Variational versus Markov random field methods for image segmentation

Variational and Markov random field (MRF) methods have been proposed for a number of tasks in image processing and early vision. The main idea of such "regularization" approaches is to pose the problem in terms of minimizing a cost function that incorporates the observations as well as smoothness and other prior assumptions. Continuous (variational) formulations have the advantages of being more amenable to analysis and more easily incorporating geometric constraints and invariants. However, discrete (MRF) formulations have computational advantages and are typically used in implementing such methods. In particular, discrete and continuous versions for the problem of image segmentation have received considerable attention from both theoretical and algorithmic perspectives. Since the variational formulations have certain analytical advantages while the MRF formulations are used in implementations, it is natural to consider whether the discrete MRF models suitably approximate the continuous variational formulations. Certain commonly used MRF models for image segmentation do not properly approximate a standard continuous formulation in the sense that the discrete solutions may not converge to a solution of the continuous problem as the lattice spacing tends to zero. We propose several modifications of the MRF formulations for which we prove convergence in the continuum limit. Although these MRF models require complex neighborhood structures, we discuss results that indicate that for MRF models with bounded number of states, the difficulties are inherent and cannot be avoided in any scheme with the desired convergence properties.