Template shape estimation: correcting an asymptotic bias

TL;DR

This paper analyzes the bias in template shape estimation across various shape data types, providing a Taylor expansion, bias correction methods, and practical guidelines for when correction is necessary.

Contribution

It introduces a geometric analysis of asymptotic bias in shape estimation, along with bootstrap correction procedures applicable to diverse shape data.

Findings

Bias in shape estimation can be quantified using a Taylor expansion.

Bootstrap procedures effectively correct the asymptotic bias.

Guidelines help determine when bias correction is necessary.

Abstract

We use tools from geometric statistics to analyze the usual estimation procedure of a template shape. This applies to shapes from landmarks, curves, surfaces, images etc. We demonstrate the asymptotic bias of the template shape estimation using the stratified geometry of the shape space. We give a Taylor expansion of the bias with respect to a parameter $\sigma$ describing the measurement error on the data. We propose two bootstrap procedures that quantify the bias and correct it, if needed. They are applicable for any type of shape data. We give a rule of thumb to provide intuition on whether the bias has to be corrected. This exhibits the parameters that control the bias' magnitude. We illustrate our results on simulated and real shape data.