An Atlas for the Pinhole Camera

TL;DR

This paper develops an algebraic geometric framework called an atlas to analyze image formation in pinhole cameras, unifying various problems in 3D computer vision through algebraic varieties and their relations.

Contribution

It introduces a novel algebraic geometric atlas for pinhole camera models, providing a unified framework for 3D vision problems and characterizing key components like triangulation.

Findings

Complete characterization of the triangulation-related atlas component

Establishment of algebraic varieties linked to image formation

Discussion of open problems and potential generalizations

Abstract

We introduce an atlas of algebro-geometric objects associated with image formation in pinhole cameras. The nodes of the atlas are algebraic varieties or their vanishing ideals related to each other by projection or elimination and restriction or specialization respectively. This atlas offers a unifying framework for the study of problems in 3D computer vision. We initiate the study of the atlas by completely characterizing a part of the atlas stemming from the triangulation problem. We conclude with several open problems and generalizations of the atlas.