The Quadrifocal Variety
Luke Oeding
TL;DR
This work reframes multi-view geometry in the language of Algebraic Geometry by modeling the quadrifocal tensor as the projection of a Grassmannian, yielding a structured, representation-theoretic description of the quadrifocal variety. It provides a detailed degree-by-degree analysis of the quadrifocal ideal, proving a substantial cubic component of minimal generators (600 in degree 3) and revealing large, highly structured higher-degree contributions up to degree $8$ (partial at $d=9$), all analyzed via $\operatorname{GL}(3)^{\times 4}$ and $\mathfrak{S}_4$ symmetry. The paper also connects the quadrifocal framework to principal-minor varieties, contractions to homography tensors, and higher-dimensional generalizations, highlighting the complexity and non-complete-intersection nature of the quadrifocal ideal. Computational methods based on symmetry (LM-algorithm) and representation theory (SchurRings) yield explicit isotypic decompositions of the ideal and produce new candidate generators, illuminating the landscape of multi-view tensors beyond trifocal. Together, these results advance the algebraic understanding of 4-view geometry and provide tools for future symbolic and numerical investigations in 3D reconstruction from multiple images.
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.