When the perspective effects are small, the problem of locating perspective epipolar lines becomes ill-conditioned. In such cases it is convenient to assume the parallel projection model of the affine camera which explicitly models the ambiguities.
The affine epipolar constraint can be described in terms of the
affine fundamental matrix as , i.e.,
(See Shapiro, Zisserman and Brady).
To derive the above, we write as
where is a general (non-singular) matrix and
is a vector. The projection equation then
and are functions only of camera parameters and the motion transformation , while explains the motion of the reference point (centroid) and depend on the translation of the object and the camera origins and .
This equation shows that associated with lies
on a line (epipolar) on the second image with offset
and direction . The unknown depth determines how far
along this line does lie. Inverting the equation we obtain
The translation invariant versions of these equations are
We can eliminate from the above equations and obtain a single
equation in terms of image measurables: