Geometric Stratification for Singular Configurations of the P3P Problem via Local Dual Space

Abstract

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.

Figures (11)

• Figure 1: Multi-solution phenomenon and singular configurations of the P3P problem. In different configurations, the number of distinct solutions of the P3P equation system varies (up to $4$ or infinite). Here $n$ denotes the number of distinct solutions and the red number denotes the multiplicity of the solution. In this paper, we investigate the geometry of singular P3P solutions as well as the geometry of complementary solutions with respect to different multiplicities.
• Figure 2: The geometry of the P3P problem.
• Figure 3: Stratification framework I.
• Figure 4: The geometry of $\mathcal{V}_{\geq2}$.
• Figure 5: The first Morley triangle.

Theorems & Definitions (25)

• Definition 1: li2022improved
• Remark 2
• Definition 3: li2022improved
• Remark 4
• Lemma 5
• proof
• Theorem 6
• Lemma 7: Proposition 12 of § 4.4 cox1997ideals
• Theorem 8
• proof