Hypoelliptic Diffusion Maps and Their Applications in Automated Geometric Morphometrics

Abstract

We introduce Hypoelliptic Diffusion Maps (HDM), a novel semi-supervised machine learning framework for the analysis of collections of anatomical surfaces. Triangular meshes obtained from discretizing these surfaces are high-dimensional, noisy, and unorganized, which makes it difficult to consistently extract robust geometric features for the whole collection. Traditionally, biologists put equal numbers of ``landmarks'' on each mesh, and study the ``shape space'' with this fixed number of landmarks to understand patterns of shape variation in the collection of surfaces; we propose here a correspondence-based, landmark-free approach that automates this process while maintaining morphological interpretability. Our methodology avoids explicit feature extraction and is thus related to the kernel methods, but the equivalent notion of ``kernel function'' takes value in pairwise correspondences between triangular meshes in the collection. Under the assumption that the data set is sampled from a fibre bundle, we show that the new graph Laplacian defined in the HDM framework is the discrete counterpart of a class of hypoelliptic partial differential operators.This thesis is organized as follows: Chapter 1 is the introduction; Chapter 2 describes the correspondences between anatomical surfaces used in this research; Chapter 3 and 4 discuss the HDM framework in detail; Chapter 5 illustrates some interesting applications of this framework in geometric morphometrics.