Reprojection error

The reprojection error is a geometric error corresponding to the image distance between a projected point and a measured one. It is used to quantify how closely an estimate of a 3D point X ^ recreates the point's true projection x . More precisely, let P be the projection matrix of a camera and x ^ be the image projection of X ^ , i.e. x ^ = P X ^ . The reprojection error of X ^ is given by d ( x , x ^ ) , where d ( x , x ^ ) denotes the Euclidean distance between the image points represented by vectors x and x ^ .

Minimizing the reprojection error can be used for estimating the error from point correspondences between two images. Suppose we are given 2D to 2D point imperfect correspondences { x i ↔ x i ′ } . We wish to find a homography H ^ and pairs of perfectly matched points x i ^ and x ^ i ′ , i.e. points that satisfy x i ^ ′ = H ^ x ^ i that minimize the reprojection error function given by

So the correspondences can be interpreted as imperfect images of a world point and the reprojection error quantifies their deviation from the true image projections x i ^ , x i ^ ′