Commutative Algebra and Algebraic Geometry: Generalized Cayley-Chow Coordinates and Computer Vision

Seminar | April 25 | 5-6 p.m. | 939 Evans Hall

 Brian Osserman, UC Davis

 Department of Mathematics

A fundamental problem in computer vision is to reconstruct the configuration of a collection of cameras from the images they have taken of a common subject. If we model a camera as a linear projection from projective 3-space to the projective plane, this problem can be rephrased algebrogeometrically in terms of recovering a subvariety of a product of projective planes. Both the equations defining these subvarieties and the relevant Hilbert scheme were studied in work of Aholt, Sturmfels and Thomas in 2011. We explain how various mathematical tools can both give new explanations for known phenomena in computer vision, and lead to some new results. The most substantive mathematical contribution is a theory of Cayley-Chow coordinates for subvarieties of products of projective spaces, generalizing the usual case of a single projective space. We find that there are nontrivial dimensional inequalities on when this generalization works, and that these inequalities explain the existence and non-existence of the "multifocal tensors" from computer vision.

This is joint work (very much still in progress) with Matthew Trager.