ECCV 2004 Tutorial T1
Date: Monday May 10, 2004 afternoon Venue:
Zofin Palace
Statistics in Projective Geometry
Instructor: Wolfgang Förstner, Bonn University
Duration: 3.5 hours
Summary:
Geometric reasoning in Computer Vision always is performed under
uncertainty. The great potential of both, projective geometry and
statistics, can be integrated easily for propagating uncertainty
through reasoning chains, for making decisions on uncertain spatial
relations and for optimally estimating geometric entities or
transformations. This is achieved by (1) exploiting the potential of
statistical estimation and testing theory and by (2) choosing a
representation of projective entities and relations which supports
this integration.
The redundancy of the representation of geometric entities with
homogeneous vectors and matrices, the nonlinearity of the
relations and the need for multiple constraints requires a careful
analysis of singularities of the uncertainty representation, of the
bias of constructing new geometric entities and of the estimation
techniques.
The half day tutorial covers the main aspects handling uncertain
geometric elements, their relations, their construction and their
optimal estimation in two and three dimensions, and deals with the
following topics:
 Representation of basic geometric and transformations in
projective geometry
 Explicit representation and interpretation of Jacobians of
multilinear relations
 Uncertainty, distributions and covariance matrices
 Singular covariance matrices and the interpretation of their
null space
 Euclidean and spherical representation and the
interpretation of their uncertainty
 Uncertainty of constructed geometric entities
 Statistical tests of geometric relations, especially
identity, incidence, parallelity, orthogonality
 Direct estimation of geometric entities
 Optimal estimation of geometric entities and their uncertainty
 Evaluation of estimation results
 Examples from grouping and orientation
Intended Audience
The introductory tutorial is meant for all researchers and
developers who are interested in the analysis of uncertain
geometric entities in 2D and 3D, especially in the context of
image analysis. Basic knowledge in linear algebra is recommended,
basic knowledge of statistics and projective geometry is
useful.
The speaker
Wolfgang Förstner is director of the Institute of Photogrammetry
of Bonn University. He studied Geodesy and obtained his PhD at
Stuttgart University in 1976. Since 1981 he works in the areas of
statistical methods in image analysis, performance
characteristics, quality of GIS, digital photogrammetry, and
computer vision.
References

W. Förstner (2004): Uncertainty and Projective Geometry. To appear in: Eduardo Bayro (Ed.): Handbook of Computational Geometry for Pattern Recognition, Computer Vision, Neurocomputing and Robotics

S. Heuel (2001): Points, Lines and Planes and their Optimal Estimation, DAGM 2001
 M. Luxen, W. Förstner (2001): Optimal Camera Orientation from Points and Straight Lines, DAGM 2001

W. Förstner (2001): Algebraic Projective Geometry and Direct Optimal Estimation of Geometric Entities, OeAgM 2001

W. Förstner et al. (2000): Statistically Testing Uncertain Geometric Relations, DAGM 2000