A function space perspective on stochastic shape evolution

TL;DR

This paper introduces a novel stochastic shape model based on Sobolev space functions, enabling parameterisation-independent shape evolution modeling with explicit basis noise, and demonstrates its properties and applications.

Contribution

It presents a new Sobolev space-based stochastic shape model with an explicit basis, advancing shape evolution modeling independent of mesh parameterisation.

Findings

Model is independent of mesh parameterisation.

Explicit orthonormal basis facilitates noise description.

Numerical examples demonstrate shape evolution simulations.

Abstract

Modelling randomness in shape data, for example, the evolution of shapes of organisms in biology, requires stochastic models of shapes. This paper presents a new stochastic shape model based on a description of shapes as functions in a Sobolev space. Using an explicit orthonormal basis as a reference frame for the noise, the model is independent of the parameterisation of the mesh. We define the stochastic model, explore its properties, and illustrate examples of stochastic shape evolutions using the resulting numerical framework.