This paper presents an efficient method for solving PnP, PnPf, and PnPfr problems, which are the problems of determining camera parameters from 2D-3D point correspondences. The proposed method is derived based on a simple usage of linear algebra, similarly to the classical DLT methods. Therefore, the new method is easier to understand, easier to implement, and several times faster than the state-of-the-art methods using Gröbner basis. Contrary to the existing Gröbner basis methods, the proposed method consists of three algorithms depending on the number of the points and the 3D point configuration. Experimental results show that the proposed method is as accurate as the state-of-the-art methods even in near-planar scenes while achieving up to three times faster.

We propose an accurate and scalable solution to the perspective-n-point problem, referred to as ASPnP. Our main idea is to estimate the orientation and position parameters by directly minimizing a properly defined algebraic error. By using a novel quaternion representation of the rotation, our solution is immune to any parametrization degeneracy. To obtain the global optimum, we use the Grobner basis technique to solve the polynomial system derived from the first-order optimality condition. The main advantages of our proposed solution lie in accuracy and scalability. Extensive experiment results, with both synthetic and real data, demonstrate that our proposed solution has better accuracy than the state-of-the-art noniterative solutions. More importantly, by exploiting vectorization operations, the computational cost of our ASPnP solution is almost constant, independent of the number of point correspondences n in the wide range from 4 to 1000. In our experiment settings, the ASPnP solution takes about 4 milliseconds, thus best suited for real-time applications with a drastically varying number of 3D-to-2D point correspondences.