Radon transform and Conformal Geometric Algebra with lines

Abstract

In this paper we apply the classic theory of harmonic analysis and the conformal geometric algebra (CGA) to evaluate the Fourier transform on the unit sphere S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> and on the rotation group SO(3). Since the images taken by omnidirectional sensors can be mapped to the sphere, the problem of attitude estimation of a 3D camera rotation can be treated as a problem of estimating rotations between spherical images. Usually, this rotation estimation problem has been solved using the Radon transform with point correlation or with line correlation. Using a catadioptric system with a parabolic mirror, 3D lines will be projected as great circles on S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> and then projected as circles on the image plane. CGA is a mathematical framework where its basic entities, spheres, circles, lines and planes can be used with incidence algebra operations to improve the line correlation in the Radon transform as a dual-circle correlation. Thus, harmonic analysis theory and conformal geometric algebra will be joined in this paper to improve the search of a 3D rotation correspondence between two omnidirectional images.