Weighted least-squares fitting of circles with variance component estimation

Although the least-squares (LS) circle fit has been widely used, the weighted LS fitting of circle is not thoroughly investigated, in particular, when the prior weight information is only partly known. Based on the Gauss–Helmert model (GHM), we first investigate the invariance on translation and rotation. The results show that though the translational invariance holds, the rotation invariance is broken except for some specific weight structures. As the main finding of this paper, we develop the VCE theory directly adapting to the nonlinear GHM representation of the circle fitting problem where the weight information is not exactly known. In the simulated example and the real applications, we show that: (1) The conclusions about the invariance of translation and rotation are validated. (2) The estimated variance components can perfectly represent the uncertainty of different point groups or different coordinate components from the statistical perspective.