Growth models for shapes

A central problem in computer vision is to detect, delineate (segment) and recognize objects in an image. One reason why this is difficult is that very little information specific to given types of objects is used during segmentation. Making use of information about an object's shape, for example, should facilitate and improve the segmentation of that object. The thrust of this thesis lies in the development of models for shape that provide an effective basis for computer-aided recovery of natural, specifically biological, shapes. We introduce a 2D discrete growth model for shape from a point on a Cartesian grid, based on notions related to biological growth. By "growth" is meant an accretionary process occurring at the boundary of the shape. We discuss two types of growth models: probabilistic models and deterministic (periodic) models. A probabilistic model on the Cartesian grid, which associates probabilities of growth with each of the eight directions, is considered. While such models empirically have been shown to describe many natural growth phenomena, complete quantitative characterizations do not yet exist. We prove that a class of models of this type is not capable of generating isotropic shapes. We introduce a new type of deterministic growth model based on the notion of "time delay". Associating a delay with each direction defines a time delay kernel (TDK); we show that such kernels produce classes of convex octagons, and that sequences of TDKs can give rise to arbitrary convex polygons. We also show that growth in a (stochastic) environment of facilitators and inhibitors, which decrease or increase the time delays respectively, appears to describe biological growth processes. As an example, we present results which suggest that simple periodic growth processes in an environment describe the gross morphology of multiple sclerosis lesions at the scale afforded by magnetic resonance images.