Limit Theorems for Object Data with Applications to 2D and 3D Image Data Analysis

Abstract

With recent advancements of technology, besides medical imaging data, digital imaging are today being used in accurate 3D scene reconstructions, significantly helping shape and shape change analysis. Since shape spaces are compact, we can compare both extrinsic means and extrinsic antimeans of probability distributions on such object spaces. Here, one develops nonparametric procedures for comparing two extrinsic antimeans for unmtched pairs of random objects on a compact manifold, using recent limit theorems for extrinsic sample antimeans w.r.t. an arbitrary embedding of such a manifold into a Euclidean space. The resulting asymptotic test statistics are detailed in the concrete case of planar Kendall shapes of finite landmark configurations, regarded as random objects on a complex projective space, that is Veronese-Whitney embedded in a space of self-adjoint matrices. Two medical imaging applications using classic data libraries in Bookstein's Morphometric Tools for Landmark Data textbook and from Dryden and Mardia's Statistical Shape Analysis are also given here.

The study of finite configurations of points modulo projective transformations, known as projective shape analysis, is the second topic of this dissertation, and has various applications in 3D machine vision. We consider a suitable projective coordinate system to analyze projective shapes. The resulting projective shape for general labeled configurations of k points in m dimensions is known to be a Cartesian product of k-m-2 copies of the axial space RP^m. Two sample tests for Veronese-Whitney means of projective shapes of k-ads in general position in RP^3 are addressed using the mean change on a Lie group, product of a finite number of copies of the group of projective quaternions. An application to 3D analysis of projective bioshape data extracted from digital camera images is given here. The methods developed here are nonparametric: for large samples one derives a chi square test for the equality of two extrinsic antimeans. If sample sizes are small, one uses nonparametric bootstrap.