Abstract-Relaxation labeling processes have been widely used in many different domains including image processing, pattern recognition, and artificial intelligence. They are iterative procedures that aim at reducing local ambiguities and achieving global consistency through a parallel exploitation of contextual information, which is quantitatively expressed in terms of a set of “compatibility coefficients. ” The problem of determining compatibility coefficients has received a considerable attention in the past and many heuristic, statistical-based methods have been suggested. In this paper, we propose a rather different viewpoint to solve this problem: we derive them attempting to optimize the performance of the relaxation algorithm over a sample of training data; no statistical interpretation is given: compatibility coefficients are simply interpreted as real numbers, for which performance is optimal. Experimental results over a novel application of relaxation are given, which prove the effectiveness of the proposed approach. Index Terms- Compatibility coefficients, constraint satisfaction, gradient projection, learning, neural networks, nonlinear

We present some new results which definitively explain the behavior of the classical, heuristic nonlinear relaxation labeling algorithm of Rosenfeld, Hummel, and Zucker in terms of the Hummel-Zucker consistency theory and dynamical systems theory. In particular, it is shown that, when a certain symmetry condition is met, the algorithm possesses a Liapunov function which turns out to be (the negative of) a well-known consistency measure. This follows almost immediately from a powerful result of Baum and Eagon developed in the context of Markov chain theory. Moreover, it is seen that most of the essential dynamical properties of the algorithm are retained when the symmetry restriction is relaxed. These properties are also shown to naturally generalize to higher-order relaxation schemes. Some applications and implications of the presented results are finally outlined.