A Geometric Algebra Model for the Image Formation Process of Paracatadioptric Cameras

Omnidirectional vision sensors capture a wide field of view than can benefit many robotic applications. One type of omnidirectional vision sensor is the paracatadioptric. A paracatadioptric sensor combines a parabolic mirror and a camera inducing an orthographic projection. This combination provides a wide field of view while maintaining the single center of projection which is a desirable property of these sensors. Furthermore, lines are projected as circles on the paracatadioptric image plane. In contrast with traditional perspective cameras the image formation process of paracatadioptric sensors is no longer linear. However in this paper we present a model which is able to linearize it. This linearization is based on the fact that the paracatadioptric projection can be represented by a sphere inversion, that belongs to the conformal group $\mathcal{R}^{n}$ which is isomorphic to the Lorentz group in $\mathcal{R}^{n+1}$. Thus a nonlinear conformal transformation can be represented with an equivalent linear Lorenz transformation, which can be represented as a versor in the CGA. Therefore the present model can be applied algebraically not only to points, but also to point-pairs, lines, circles in the same way to all them and in a linear form. The benefits of the proposed method will be reflected on the development of complex applications that use paracatadioptric sensors.