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Maseeh Colloquium: Object-Image Correspondence for Algebraic Curves under Projections
Friday, March 8, 2013 - 3:15pm

The Maseeh Mathematics and Statistics Colloquium Series* 

Irina Kogan, Ph.D.
Department of Mathematics, North Carolina State University

Object-Image Correspondence for Algebraic Curves under Projections

Friday, March 8th, 2013 at 3:15pm 
Urban Center room 204
(Refreshments served at 2:45 in presentation room)

I will present a novel algorithm for deciding whether a given planar curve is an image of a given spatial curve, obtained by a central or a parallel projection with unknown parameters. The motivation comes from the problem of establishing a correspondence between an object and an image, taken by a camera with unknown position and parameters.  A straightforward approach to this problem consists of setting up a system of conditions on the projection parameters and then checking whether or not this system has a solution. The computational advantage of the algorithm presented here, in comparison to algorithms based on the straightforward approach, lies in a significant reduction of a number of real parameters that need to be eliminated in order to establish existence or non-existence of a projection that maps a given spatial curve to a given planar curve. Our algorithm is based on projection criteria that reduce the projection problem to a certain modification of the equivalence problem of planar curves under affine and projective transformations.  To solve the latter problem we make an algebraic adaptation of signature construction that has been used to solve the equivalence problems for smooth curves. We introduce a notion of a classifying set of rational differential invariants and produce explicit formulas for such invariants for  the actions of the projective and the affine groups on the plane. This is a joint work with Joseph Burdis and Hoon Hong.

* Sponsored by the Maseeh Mathematics and Statistics Colloquium Series Fund and the Fariborz Maseeh 
Department of Mathematics & Statistics, PSU. This event is free and open to the public.