Algebraic Relations and Triangulation of Unlabeled Image Points

In multiview geometry when correspondences among multiple views are unknown the image points can be understood as being unlabeled. This is a common problem in computer vision. We give a novel approach to handle such a situation by regarding unlabeled point configurations as points on the Chow variety $\text{Sym}_m(\mathbb{P}^2)$. For two unlabeled points we design an algorithm that solves the triangulation problem with unknown correspondences. Further the unlabeled multiview variety $\text{Sym}_m(V_A)$ is studied.

Pictures of Combinatorial Cubes

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 $\mathbb{R}^3$.

Rigid Multiview Varieties

The multiview variety from computer vision is generalized to images by $n$ cameras of points linked by a distance constraint. The resulting five-dimensional variety lives in a product of $2n$ projective planes. We determine defining polynomial equations, and we explore generalizations of this variety to scenarios of interest in applications.