Template estimation in computational anatomy: Fr\'echet means in top and quotient spaces are not consistent

TL;DR

This paper investigates the inconsistency of Fréchet mean-based template estimation in quotient spaces, revealing that noise and group actions often lead to biased and unreliable results in computational anatomy.

Contribution

It identifies conditions causing inconsistency in Fréchet mean estimators in quotient spaces and quantifies the bias related to noise levels.

Findings

Fréchet mean estimators are often inconsistent in quotient spaces.

Inconsistency arises when the noise distribution's support is large.

Bias increases with higher noise levels, affecting estimation reliability.

Abstract

In this article, we study the consistency of the template estimation with the Fr\'echet mean in quotient spaces. The Fr\'echet mean in quotient spaces is often used when the observations are deformed or transformed by a group action. We show that in most cases this estimator is actually inconsistent. We exhibit a sufficient condition for this inconsistency, which amounts to the folding of the distribution of the noisy template when it is projected to the quotient space. This condition appears to be fulfilled as soon as the support of the noise is large enough. To quantify this inconsistency we provide lower and upper bounds of the bias as a function of the variability (the noise level). This shows that the consistency bias cannot be neglected when the variability increases.