Intrinsic Inference on the Mean Geodesic of Planar Shapes and Tree Discrimination by Leaf Growth

TL;DR

This paper develops a statistical test for the mean geodesic of planar shapes considering non-Euclidean geometry, enabling leaf growth analysis and tree discrimination, with proven consistency and practical algorithms.

Contribution

It introduces a new test for the mean geodesic in shape space, with strong theoretical foundations and applications to leaf growth and tree discrimination.

Findings

Verified the geodesic hypothesis for leaf growth.

Discriminated genetically different trees based on leaf shape growth.

Provided algorithms for computing the Ziezold mean geodesic.

Abstract

For planar landmark based shapes, taking into account the non-Euclidean geometry of the shape space, a statistical test for a common mean first geodesic principal component (GPC) is devised. It rests on one of two asymptotic scenarios, both of which are identical in a Euclidean geometry. For both scenarios, strong consistency and central limit theorems are established, along with an algorithm for the computation of a Ziezold mean geodesic. In application, this allows to verify the geodesic hypothesis for leaf growth of Canadian black poplars and to discriminate genetically different trees by observations of leaf shape growth over brief time intervals. With a test based on Procrustes tangent space coordinates, not involving the shape space's curvature, neither can be achieved.