J Math Imaging Vis (2012) 43:214–226 DOI 10.1007/s10851-011-0305-z A Geometric Algebra Model for the Image Formation Process of Paracatadioptric Cameras C. López-Franco · N. Arana-Daniel Published online: 6 July 2011 © Springer Science+Business Media, LLC 2011 Abstract 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 combi- nation 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 para- catadioptric sensors is no longer linear. However in this pa- per 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 R n which is isomorphic to the Lorentz group in 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 al- gebraically 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 de- velopment of complex applications that use paracatadioptric sensors. Keywords Omnidirectional vision · Geometric algebra C. López-Franco ( ) · N. Arana-Daniel Blvd. Marcelino García Barragán # 1421, CP 44430, Guadalajara, Jalisco, México e-mail: carlos.lopez@cucei.udg.mx N. Arana-Daniel e-mail: nancy.arana@cucei.udg.mx 1 Introduction Conventional cameras suffer from a restricted ﬁeld of view. Many applications in vision-based robotics, such as mobile robot navigation can beneﬁt from an enhanced ﬁeld of view provided by omnidirectional cameras. One effective way to increase the ﬁeld of view is to use mirrors in combination with conventional cameras. The ap- proach of combining mirrors with conventional cameras to enhance sensor ﬁeld of view is referred as catadioptric image formation [5]. One effective way to increase the ﬁeld of view of traditional cameras is to combine mirrors with conven- tional imaging systems. The obtained sensors are referred as catadioptric sensors. A desirable property of these sensors is the single center of projection. The complete class of mirrors that satisfy such restriction where analyzed by Baker and Nayar [1]. In [16] the authors deal with the epipolar geom- etry of two catadioptric sensors. Later, Geyer and Daniilidis [9] presented a general model for central catadioptric im- age formation. Also, a representation of this general model using the CGA was shown in [3]. In contrast with previ- ous works where the paracatadioptric projection is deﬁned only for points or a parametric representation of geometric entities, the present work introduces a linear model which can handle algebraically the paracatadioptric projection of points, point-pairs, lines and circles. Vision based servoing schemes are effective methods to control robot motion from camera observations [8] and [14]. Visual servoing applications can beneﬁt from sensors pro- viding large ﬁelds of view. The present work is mainly con- cerned with the use of projected lines extracted from cen- tral catadioptric images as input of a visual servoing control loop. The paracatadioptric image of a line is in general a cir- cle but sometimes it can be a line. This is something that should be taken into account to avoid a singularity in the visual servoing task.