Pictures of Combinatorial Cubes

TL;DR

This paper investigates the failure of the 8-point algorithm in reconstructing fundamental matrices from images of combinatorial cube vertices, and proposes an improved algorithm for such cases, analyzing the conditions for successful reconstruction.

Contribution

It demonstrates the limitations of the 8-point algorithm with combinatorial cube projections and introduces an improved method for these specific configurations.

Findings

8-point algorithm fails for combinatorial cube projections

Proposed an improved algorithm for 7- and 8-point cases

Analyzed focal point regions for successful reconstruction

Abstract

We prove that the 8-point algorithm always fails to reconstruct a unique fundamental matrix $F$ independent on the camera positions, when its input are image point configurations that are perspective projections of the vertices of a combinatorial cube in $\mathbb{R}^3$. We give an algorithm that improves the 7- and 8-point algorithm in such a pathological situation. Additionally we analyze the regions of focal point positions where a reconstruction of $F$ is possible at all, when the world points are the vertices of a combinatorial cube in $\mathbb{R}^3$.