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Perspective-$n$-Point (P$n$P) stands as a fundamental algorithm for poseestimation in various applications. In this paper, we present a new approach tothe P$n$P problem with relaxed constraints, eliminating the need for precise 3Dcoordinates or complete calibration data. We refer to it as AP$n$P due to itsability to handle unknown anisotropic scaling factors of 3D coordinates oralternatively two distinct focal lengths in addition to the conventional rigidtransformation. Through algebraic manipulations and a novel parametrization,both cases are brought into similar forms that distinguish themselves primarilyby the order of a rotation and an anisotropic scaling operation. AP$n$P thenboils down to one unique polynomial problem, which is solved by the Gr\"obnerbasis approach. Experimental results on both simulated and real datasetsdemonstrate the effectiveness of AP$n$P as a more flexible and practicalsolution to camera pose estimation. Code: https://github.com/goldoak/APnP.

AP$n$P: A Less-constrained P$n$P Solver for Pose Estimation with Unknown Anisotropic Scaling or Focal Lengths