Average degree of the essential variety

TL;DR

This work analyzes the average number of real solutions to the relative pose problem arising from intersecting the essential variety $\mathcal E$ with random codimension-5 linear spaces in $\mathbb{RP}^8$, a core step in the 5-point algorithm for calibrated cameras. It establishes two natural distributions for the random linear spaces: (i) a uniform Grassmannian model yielding an exact average of 4 real intersections, and (ii) a vision-inspired model $\psi$ from five random correspondences, for which the average is tied to a random-determinant quantity and empirically approximates $3.95$. A general integral framework (main3) expresses the average under $\psi$ for arbitrary densities $\Psi$, and Vitale’s zonoid theory recasts the same average in terms of the volume of the essential zonoid $K$, with a computable lower bound near $0.93$ that highlights the gap to the observed $\approx 3.95$. Through two proofs of the exact volume of $\mathcal E$—showing $\operatorname{vol}(\mathcal E)=4\operatorname{vol}(\mathbb{RP}^5)$—and a detailed analysis of the incidence variety and Jacobians, the paper provides both exact geometric constants and practical Monte Carlo estimates that align with, and justify, the observed prevalence of real solutions in typical datasets. The results have direct implications for the expected difficulty of solving the 5-point relative-pose equations in real-world vision tasks and connect algebraic geometry with practical computer-vision performance promises.

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 $(3.95 - 0.05,\ 3.95 + 0.05)$.

Figures (3)

• Figure 1: The two pie charts show the outcome of the following two experiments. We sampled $N=1000$ random linear spaces, once with distribution $\mathrm{Unif}(\mathbb G)$ (the left chart) and once with distribution $\psi$ (the right chart). Then, we computed $\mathcal{E}\cap L$ by solving the system of polynomial equations with the software HomotopyContinuation.jlHC.jl. The charts show the empirical distribution of real zeros and the corresponding empirical means in these experiments.
• Figure 2: The pie chart shows the outcome of the following experiment, similar to the experiments in Figure \ref{['fig:exp']}. We sampled $N=1000$ pairs $(\mathbf{x}_i,\mathbf{y}_i)_{i=1}^5$, where the $\mathbf{x}_i$ and $\mathbf{y}_i$ are sampled i.i.d. from the box $[-5,5]\times [-5,5]\subset \mathbb R^2$. Then, we computed $\mathcal{E}\cap L$ by solving the system of polynomial equations with the software HomotopyContinuation.jlHC.jl. The chart shows the empirical distribution of real zeros and the corresponding empirical means in these experiments.
• Figure 3: The polytope $P$ from \ref{['def_P']}.

Theorems & Definitions (25)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Corollary 2.3
• Lemma 2.4
• proof