Fast and almost unbiased weighted least squares fitting of circles

The algebraic fitting of circles was first proposed by Delogne (1972) and Kasa (1976). We extend the work by Delogne (1972) and Kasa (1976) and propose fast and almost unbiased weighted least squares to best fit circles. The simulations have shown that two non-iterative versions of the bias-corrected weighted LS method are fast and almost unbiased and perform much better than the naive weighted LS method (without applying any bias correction), the ordinary LS-based methods and the gradient-based weighted LS method in terms of both biases and mean squared errors, no matter whether fitting of circles is strongly or weakly constrained geometrically. Nevertheless, a weak geometrical constraint results in a poor fitting of circles. We accordingly propose the regularized variants of the bias-corrected weighted LS method to fit circles from data with a weak geometrical constraint. The simulations have also shown that the two non-iterative regularized variants fit circles satisfactorily well and consistently perform the best among all the regularization methods under study for ill-conditioned problems of fitting circles.

Automated summary

The algebraic fitting of circles was originally proposed by Delogne (1972) and Kasa (1976). In this study, we extend their work and introduce fast and nearly unbiased weighted least squares methods to best fit circles. Simulation results demonstrate that the two non-iterative bias-corrected weighted LS methods outperform the naive weighted LS method, ordinary LS-based methods, and gradient-based weighted LS method in terms of both biases and mean squared errors, regardless of strong or weak geometric constraints. However, a weak geometric constraint leads to poor circle fitting. To address this, we propose regularized variants of the bias-corrected weighted LS method to fit circles with weak geometric constraints. Simulations also reveal that these two non-iterative regularized variants achieve satisfactory circle fitting and consistently perform the best among all regularization methods for ill-conditioned circle fitting problems.