-—À he M 
ISPRS Commission III, Vol.34, Part 3A ,Photogrammetric Computer Vision“, Graz, 2002 
  
  
  
  
  
  
  
  
  
Projections Gus projection UTM projection 
A= 15 A =2.0° À = 0.0° à. = 3.09 À — 3.5? 
Planar error € 10 ummy, | ¢ <729 mm c <391 mm c < 273 mm c < 547 mm c < 168 mm 
Height error €« 0.1960-Hg | c7 21 mm c» 39 mm c 735 mm c>28 mm c >90 mm 
  
  
  
  
Table 5: Values for the principal distance c for which the additional planar and height errors are less than 10 um in the 
image resp. 0.1%o of the flight height, if the roll (5*°") and pitch (3%°") of the images are neglected and if method M3 is used. 
  
  
  
  
  
X 70,0 m ].5? 1-20 A=3.0° Y 3.59 
X7 0 km X7 112 km X,7 150 km Xo 220 km X, 290 km 
2.|X - X] 140 km 44 km 32 km 22 km 16 km 
  
  
  
  
  
Table 6: Maximum values for the lateral extension of the area of interest, so that the remaining height 
error AHy (equation (6)) when neglecting P3 is below 0.06%o(H; — Hy) — valid for Gauss-Krüger and UTM. 
neglected. If the project area is very large and P3 can no longer 
be neglected, then t must be computed for each single PRC (— 
Tiocal) and applied for the height correction. 
This method M2, however, has the disadvantage that the 
artificial introduction of t must be finally removed in all 
resulting heights in order to get (ellipsoidal) heights 
corresponding to the definition of the NPS. This is inevitable, if 
the heights determined during direct georeferencing need to be 
compared with e.g. terrestrial measured ground truth. However, 
it can be imagined that this kind of work is carried out by the 
AT package itself during an ‘extended’ Earth curvature 
correction. The user always sees ellipsoidal heights, which are 
corrected by Tjop4 Or Tjj; each time before an adjustment is 
computed. The heights after such an adjustment are immediately 
removed by the effect of t and are stored in the respective 
memories. This ‘extended’ Earth curvature correction then 
needs to know which type of map projection needs to be applied 
and how the control points were reduced in advance. 
In method M3 the ellipsoidal heights introduced in the 
adjustment remain unchanged, but the principal distance is 
corrected (actually falsified). Figure 4 shows how this is done. 
Cartesian: NPS: 
4 An 23 
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Hi 
"N IN 
H 
T | | e 
Figure 4: Change of the principal distance 
Using the very simple relations in Figure 4 we get: 
c'=—c (5) 
Similar to M2 depending on the area of interest one can use for 
all images the same altered principal distance C’ global (Via Tglobal) 
or use for each image an individual value cic (via Trocat)- And 
again it can be seen, that this work is done by an ‘extended’ 
correction of the Earth curvature. 
It must be pointed out, that M2 and M3 are just approximate 
solutions? for the given problem, since the change of t within 
the area covered by one image is not taken into concern. 
The degree of approximation using M3 is further decreased if 
the images are rather oblique, since M3 holds rigorously true 
only for exact normal (vertical) images. If the roll and pitch 
angles of the images are neglected additional errors in the 
planar and height coordinates in the determined points on the 
ground are induced. Table 5 holds the values for the principal 
distance c for which these additional errors are less than 10 um 
in the image resp. 0.1%o of the flight height when using method 
M3. The deviations in roll resp. pitch from the exact vertical 
viewing direction were assumed to be 55" resp. 35%, cf. [Kraus 
1996]. 
This height problem is relevant only when the original GPS/INS 
measurements are taken as the images’ XOR for the restitution 
in the NPS. If the GPS/INS data together with TP 
measurements are used to perform a so-called integrated AT 
[Heipke et al. 2001] where certain system parameters of 
GPS/INS might be corrected then this height problem remains 
relevant as long as no height control points on the ground are 
introduced into the AT. In case of given height control the 
operator (unaware of the real reason) would encounter large 
Zmap errors and then would either introduce a vertical shift 
parameter for the GPS observations (of each flight height — in 
case of different scales with the same camera) or allow the 
principal distance to be corrected. In both cases problem P2 
would be solved (comparable to using Tis in M2 Or C giopai in 
M3). Problem P3, however, would still remain uncorrected. 
And its effect on the height of a determined point N would be: 
  
T 
A wr ug Pl (Hp - Hy) (6) 
global T global 
AHN.giopai i$ larger than 0.06760 (Hg — Hy) if (using equation (1)): 
‚06%oR? 
Ix - xo] Some with T global = T(X0) (7) 
0 
|X — Xo]| represents half of the lateral extension of the area of 
interest. Table 6 gives an overview of (X — X9) depending on X, 
and corresponding to equation (6). 
  
? If during M2 ti, is applied for each individual unknown 
object point (iteratively using approximate values) then M2 
would be a rigorous method. 
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