Shape is a fundamental property of objects, studied at least since Aristotle. In the modern era, the famous treatise by D’Arcy Thompson, “On Growth and Form”, established the idea of shape and shape deformations as a fascinating and important topic of study. Our ability to describe shapes mathematically, to measure the differences between them, and to model them statistically, is now key to many current areas of science and technology, from molecular biology to computer vision.
Kendall’s shape space for triangles in two dimensions (Klingenberg 2016).
This project focuses on finite-dimensional representations of shape, specifically using configurations of so-called ‘landmark’ points on an object. The idea of ‘shape spaces’ then makes precise the concept of shape as a geometrical property independent of location, orientation, and size: if $M$ is the space of configurations (e.g. of $k$ landmark points in $\mathbb{R}^{m}$), the shape space is derived by identifying points related by translations, rotations, and scalings; mathematically, it is the quotient manifold $S = M/G$, where $G$ is the group of similarity transformations. The resulting spaces are interesting mathematical objects in their own right: nonlinear and sometimes singular. One of the simplest examples is the shape space of triangles in $\mathbb{R}^{2}$, which is topologically a disc.
Statistical shape analysis then proceeds by defining probability distributions on shape space, a nontrivial exercise due to the nature of the spaces: even elementary ideas like the sample mean require generalization to these more complex geometries.
The goal of this project is to explore the geometry of shape spaces, methods for performing statistical shape analysis, and their applications to concrete problems.
The group project will establish the mathematical foundations of statistical shape analysis, focusing on Kendall's shape spaces and the Procrustes distance.
By the end of the group project, you will have learned:
The project will revolve around reading, alongside coding in R or Python. Students will demonstrate their understanding by proving key properties, implementing analysis methods, analysing datasets, and clearly communicating the material in written and oral formats.
The individual project will build upon the differential geometry and statistical models developed in the group phase to tackle more advanced frameworks and modern applications within finite-dimensional shape spaces. A few examples of topics you will be able to investigate are: