To compute the rectifying projection matrices
and
, we set up a linear, homogeneous system of equations formed by the
constraints sufficient to guarantee rectification [
1
], and incorporate explicitly quadratic constraints on the entries of
and
to ensure a nontrivial solution. The constrained system and its
equations are detailed in this section. In the following, we shall write
as follows:
Projection matrices are defined up to a scale factor. The common choice
to block the latter is
and
[
1
], unfortunately, brings about two problems: first, the intrinsic
parameters become dependent on the choice of the world coordinate system
[
5
]; second, the resulting projection matrices do not satisfy the
conditions guaranteeing that meaningful calibration parameters can be
extracted from their entries of the matrices [
4
], that is (for example for
),
To obviate the problems mentioned, we enforce
The second equation in (
10
) and its equivalent for
are actually implied by the system defining our algorithm (proof omitted
for reasons of space).
The optical centers of the rectified projections,
and
, must be the same as those of the original projections:
Eq. ( 12 ) gives six linear constraints:
The two rectified projections must share the same focal plane, i.e.
The vertical coordinate of the projection of a 3-D point onto the rectified retinal plane must be the same in both image, i.e:
Using Eq. ( 14 ) we obtain the constraints
Notice that the equations written to this point are sufficient to guarantee rectification (indeed some authors stop here, e.g. [ 1 ]), but not a unique solution : the orientation of the rectified retinal plane and the intrinsic parameters are still free. Our algorithm constrains these quantities explicitly, as follows.
We choose the rectified focal planes to be parallel to the intersection of the two original focal planes, i.e.
where
and
are the third rows of
and
respectively. Notice that the dual equation
is redundant thanks to Eq. (
14
).
The intersections of the retinal plane with the planes
and
correspond to the
v
and
u
axes, respectively, of the retinal reference frame. As this reference
frame to be orthogonal, the planes must be perpendicular, hence, using
Eq. (
16
),
Given a full-rank
matrix satisfying constraints (
10
), the principal point
is given by [
4
]:
We set the two principal points to (0,0) and use Eqs. ( 14 ) and ( 16 ) to obtain the constraints
The horizontal and vertical focal lengths in pixels, respectively, are given by
By setting the values of
and
, for example, to the values extracted from
, we obtain the constraints
which, by virtue of the equivalence
and Eq. (
20
), can be rewritten as
Adrian F Clark
