# Ideals of the Multiview Variety

## TL;DR

This paper investigates the algebraic structure of the multiview variety in computer vision, establishing when certain polynomial sets generate its ideal and clarifying relationships among various proposed ideals.

## Contribution

It proves that bifocal and trifocal polynomials generate the multiview ideal under distinct foci and clarifies algebraic relationships among different polynomial ideals in multiview geometry.

## Key findings

- Bifocal and trifocal polynomials generate the multiview ideal with distinct foci.
- The multiview ideal is obtained by saturating bifocal polynomials when foci are noncoplanar.
- All considered ideals coincide when dehomogenized, describing the space of finite images.

## Abstract

The multiview variety of an arrangement of cameras is the Zariski closure of the images of world points in the cameras. The prime vanishing ideal of this complex projective variety is called the multiview ideal. We show that the bifocal and trifocal polynomials from the cameras generate the multiview ideal when the foci are distinct. In the computer vision literature, many sets of (determinantal) polynomials have been proposed to describe the multiview variety. We establish precise algebraic relationships between the multiview ideal and these various ideals. When the camera foci are noncoplanar, we prove that the ideal of bifocal polynomials saturate to give the multiview ideal. Finally, we prove that all the ideals we consider coincide when dehomogenized, to cut out the space of finite images.