Recovering Circles and Spheres from Point Data

Methods for fitting circles and spheres to point sets are discussed. LADAR (LAser Detection And Ranging) scanners are capable of generating ‘point clouds’ containing the (x, y, z) coordinates of up to several millions of points reflecting the laser signals. In particular, coordinates collected off objects such as spheres may then be used to model these objects by fitting procedures. Fitting amounts to minimizing what is called here a “gauge function,” which quantifies the quality of a particular fit. This work analyzes and experimentally examines the impact of the choice of three such gauge functions. One of the resulting methods, termed here as “algebraic” fitting, formulates the minimization problem as a regression. The second, referred to as “geometric” fitting, minimizes the sum of squares of the Euclidean distances of the data points from the tentative sphere. This method, based on orthogonal distance minimization, is most highly regarded and widely used. The third method represents a novel way of fitting. It is based on the directions in which the individual data points have been acquired.