Stochastics of shapes and Kunita flows

TL;DR

This paper formalizes stochastic processes of evolving shapes, linking them to Kunita flows, and discusses methods for statistical inference using bridge sampling techniques.

Contribution

It introduces a formal framework for stochastic shape processes compatible with shape structures and connects them to Kunita flows, including a survey and inference methods.

Findings

Defined properties for stochastic shape processes

Linked shape processes to Kunita flows

Demonstrated use of bridge sampling for inference

Abstract

Stochastic processes of evolving shapes are used in applications including evolutionary biology, where morphology changes stochastically as a function of evolutionary processes. Due to the non-linear and often infinite-dimensional nature of shape spaces, the mathematical construction of suitable stochastic shape processes is far from immediate. We define and formalize properties that stochastic shape processes should ideally satisfy to be compatible with the shape structure, and we link this to Kunita flows that, when acting on shape spaces, induce stochastic processes that satisfy these criteria by their construction. We couple this with a survey of other relevant shape stochastic processes and show how bridge sampling techniques can be used to condition shape stochastic processes on observed data thereby allowing for statistical inference of parameters of the stochastic dynamics.