The metric constraints for perspective cameras are also derived based on the orthogonality of the camera orientation matrix ; Suppose that we have obtained projective camera motion by factoring the measurement matrix. Since the intrinsic camera parameters are assumed to be known, we can set . Then, substituting (1) into (13) and eliminating , we have
where stands for equality up to an unknown non-zero scale. Let , and are three rows of , and be a 43 matrix composed from the first three columns of . Then (25) can be rewritten as
which gives 5F linear homogeneous equations in terms of 10 elements of the 44 symmetric matrix and can be solved for in the least-squares sense. Since is a 43 matrix, cannot be full rank and one of its eigenvalues is zero. Therefore its eigenvalue decomposition has the form: where , and are non-negative eigenvalues of , and is a 44 orthogonal matrix. Letting be a 43 matrix taking first three columns of , we then obtain as .
Since the fourth column of affects only the translation and isometric scaling, it can be set to an arbitrary non-zero 4-vector. Thus, we have determined a projective transformation with which projective motion and shape are upgraded to Euclidean descriptions.