A Minimal Solution to the Perspective-3-Point Problem for the Camera with Unknown Focal Length and Two Degrees-of-Freedom Rotation.

This paper presents a minimal solution to the absolute camera pose based on Perspective-3-Point problem for the camera with two degrees-of-freedom rotation and unknown focal length (P3P+f). First, according to the imaging equation from the control points to the corresponding image points, the minimal problem is converted into solving a system of four nonlinear algebraic equations in four unknowns. Then, using the classical resultant elimination method, the problem is cast into solving a six-degree polynomial equation in a single variable. Meanwhile, we propose an algorithm that may obtain directly the six-degree polynomial equation. We show that when three control points are in general position, the upper bound of the number of solutions to our P3P+f problem is 6. We also show the conditions in which the upper bounds of the solutions are two, four and infinite. Finally, we give some examples to validate our results and the accuracy, robustness and efficiency of our P3P+f solver are evaluated by testing on synthetic and real data.