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Figure 6: Area of convergence for Figure 7: Area of convergence for 
the condition (2.1-1) the condition (2.1-3) 
The figures 6 and 7 show the surprisingly small area of convergence for the equation (2.1-3) in contrast to the one 
according to (2.1-1). The circled grid points can be used as approximate values for linearisation. The figure for equation 
(2.1-2) was omitted because the behaviour is very similar to figure 7. Special notice should be taken of the non- 
connected area of convergence in the figure for the condition (2.1-3). 
This method gives very precise figures for the necessary accuracy. 
3 IMPLEMENTATION 
For a first test of the theory the procedure was implemented in the adjustment-software ORIENT used for 
photogrammetric purposes. The algorithm is designed to apply the condition of contour for points on algebraic implicit 
surfaces up to the 9" degree or to surfaces that can be described implicitly with a maximum of four parameters [Kager 
2000]. In certain cases multiple iterations with one equation have to be carried out to achieve convergence with the 
desired condition. A suitable combination or a changing selection of one or more of the above criteria leads to the 
desired convergence. 
4 TESTS 
Testruns with numerous objects like quadrics, tori and rotation-symmetric free-form surfaces lead to the expected 
results: the additional over-determination lead to an increased reliability and accuracy of the desired parameters. In case 
of ellipsoids of revolution a special increase in the accuracy for the parameters of symmetry could be detected. 
Unfortunately, no results from industrial applications can be presented yet because they are just being planned for at the 
present time. 
5 CONCLUSION AND OUTLOOK 
The aim of image orientation and object reconstruction using points on contours Was achieved. The theory was adapted 
to different representations for the surface and translated into algorithms for practical purposes. The first major 
application is the measurement of a pipeline-system where the advantages of a limited number of control points and the 
automatic detection of points on contours are especially applicable. 
6 REFERENCES 
[Kager, 2000] Kager, H., 2000. Adjustment of Algebraic Surfaces by Least Squared Distances, International Archives 
of Photogrammetry and Remote Sensing, ISPRS XIX, Amsterdam. 
[Kraus, 1994] Kraus, K., 1994. Photogrammetrie, Band 1, Grundlagen und Standardverfahren, Verlag Dümmler, Bonn. 
[Legenstein, 1997] Legenstein, D., 1997. Bildorientierung und Objektrekonstruktion mit Hilfe von Punkten auf 
Umriflinien, Diplomarbeit, Institute of Photogrammetry and Remote Sensing. 
  
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 479