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
