Shape is a fundamental property of objects, and the famous treatise by D’Arcy Thompson, “On Growth and Form”, established the idea of shape 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.
Many different mathematical approaches have been developed for the description of shape and its variability. Shapes have been represented in different ways: using landmark points; as closed curves and surfaces; as level sets; and by symmetry axes. The idea of ‘shape spaces’, quotients of these representation spaces by groups of translations, rotations, and scalings, makes precise the concept of shape as a geometrical property independent of location, orientation, and size.
These representation and shape spaces are complex: nonlinear, sometimes singular, often infinite-dimensional. Shape variability can be described by probability distributions defined on these spaces, but also by distributions on the diffeomorphism groups of these space, defining shape variability in terms of deformations (and in fact this was D’Arcy Thompson’s approach).
More recently, deep learning has been applied to shape analysis. Most neural networks implicitly treat their input as belonging to a linear space, and adapting them to nonlinear inputs requires some effort and introduces interesting mathematical structures.
The applications of shape analysis are manifold. The ability to recognize an object in an image often depends crucially on a model of the shape of that object. For example, what is the object in the image above? (Scroll down down for the answer.) As a result, shape models, however constructed, are crucial for the analysis of image and video data, a vital information resource in many domains, and also link to psychological theories of perception.
The different approaches to shape modelling use a wide range of fascinating mathematics. This project could involve an in-depth study of one approach, a comparison of different approaches, or could take a more computational direction, studying and implementing shape models as applied to image or other data.
Statistical Inference 2, and in particular Bayesian statistics, would be useful. Depending on the direction of the project, some knowledge of differential and Riemannian geometry and group theory would be an advantage. For the more practical directions, it is important have a working knowledge of a programming language.
There are very many approaches to shape modelling. Here is a list of papers and books on various aspects of shape modelling and statistical shape analysis. It might be a good idea to look at the introductory chapters or sections of these, and browse through the rest to get an idea of the topic.