Statistical modeling and geometry of shapes

Date of Completion

January 2001






Objects are distinguished from each other amongst other things by their shapes. The thesis is concerned with the shape of an object, in two different contexts. The first approach is to describe shape by locating a finite number of points from the object which we call landmarks. Then inference about average shape or difference in average shape between populations of shapes is invoked by considering a statistical distribution on the landmarks, that describes shape. We develop a general class of complex elliptical distributions on a complex sphere. We study some properties of this class of distributions and apply the distribution theory for modeling shapes in two dimensions. Maximum likelihood and Bayesian methods of estimation are developed for modal shape in one population as well as difference in shape for two populations. We also provide a Bayesian classification procedure for shapes. ^ The second approach is different, in that the shape of the object is not known and hence landmarks cannot be determined. In this case we assume that we have a set of points drawn uniformly from the true shape, and we try to estimate the shape using these points. We study the problem by utilizing the notion of support function of a convex body. Assuming that the body under investigation is smooth, e.g. ellipse or amoeba shaped, we obtain improved estimates of the convex body. We study asymptotic properties of these estimates using geometry of shapes. ^