Geometry and Kinematics with Uncertain Data

Abstract

In Computer Vision applications, one usually has to work with uncertain data. It is therefore important to be able to deal with uncertain geometry and uncertain transformations in a uniform way. The Geometric Algebra of conformal space offers a unifying framework to treat not only geometric entities like points, lines, planes, circles and spheres, but also transformations like reflection, inversion, rotation and translation. In this text we show how the uncertainty 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 Algebra 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 cameras with parabolic mirrors within this framework. This extends previous work by Förstner 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.