Direct optimization of frame-to-frame rotation

Abstract

This work makes use of a novel, recently proposed epipolar constraint for computing the relative pose between two calibrated images. By enforcing the co planarity of epipolar plane normal vectors, it constrains the three degrees of freedom of the relative rotation between two camera views directly-independently of the translation. The present paper shows how the approach can be extended to n points, and translated into an efficient eigenvalue minimization over the three rotational degrees of freedom. Each iteration in the non-linear optimization has constant execution time, independently of the number of features. Two global optimization approaches are proposed. The first one consists of an efficient Levenberg-Marquardt scheme with randomized initial value, which already leads to stable and accurate results. The second scheme consists of a globally optimal branch-and-bound algorithm based on a bound on the eigenvalue variation derived from symmetric eigenvalue-perturbation theory. Analysis of the cost function reveals insights into the nature of a specific relative pose problem, and outlines the complexity under different conditions. The algorithm shows state-of-the-art performance w.r.t. essential-matrix based solutions, and a frame-to-frame application to a video sequence immediately leads to an alternative, real-time visual odometry solution.