Euler Integration with Applications to Statistical Shape Analysis and Imaging

Abstract

This dissertation focuses on use of Euler calculus in statistical shape analysis. The shapes have no metric structure, so to analyze them we need to define such a structure. Classical work on this has been done by Kendall, who imposed a metric structure by introducing landmarks on the shapes. These are points on the shape that have corresponding counterparts across the shapes. Such representation has two drawbacks: It does not use all the information about the shape and the choice of landmarks and correspondences is a challenging task that does not always have the right answer. One can create a digital version of Kendall’s shape spaces by looking at diffeomorphisms between the shapes. The obvious limitation in this method is that the shapes need to be diffeomorphic. The subject of this dissertation is a more general construction that makes use of the idea of integrating against the Euler characteristic. For shape analysis, this method was first proposed by Turner et al. In Chapter 2 we introduce an extension of the Euler calculus shape analysis framework for continuous type data. We show these lifted transforms retain the most important properties of the discrete transform, making them very well suited for statistical applications. We provide the necessary theoretical results as well as demonstrate the utility of this approach on real and simulated data. In Chapter 3 we present a first ever subshape selection pipeline that does not rely on diffeomorphisms nor landmarks. This is achieved by first transform the shapes with the aforementioned tools to a space where distances and inner products can be defined. With these tools we solve the statistical problem of feature selection. Finally, we pull back this evidence on the original shape space by an evidence reconstruction procedure. We providea detailed study of the method on simulated data and apply it on a problem in geometric morphometrics.