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 4
3 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 4
4 symmetric matrix 
and can be
solved for 
in the least-squares sense.
Since 
is a 4
3 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 4
4 orthogonal matrix. Letting 
be a 4
3 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.
