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International cooperation and technology transfer
Mussio, Luigi

G. Fangi - Ancona University- Italy
email: fangi@pocsi. unian. it
Carla Nardinocchi — Parma University — Italy
Email: nardinoc@dits. ing. uniromal.it
KEY WORDS: Photogrammetry, non-metric cameras, calibration, distortion
Abstract: We propose a simple procedure for the determination of the radial distortion for non-metric cameras.
Normally the distortion characteristics are estimated together with the other interior orientation parameters both
with laboratory equipment via goniometer, both by block bundle adjustment with the so-called additional
parameters. The self-calibration approach is followed for non-metric cameras also, with the difference that any
orientation is in practice a new calibration. Often adding the parameters for the distortion brings the solution system
to numerical problems especially when the geometric configuration is weak. It happens with narrow angle of field
lenses, of poor distribution of control points, leading to the impossibility of the estimation for the distortion.
Therefore we intend to estimate the main distortion - the radial component - a part from the orientation. The
proposed approach is intended to be a first step for the full camera calibration, divided in distortion determination
and interior orientation. We neglect the variation of the distortion with the focussing distance. The distortion is
deeply linked to the other interior orientation parameters. We will find that particular interior orientation set fitting
into the computed distortion curve. The ideal distortion-free camera corresponds to the pinhole model. The
distortion is the systematic difference from the real camera to the pinhole model. The equations of the strict
projection are the collinearity equations, or, in an implicit form, the Direct Linear Transformations. A particular
case of the DLT are the equations of the homografic transformations. In order to avoid numerical problems with the
DLT and additional parameters, we use the special case of DLT, that is the homografic transformation. The
transformation residuals are the components of the distortion vector. We tested the procedure to estimate the radial
distortion for a already calibrated semi-metric camera, a Rollei 6008 with a 40mm Distagon lens, using a picture of
an high accuracy plotted grid.
The radial distortion of non metric cameras is
normally corrected by adding additional parameters
to the perspective transformation or DLT (Abdel-
Aziz 1971, 1973 ):
The distortion parameters can be expressed as
function of the so-called additional parameters, being
k;, the coefficients of radial distortion, Pj those of the
asymmetrical distortion, 0 the rotation angle of the
axis where the asymmetrical distortion, or
decentering lens distortion, becomes null. There exist
many distortion models up to the very sophisticated
24 additional parameters function of Mueller, Bauer,
Jacobsen (Kruck, 1985). But normally the distortion
is distinguished in two parts: the radial distortion and
then tangential one. A complete treatment of lens
distortion modélisation can be found in (Fryier,
4- a 2 Y + afiZ + a^
x + dx =
QqX + ûjqK + d^\Z 1
dx = dx'+dx"= f\x,y,x M ,yM,k l ,k 2 ,k 3 ,...) +
+ g'(x,y,x M ,y M ,Q,p h p 2 ,p 3 ,...)
Adding additional parameters to the projectivity
equations has several drawbacks: very many control
points are required, while when the points are not
well distributed the estimation of the distortion is
poor, due also to the high correlation between the
DLT or collinearity equations parameters and those
of the distortion. The introduction of the distortion
parameters brings to numerical problems for the
stability of the solution system (Faig and Shih 1986),
especially with narrow angle lenses (Fangi, 1990).
The long focal lenses non-metric cameras are
impossible to be corrected for radial distortion.
dy = dy'+dy" = f'\x,y,x M ,y M ,k v k 2 , k 3 ,...) +
+ g" (*, y,x M , y M 9, P\, p 2 > Py>•••)
• x, y are the comparator co-ordinates
• X, Y, Z are the terrain co-ordinates
• aj are the transformation parameters
• dx ’ and dy ’ are the components of radial
distortion and
• dx” and dy” those of the tangential distortion.
The present research has been financed by Cofin97