Stochastically Optimal Epipole Estimation in Omnidirectional Images with Geometric Algebra

Abstract

We consider the epipolar geometry between two omnidirectional images acquired from a single viewpoint catadioptric vision sensor with parabolic mirror. This work in particular deals with the estimation of the respective epipoles. We use conformal geometric algebra to show the existence of a 3×3 essential matrix, which describes the underlying epipolar geometry. The essential matrix is preferable to the 4×4 fundamental matrix, which comprises the fixed intrinsic parameters, as it can be estimated from less data. We use the essential matrix to obtain a prior for a stochastic epipole computation being a key aspect of our work. The computation uses the well-tried amalgamation of a least-squares adjustment (LSA) technique, called the Gauss-Helmert method, with conformal geometric algebra. The imaging geometry enables us to assign distinct uncertainties to the image points which justifies the considerable advantage of our LSA method over standard estimation methods. Next to the stochastically optimal position of the epipoles, the method computes the rigid body motion between the two camera positions. In addition, our text demonstrates the effortlessness and elegance with which such problems integrate into the framework of geometric algebra.