Companion Surface of Danger Cylinder and its Role in Solution Variation of P3P Problem
This provides incremental insights into the P3P multi-solution phenomenon, relevant for researchers in computer vision and PnP studies.
The paper tackles the relationship between the danger cylinder and multiple-solution phenomena in the P3P problem, showing that when the optical center lies on the danger cylinder, it leads to a double solution and two other solutions, with the optical centers of the latter forming a new surface called the Companion Surface of Danger Cylinder (CSDC), which acts as a delimiter causing solution number changes by 2.
Traditionally the danger cylinder is intimately related to the solution stability in P3P problem. In this work, we show that the danger cylinder is also closely related to the multiple-solution phenomenon. More specifically, we show when the optical center lies on the danger cylinder, of the 3 possible P3P solutions, i.e., one double solution, and two other solutions, the optical center of the double solution still lies on the danger cylinder, but the optical centers of the other two solutions no longer lie on the danger cylinder. And when the optical center moves on the danger cylinder, accordingly the optical centers of the two other solutions of the corresponding P3P problem form a new surface, characterized by a polynomial equation of degree 12 in the optical center coordinates, called the Companion Surface of Danger Cylinder (CSDC). That means the danger cylinder always has a companion surface. For the significance of CSDC, we show that when the optical center passes through the CSDC, the number of solutions of P3P problem must change by 2. That means CSDC acts as a delimitating surface of the P3P solution space. These new findings shed some new lights on the P3P multi-solution phenomenon, an important issue in PnP study.