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 non-linearity 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 multi-linear 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.