Robert T. Collins,
"Model Acquisition Using Stochastic Projective Geometry,"
*Ph.D. Thesis,* Department of Computer Science,
University of Massachusetts, September 1993. Also published
as UMass Computer Science Technical Report TR95-70.

## Abstract

This thesis presents a methodology for scene
reconstruction based on the principles
of projective geometry, yet dealing
with uncertainty at a fundamental level.
Uncertainty in geometric features is
represented and manipulated using probability
density functions in projective space, allowing
geometric constructions to be carried out via
statistical inference.
The main contribution of this thesis is the development
of * stochastic projective geometry*, a formalism
for performing uncertain geometric reasoning during
the scene reconstruction process.
The homogeneous coordinates of points and lines in
the projective plane are represented by antipodal pairs of
points on the unit sphere, and geometric uncertainty in their
location is represented using Bingham's distribution.
Geometric reasoning about homogeneous coordinate
vectors then reduces to well-defined manipulations
on probability density functions.
For example, a Bayesian approach to evidence combination
on the sphere is presented for fusing noisy
homogeneous coordinate observations constrained
by known projective incidence relations.
The result is an uncertainty calculus
in projective space analogous to the
Gaussian uncertainty calculus in affine space.
The main strength of the Gaussian calculus is
maintained, namely its uniform treatment of uncertainty
in all stages of the geometric reasoning process.
At the same time, the limitations of the Gaussian
density function as a representation of uncertainty
in projective space are removed.

The effectiveness of stochastic projective geometry
for dealing with noisy projective relationships
is demonstrated on three geometric vision problems:
deriving line and plane orientations using
vanishing point analysis, partitioning scene features
into planar patches using line correspondence stereo,
and extending a partial model of planar surface
structure using projective invariants.

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