The Quadrifocal Variety

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 $\mathrm{GL}(3{)}^{\times 4}$ 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.

20 pages, updates to references, introduction, and main conjecture, and added ancillary files. The article has been accepted to LAA

References in corpus (6)

• Hyperdeterminantal relations among symmetric principal minors
• Equations for the fifth secant variety of Segre products of projective spaces
• Rigid Multiview Varieties
• Certifying the Existence of Epipolar Matrices
• A QCQP Approach to Triangulation
• On the Existence of Epipolar Matrices

Cited by in corpus (4)

• An Atlas for the Pinhole Camera
• Trifocal Relative Pose from Lines at Points and its Efficient Solution
• Hyperdeterminants from the $E_8$ Discriminant
• Graph States and the Variety of Principal Minors