Shape Splines and Stochastic Shape Evolutions: A Second Order Point of View

TL;DR

This paper introduces a second order spline interpolation framework for shape sequences using Riemannian geometry and develops a stochastic model for random shape evolutions, extending to infinite-dimensional shape spaces.

Contribution

It proposes a novel second order spline interpolation method on Riemannian manifolds and a stochastic model for shape evolution, advancing statistical analysis of shape sequences.

Findings

Developed a Riemannian cubic spline interpolation method for shape data.

Designed a stochastic model for random shape evolutions based on control variables.

Framework can be extended to infinite-dimensional shape spaces.

Abstract

This article presents a new mathematical framework to perform statistical analysis on time-indexed sequences of 2D or 3D shapes. At the core of this statistical analysis is the task of time interpolation of such data. Current models in use can be compared to linear interpolation for one dimensional data. We develop a spline interpolation method which is directly related to cubic splines on a Riemannian manifold. Our strategy consists of introducing a control variable on the Hamiltonian equations of the geodesics. Motivated by statistical modeling of spatiotemporal data, we also design a stochastic model to deal with random shape evolutions. This model is closely related to the spline model since the control variable previously introduced is set as a random force perturbing the evolution. Although we focus on the finite dimensional case of landmarks, our models can be extended to infinite dimensional shape spaces, and they provide a first step for a non parametric growth model for shapes taking advantage of the widely developed framework of large deformations by diffeomorphisms.