Compatibility of Fundamental Matrices for Complete Viewing Graphs

TL;DR

This paper investigates the conditions under which a set of fundamental matrices from multiple camera pairs can be globally compatible, simplifying previous criteria and providing explicit polynomial conditions for camera reconstruction.

Contribution

It shows that eigenvalue conditions are redundant in generic cases, and introduces explicit polynomial conditions for compatibility based on fundamental matrices and epipoles.

Findings

Eigenvalue condition is redundant in generic and collinear cases.

Quadruple-wise compatibility guarantees global compatibility.

For four cameras, compatibility is characterized by triple-wise conditions plus one global equation.

Abstract

This paper studies the problem of recovering cameras from a set of fundamental matrices. A set of fundamental matrices is said to be compatible if a set of cameras exists for which they are the fundamental matrices. We focus on the complete graph, where fundamental matrices for each pair of cameras are given. Previous work has established necessary and sufficient conditions for compatibility as rank and eigenvalue conditions on the n-view fundamental matrix obtained by concatenating the individual fundamental matrices. In this work, we show that the eigenvalue condition is redundant in the generic and collinear cases. We provide explicit homogeneous polynomials that describe necessary and sufficient conditions for compatibility in terms of the fundamental matrices and their epipoles. In this direction, we find that quadruple-wise compatibility is enough to ensure global compatibility for any number of cameras. We demonstrate that for four cameras, compatibility is generically described by triple-wise conditions and one additional equation involving all fundamental matrices.