Though perspective projections give
accurate models for a wide range of existing cameras, the mapping from
the object point 
to the image point 
is
non-linear due to the scaling factor 
. In order to make
the projection model more mathematically tractable, affine cameras
have been introduced.
Now we pick up one arbitrary point 
in 3D space
and let 
be the image of
observed by i-th camera. We call
a reference point. We also introduce a
2
3 matrix 
and a 3-vector 
which
are composed from first two rows and third row of
respectively. Then the depths of
and 
for i-th camera are given
by 
and 
respectively with
. Using these
entities, we approximate the perspective projection
(1) as
where we assumed that 
and 
. Translating both image and object coordinates by
and
, we have
an affine camera model:
which is a first-order approximation obtained from Taylor expansion of
the perspective camera model around the reference point
. The reference point may be chosen from the
object points 
or, in many cases, be
set to their centroid, i.e.
. In the
latter, the image of 
in i-th camera
coincides, to a first-order approximation, with the centroid of the
images of the object points, that is,
.
If the intrinsic camera parameters in 
are known, we can
set 
by using appropriate canonical image
coordinates. Under this condition, three special types listed below are
important instances of the affine camera;
, the
affine projection matrix 
in (2) becomes
which is refered to as a paraperspectve camera model.
.
If we adopt cruder approximation
instead, a weak perspective camera model
results:
. Then we
have an orthographic camera model:
where the distances of the reference point from the cameras are normalized to unity.
