Luke Oeding: Algebraic Vision, The Quadrifocal Variety.

Part I: Multifocal tensors from an algebraic perspective

Multi-view Geometry is a branch of Computer Vision that aims to reconstruct a high dimensional image from several lower dimensional images. I will review this construction from an Algebraic Geometry perspective. I will explain how multi-focal tensors may be constructed as equivariant projections of the Grassmannian.  In particular, the quadrifocal variety is a 39-dimensional algebraic subvariety of the 80-dimensional projective space of tensors of format $3\times 3 \times 3 \times 3$. As a special case of multi-focal tensors, I'll show how the (seemingly unrelated) principal minor assignment problem arises by considering several flatlander cameras.

Part II: Large scale computations with symmetry

I will discuss how to use symmetry to study the ideal of polynomial equations vanishing on the quadrifocal variety. Despite being of high dimension and codimension, it is still possible to compute its ideal up to degree 6 in terms of representations of $\GL(3)^{\times 4}$. These computations are performed in Maple by explicitly constructing a basis of the highest weight space for each irreducible representation that occurs in the polynomial ring (in low degrees).  

Further analysis using Macaulay2 (and the package "SchurRings") allows us to rule out certain syzygies, giving a lower bound for the number of minimal generators.  Led by these computations we conjecture that the ideal of the quadrifocal variety is minimally generated in degree at most 6.