提出四点透视问题的快速精确解法，计算效率提升一个量级。
A polynomial formula for the perspective four points problem
- 通过变量分离，将透视问题转化为可解析求解的绝对定向问题。
- 在真实噪声下精度与现有方法相当，速度提升约10倍。
- 适合需要高效鲁棒估计的计算机视觉应用，如RANSAC优化。
本文针对透视n点问题中的n=4情形，提出一种快速且精确的解法。给定四个三维点及其在相机成像平面上的对应二维点，新方法首先构造一组位于相机到二维点射线上的新三维点，使这些点间的六对距离尽可能接近原三维点的距离。该步骤将透视问题转化为绝对定向问题，可通过显式公式求解。为实现此目标，采用对称性最强的坐标系：三维侧使用点对间平方距离，二维侧则将其中一个点旋转至光轴后计算点积。最终解法借助计算机代数系统推导得出。相比现有最优算法，本方法速度快约一个量级，且在真实噪声下保持相近精度；其核心降维步骤比其他视角求解器快两个量级，极大提升了RANSAC中初始样本剔除的效率。
原文摘要 · Abstract (English)
We present a fast and accurate solution to the perspective $n$-points problem, by way of a new approach to the n=4 case. Our solution hinges on a novel separation of variables: given four 3D points and four corresponding 2D points on the camera canvas, we start by finding another set of 3D points, sitting on the rays connecting the camera to the 2D canvas points, so that the six pair-wise distances between these 3D points are as close as possible to the six distances between the original 3D points. This step reduces the perspective problem to an absolute orientation problem, which has a solution via explicit formula. To solve the first problem we set coordinates which are as orientation-free as possible: on the 3D points side our coordinates are the squared distances between the points. On the 2D canvas-points side our coordinates are the dot products of the points after rotating one of them to sit on the optical axis. We then derive the solution with the help of a computer algebra system. Our solution is an order of magnitude faster than state of the art algorithms, while offering similar accuracy under realistic noise. Moreover, our reduction to the absolute orientation problem runs two orders of magnitude faster than other perspective problem solvers, allowing extremely efficient seed rejection when implementing RANSAC.
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