SCIENTIA SINICA Informationis, Volume 47 , Issue 4 : 482-491(2017) https://doi.org/10.1360/N112016-00198

Generality of the multi-solution phenomenon in the P3P problem

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  • ReceivedAug 18, 2016
  • AcceptedNov 24, 2016
  • PublishedFeb 13, 2017


The P3P problem is a significant single-view-based pose estimation method, and its multi-solution phenomenon has been the focus of investigation since its inception. In this work, we demonstrate from a geometrical point of view that when the distance from the camera's optical center to the three control points is relatively large, the corresponding P3P problem cannot have a unique solution in general; alternatively, or it should have at least two solutions satisfying the three original constraints. A further contribution of this study is the finding that the intersection of the three 3D pumpkin surfaces in the P3P problem can be generally deduced from the intersection of the three 2D intersecting curves of the three pumpkin surfaces with the control point plane, which provides a new way to investigate the multi-solution phenomenon. Our results may be significant for both P3P theorists and practitioners.

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图A1为当光心位于由三个控制点决定的六个南 瓜面外部但控制点三角形ABC为钝角三角形时的 情形(角C为钝角, 3种不同颜色表示的交线与图5相同), 从图A1可见, 此时这些南瓜面与控制平面交线之间的相交性质, 与图5所示没有区别. 即定理1当控制点三角形为钝角三角形时仍成立.

图A2为当光心位于由3个控制点决定的6个 南瓜面内部且控制点三角形ABC为锐角三角形时的情形($ \alpha < \angle {\rm A} $, $\gamma < \angle {\rm C} $ 但 $ \beta > \angle {\rm B} $,). 从图A2 可见, 此时这些南瓜面与控制平面交线之间的相交性质 与图5所示发生了根本性变化, 此时证明定理1途径不再有效.