Closed-form solution of absolute orientation using orthonormal matrices

Finding the relationship between two coordinate systems using pairs of measurements of the coordinates of a number of points in both systems is a classic photogrammetric task. It finds applications in stereophotogrammetry and in robotics. We present here a closed-form solution to the least-squares problem for three or more points. Currently, various empirical, graphical and numerical iterative methods are in use. Derivation of a closed-form solution can be simplified by using unit quaternions to represent rotation, as was shown in an earlier paper1. Since orthonormal matrices are more widely used to represent rotation, we now present a solution using 3×3 matrices. Our method requires the computation of the square-root of a symmetric matrix. We compare the new result with an alternate method where orthonormality is not directly enforced. In this other method a best fit linear transformation is found and then the nearest orthonormal matrix chosen for the rotation. We note that the best translational offset is the difference between the centroid of the coordinates in one system and the rotated and scaled centroid of the coordinates in the other system. The best scale is equal to the ratio of the root-mean-square deviations of the coordinates in the two systems from their respective centroids. These exact results are to be preferred to approximate methods based on measurements of a few selected points.