Inference on subspheres model for directional data

TL;DR

This paper develops statistical methods for estimating and analyzing the deformation of 3D objects modeled as subspheres on a polysphere, with proven consistency and asymptotic properties.

Contribution

It introduces a novel framework for estimating subspheres representing rotational deformations using generalized Fréchet means, with theoretical guarantees.

Findings

Establishes consistency of the estimators.

Proves asymptotic normality of the estimators.

Provides a mathematical foundation for modeling rotational deformations.

Abstract

Modeling deformations of a real object is an important task in computer vision, biomedical engineering and biomechanics. In this paper, we focus on a situation where a three-dimensional object is rotationally deformed about a fixed axis, and assume that many independent observations are available. Such a problem is generalized to an estimation of concentric, co-dimension 1, subspheres of a polysphere. We formulate least-square estimators as generalized Fr\'{e}chet means, and evaluate the consistency and asymptotic normality.