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;
which is refered to as a paraperspectve camera model.
where the distances of the reference point from the cameras are normalized to unity.