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for the range Ah = 1m and |dZI<0.5m for Ah = 4 km, both cases being computed for the maximum side view angle 27? 
of the imaging sensor." Baltsavias and Stallmann (1992) tested the geometric accuracy of polynomial mapping 
functions (PMFs) with a SPOT stereo pair. 136 points were used with a height range of 350-2100m. When using 
different versions of Kratky's SPOT model the accuracy in X, Y or Z on control points varied from 1.0m to 4.7m and 
from 6.3m to 16.0m on the check points. In comparing the computed co-ordinates of 136 points by a rigorous 
solution and with polynomial mapping functions, the difference did not exceed 1m in object and 1um in image space, 
thus verifying Kratky's results. Baltsavias and Stallmann also show that these functions can be used in image 
matching and orthoimage generation and that a good approximation of the epipolar line can be derived using PMFs 
Madani (1999) reports on the use of rational functions with the orientation of 12 SPOT scenes. The model was set up 
using a rigorous analytical triangulation adjustment with 14 control points and 12 check points. This gave RMS 
values on the check points of 8.5m in planimetry and 6.8m in elevation. To quote from Madani (1999): 
"A total of 50 ground (control/pass) points were used in each stereopair for accuracy analysis. Using the image 
.co-ordinates of each stereo pair, the corresponding ground co-ordinates were computed via the RFs using least 
squares. The RMS radial differences between the originally given and the computed ground co-ordinates of 
identical points was 0.18m and the RMS image parallax was about 0.15pm. The maximum radial difference was 
0.53m and the maximum parallax was about 0.2um. these results show that the RFs expressed these SPOT scenes 
very well. 
"In the second part of the RF accuracy analysis, DTM points were automatically generated for these two stereo 
pairs. Z values of about 50 points per stereo pair were interpolated from the corresponding DTM data. The RMS 
Z-difference, between given/computed and interpolated points, for the first pair was 11.5m and for the second pair 
9.8m. Again these results show that properly selected rational function co-efficients, can be used in the real time 
loop of digital photogrammetric systems." 
Ramon Alamus (2000) from the Cartographic Institute of Catalonia (ICC) has tested the use of rational functions with 
MOMS data. This was in mode A, (2 stereo plus a high resolution channel), acquired during the MOMS-02-D2 
mission on the shuttle in 1993. The scene covers an area of 120km x 40km over the Andes between Chile and Bolivia. 
About 50 ground points were identified from Aerial photography. A strict model of the camera and trajectory were 
adjusted using the GeoTeX-ACX ICC software. 
The mathematical model took into account position and attitude of the spaceborne platform (the Shuttle). On the 
images there were over 1000 tie points (courtesy of Institut fiir Optoelectronic - DLR, Oberpfaffenhofen). In the 
adjustment only 4 ground points of the 50 identified there were used. The rest were used as check points. Results 
achieved using these data are: 8m rms error in planimetry and 17m rmse in height. 
The rigorous model has been interpolated by rational functions using Intergraph software developed for ICC. These 
RFs have been used to stereoplot a MOMS pair (Channels ST6 and ST7, the low resolution channels pixel size of 13.5 
m). A set of 290 points uniformly distributed on the scene were selected and measured as part of the aerial 
triangulation procedure using the Intergraph stereoplotter (ISSD) and then these were remeasured on the ground. The 
co-ordinates of the manually identified points were checked against the adjusted co-ordinates of those points. 
The result are 286 points were gave rms errors of 7.9m in X, 8.1m in Y and 12.8m in Z. These errors are similar in 
planimetry to those achieved in the fit to the rigorous model, and better than the fit in height. 
The results from Madani and Alamus indicate that the rational functions available in the Intergraph software can be 
used without loss of accuracy. Neither report makes any mention of stability of the solution. 
Papapanagiotou (2000), from the University of the Aegean has developed a general model for simulating stereo-image 
geometry using polynomials. The polynomials have some similarities with the USM, for example the denominator 
polynomial is always omitted, the polynomials are of variable order and the ground co-ordinates can be in any defined 
ground co-ordinate system. There are also some aspects of the model that are totally different: 
l. The terms of the polynomials don't have to be the ground co-ordinate values. The image co-ordinates of one 
image are used and the ground elevation. This is similar to Kratky's polynomial mapping functions (Kratky, 
1989). Other combinations are also possible for example, three out of the four image co-ordinates. 
ro 
A rigorous model is not used to obtain control and check point data. The polynomials are constructed purely on 
the control and check data available from other sources e.g. maps. This is to say, no rigorous model has to be at 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 259