The Quadrifocal Variety
This work addresses foundational challenges in computer vision and algebraic geometry for researchers in these fields, but it is incremental as it builds on existing multi-focal tensor theory.
The paper tackles the problem of understanding the quadrifocal variety in multi-view geometry by computing its ideal up to degree 8 using algebraic geometry and representation theory, resulting in a lower bound for the number of minimal generators.
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.