# Geodesic analysis in Kendall's shape space with epidemiological applications

## TL;DR

This paper develops analytical tools for geodesic analysis in Kendall's shape space, enabling efficient Riemannian optimization and applying it to longitudinal epidemiological shape data for early osteoarthritis prediction.

## Contribution

It introduces analytical expressions for Jacobi fields and parallel transports in Kendall's shape space, significantly reducing computational costs for geodesic regression.

## Key findings

- Identified shape differences between osteoarthritis and control groups
- Demonstrated efficient geodesic regression in shape analysis
- Enabled early prediction of knee osteoarthritis using shape data

## Abstract

We analytically determine Jacobi fields and parallel transports and compute geodesic regression in Kendall's shape space. Using the derived expressions, we can fully leverage the geometry via Riemannian optimization and thereby reduce the computational expense by several orders of magnitude over common, nonlinear constrained approaches. The methodology is demonstrated by performing a longitudinal statistical analysis of epidemiological shape data. As an example application we have chosen 3D shapes of knee bones, reconstructed from image data of the Osteoarthritis Initiative (OAI). Comparing subject groups with incident and developing osteoarthritis versus normal controls, we find clear differences in the temporal development of femur shapes. This paves the way for early prediction of incident knee osteoarthritis, using geometry data alone.