Uncertain Geometry with Circles, Spheres and Conics

Spatial reasoning is one of the central tasks in Computer Vision. It always has to deal with uncertain data. Projective geometry has become the working horse for modelling multiple view geometry, while modelling uncertainty with statistical tools has become a standard. Geometric reasoning in projective geometry with uncertain geometric elements has been advocated by Kanatani in the early 90’s, and recently made transparent and generalized to basic entities in projective geometry including transformations by Förstner and Heuel, exploiting the multilinearity of nearly all relations, such as incidence and identity, which results from the underlying Grassmann-Cayley algebra (cf. [21, 8, 7]). This paper generalizes geometric reasoning under uncertainty towards circles, spheres and conics, which play a role in many computer vision applications. In particular it will be shown how within the Clifford algebra of conformal space, as introduced by Hestenes et al. [11, 16], circles can be constructed from three uncertain points in 3D-Euclidean space, while propagating the covariance matrices of the points. This then enables us to obtain and visualize the uncertainty of the resulting circle. We also introduce the Clifford algebra over the vector space of 2D-conics, which allows us to apply the same error propagation procedures as for the Clifford algebra of conformal space.