lan Dowman 
  
hand, but the cost is additional ground data - usually more that 25 points. Using a rigorous model constrains the 
polynomial accuracy to that offered by the model used. 
3. The order and form of the polynomials is automatically computed for each stereo pair under study. This is 
achieved by a trial and error method based on check point RMSE which completes in a few seconds. This method 
has some similarities with the Group Method of Data Handing (GMDH) used in the so called "polynomial neural 
networks". 
Papapanagiotou,'s test results indicate that similar, if not better, accuracy can be obtained using the above method. The 
model has been tested with two stereo pairs one SPOT and one MOMS-2P. For the SPOT stereo pair the same RMSE 
for the check points was obtained as with Kratky's strict model in X and Y. The height (Z) RMSE was by 1m less. For 
the MOMS stereo pair the RMSE obtained was better than Kratky's model in almost all cases (linear and quadratic). 
The method was also tested on two small parts of a second stereo pair. The total RMSE obtained for the check points 
was 4.8m and 5.5m, respectively. Another indication of the great flexibility of the proposed model (and polynomial 
models in general) is that the results obtained for a pair of aerial photographs scanned by an ordinary A4 DTP scanner 
without any prior calibration was similar to the maximum accuracy provided by a map sheet of scale 1:5,000 used to 
extract ground data. 
Papapanagiotou does not argue that polynomials can replace rigorous models and agrees about problems of instability 
and extrapolation errors, but he believes that they don't lack in accuracy. Polynomials do not have to be complex and 
are simple and easily understandable by non-experts. Another conclusion was that if tests are made with controlled 
erroneous data, the accuracy deteriorates rather quickly but in all cases the erroneous points can be identified and 
corrected or ignored. 
As indicated above BAE (Whiteside, 1999) firmly believes that the Universal Sensor Model (USM) is as accurate as 
anyone might want, when fitted to any rigorous image geometry model using well designed software, such as existing 
BAE software. The USM has been tested with a wide variety of images and image types, and the maximum fitting 
error has never been greater than about 0.25 pixel spacing. Typical results are shown in Table 1 (Craig, unpublished 
data, 1997). 
  
  
  
  
  
  
Sensor Image size Ground sample Polynomial 90% fit errors 
: distance order 4 
(pixels) (pixels) 
(x-y-z) 
SPOT 1A Pan 6000 x 6000 10m 2-2-1 0.05 
Landsat TM 7000 x 6000 30m 2-2-1 0.01 
Commercial 18000 x 18000 0.3m 4-4-2 0.2 
frame camera 
RADARSAT fine 9000 x 8000 6m 3-3-2 0.06 
beam 
  
  
  
  
  
  
  
Table 1. Results of testing the USM at BAE SYSTEMS 
Neither image segmentation nor correction tables are needed. The 90% fit error is defined as the 90% level of fit error 
samples, the image pixel difference between the rigorous model and polynomial evaluated at common ground points. 
For each sensor, the resultant 90% fit error as well as the polynomial order was a function of the 90% fit tolerance. It 
was set outomatically to the smaller 0.25 pixels and (1m/GSD) pixels. Other factors influenced the polynomial order 
as well. In particular, the commercial frame camera's wide field of view, relative to the space sensor's was a major 
contributor to the high polynomial order. In addition, the USM format allows for (but does not require populating) all 
cross terms (see equation [3]) where other RFs do not. That is, even though the SPOT fit used only order 2 in x, order 
2 in y, and order 1 in z, there are terms that have a combined order of up to 2+2+1=5. 
XY ia [3] 
i=0 j=0 k=0 
t3 
2 1 
= Mags y! y? and 
i-0 jz0 k=0 
These tests show that the accuracy possible with RFs is quite suitable for most applications. They do not show any 
disadvantages although information is not available on how exhaustive the tests were. In other words, the tests do not 
demonstrate that the disadvantages claimed are not well founded. 
  
260 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.