The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
It should be mentioned that the center of the ellipse does not 
exactly coincide with the projected center of the circular 
marker. This small eccentricity is a function of the viewing 
angle, marker size and focal length (Dold, 1997). The 
evaluation of this formula (see Fig. 5) showed that maximum 
eccentricity values of 1.25 pm can be expected for the 2D target 
(80mm lens) and even lower values (below 0.25pm) for the 3D 
test field. The eccentricity error was therefore neglected for the 
purposes of our study. 
Eccentricity error 
Figure 5. Eccentricity error of the ellipse operator 
Another error source concerning the ellipse operator was also 
investigated. This error occurs when large radial distortion is 
present in the images and the markers are imaged at a large 
image scale. In this case distortion may change even within a 
single imaged marker, which causes a non-linear deformation of 
the ellipse, resulting in a positioning error. As can be seen from 
Fig. 6, this error is estimated to be well below 0.5pm for all 
lenses and can therefore also be neglected in this study. 
Distorted ellipse error 
Figure 6. Influence of large distortion on the ellipse operator 
The following steps are necessary in order to fully automate the 
process of marker measurement for a planar target (cp. 
Langauer, 2008): 
1. Localization of (coded) markers in the image 
2. Determination of a 2D projective transformation 
between image and object coordinates 
3. Estimation of radial distortion (optional) 
4. Calculation of approximate positions for each marker 
5. Precise measurement using the ellipse operator 
6. Determination of approximate EO parameters using a 
robust resection algorithm (Killian, 1955: 97-104) 
This workflow was successfully implemented in a MATLAB 
tool (see Fig. 7). Image measurements for all images of the 
calibration project are exported to an ASCII file for further 
evaluation in the bundle adjustment. 
Figure 7. MATLAB tool for automated 2D measurements 
The following table gives an overview of the H3D calibration 
projects performed using the planar target: 
Date 
Lens 
Focus 
# Images 
# Points 
June 25, 2007 
35mm 
OO 
16 
2450 
50mm 
OO 
20 
3150 
80mm 
oo 
19 
2900 
July 1,2007 
35mm 
OO 
17 
2800 
50mm 
00 
18 
3000 
80mm 
00 
18 
2800 
March 19, 2008 
35mm 
oo 
31 
5200 
50mm 
oo 
23 
3660 
80mm 
oo 
22 
2950 
35mm 
1.5m 
28 
4660 
50mm 
2m 
24 
3800 
80mm 
2m 
19 
2100 
Table 1. Statistics of the 2D marker measurements 
Automation of marker measurements is much more complicated 
for a 3D test field. We therefore decided to use a semi- 
automated approach where a limited number of manual 
measurements must be made for each image. Again, the 
necessary steps were implemented in a MATLAB tool (see Fig. 
8): 
1. Manual measurement of at least four markers 
2. Determination of approximate EO parameters using 
the robust Muller/Killian resection algorithm 
(modified after Killian, 1955: 171-179) 
3. Determination of refined EO and IO parameters and 
radial distortion by a single image bundle adjustment 
(using additional manual measurements) 
4. Re-projection of all visible markers into the image 
gives approximate positions for the ellipse operator 
5. Automated precise marker measurement using the 
ellipse operator 
Steps 3 to 5 can be performed iteratively until all markers have 
been successfully measured. It is also possible to carry out steps 
1 to 4 for all images of the calibration project in advance and 
run the (time consuming) automated marker measurement as a