A Hierarchical Geodesic Model for Longitudinal Analysis on Manifolds

TL;DR

This paper introduces a hierarchical geodesic model for analyzing longitudinal manifold-valued data, specifically applied to shape spaces, improving computational efficiency and enabling hypothesis testing and trend estimation.

Contribution

It develops a novel variational time discretization of geodesics on shape spaces using a functional-based metric, enhancing analysis of longitudinal shape data.

Findings

Efficient geodesic computation in shape spaces.

Successful application to 2D rat skull and 3D osteoarthritis shapes.

Enabled hypothesis testing and trend estimation in longitudinal shape analysis.

Abstract

In many applications, geodesic hierarchical models are adequate for the study of temporal observations. We employ such a model derived for manifold-valued data to Kendall's shape space. In particular, instead of the Sasaki metric, we adapt a functional-based metric, which increases the computational efficiency and does not require the implementation of the curvature tensor. We propose the corresponding variational time discretization of geodesics and employ the approach for longitudinal analysis of 2D rat skulls shapes as well as 3D shapes derived from an imaging study on osteoarthritis. Particularly, we perform hypothesis test and estimate the mean trends.