On inverse problems and finite elements in shape analysis

Abstract

Shapes are nonlinear, multifarious objects and it is a nontrivial task to attach meaning to statements about their comparison. A central problem in the field of shape analysis is therefore to make sense of the notion of a metric on an abstract shape space. Through the lens of infinite-dimensional Riemannian geometry one can endow such spaces with different metrics stemming from nonrigid group actions to compute so-called geodesic matchings between shapes that share diffeomorphic or homeomorphic orbits. Owing to shape analysis being a relatively new field in the history of mathematics, several open questions require answering to reconcile theoretical development with numerical approximation. This project sets out to unite the framework of diffeomorphometry with linear functional analysis amenable to finite element methods. Shapes in this project will be images, curves or even measures, and we derive and compute convergent numerical approximations to the geodesic equations related to the motion on the aforementioned groups.