New algebraic tools for classical geometry
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Paracatadioptric Camera Calibration
IEEE Transactions on Pattern Analysis and Machine Intelligence
Theory of Robot Control
Epipolar Geometry of Panoramic Cameras
ECCV '98 Proceedings of the 5th European Conference on Computer Vision-Volume I - Volume I
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ECCV '00 Proceedings of the 6th European Conference on Computer Vision-Part II
A Theory of Catadioptric Image Formation
ICCV '98 Proceedings of the Sixth International Conference on Computer Vision
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Journal of Mathematical Imaging and Vision
Geometric Algebra with Applications in Engineering
Geometric Algebra with Applications in Engineering
Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry
Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry
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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.