CI PA 2003 XIX th International Symposium, 30 September- 04 October, 2003, Antalya, Turkey
Fig. 3. Barrel (blue) and pincusion (red).
This formulation could make the student think of “principal
distance” as an invariable parameter. The great importance
given to the focal length inside the photogrammetry context
contributes to increase this risk. The focal length plays, without
any doubt, a leading role not only in mathematic models such as
co-linearity condition or co-planarity, but also during setting up
of a stereoplotter and in photogrammetric projects planning.
The high accuracy that, as it is assumed, underlies in a
calibration certificate (in which principal distance is usually
expressed in a magnitude order of microns) highlights the pre
eminence of focal length as the most basic parameter in
Photogrammetry.
But being true that the principal length value must be unique,
one must notice that this parameter is directly correlated to
other parameters and together they define the internal
characteristics of the camera, so that a change or lock of
anyone of them must have effects on the others.
The problem, as we all known, is that while one can observe
and thus measure the value of the incidence angle (alpha) and
the radial distance resulting from it (r), it is impossible to
measure the principal distance. On the contrary, we infer the
knowledge of the principal length from what we want to know:
the radial distortion that has already been defined as a function
of the same distance. So nothing is that well-defined.
The principal length is the distance between the image nodal
point and the image plane which is located in a certain point
where both the actual and the theoretical image-points are the
same. But, in fact, we can’t neither know at what distance that
occurs (where radial distortion is null), nor find it useful in
operative terms. In any case an additional criterion is needed.
As Brown says there are three possibilities.
a) We can assume that the principal distance makes null
the radial distortion at a fixed radial distance RO
b) We can solve for a principal distance that makes
minimum the summation of squares of deviations.
c) Or we can search for the principal length that makes
equal absolute values of maximum and minimum
deviation.
Another way of talking about radial distortion has been used by
Albertz, Kraus or Burnside. For them, the radial distortion can
be considered as the variation of the principal distance, as a
function of the incidence angle of rays (we do prefer this
version due to its didactic value). This distance should be given
as a nominal value by the calibrator so the assumption of a
fixed physical dimension is better avoided. In this way it is
understood that this parameter depends on a predefined
specification and gives as a result an specific distribution of
radial distortion values.
r = (f + 4/)* tan a
Following Hallerf s notation
[2]
that we rather prefer instead of the more commonly seen:
r — /*tan(tz + Aa) [3]
Even when the nodal point actually exists, and in the image
plane there is a certain region of points that are perspective
rules compliant, it is useless to search for their position. The
reason of this is that any elected value of the principal distance
is a good choice if it is taken into account that the discrepancy
the actual principal distance and the preset one. This
discrepancy brings on a certain distribution of the displacement
(Ar) of every image point from its ideal radius (r’) as a
consequence.
Figure 4. Scheme of interdependence between/and dr.
From a practical point of view, there is no need to use the
concept of “true principal distance”, but to know the diverse
distribution of radial distortion (drl, dr2, ... dri) associated to
their corresponding focal lengths (f 1, f2,... f i) as functions of
incidence angle.
What we pretend to show is that a given physical point always
has the same image point, associated with a residual that
function of its distance to the principal point (best symmetry
point). In other words: for a certain object point coordinates,
there only exists a unique pair of image coordinates which
corresponds to it (orientations are supposed fixed) but there are
virtually infinite combinations of principal length and distortion
that make colinearity condition complied.
The application that we have developed offers a workspace in
which the user can simulate the effects of radial distortion on a
test pattern, seeing its connection with the focal length. It
emulates in someway a multi-collimator; a grid of points project
light rays through a virtual lens (the lens axis is supposed to be
normal to the grid plane or w = f = 0). Fig. 5.
