regularization terms promoting solutions with desired properties. This formulation ends up being an integer optimization problem that, apart from a few exceptions, is NP-hard and thus impossible to solve exactly. This roadblock has stimulated active research aimed at computing "good" approximations to the solutions of those integer optimization problems. Relevant lines of attack have focused on the representation of the regions (i.e., the partition elements) in terms of functions, instead of subsets, and on convex relaxations which can be solved in polynomial time. In this paper, inspired by the "hidden Markov measure field" introduced by Marroquin et al. in 2003, we sidestep the discrete nature of image segmentation by formulating the problem in the Bayesian framework and introducing a hidden set of real-valued random fields determining the probability of a given partition. Armed with this model, the original discrete optimization is converted into a convex program. To infer the hidden fields, we introduce the Segmentation via the Constrained Split Augmented Lagrangian Shrinkage Algorithm (SegSALSA). The effectiveness of the proposed methodology is illustrated with simulated and real hyperspectral and medical images.