An Algorithm for Least-Squares Circle Fitting to Data with Specified Uncertainty Ellipses

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Abstract
The problem considered here is that of fitting a circle to a set of measured data points specified in terms of their cartesian coordinates. It is assumed that the data adequately represents a circle and that associated with each data point there is an uncertainty ellipse describing the measurement error. A weighted nonlinear least-squares problem is formulated in order to determine unbiased estimates of the centre and radius of the circle which bests fits the given data. The resulting problem displays structure which is exploited when the Gauss-Newton algorithm is used to obtain a solution. In addition to estimates of the circle parameters the algorithm produces error ellipses for the centre of the circle and any point on its circumference.

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