Designing Spatially Coherent Minimizing Flows for Variational Problems Based on Active Contours

Abstract

This paper tackles an important aspect of the variational problems involving active contours, which has been largely overlooked so far: the optimization by gradient flows. Classically, the definition of a gradient depends directly on the choice of an inner product structure. This consideration is largely absent from the active contours literature. Most authors, overtly or covertly, assume that the space of admissible deformations is ruled by the canonical L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> inner product. The classical gradient flows reported in the literature are relative to this particular choice. In this paper, we investigate the relevance of using other inner products, yielding other gradient descents, and some other minimizing flows not deriving from any inner product. In particular we, show how to induce different degrees of spatial coherence into the minimizing flow, in order to decrease the probability of getting trapped into irrelevant local minima. We show with some numerical experiments that the sensitivity of the active contours method to initial conditions, which seriously limits its applicability and its efficiency, is alleviated by our application-specific spatially coherent minimizing flows