An averaged projected Robbins-Monro algorithm for estimating the parameters of a truncated spherical distribution

TL;DR

This paper introduces a projected Robbins-Monro algorithm with averaging for accurately estimating the center and radius of a sphere from noisy 3D point cloud data, including truncated spheres.

Contribution

It presents a novel stochastic approximation algorithm with proven convergence and asymptotic properties for sphere parameter estimation from noisy point clouds.

Findings

Algorithm demonstrates effective convergence on simulated data.

Provides asymptotic normality and convergence rates.

Shows efficiency for small to moderate sample sizes.

Abstract

The objective of this work is to propose a new algorithm to fit a sphere on a noisy 3D point cloud distributed around a complete or a truncated sphere. More precisely, we introduce a projected Robbins-Monro algorithm and its averaged version for estimating the center and the radius of the sphere. We give asymptotic results such as the almost sure convergence of these algorithms as well as the asymptotic normality of the averaged algorithm. Furthermore, some non-asymptotic results will be given, such as the rates of convergence in quadratic mean. Some numerical experiments show the efficiency of the proposed algorithm on simulated data for small to moderate sample sizes.