P3P问题解分布的临界曲面研究

P3P Problem, or the Perspective-Three-Point Problem, is a very important single-image based pose estimation method for any real applications, multi-solution phenomenon,i.e., whether the unique solution exists, and if not, how many solutions are possible must be addressed, is a key aspect of the P3P problem, and has not been well understood till now. This project is to investigate this multi-solution phenomenon by determining so-called “critical surface of solutions”. Here the critical surfaces are meant that the whole 3D space is partitioned into different regions by such surfaces that all the P3P problems whose optical centers lie in the same region must have the same number of solutions, and when the optical center passes through the surfaces, the number of the solutions of the corresponding P3P problem must change. Our recent work shows that the danger cylinder, its companying surface and the 6 toroids associated with the 3 control points must be part of the critical surfaces, but we cannot prove whether these surfaces constitute the complete set. This project will focus on the investigation on the complete set of critical surfaces, by which some geometrical interpretation of the multi-solution phenomenon could be provided, and hopefully it will also bridge the gap between the geometrical approach and the algebraic approach in the literature. The investigation is not only of theoretical interests and significance, it could also act as theoretical guidance for single-image based pose estimation practitioners.

P3P问题是基于单幅图像进行物体定位的重要方法，多解现象，即“是否有唯一解”及“解不唯一时可能解的个数”是P3P问题在具体应用中需解决的基本问题，但至今仍没有得到很好的解决。本项目通过确定“P3P问题解分布的临界曲面”的途径，探索研究P3P问题的多解现象。这里的“临界曲面”指由P3P问题的三个控制点确定的特殊曲面，它们可以将三维空间分割成不同的区域，当光心在同一区域时，所有对应的P3P问题的解的个数都相同；当光心穿越这些曲面时，解的个数必然变化。我们的近期工作表明，临界曲面至少包含危险圆柱面及其伴随曲面和6个超环面，本项目将对这些曲面是否构成临界曲面的完备集开展系统的研究，获得具有明确几何意义的多解条件和解的分布状况，建立起从代数到几何的桥梁。本项目从几何途径阐明P3P问题多解现象的本质，不仅对P3P问题多解现象的研究具有重要的理论价值，也对基于单幅图像的视觉定位应用具有一定的指导作用。

P3P问题是PnP问题的最小单元，是计算机视觉中利用3对点对来进行单目相机定位的经典且常用的方法之一，尤其适用于受限场合、大范围连续定位、鲁棒定位等场景。本项目的主要研究内容是P3P问题的多解现象，尤其着重于多解情形下的几何解释。目前取得的主要结论包括:1) 提出了P3P 问题的解与本质二次曲线对交点之间的对应关系; 2)严格证明了P3P问题的多解现象是普遍存在的，其唯一解只能存在于由6个超环面(the spindle tori)围成的闭区域内; 并证明了其中3个超环面扮演了临界曲面的角色，而3个控制点则起到了奇点的作用。3)证明了当存在四个解，且其中一对形成共边解，一对形成共点解时，这两对解存在伴生现象，且推导出两个共边解对应的摄像机的光心都落在一个垂直平面上，而两个共点解对应的光心都落在扭危险曲面(the skewed danger cylinder)上。4）推导出危险圆柱面的伴随曲面的代数表达，它是一个关于光心坐标的12次的多项式方程。同时证明了该伴随曲面也是一种临界曲面，即在该曲面的两侧，解的个数会发生变化，增加或减少两个。这些结论本身不仅具有基础理论意义，而且在控制点的布局方面具有现实的指导意义。