Georgia Institute of Technology College of Sciences

In Computer Vision and multi-view geometry one considers several cameras in general position as a collection of projection maps. One would like to understand how to reconstruct the 3-dimensional image from the 2-dimensional projections. [Hartley-Zisserman] (and others such as Alzati-Tortora and Papadopoulo-Faugeras) described several natural multi-linear (or tensorial) constraints which record certain relations between the cameras such as the epipolar, trifocal, and quadrifocal tensors. (Don't worry, the story stops at quadrifocal tensors!) A greater understanding of these tensors is needed for Computer Vision, and Algebraic Geometry and Representation Theory provide some answers.I will describe a uniform construction of the epipolar, trifocal and quadrifocal tensors via equivariant projections of a Grassmannian. Then I will use the beautiful Algebraic Geometry and Representation Theory, which naturally arrises in the construction, to recover some known information (such as symmetry and dimensions) and some new information (such as defining equations). Part of this work is joint with Chris Aholt (Microsoft).