Uncertain Geometric Entities and Operators

In computer vision, one has to deal almost invariably with uncertain data and thus also with uncertain geometric entities. Appropriate methods to deal with this uncertainty therefore play an important role. Building mainly on the results presented in Chap. 5, the construction and estimation of geometric entities and transformation operators are discussed in the following.

A particular advantage of the approach presented here stems from the linear representation of geometric entities and transformations and from the fact that algebraic operations are simply bilinear functions. This allows the simple construction of geometric constraints using the symbolic power of the algebra, which can be expressed equivalently as multi-linear functions, such that the whole body of linear algebra can be applied. The solutions to many problems, such as the estimation of the best t of a line, plane, circle, or sphere through a set of points, or the best rotation between two point sets (in a least-squares sense), reduce to the evaluation of the null space of a matrix. By applying the Gauss{Helmert model (see Sect. 5.9), it is also possible to evaluate the uncertainty of the estimated entity.

This chapter builds on previous work by Forstner et al. [74] and Heuel [93], where uncertain points, lines, and planes were treated in a unied manner. The linear estimation of rotation operators in geometric algebra was rst discussed in [146], albeit without taking account of uncertainty. The description of uncertain circles and 2D conics in geometric algebra was introduced in [136], and the estimation of uncertain general operators was introduced in [138]. Applications and extensions of these initial developments were presented in [139].