Bi-invariant Two-Sample Tests in Lie Groups for Shape Analysis

TL;DR

This paper introduces bi-invariant two-sample tests for shape analysis in Lie groups, enabling unbiased, consistent comparison of shapes without dependence on reference choices, validated on hippocampal shape data.

Contribution

It generalizes Hotelling's T^2 and Bhattacharayya distance to Lie groups, creating bi-invariant, nonparametric two-sample tests for shape analysis.

Findings

Successfully detected shape differences in hippocampal data

Ensured tests are bi-invariant and consistent

Validated method on real-world shape data

Abstract

We propose generalizations of the Hotelling's $T^2$ statistic and the Bhattacharayya distance for data taking values in Lie groups. A key feature of the derived measures is that they are compatible with the group structure even for manifolds that do not admit any bi-invariant metric. This property, e.g., assures analysis that does not depend on the reference shape, thus, preventing bias due to arbitrary choices thereof. Furthermore, the generalizations agree with the common definitions for the special case of flat vector spaces guaranteeing consistency. Employing a permutation test setup, we further obtain nonparametric, two-sample testing procedures that themselves are bi-invariant and consistent. We validate our method in group tests revealing significant differences in hippocampal shape between individuals with mild cognitive impairment and normal controls.