with respect to the principal point. Then the standard 
observation equation (1) yield 
X here fX, 7X tr, «Y Y tf AZ, - 20) ds 
2 X's fis (X, 7 X9) tr (Y, - Y) rn, (Z, - Z,) (1) 
H2 (x, 40) +R y -h)tn (Z, —Zo) 
* dy' 
hs Xx, — Xo) try «Y TH); HZ, - 20) 
  
yz yt: 
Camera modelling with image-variant parameters causes three 
more parameters per image to be estimated within the bundle 
adjustment. Hence the number of unknown grows up to nine per 
image. These parameters describe the variation of the principal 
x'— (X'y-FAx';) - (e Ac;): 
D x X0) 15 "(Y -h*r AZ: -Z,) 
distance and the shift of the principal point, hence the possible 
displacement and rotation of the lens with respect to the image 
sensor are compensated by this approach using extended 
standard observation equation (2). 
The variation of the principal point affects the lens distortion 
with respect to the image plane. Consequently this can no 
longer be modelled as a function of image coordinates rather 
than a function of imaging angle. Additionally, the local shift of 
principal point influences the real effect of distortion for each 
image position. 
  
  
+ dx! 
Ha X AG) 5. {4 +R, «27 (mam) 
' ' ' n» 
y'- Q'o*Ay'; )-(e t Aa): 
(2) 
X, 7 Xo) t 0; (Y, - Y) e; '(Z, 7 Zo) 
  
i -],number of images 
The expected variations of principal distance and principal point 
are estimated in the range of a few hundreds of a millimetre. 
Therefore these parameters are introduced as observed 
unknowns to the bundle adjustment weighted according to the a 
priori accuracy chosen by the user. Using this proceeding the 
bundle adjustment results do not become “weak” and smearing 
effects caused by correlation between other parameters can be 
avoided. 
2.2 Finite elements correction grid 
In order to consider the remaining effects — non-variant effects — 
a finite-element correction grid based on anchor points is 
implemented (Fig. 1). In this case the non-variant effects are 
given by all lens and sensor-based influences that are not 
considered by radial-symmetric distortion parameters. In 
addition this correction grid covers possible sensor unflatness 
and influences which are usually not taken into account by 
conventional calibration models. 
  
Figure 1: Principle of the correction grid applied for a digital 
camera (Kodak DCS) 
Each grid point provides corresponding corrections as plane 
vectors. The correction values for a measured image point are 
interpolated according to a linear equation (3; Fig. 2). 
+1 
  
Figure 2: Interpolation within the correction grid 
TV A 
Jis (Xp = Xo) +r (¥, - Y) 3 (Z, ~ Z,) Groeten) 
Yin Am Yr N n): koi 
HC) Ka] 
tQ -x:y) kt; ju] 
tx yk 
(3) 
x[i+1,7+1] 
Here x, denotes the correction of the measured image 
coordinate (x), the coordinates X1Yı describe the local position of 
the measured image point inside the grid element and the 
elements k[ij], [1j], klij+1], k[i+1,/+1] identify the 
participating grid points. In analogy the similar equation results 
for the image coordinate (y). The collinearity equations are 
extended by the terms described above. 
Separating random measuring errors from real sensor 
deformations and not modelled imaging errors of the lens, 
curvature constraints (4) are added as pseudo observations 
(Kraus 2000). 
0= (Kt ju] i ki) oF (Eu j] WS Ki ja) (4) 
07051 7 Ep Gp Epp 
(similar function for kn 
These equations are applied inside the correction grid in 
horizontal and vertical direction. This leads to a new group of 
observations within the equation system. The equations are 
introduced with an appropriate accuracy (globally estimated a 
priori weight) depending on the estimated unflatness 
(roughness) of the correction grid and the actual number of 
image measurements for one grid element of one set of images. 
Additionally, these constraints avoid possible singularities of 
the adjustment as they might occur for grid elements without 
measured image position. 
3. INVESTIGATIONS AND RESULTS 
The exposed extended mathematical model for camera 
calibration ought to be tested and determined by different data 
sets. The German guideline for acceptance and reverification 
test of optical 3D measuring Systems proposes to have a 
testfield with a range of 2m x 2m x 1.5m (VDI 2001). The 
maximum permissible error of length measurement needs to be 
tested by at least seven different measuring lines distributed in a 
29. 
  
  
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