Gröbner Basis Methods for Minimal Problems in Computer Vision

Abstract: A method is presented for building solvers for classes of multivariate polynomial equations. The method is based on solving an analogous template problem over a finite field, and then using the elimination order established for this problem for the original class of problems. A strength of this method is that this permits pivoting in the elimination. Solvers for several minimal problems in computer vision are presented. Relative pose is solved both for a generalised camera, and for a camera with unknown focal length, both in two positions with six visible points. A solver for optimal triangulation in three images is presented. Model-free calibration for pinhole cameras is investigated. It is shown that for a smooth deformation of the image plane, the image plane can be projectively reconstructed from two flow-fields from purely translating cameras. Methods for hand-eye calibration using the multilinear constraints and vehicle-eye for laser-scanner based navigation systems are presented.