Robert T. Collins, 
"A Bayesian Analysis of Projective Incidence,"
in Applications of Invariance in Computer Vision II, 
Ponta Delgada, Azores, October 1993, pp. 151-163.


The theorems of projective geometry were developed with mathematically precise objects in mind. In contrast, a practical vision system must deal with errorful measurements extracted from real image sensors. A more robust form of projective geometry is needed, one that allows for possible imprecision in its geometric primitives. In this paper, uncertainty in projective elements is represented and manipulated using probability density functions in projective space. Projective n-space can be visualized using the surface of a unit sphere in (n+1)-dimensional Euclidean space. Each point in projective space is represented by antipodal points on the sphere. This two-to-one map from the unit sphere to projective space enables probability density functions on the sphere to be interpreted as probability density functions over the points of projective space. Standard constructions of projective geometry can then be augmented by statistical inferences on the sphere. In particular, a Bayesian analysis is presented for fusing multiple noisy observations related by known projective incidence relations.

Full Paper

Click here for full paper (95850 bytes, compressed using gzip).