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.
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