Average degree of the essential variety

TL;DR

This paper computes the expected number of real solutions when intersecting a linear space with the essential variety in computer vision, revealing an average of about 4 real solutions under certain distributions.

Contribution

It provides the first expected value calculations for real intersection points of the essential variety with random linear spaces, relevant for 3D reconstruction.

Findings

Expected real intersection points under orthogonal invariance: 4

Monte Carlo estimates for computer vision distribution: approximately 3.95

Results inform the likelihood of real solutions in 3D pose estimation

Abstract

The essential variety is an algebraic subvariety of dimension $5$ in real projective space $\mathbb R\mathrm P^{8}$ which encodes the relative pose of two calibrated pinhole cameras. The $5$-point algorithm in computer vision computes the real points in the intersection of the essential variety with a linear space of codimension $5$. The degree of the essential variety is $10$, so this intersection consists of 10 complex points in general. We compute the expected number of real intersection points when the linear space is random. We focus on two probability distributions for linear spaces. The first distribution is invariant under the action of the orthogonal group $\mathrm{O}(9)$ acting on linear spaces in $\mathbb R\mathrm P^{8}$. In this case, the expected number of real intersection points is equal to $4$. The second distribution is motivated from computer vision and is defined by choosing 5 point correspondences in the image planes $\mathbb R\mathrm P^2\times \mathbb R\mathrm P^2$ uniformly at random. A Monte Carlo computation suggests that with high probability the expected value lies in the interval $(3.95 - 0.05,\ 3.95 + 0.05)$.