Least Squares Fitting of Quadratic Curves and Surfaces

In computer vision one often fits ellipses and other conics to observed points on a plane or ellipsoids/quadrics to spacial point clouds. The most accurate and robust fit is obtained by minimizing geometric (orthogonal) distances, but this problem has no closed form solution and most known algorithms are prohibitively slow. We revisit this issue based on recent advances by S. J. Ahn, D. Eberly, and our own. Ahn has sorted out various approaches and identified the most efficient one. Eberly has developed a fast method of projecting points onto ellipses/ellipsoids (and gave a proof of its convergence). We extend Eberly’s projection algorithm to other conics, as well as quadrics in space. We also demonstrate that Eberly’s projection method combined with Ahn’s most efficient approach (and using Taubin’s algebraic fit for initialization) makes a highly efficient fitting scheme working well for all quadratic curves and surfaces. ∗E-mail address: [email protected]; [email protected] 2 N. Chernov and H. Ma