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 44 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.