用几何变换简化三点透视问题求解，速度更快且稳定。
A Conic Transformation Approach for Solving the Perspective-Three-Point Problem
- 将两圆锥曲线映射到新坐标系，使一曲线化为标准抛物线，简化求交。
- 虽导出四次方程，但计算系数快，仅需实数解，避免复数运算开销。
- 实测速度优于现有方法，适合对精度和效率要求高的视觉定位场景。
我们提出一种基于圆锥变换的视角三点（P3P）问题求解方法。与现有最优解法不同，后者通过相交两条圆锥曲线并构造退化圆锥来寻找交点，我们的方法基于新公式：将两条圆锥曲线映射至新坐标系，使其中一条变为标准抛物线的规范形式。这使得一个变量可表示为另一个变量的函数，从而大幅简化圆锥曲线交点求解过程。此外，多项式系数计算迅速，只需确定实数交点，无需使用计算代价高昂的复数运算。尽管当前最优方法将圆锥交点问题归约为求解单变量三次方程，而本方法虽产生四次方程，仍因新公式的简化优势实现更快求解。大量实验表明，该方法在保持与最先进方法相当的鲁棒性和稳定性的同时，显著提升速度。
原文摘要 · Abstract (English)
We propose a conic transformation method to solve the Perspective-Three-Point (P3P) problem. In contrast to the current state-of-the-art solvers, which formulate the P3P problem by intersecting two conics and constructing a degenerate conic to find the intersection, our approach builds upon a new formulation based on a transformation that maps the two conics to a new coordinate system, where one of the conics becomes a standard parabola in a canonical form. This enables expressing one variable in terms of the other variable, and as a consequence, substantially simplifies the problem of finding the conic intersection. Moreover, the polynomial coefficients are fast to compute, and we only need to determine the real-valued intersection points, which avoids the requirement of using computationally expensive complex arithmetic. While the current state-of-the-art methods reduce the conic intersection problem to solving a univariate cubic equation, our approach, despite resulting in a quartic equation, is still faster thanks to this new simplified formulation. Extensive evaluations demonstrate that our method achieves higher speed while maintaining robustness and stability comparable to state-of-the-art methods.
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