y psx 
Figure 14 Pixel and image coordinate systems. 
Additional parameters modelling the interior orientation, 
a scale factor in x, a shear, radial symmetric lens distor- 
tion, and decentering distortion was formulated as: 
Ax = Ax,~Z Ac—X s +5 a+Xr’K, + 3) 
Xr K, - XifK, --. (1? 2%) P, + 2XyP, 
Ay = Ay,-2 ActKa yr K, 4 (4) 
yr K, t yrfK, - 2xyP, 4 (1^ 4 2y?) P; 
with: 
X = X—-X,, y = ya 15 Jg (5) 
Ax, Ayp, Ac .change of interior orientation elements 
Sgusvessoensensosnre scale factor in x 
BD oocesnmvennennitives shear 
K;, Ky, Kj ....first three parameters of radial symmetric 
distortion 
Pp Pom first two parameters for decentering distor- 
tion 
The location of the principal point is not specified for 
most CCD-cameras and varies from camera to camera 
and depends on the configuration of the frame grabber. 
The scale factor in x is required to model the imprecise 
specification of the sensor element spacing and addition- 
al imprecisions introduced with PLL line-synchroniza- 
tion. In latter case the pixel spacing in x must be 
computed from the sensor element spacing, the sensor 
clock frequency and the sampling frequency with: 
fem 
psx = ss" (6) 
sampling 
with: 
DSX ..) eos sanos pixel spacing in x 
C MAMANS sensor element spacing in x 
f eor * sensor clock frequency 
sampling 77 sampling frequency of frame grabber 
   
     
   
    
    
    
       
    
    
  
   
The shear (a) must be included to compensate for the 
geometric deformation induced by PLL line-synchroni- 
zation as discussed above. 
3.3 Accuracy with Pixelsynchronous Frame 
Grabbing 
The results of several versions of bundle adjustments and 
accuracy verifications are given in Table 2. The large in- 
fluence of the modelling of systematic errors is demon- 
strated with versions 1 and 2. Version 1 is computed with 
the initial values for the pixel-to-image coordinate trans- 
formation and the interior orientation and without addi- 
tional parameters. The initial values of the pixel-to- 
image coordinate transformation (equations (1) and (2)) 
were derived from the camera and frame grabber specifi- 
cations and settings. The camera constant was computed 
from the focal length and the focussing distance by the 
lens formula. Version 2 uses updated values for the trans- 
formation and the additional parameters of equations (3) 
and (4). The large improvement of the accuracy by fac- 
tors reaching 70 is primarily due to the modelling of dis- 
tortion. Table 1 gives the initial and adjusted values for 
the pixel spacing and the interior orientation elements. It 
can be seen that the changes to the pixel spacing and the 
camera constant are in this case rather small. The change 
in the location of the principal point corresponds to 3 
pixel in x and 10 pixel in y. The precision of the location 
of the principal point and the camera constant are each 
determined with a precision better than one micrometer. 
The large distortion of the lens, exceeding 10 pixels at a 
radius of 3.6 mm, is apparent as a bending of the testfield 
rods in Figure 13. 
  
  
  
  
  
  
  
  
  
  
  
Parameter mua Juss mr] 
value value deviation 
Fixel spacing 1n x 0.011| 0.0109995| 0.0000036 
[mm/pixel] : 
Principal point 1n x 0.0 0.0323 0.00031 
[mm] 
Frincipal point in y 0.0 0.1191 0.00025 
[mm] | : 
Camera constant 9.0 8.9821 0.00018 
[mm] 
Table 1 Initial and adjusted values of pixel spacing 
and interior orientation elements with their 
standard deviations (from version 2). 
The results of version 2 indicate a large discrepancy be- 
tween the theoretical precision estimates (Ox, Oy, Oz) 
and the empirical accuracy measures (Ux, Ly, Uz). Large 
degradations were found to be due to local intensity vari- 
ations. It was shown in an investigation that the effect of 
these gradients, due to shadows and differing reflectivity 
of surrounding areas of the testfield, could be decreased 
by using a smaller template (5 x 5 instead of 7 x 7). The 
results of these measurements are given in version 3. The 
accuracy in image space was thus improved by a factor 
of 1.3. The smaller template nevertheless only decreases 
but does not eliminate the effect of the local illumination 
gradients. It was thus decided to remove the targets with 
strong illumination gradients. This led to another im- 
provement by factors up to 1.4 as indicated by the com- 
parison of versions 3 and 4.