On the contrary, the use of convergent photos (photos 1 and 2) 
resulted in the elimination of the correlation between y, and 
Y,'s and alleviated the correlation between camera constant c 
and Z,'s ofthe convergent photos (Table 3.2); but involved on 
the other hand x rotations in the near dependency involving 
D uy; and o rotations. 
Table 3.2. Correlated Parameters in the Case of Calibration 
over Flat Terrain with Convergent Photos. 
constraints on exterior orientation parameters. Prior 
information here is introduced as appropriate weighting of the 
unknown parameters leading to what is generally called indirect 
observations or quasi-observations. 
Introduction however of constraints on the interior orientation 
parameters ( c, x, and y,) succeded in isolating y, from the 
near dependency involving P, and Omega rotations; it remains 
only two near dependencies as in Table 3.5. 
Table 3.5. Correlated Parameters in the Case of 
  
Condition Correlated Prameters 
Indices (Variance-Decomposition Proportions) 
  
4.9x1010 | P2 (1.00) y, (63) x, (61) , (64) $5 (72) &4 (.60) 
Calibration with Elevation Differences on Ground 
Control, Orthogonal Kappa, Convergent Photos and 
Constraints on Interior Orientation Parameters. 
  
  
12x10 |Bıl87) Ka(97) Ks 99) 
  
3.6x104 C (85) Zo,(.60) Zo,(.64) Zo,(.85) Zo,(.85) 
Condition Correlated Prameters 
Indices (Variance-Decomposition Proportions) 
  
8.0x109 | P2(1.00) 0,(36) 0,(.61) @3(.68) @4(.68) 
  
  
2 5x10% x,(.65) Xo,(90) Xo,(90) ¢,(.65) ¢,(.60) 
  
  
  
&3x106. | Ki(76) K5(94) K4(98) 
  
  
  
  
* Inthe case of camera calibration with elevation differences 
on ground control (Dh/H = 30%), the variance decomposition 
showed that the highest condition index (3.7x10") is still 
induced by the correlation involving P,, y, and Omega 
rotations (Table 3.3). This also has alleviated the correlation 
between camera constant c and Z,'s; but as c is freed it 
becomes part of the correlation involving K,, K, and K,. 
Table 3.3. Correlated Parameters in the Case of 
Calibration with Elevation Differences on Ground 
Control. 
  
Condition Correlated Prameters 
Indices (Variance-Decomposition Proportions) 
  
3751010 | P (1.00) y, (.69) o, (.88) c; (.87) 05 (.90) o4 (.90) 
  
   
1.1x107 |C(SI) K,(86) K3(.95) K3(97) 
  
  
   
30x104 | P (85) e,(71) 9,(52) 9,071) 9,(64) 
  
   
  
  
1.1x104 | C(30) Zo,(87) Zo,(88) Zo,(86) Zo,(84) 
  
  
    
   
   
    
  
  
The introduction of orthogonal Kappa rotations on exposures 
and the use of convergent photos has eliminated or alleviated 
all the correlations except those involving respectively K,, 
K, and K,, and P, and y, (Table 3.4). 
Table 3.4. Correlated Parameters in the Case of 
Calibration with Elevation Differences on Ground 
Control, Orthogonal Kappa and Convergent Photos.. 
  
    
Condition Correlated Prameters 
Indices (Variance-Decomposition Proportions) 
  
   
1.1x1010 | P» (1.00) y, (.71) 6, (62) o, (.79) $5 (.82) o4 (.83) 
  
     
1.0x107 | K1(83) K2(.96) K3(.99) 
  
   
  
83x10* | C(20) Zo,(50) Zo,(36) Zo,(.68) Zo,(.72) 
  
  
  
     
    
     
    
    
* [n the case of calibration with elevation differences on 
ground control, the introduction of prior information on 
exterior orientation parameters did not bring any change to the 
existing pattern of correlations. It seems that elevation 
differnces on control encompasses the information brought by 
184 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
* The application of variance decomposition to the design 
matrix resulting from the calibration based on real data showed 
the same pattern of near dependencies and the same correlated 
parameters as those found with the simulated data. 
4. CONCLUSION 
The testing done with simulated and real data for different 
geometric configurations indicated that the variance 
decomposition proportions method is a valuable analytical tool 
for the identification of parameters involved in functional 
groupings. 
In the case of non linear models, which is usually the case in 
photogrammetry, the design matrix is changing because it has 
to be updated after each iteration. In order to reduce the 
computational burden, the variance decomposition needs to be 
applied only to the initial design matrix if realistic first 
approximations are introduced as initial values. 
In the testing all condition indices were considered; but the 
results of the calibration showed that estimates of parameters 
involved in near dependencies associated with condition indices 
smaller than 10° are not degraded. 
On the other hand, Belsley et al (1980) advocated the scaling of 
the design matrix to a unit column norm before applying the 
variance decomposition. In this paper the variance 
decomposition was applied to an unscaled design matrix 
because the scaling can undo ill-conditioning associated with 
features. such as mixed units, which may mask the real 
correlations between parameters. 
REFERENCES 
Belsley, D.A., Kuh, E., and Welsch, R.E., 1980. Regression 
diagnostics: identifying influential data and sources of 
collinearity. John Wiley and Sons, New York. 
Brown, D.C., 1969. Advanced methods for the calibration of 
metric cameras. DBA Report presented at the symposium on 
computational photogrammetry, Syracuse University. 
Ettarid, M., 1992. Investigation into parameter functional 
groupings associated with dynamic camera calibration. Ph.D. 
     
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