Applications of Shape Metrics: From Classifying Ancient Pots to Monitoring Endangered Birds

Broadly, the aim of this PhD is to explore methods of analysing images with an
emphasis on a significant problem in shape analysis: pairwise shape registration.
Pairwise registration of shapes found in images, using difieomorphic mapping meth-
ods, seek to find a difieomorphism that acts like a sheer sheet of elastic which
stretches and shrinks the pair of shapes in order to align them. Though they have
been vastly studied in recent years, the study of their applications on real-world
data has been infrequent. The objectives of this PhD are to create a detailed back-
ground of difieomorphic mapping methods for pairwise registration, and study its
applications on real-world data with two projects. The first is a grand study of the
shapes of Ancient Greek vases with the aim of automatic classification of each vase,
ranging from amphorae to cups. This project is in collaboration with Imperial Col-
lege London. While the second project, in collaboration with Victoria University
of Wellington, investigates the use of registration in classifying birds from birdsong
data. Furthermore, we hope that a by-product of the PhD would be a comprehen-
sive manual on applying shape analysis methods to real-world data, which can be
used by researchers from a variety of fields, exploring a myriad of data-sets.
Our first project involves working with the Beazley Archive, a large collection of
images of ancient vases, collated by multiple museums across the globe. We will
be extracting an outline contour of the vases in the images. Henceforth, we will
use pairwise registration and compute distances between the pots, on the relevant
shape space. This distance matrix can then be incorporated in classification algo-
rithms and used to build a phylogeny, an evolutionary tree depicting the relation-
ships between the classes of vases. Furthermore, by using pairwise registration, we
will perform Principal Geodesic Analysis to build a statistical shape model, SSM,
displaying the variability of vase shapes. Though building SSMs using Principal
Component Analysis, PCA, has been a popular tool in analysing shape variability,
replacing this with Principal Geodesic Analysis, a variant of PCA in a manifold
setting, has not been studied in shape analysis literature.
The second application is joint work with the AviaNZ Project, a collaboration
between mathematicians, conservation-biologists and more, in order to measure
the abundance of birds from birdsong data. When recording birdsong, the data
can be plotted onto spectrograms which display the energy at multiple frequencies
over time. Once we have these spectrograms, we can extract high energy time
frequency contours. Hence, we reach our aim of registering certain frequencies seen
in birdsong data. To do this, we introduce a novel mathematical framework for
registering curves in a product space. The registered frequencies will subsequently
be incorporated in further classification analysis.