# Using Uncertain Projective Geometry for Object Extraction from
Multiple Views

### Bio

Stephan Heuel studied Computer Science at the University of Bonn and
graduated in 1997 with a diploma degree and is now a scientific
assistant and PhD student at the Institute for Photogrammetry,
Bonn. He was a visiting researcher at IRIS, University of Southern
California in 1994/95 and at Siemens Corporate Research in Princeton,
NJ in 1999/2000.

### Abstract

In the last decade algebraic projective geometry have been
successfully used to solve various computer vision
problems. Projective Geometry has the advantage of a simple and
consistent representation of geometric objects and transformations in
2D and 3D.

In particular, projective geometry can be used to do "geometrical
reasoning", which means: (i) construction of geometric entities using
join and intersection of given entities and (ii) Testing spatial
relationships between the entities. However, observed entities such as
image points or image lines are uncertain, since they are observations
or measurements with some inaccuracy. The inaccuracy of entities has
to taken into account when doing geometrical reasoning, therefore
projective geometry has to be extended to include notions of
uncertainty, enabling "statistical geometric reasoning".

We present a calculus for projective points, lines and planes in 2D
and 3D with the following properties: (i) simple representation of the
uncertainty of geometric entities, (ii) rules for direct and
over-constrained constructions of new entities, (iii) statistical
hypotheses tests for spatial relationships. An application of this
calculus is the reconstruction of polyhedral objects from multiple
images, for which we solve the following subtasks: (a) matching of 2D
image line segments and points from different images, (b) optimal
reconstruction of 3D segments using matched image features, (c)
grouping of 3D edges to 3D corners. An important property of this
algorithm is the fact that no data-dependent thresholds are used other
than significance values for hypotheses tests.

Maintained by
Philippos Mordohai