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International cooperation and technology transfer
Mussio, Luigi

Gabriele Fangi - Ancona University - Italy - fangi@popcsi.uman.it
Key words: non-metric cameras, close-range photogrammetry, distortion, calibration
The camera calibration methods are essentially distinguished in calibration in the laboratory by optical goniometers and calibration
in the field by bundle adjustment. In a previous work (Fangi, Nardinocchi, 9, 1999), we set up the so-called grid method for the
determination of the camera radial distortion curve. The distortion is obtained as the non-linear part of the homographic
transformation from object plane the image plane. The homographic transformation is a special case of the Direct Linear
Transformation, where the object space is reduced to an object plane. The principal point is estimated by intersection of the vectors
of the transformation residuals. Since the determination is separated from full interior orientation calibration, the estimate is not
very accurate. When the lens is distortion free no principal point can be estimated. The reason for separate the computation of the
distortion from the interior orientation is that sometimes, especially with narrow angle lens, numerical problems arise, leading to the
instability of the system solution. Therefore a simple method for full calibration, making use of what we called control directions, has
been set-up. First the control direction is defined as known direction toward an unknown point, measured directly by means of a
theodolite whose centre coincides with the exterior nodal point of the camera to be calibrated. In the collinearity equations the
exterior orientation parameters reduce to three. The grid method is then coupled to the control directions method, in the sense that a
first estimate of the principal point position can be achieved by this last method, followed by the distortion determination, and finally
again by bundle adjustment. Some tests are described and shown.
The camera calibration methods are essentially distinguished
in calibration in the laboratory by optical goniometers
(Hakkarainen, 1978, Taymann, 1978) and calibration in the
field by bundle adjustment (Tayman, 1978, Tayman
Ziemann, 1984, Ziemann, 1986). Normally for non-metric
cameras the determination of the radial and tangential
distortion is carried out contemporary to the interior
orientation (Abdel-Aziz, 1973, Adams, 1981, Fryer, 1989,
Kruck, 1985). The distortion varies not only with the
focussing distance (Fraser, 1982), but also in the depth of
field (Fryer, 1986, Magill 1955). The configuration of the
control points can be near to the critical one ( Faig , Shih,
1986) and the estimation of the interior orientation brings to
the instability of the system solution. This holds especially
with marrow angle lenses. A possible solution is to use the
known terrain co-ordinates of the projection centre as control
information (Fangi, 1990). Nevertheless the contemporary
estimation of the distortion and of the interior orientation
parameters gives high numerical problems. The separation in
two steps of the calibration procedure can cope with these
numerical problems. The determination of the radial
distortion by the grid method (Fangi, Nardinocchi, 1999)
neglects the possible variations of the distortion. Now here
the calibration by control direction is proposed as second step
of the full camera calibration.
1. The control directions
In the collinearity equations
x = x .r 1 (X-X 0 ) + r 4 (y-y 0 ) + r 7 (Z-Z 0 )
M /3(X-X 0 ) + r 6 (y-y 0 ) + r 9 (Z-Z 0 )
,r 2 (X-X 0 ) + r 6 (y-y 0 ) + /- 8 (Z-Z 0 )
r 3(^ - ^o) + r 6 - ^o) + r 9 (Z -Zq)
the quantities (X-X 0 ), (Y-Y 0 ), (Z-Z 0 ) are
proportional to the direction cosines of the projective line
going from the projection centre to the object point.
This present research has been financed by Cofin97
cos(A7?) = (X Xq) = sin(0 0 + r)
COS07?) = (y ~ Yo) = cos(fl 0 + r) (2)
cos (ZR) = ———- = ctan(0 o + r)
The directions (horizontal, r and zenital co) can be measured
with a theodolite whose centre coincides with the projection
centre O (figure 1). The control direction can be defined as
the direction toward an unknown point P(X,Y,Z) but visible
in the image as P’(*,y).
Fig.l- The measured quantities by theodolite
The Orientation parameters are for exterior Orientation: a*,
oty, a*, rotations about the co-ordinate axes and for interior
Orientation the principal distance c, and the image co
ordinates x m , y m of the principal point M. In this occasion a*
= 0 O . is the so-called constant of station or zero bearing. He
can take full advantage of the terrestrial photogrammetry.
The operator has the advantage to have more freedom in the
selection of control directions than in the control points.
We don’t know exactly were the projection centre is placed.
The larger the object-camera distance the smaller the
eccentricity errors; 1/100 mm over 100mm corresponds to
10cm over 1000 m (figures 2, 3).