Hey! You got your algebraic geometry in my computer vision!

Abstract

We consider problems in the intersection of algebraic geometry and computer vision. In particular, we study algebraic varieties associated with the camera resectioning problem. We characterize these resectioning varieties' multigraded vanishing ideals using Gr\"{o}bner basis techniques. As an application, we derive and re-interpret celebrated results in geometric computer vision related to camera-point duality. We also clarify some relationships between the classical problems of optimal resectioning and triangulation, state a conjectural formula for the Euclidean distance degree of the resectioning variety, and discuss how this conjecture relates to the recently-resolved multiview conjecture. Additionally, we work toward a compactification of the moduli of camera pairs that allows for a continuous geometric interpolation between concentric and non-concentric pairs. Furthermore, we give a new construction of the space of essential matrices from first principles. This construction enables us to re-prove the fundamental results of Demazure and to re-prove the recent description of the essential variety due to Kileel--Fløystad--Ottaviani. We also describe a new five-point solver for generic data based upon generic Gröbner bases for symmetric bilinear forms. Finally, we perform some intersection-theoretic calculations to provide basic geometric invariants of the classical multiview variety as well as the universal imaging variety.