Geometric Stratification for Singular Configurations of the P3P Problem via Local Dual Space
This work addresses a specific issue in computer vision and photogrammetry for researchers and practitioners dealing with camera calibration and 3D reconstruction, but it is incremental as it builds on existing algebraic frameworks to refine geometric understanding.
This paper tackles the problem of singular configurations in the P3P (Perspective-Three-Point) problem, which can lead to ambiguous or infinite solutions in camera pose estimation, by providing a complete geometric stratification using local dual space, resulting in classifications based on multiplicity thresholds (e.g., for μ≥4, the camera center lies on the circumcircle, corresponding to infinite solutions).
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.