Consistent estimation of a mean planar curve modulo similarities

TL;DR

This paper analyzes the consistency of a smoothed Procrustean mean curve estimator for a set of random planar curves, considering the effects of data dimension and sample size, with numerical validation.

Contribution

It provides a theoretical analysis of the convergence properties of the Procrustean mean estimator under deformable models with nuisance parameters.

Findings

Convergence of the estimator depends on data dimension and sample size.

Numerical experiments support theoretical results.

The method effectively estimates the mean curve despite nuisance parameters.

Abstract

We consider the problem of estimating a mean planar curve from a set of $J$ random planar curves observed on a $k$-points deterministic design. We study the consistency of a smoothed Procrustean mean curve when the observations obey a deformable model including some nuisance parameters such as random translations, rotations and scaling. The main contribution of the paper is to analyze the influence of the dimension $k$ of the data and of the number $J$ of observed configurations on the convergence of the smoothed Procrustean estimator to the mean curve of the model. Some numerical experiments illustrate these results.