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x 10 Residuals in planimetry 
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Figure 5. Residuals in planimetry (top) and height (bottom) for 
all 41 GCPs. 
4.2 Procedure 2: Rational functions model 
The alternative approach is based on the RPC (Rational 
Polynomial Coefficients) model. The idea is to describe the 
camera model contained in the metadata file with suitable 
rational functions and apply a bock adjustment to correct for 
remaining systematic errors (Zhang et al., 2004). The procedure 
consists of two main steps: 
RPC model estimation. After generating a 3D grid of 
points using the given camera model parameters, the 
ephemeris and the attitude data attached in the metadata 
file, the RPC coefficients are determined by a least-squares 
approach and without GCPs. For details see (Tao et al., 
2001). The Equations used for this are rational functions: 
NUM (9, À. h) 
DEN (9, A, h) 
NUM , (Q9, À.h) Ch 
DEN ,( 9. À.h) 
xz RPC.(9,A,h) — 
peRPC. Ah) 
Here (@,\,h) are normalised object-space geographic 
coordinates (latitude, longitude and height) and (x, y) are 
normalised image coordinates, in line and column 
direction. NUMx, NUMy, DENx and DEMy are 3 order 
polynomials on (@,Ah), resulting in 67 unknown 
parameters for each image. The 3D grid of object points is 
generated from the image-space coordinates, for a set of 
elevation levels. The RPCs were computed for the whole 
ba 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B1. Istanbul 2004 
test HRS scenes with an internal fitting accuracy of 0.07 
pixels (RMSE) and 0.23 pixels maximum difference. 
Block adjustment with computed RPC model. After the 
RPC generation in step 1, a block adjustment was 
performed in order to estimate 6 parameters for each image 
(affine transformation) to remove remaining systematic 
errors. As mathematical model of the adjustment, we used 
the method proposed by (Grodecki et al., 2003). The 
method is an affine transformation: 
X*agtayxta»y - RPC,(Q, Ah) 
ytbo*byixebyy z RPC,(Q. A h) 
where ao, a;, a; and b, b;, b, are the adjustment 
parameters for an image, (x, y) and (À, €, /) are image and 
object coordinates of a GCP or a tie point. 
Using this adjustment model, we expect that a; and 5; will 
absorb any shifts and misalignments in the position and 
attitude, and the residual interior orientation errors in 
image line and sample directions. The parameters a,, a>, 
b,, b; are used to absorb the effects of on-board drift 
errors. 
The adjustment results are shown in Table 4 and Figure 6. 
  
  
  
  
  
  
  
  
  
  
Table 4. RMSE for all points with the RPC orientation method. 
Number of RMSE in RMSE in RMSE in 
GCPs + CPs East (m) North (m) | Height (m) 
4+37 5.28 3.87 2.64 
8 + 33 5.63 3.96 2.38 
41+0 4.63 3.66 2.21 
x10 : Residuals in planimetry 
5.34 
10m 
532 
i 
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45 451 452 453 454 455 455 
East vin 
x10 Residuals in height 
su} 10m 
533 
& 532: 
3 
£31 
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$529: Sue ed ES A i E 
45 4.51 4.52 4.53 454 455 456 
Eest x 10 
Figure 6. Results in planimetry (top) and height (bottom) from 
block adjustment with the RPC model and 41 GCPs.