A statistical analysis of the Delogne-Kåsa method for fitting circles

In this paper, we examine the problem of fitting a circle to a set of noisy measurements of points on the circle’s circumference. Delogne (Proc. IMEKO-Symp. Microwave Measurements 1972, 117-123) has proposed an estimator which has been shown by Kåsa (IEEE Trans. Instrum. Meas. 25, 1976, 8-14) 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 3 and the variance exists when this number is greater than 4. 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.