揭示P3P问题奇异位形的几何结构，明确相机位置与解的数量关系。
Geometric Stratification for Singular Configurations of the P3P Problem via Local Dual Space
- 基于局部对偶空间构建代数计算框架，系统分析奇异位形。
- 当相机中心在危险圆上时，解的数量≥4，对应无穷多解。
- 适用于计算机视觉中三维重建与相机定位的理论研究者。
本文研究P3P问题的奇异配置。通过局部对偶空间，提出一个系统的代数-计算框架，对相机中心O的多重性μ≥2、μ≥3、μ≥4的情形进行完整的几何分层：当μ≥2时，O位于'危险圆柱'上；当μ≥3时，O位于与第一个Morley三角形或外接圆相关的三条母线上；当μ≥4时，O位于外接圆上，此时存在无穷多P3P解。此外，还研究了与奇异配置O对应的补配置O′的几何分层：当μ≥2时，O′位于与危险圆柱相关的Deltoidal曲面上；当μ≥3时，O′位于该曲面的三条尖点曲线上。
原文摘要 · Abstract (English)
This paper investigates singular configurations of the P3P problem. Using local dual space, a systematic algebraic-computational framework is proposed to give a complete geometric stratification for the P3P singular configurations with respect to the multiplicity $μ$ of the camera center $O$: for $μ\ge 2$, $O$ lies on the ``danger cylinder'', for $μ\ge 3$, $O$ lies on one of three generatrices of the danger cylinder associated with the first Morley triangle or the circumcircle, and for $μ\ge 4$, $O$ lies on the circumcircle which indeed corresponds to infinite P3P solutions. Furthermore, a geometric stratification for the complementary configuration $O^\prime$ associated with a singular configuration $O$ is studied as well: for $μ\ge 2$, $O^\prime$ lies on a deltoidal surface associated with the danger cylinder, and for $μ\ge 3$, $O^\prime$ lies on one of three cuspidal curves of the deltoidal surface.
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