Shape Modelling

The famous treatise by D'Arcy Thompson, “On Growth and Form”, established the idea of shape as a fascinating and important topic of study. Since then, many different approaches have been developed for the description of shape and its variability. A multitude of different shape representations have been used, among them landmark points, closed curves and surfaces, level sets, and symmetry axes. Shape spaces, quotients of representation spaces by a similarity group, make precise the idea of shape as a geometrical property independent of location, orientation, and scale. Shape variability has been described by probability distributions defined on representation or shape spaces (which often involves defining distributions on manifolds), but also by diffeomorphisms of the space in which the shape 'lives' (and in fact this was D'Arcy Thompson's approach).

The applications of shape models are many. The ability to recognize an object in an image often depends crucially on a prior 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 are crucial for the analysis of image 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 data.

Statistical Concepts II, 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 might also help to know a little about 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.