CIP A 2003 XIX th International Symposium, 30 September - 04 October, 2003, Antalya, Turkey
In this paper we want to focus on three aspects concerning this
question:
1. How do we manage with the high degree of geometric
distortions (particularly the radial symmetric distortion)
when we apply direct image resampling techniques such as
rectification?
2. How will we set up internal camera parameters including
those related to lens distortions when we make use of
different programs? And being more specific, how will we
manage all the different mathematic schemes that are
involved in the description of the same phenomenon?
3. In case of being the distortion well-known, what is the most
efficient way of working? Should be a good practice to
eliminate it as a first stage by image re-sampling prior to
any further process such as re-projection or restitution? Or
would it be better to have it mathematically modelled doing
the re-sampling as a unique stage that maps for instance
distortion and rectification at the same.
It is obvious that, nowadays, we are more focused on a certain
type of cameras, those that have been leading the reconciliation
of non-skilled people and Photogrammetry. In a general sense,
the boom of this science as a powerful instrument for heritage
recording has been made possible thanks to the irruption of
digital cameras on the scene.
2. INTRODUCING “LDS”
Lens Distortion Simulator is a computer application that will
bring some light to some points that are often found obscure by
many users, and particularly by students when they face this
question. The following lines will act as a review of some key
points. We will just focus on the radial symmetric distortion and
not so much on the tangential and asymmetric due to the higher
dimensional magnitude and conceptual importance of the first
one as it affects the principal distance concept. For this reason,
only radial distortion simulation is being implemented with the
main purpose of simulating its consequences on images; this is
something that we consider very didactic. On the other hand, it
will allow to know which range applies to those parameters and
at what levels they have noticeable effects.
Figure 1. The simple fact.
In the classic Optics language, as shows the figure 1, the
distortion is defined as the inconstancy of lateral magnification.
But even if the classic definition is easy to learn, as it explains
the differences in terms of point image-coordinates between
real and theoretical locations (those resulting from the
perspective laws compliance), the complexity comes from the
diverse ways of expression of this difference sometimes as an
error and sometimes as a correction. (It depends on authoring).
These displacements, that can be easily understood, become
hard, dense and opaque when the user finds that different
programs use different nomenclature and parameterisation. As a
result sometimes the user finds expressions of error while in
others terms of corrections. But all definitions are in fact the
same; all models define the displacement of points from their
bundle perspective rules compliant positions, but using different
notation to this same fact.
3. THE CONCEPT OF RADIAL SYMMETRIC
DISTORTION.
It is important to take into consideration that the concept of
radial distortion itself does not offer a clear panorama at all, at
least from a didactic point of view
Many authors such as Bonneval, Ghosh, Moffit, or Mikhail,
define radial distortion as the deviation of light rays during lens
crossing.
It seems that such definition comes from the didactic need of
making the model fit into a perfect projective scheme where the
projection centre can be exactly located in a certain point (like
pinhole camera), or instead of this, it could also be considering
the nodal image point being equally well-located and
determinable.
In this way, the lense’s radial distortion is measured as a
distance or separation (along radial directions contained in the
image plane) between the actual positions and their ideal
corresponding ones. So that an ideal ray trajectory and its
resulting image spot are exactly determined by the value of the
principal distance (f) and the incidence angle (a).
Image Plane
Figure 2. Scheme of ray deviation.
Under such assumptions, the radial distortion is expressed in
terms of residuals or errors: actual position - theoretical
position.
dr = r - r' = r - f*tana HI
This expression of “error” gives us the sign criterion: It will be
considered positive the outgoing way and negative the opposite
one. We can see its results in the following figure: The red
figure corresponds to a positive distorted image (“pincushion”)
of a perfect square shape, while the blue one results from a
negative distortion (“barrel”).
