Continuous-Time Relaxation Labeling Processes

We study the dynamical properties of two new relaxation labeling schemes described in terms of di!erential equations, and hence evolving in continuous time. This contrasts with the customary approach to de"ning relaxation labeling algorithms which prefers discrete time. Continuous-time dynamical systems are particularly attractive because they can be implemented directly in hardware circuitry, and the study of their dynamical properties is simpler and more elegant. They are also more plausible as models of biological visual computation. We prove that the proposed models enjoy exactly the same dynamical properties as the classical relaxation labeling schemes, and show how they are intimately related to Hummel and Zucker's now classical theory of constraint satisfaction. In particular, we prove that, when a certain symmetry condition is met, the dynamical systems' behavior is governed by a Liapunov function which turns out to be (the negative of) a well-known consistency measure. Moreover, we prove that the fundamental dynamical properties of the systems are retained when the symmetry restriction is relaxed. We also analyze the properties of a simple discretization of the proposed dynamics, which is useful in digital computer implementations. Simulation results are presented which show the practical behavior of the models. ( 2000 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved.