Abstract

In Computer Vision applications, one usually has to work with un- certain data. It is therefore important to be able to deal with uncertain geome- try and uncertain transformations in a uniform way. The Geometric Algebra of conformal space offers a unifying framework to treat not only geometric enti- ties like points, lines, planes, circles and spheres, but also transformations like reflection, inversion, rotation and translation. In this text we show how the un- certainty of all elements of the Geometric Algebra of conformal space can be appropriately described by covariance matrices. In particular, it will be shown that it is advantageous to represent uncertain transformations in Geometric Alge- bra as compared to matrices. Other important results are a novel pose estimation approach, a uniform framework for geometric entity fitting and triangulation, the testing of uncertain tangentiality relations and the treatment of catadioptric cam- eras with parabolic mirrors within this framework. This extends previous work by F ¨orstner and Heuel from points, lines and planes to non-linear geometric entities and transformations, while keeping the linearity of the estimation method. We give a theoretical description of our approach and show exemplary applications.