Further statistical analysis of circle fitting

This study is devoted to comparing the most popular circle fits (the geometric fit, Pratt's, Taubin's, Kasa's) and the most recently developed algebraic circle fits: hyperaccurate fit and HyperLS fit. Even though hyperaccurate fit has zero essential bias and HyperLS fit is unbiased up to order sigma(4), the geometric fit still outperforms them in some circumstances. Since the first-order leading term of the MSE for all fits are equal, we go one step further and derive all terms of order sigma(4), which come from essential bias, as well as all terms of order sigma(4)/n, which come from two sources: the variance and the outer product of the essential bias and the nonessential bias. Our analysis shows that when data are distributed along a short circular arc, the covariance part is the dominant part of the second-order term in the MSE. Accordingly, the geometric fit outperforms all existing methods. However, for a long circular arc, the bias becomes the most dominant part of the second-order term, and as such, hyperaccurate fit and HyperLS fit outperform the geometric fit. We finally propose a 'bias correction' version of the geometric fit, which in turn, outperforms all existing methods. The new method has two features. Its variance is the smallest and has zero bias up to order sigma(4). Our numerical tests confirm the superiority of the proposed fit over the existing fits.