Multiple-solutions RANSAC for finding axes of symmetry in fragments of objects

Abstract

The problem of “finding best lines passing through a set of straight lines” has appeared in applications such as archaeological pottery analysis, precision manufacturing, and 3D modelling. In these applications, an instance of this problem is finding the symmetry axis of a symmetrical object from a set of its surface normal lines. We show that the mentioned instance of the problem may have two meaningful local minima, one of which is the symmetry axis, a fact that has been neglected in the literature. A multiple-solutions RANSAC algorithm is proposed for finding initial estimates of both local minima in the presence of outliers. Then, a coordinate-descent algorithm is presented that starts from these initial estimates and finds the local minima of the problem. The proposed coordinate-descent method does not involve any line search procedure, and its convergence is guaranteed. We also provide a proof for the rate of the convergence.