||<p>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.</p><p>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.</p>