Congruences and Concurrent Lines in Multi-View Geometry

TL;DR

This paper introduces a unified geometric framework for various camera models in computer vision, focusing on congruences and concurrent lines to derive constraints for multi-view image correspondences.

Contribution

It develops a new abstraction of cameras as mappings between projective space and line congruences, encompassing many camera types and deriving new geometric constraints.

Findings

Derived equations for the concurrent lines variety.

Unified framework for different camera models.

Constraints for multi-view image correspondences.

Abstract

We present a new framework for multi-view geometry in computer vision. A camera is a mapping between $\mathbb{P}^3$ and a line congruence. This model, which ignores image planes and measurements, is a natural abstraction of traditional pinhole cameras. It includes two-slit cameras, pushbroom cameras, catadioptric cameras, and many more. We study the concurrent lines variety, which consists of $n$-tuples of lines in $\mathbb{P}^3$ that intersect at a point. Combining its equations with those of various congruences, we derive constraints for corresponding images in multiple views. We also study photographic cameras which use image measurements and are modeled as rational maps from $\mathbb{P}^3$ to $\mathbb{P}^2$ or $\mathbb{P}^1\times \mathbb{P}^1$.