Robust multiple homography estimation: An ill-solved problem

The estimation of multiple homographies between two piecewise planar views of a rigid scene is often assumed to be a solved problem. We show that contrary to popular opinion various crucial aspects of the task have not been adequately emphasised. We are motivated by a growing body of literature in robust multi-structure estimation that purports to solve the multi-homography estimation problem but in fact does not. We demonstrate that the estimation of multiple homographies is an ill-solved problem by deriving new constraints that a set of mutually compatible homographies must satisfy, and by showing that homographies estimated with prevailing methods fail to satisfy the requisite constraints on real-world data. We also explain why incompatible homographies imply inconsistent epipolar geometries. The arguments and experiments presented in this paper signal the need for a new generation of robust multi-structure estimation methods that have the capacity to enforce constraints on projective entities such as homography matrices.