Energy Minimization and Relaxation Labeling

Recently, there has been increasing interest in Markov random eld (MRF) modeling for solving a variety of computer vision problems formulated in terms of the maximum a posteriori (MAP) probability. When the label set is discrete, such as in image segmentation and matching, the minimization is combinatorial. The objective of this paper is twofold: Firstly, we propose to use the continuous relaxation labeling (RL) as an alternative approach for the minimization. The motivation is that it provides a good compromise between the solution quality and the computational cost. We show how the original combinatorial optimization can be converted into a form suitable for continuous RL. Secondly, we compare various minimization algorithms, namely, the RL algorithms proposed by Rosenfeld et al. and by Hummel and Zucker, the mean eld annealing of Peterson and Soderberg, simulated annealing of Kirkpatrick, the iterative conditional modes (ICM) of Besag and an annealing version of ICM proposed in this paper. The comparisons are in terms of the minimized energy value (i.e. the solution quality), the required number of iterations (i.e. the computational cost), and also the dependence of each algorithm on heuristics.