Congruences and Concurrent Lines in Multi-View Geometry
This work addresses the foundational problem of camera modeling and geometric constraints in computer vision, offering a novel abstraction that could impact various camera types, though it appears incremental in its mathematical formulation.
The paper tackles the problem of multi-view geometry in computer vision by introducing a new framework that models cameras as mappings between projective spaces, ignoring image planes and measurements, to derive constraints for corresponding images across multiple views.
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 $\mathbb{P}^1\times \mathbb{P}^1$.