Solutions and evaluations for fitting of concentric circles

Many practical applications involve the basic problem of fitting a number of data points to a pair of concentric circles, including coordinate metrology, petroleum engineering and image processing. In this paper, two versions of the Levenberg-Marquardt (LM) are applied to obtain the maximum likelihood estimator of the common center and the radii for the concentric circles. In addition, two numerical schemes for conic fitting are extended to the concentric circles fitting problem, as well as several algebraic fits are proposed. This paper shows analytically that the MLE and the numerical schemes are statistically optimal in the sense of reaching the Kanatani-Cramér-Rao (KCR) lower bound, while the other algebraic fits are suboptimal. Our results are confirmed by several numerical experiments.