On the consistency of Fr\'echet means in deformable models for curve and image analysis

TL;DR

This paper introduces a new class of deformable models for high-dimensional curves and images, focusing on the statistical properties of Fréchet means for estimating mean patterns and deformation parameters.

Contribution

It studies the asymptotic behavior of Fréchet mean estimators in deformable models, considering the effects of sample size and number of observations.

Findings

Establishes conditions for consistency of Fréchet mean estimators.

Demonstrates finite sample performance through numerical experiments.

Abstract

A new class of statistical deformable models is introduced to study high-dimensional curves or images. In addition to the standard measurement error term, these deformable models include an extra error term modeling the individual variations in intensity around a mean pattern. It is shown that an appropriate tool for statistical inference in such models is the notion of sample Fr\'echet means, which leads to estimators of the deformation parameters and the mean pattern. The main contribution of this paper is to study how the behavior of these estimators depends on the number n of design points and the number J of observed curves (or images). Numerical experiments are given to illustrate the finite sample performances of the procedure.