A Hilbert Scheme in Computer Vision

TL;DR

This paper explores the algebraic structure of multiview geometry in computer vision, providing a universal Groebner basis for multiview ideals and analyzing the associated Hilbert scheme to understand camera configurations.

Contribution

It introduces a universal Groebner basis for multiview ideals and studies the multigraded Hilbert scheme component related to camera configurations.

Findings

Determined a universal Groebner basis for the multiview ideal.

Analyzed the family of multiview varieties as a component of a Hilbert scheme.

Provided a combinatorial study of ideals on that Hilbert scheme.

Abstract

Multiview geometry is the study of two-dimensional images of three-dimensional scenes, a foundational subject in computer vision. We determine a universal Groebner basis for the multiview ideal of n generic cameras. As the cameras move, the multiview varieties vary in a family of dimension 11n-15. This family is the distinguished component of a multigraded Hilbert scheme with a unique Borel-fixed point. We present a combinatorial study of ideals lying on that Hilbert scheme.