Extension to Infinite Dimensions of a Stochastic Second-Order Model associated with the Shape Splines

TL;DR

This paper extends a stochastic second-order shape model from finite to infinite dimensions, ensuring mathematical consistency and applicability to biological shape variability in medical imaging.

Contribution

It introduces an infinite-dimensional stochastic shape model, proves non-explosion of solutions, and demonstrates convergence from finite to infinite-dimensional settings.

Findings

Solutions do not blow up in finite time

Strong convergence from finite to infinite dimensions

Model applicable to diverse shapes and dimensions

Abstract

We introduce a second-order stochastic model to explore the variability in growth of biological shapes with applications to medical imaging. Our model is a perturbation with a random force of the Hamiltonian formulation of the geodesics. Starting with the finite-dimensional case of landmarks, we prove that the random solutions do not blow up in finite time. We then prove the consistency of the model by demonstrating a strong convergence result from the finite-dimensional approximations to the infinite-dimensional setting of shapes. To this end we introduce a suitable Hilbert space close to a Besov space that leads to our result being valid in any dimension of the ambient space and for a wide range of shapes.