Abstract
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.
Highlights
AMS 2000 subject classifications: Primary 60K35, 60K35; secondary 60K35
We employ higher-order error analysis and show exactly why and by how much the geometric circle fit outperforms the algebraic circle fits in accuracy; we compare the precision of different algebraic fits
A standard approach to fitting circles to 2D data is based on orthogonal least squares, it is called geometric fit, or orthogonal distance regression (ODR)
Summary
We adopt a standard functional model in which data points (x1, y1), . . . , (xn, yn) are noisy observations of some true points (x1, y1), . . . , (xn, yn), i.e. We adopt a standard functional model in which data points (x1, y1), . (xn, yn) are noisy observations of some true points (x1, y1), . The true points (xi, yi) are supposed to lie on a ‘true circle’, i.e. satisfy (xi − a)2 + (yi − ̃b)2 = R2, i = 1, . Where (a, ̃b, R) denote the ‘true’ (unknown) parameters. Φn are regarded as fixed unknowns and treated as additional parameters of the model (called incidental or latent parameters). In our paper δi and εi have common variance σ2, i.e. our noise is homoscedastic. In many studies the noise is heteroscedastic [25, 35], i.e. the normal vector (δi, εi) has point-dependent covariance matrix σ2Ci, where Ci is known and depends on i, and σ2 is an unknown factor. Our analysis can be extended to this case, too, but the resulting formulas will be somewhat more complex, so we leave it out
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