A statistical analysis of least-squares circle-centre estimation

Abstract

In this paper, we examine the problem of fitting a circle to a set of noisy measurements of points on the circle's circumference. An estimator based on standard least-squares techniques has been proposed by DELOGNE which has been shown by KÅSA to be convenient for its ease of analysis and computation. Using CHAN's circular functional model to describe the distribution of points, we perform a statistical analysis of the estimate of the circle's centre, assuming independent, identically distributed Gaussian measurement errors. We examine the existence of the mean and variance of the estimator for fixed sample sizes. We find that the mean exists when the number of sample points is greater than 2 and the variance exists when this number is greater than 3. We also derive approximations for the mean and variance for fixed sample sizes when the noise variance is small. We find that the bias approaches zero as the noise variance diminishes and that the variance approaches the CRAMÉR-RAO lower bound. We also show this through Monte-Carlo simulations.

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