Algebra and Geometry of Camera Resectioning

TL;DR

This paper explores the algebraic structures underlying camera resectioning, using Gr"obner basis techniques to analyze associated varieties and connect classical geometric vision problems.

Contribution

It introduces a novel algebraic framework for camera resectioning, characterizes related varieties, and links classical problems with modern algebraic geometry.

Findings

Characterization of resectioning varieties using Gr"obner bases

Re-interpretation of camera-point duality results

Proposed conjecture for Euclidean distance degree of resectioning variety

Abstract

We study algebraic varieties associated with the camera resectioning problem. We characterize these resectioning varieties' multigraded vanishing ideals using Gr\"obner 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.