Unbiased Least Squares Fitting of Circular Arcs

Previous solutions to the problem of obtaining a least squares fit to a circular arc are discussed. The existence of severe bias in closed form solutions and non-convergence in iterative solutions for shallow arcs is noted. A straightforward and economical iterative procedure is developed which is shown to be stable and have rapid convergence to an unbiased least squares fit on a wide range of synthetic data. The random error in the parameters of these fits is measured and compared with theoretical predictions. The procedure is shown to operate up to the limit of the validity of circular arc fitting. The term well-defined is introduced to describe arcs within this limit. Example applications to image data show the utility of the method, and the inadequacy of previous solutions, in real image analysis tasks.