Multiprojective Geometry of Compatible Triples of Fundamental and Essential Matrices
This work provides a complete algebraic characterization of compatible fundamental matrix triples, improving upon incomplete previous constraints for researchers in geometric computer vision.
This paper characterizes the variety of compatible fundamental matrix triples by computing its multidegree and multihomogeneous vanishing ideal, answering a question posed by Bråtelund and Rydell. The authors discovered a new set of simple quartic constraints that vanish on compatible fundamental matrix triples, which also locally cut out the variety of compatible essential matrix triples.
We characterize the variety of compatible fundamental matrix triples by computing its multidegree and multihomogeneous vanishing ideal. This answers the first interesting case of a question recently posed by Bråtelund and Rydell. Our result improves upon previously discovered sets of algebraic constraints in the geometric computer vision literature, which are all incomplete (as they do \emph{not} generate the vanishing ideal) and sometimes make restrictive assumptions about how a matrix triple should be scaled. Our discussion touches more broadly on generalized compatibility varieties, whose multihomogeneous vanishing ideals are much less well understood. One of our key new discoveries is a simple set of quartic constraints vanishing on compatible fundamental matrix triples. These quartics are also significant in the setting of essential matrices: together with some previously known constraints, we show that they locally cut out the variety of compatible essential matrix triples.