Consistent sets of lines with no colorful incidence

TL;DR

This paper investigates how incidences among colored lines in multi-dimensional space influence the existence of higher-order concurrences, with implications for 3D reconstruction in computer vision.

Contribution

It introduces a new perspective on incidences among colored lines and explores conditions under which concurrences of certain color combinations imply more complex concurrences.

Findings

Identifies conditions linking k-color concurrences to (k+1)-color concurrences.

Provides theoretical insights relevant for 3D reconstruction algorithms.

Establishes foundational results connecting line incidences and color combinations.

Abstract

We consider incidences among colored sets of lines in $\mathbb{R}^d$ and examine whether the existence of certain concurrences between lines of $k$ colors force the existence of at least one concurrence between lines of $k+1$ colors. This question is relevant for problems in 3D reconstruction in computer vision.