Line Multiview Varieties

TL;DR

This paper explores the algebraic structure of line correspondences in multiview geometry, defining the line multiview variety, analyzing its properties, and evaluating its robustness in triangulation tasks.

Contribution

It introduces the line multiview variety, characterizes it as a determinantal variety for generic cameras, and provides a comprehensive algebraic description and analysis.

Findings

Line multiview variety is a determinantal variety for generic camera matrices.

Complete set-theoretic description of the variety for any camera arrangement.

Experimental results on Euclidean distance degree and noise robustness in line triangulation.

Abstract

We present an algebraic study of line correspondences for pinhole cameras, in contrast to the thoroughly studied point correspondences. We define the line multiview variety as the Zariski closure of the image of the map projecting lines in 3-space to tuples of image lines in 2-space. We prove that in the case of generic camera matrices the line multiview variety is a determinantal variety and we provide a complete set-theoretic description for any camera arrangement. We investigate basic properties of this variety such as dimension, smoothness, and multidegree. Finally, we give experimental results for the Euclidean distance degree and robustness under noise for the triangulation of lines.