The geometry and topology of projective shape space

John Kent (May 22, 2012)

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Abstract

Projective geometry underlies the way in which information about a 3d scene can be deduced from (one or more) 2d camera views. A key concept in projective geometry is that of a projective invariant for a configuration of collinear or coplanar points. The collection of information in the projective invariants can be termed the "projective shape" of a configuration. In this talk we use a spherical camera and adapt ideas from the Procrustes approach to similarity shape analysis to give a standardized representation for projective shapes. The resulting geometry faciliates metric comparisons between different projective shapes. The resulting topology leads to a clear understanding of the singularities in projective shape space.

Finally, the details behind the standardization lead to a distinction between four variants of projective shape space depending on the "type" of camera: oriented vs. non-oriented and directional vs. axial.