论文标题
基本品种的平均程度
Average degree of the essential variety
论文作者
论文摘要
基本品种是尺寸的代数子变量$ 5 $在实际投影空间中$ \ mathbb r \ mathrm p^{8} $，它编码了两个校准的针孔摄像机的相对姿势。计算机视觉中的$ 5 $ - 点算法计算了基本品种与线性空间编成$ 5 $的相交中的实际点。基本品种的程度为$ 10 $，因此该交集总体上由10个复杂点组成。 当线性空间是随机的时，我们计算预期的实际交叉点数。我们专注于线性空间的两个概率分布。在正交组$ \ mathrm {o}（9）$上作用于$ \ mathbb r \ mathrm p^{8} $的线性空间上的第一个分布是不变的。在这种情况下，实际交叉点的预期数量等于$ 4 $。第二个分布是从计算机视觉的动机，并通过在图像平面中选择5点对应关系来定义，$ \ Mathbb r \ Mathrm P^2 \ times \ times \ Mathbb r \ Mathrm P^2 $随机均匀地均匀。蒙特卡洛计算表明，具有高概率的预期值在于间隔$（3.95-0.05，\ 3.95 + 0.05）$。
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 $(3.95 - 0.05,\ 3.95 + 0.05)$.
