Migration Processes I: The Continuous Case

In this paper the general concept of a migration process (MP) is introduced; it involves iterative displacement of each point in a set as function of a neighborhood of the point, and is applicable to arbitrary sets with arbitrary topologies. After a brief analysis of this relatively general class of iterative processes and of constraints on such processes, we restrict our attention to processes in which each point in a set is iteratively displaced to the average (centroid) of its equigeodesic neighborhood. We show that MPs of this special class can be approximated by “reaction-diffusion”-type PDEs, which have received extensive attention recently in the contour evolution literature. Although we show that MPs constitute a special class of these evolution models, our analysis of migrating sets does not require the machinery of differential geometry. In Part I of the paper we characterize the migration of closed curves and extend our analysis to arbitrary connected sets in the continuous domain (Rm) using the frequency analysis of closed polygons, which has been rediscovered recently in the literature. We show that migrating sets shrink, and also derive other geometric properties of MPs. In Part II we will reformulate the concept of migration in a discrete representation (Zm).