The Quadrifocal Variety

TL;DR

This paper explores the algebraic geometry of multi-view geometry, constructing multi-focal tensors as equivariant projections and analyzing the quadrifocal variety's ideal using advanced representation theory.

Contribution

It introduces a novel algebraic geometric framework for multi-view geometry and computes the ideal of the quadrifocal variety up to degree 8, advancing theoretical understanding.

Findings

Computed the ideal of the quadrifocal variety up to degree 8

Connected multi-focal tensors to the principal minor assignment problem

Provided a lower bound for the number of minimal generators

Abstract

Multi-view Geometry is reviewed from an Algebraic Geometry perspective and multi-focal tensors are constructed as equivariant projections of the Grassmannian. A connection to the principal minor assignment problem is made by considering several flatlander cameras. The ideal of the quadrifocal variety is computed up to degree 8 (and partially in degree 9) using the representations of $\operatorname{GL}(3)^{\times 4}$ in the polynomial ring on the space of $3 \times 3 \times 3 \times 3$ tensors. Further representation-theoretic analysis gives a lower bound for the number of minimal generators.