An image point
back projects to a ray defined by
and the camera center. Calibration relates the image point
to the ray's direction.
Suppose points on the ray are written as
in the camera Euclidean frame. Then these points map to the point
Thus,
is the (affine) transformation between
and the ray's direction
measured
in the cameras Euclidean frame.
The angle between two rays
and
corresponding
to image points
and
may be obtained as
(by the cosine formula)
The above shows that if
is known (camera is calibrated),
then the angle between
rays can be computed from their corresponding image points. A
calibrated camera is like a 2D protractor.
An image line
defines a plane through the camera center
with normal direction
.
Proof.
Points
on
back projects to directions
which are orthogonal to the
plane normal. Hence,
. Since points on
satisfy
we have
that
.