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.


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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