Accurate Non-Iterative 𝑂(𝑛) Solution to the P𝑛P Problem

We propose a non-iterative solution to the PnP problem-the estimation of the pose of a calibrated camera from n 3D-to-2D point correspondences—whose computational complexity grows linearly with 𝑛<. This is in contrast to state-of-the-art methods that are 𝑂(𝑛
⁵
) or even 𝑂(𝑛
⁸
), without being more accurate. Our method is applicable for all 𝑛≥4 and handles properly both planar and non-planar configurations. Our central idea is to express the 𝑛 3D points as a weighted sum of four virtual control points. The problem then reduces to estimating the coordinates of these control points in the camera referential, which can be done in 𝑂(𝑛) time by expressing these coordinates as weighted sum of the eigenvectors of a 12 × 12 matrix and solving a small constant number of quadratic equations to pick the right weights. The advantages of our method are demonstrated by thorough testing on both synthetic and real-data.