Pictures of Combinatorial Cubes
This addresses a specific failure case in computer vision for 3D reconstruction from images, offering an incremental improvement for handling cube-like point configurations.
The paper demonstrates that the 8-point algorithm fails to uniquely reconstruct the fundamental matrix from images of combinatorial cube vertices, regardless of camera positions, and provides an improved algorithm for this pathological case. It also analyzes camera position regions where reconstruction is possible.
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 $\mathbb{R}^3$.