EPNP

EPNP stands for Efficient Perspective-n-Point, a method in computer vision for estimating a calibrated camera’s pose from 2D–3D correspondences. Given a set of known 3D points X_i in the world and their observed 2D projections u_i in the image, EPNP computes the rotation R and translation t that align the world frame with the camera frame, intrinsics being known (calibrated camera).

The central idea of EPNP is to express each 3D point X_i as a barycentric combination of four virtual control points C_j in the world coordinates: X_i = sum_j alpha_ij C_j, where the barycentric weights alpha_ij depend only on the chosen control points and sum to one. The four control points are treated as variables to be recovered in the camera coordinate frame, and the projection equations become linear in these variables. By assembling equations from all correspondences, the method reduces the problem to solving a small linear system, from which the camera-space coordinates of the four control points are obtained. Once these are known, standard rigid-motion estimation (for example, a Procrustes solution) yields R and t that best align the control points between world and camera frames. The final pose can then be refined with non-linear optimization if desired.

Key characteristics include its non-iterative, closed-form nature and linear complexity with respect to the number of correspondences, making EPNP fast for large n. It is commonly used when a fast, reasonably accurate pose estimate is needed and can serve as a robust initial solution for subsequent refinement. Limitations include sensitivity to outliers and, like other PnP methods, reliance on a calibrated camera and well-distributed 3D points to ensure numerical stability.