We study the problem of fitting circles (or circular arcs) to data points observed with errors in both variables. A detailed error analysis for all popular circle fitting methods -geometric fit, Kåsa fit, Pratt fit, and Taubin fit -is presented. Our error analysis goes deeper than the traditional expansion to the leading order. We obtain higher order terms, which show exactly why and by how much circle fits differ from each other. Our analysis allows us to construct a new algebraic (non-iterative) circle fitting algorithm that outperforms all the existing methods, including the (previously regarded as unbeatable) geometric fit.
Abstract:The technique of "renormalization" for geometric estimation attracted much attention when it appeared in early 1990s for having higher accuracy than any other then known methods. The key fact is that it directly specifies equations to solve, rather than minimizing some cost function. This paper expounds this "non-minimization approach" in detail and exploits this principle to modify renormalization so that it outperforms the standard reprojection error minimization. Doing a precise error analysis in the most general situation, we derive a formula that maximizes the accuracy of the solution; we call it hyper-renormalization. Applying it to ellipse fitting, fundamental matrix computation, and homography computation, we confirm its accuracy and efficiency for sufficiently small noise. Our emphasis is on the general principle, rather than on individual methods for particular problems.
This study is devoted to comparing the most popular circle fits (the geometric fit, Pratt's, Taubin's, Kåsa'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 σ 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 σ 4 , which come from essential bias, as well as all terms of order σ 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 σ 4 . Our numerical tests confirm the superiority of the proposed fit over the existing fits.
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