with the j? singular 
(2.9) 
meter x; associated 
(2.10) 
all parameters with 
is called the variance- 
ters 
tion of an exact linear 
indicative of a near 
be as many near 
lues. 
9) A;'s appear in the 
> components of the 
Il singular values will 
its; which will lead to 
' can be said to be 
gh proportion of their 
wall singular value (or 
position will enable 
jendencies (multiple 
he number of high 
in these multiple 
> proportion of their 
ndition indiex. 
1ld be considered as a 
should be considered 
to base this decision. 
tion of the variance, 
on to be large when it 
riance of a parameter. 
x is however more 
lered a high condition 
rticular application , 
ng for other type of 
idition indices will be 
hed during the testing 
3. CASE STUDY: IN-FLIGHT CAMERA CALIBRATION 
In-flight camera calibration is a typical and an oldfashioned 
problem where the model is extended to account for 
perturbations. 
The model used is the well-known collinearity condition 
equations extended to include variations in the interior 
orientation parameters expressed as: 
F(x) - (x-x,) t (x-x, (K,r &K,r +Kor +...) + 
{BE +2x )+ iy Ht Pres _eX A =0 
(2.11) 
F() - (y-y,) * (y- y,K,r *K,r + Kr +.) + 
pm +P(r + 2y) {1+ Pr ht eyed 
where: 
X'- ajg(X - X,) t aj (Y - Y) * a43(2 7 Z4) 
Y'2 aj (X - X,) £ az3(Y - Y) * a23(Z - Z4) (2.112) 
Z'-agí(X- X,) t ag(Y - Yo) * a33(Z - Z4) 
Fe —2 —2 — — 
r=yx ty iX-2X-Xyy 7 Y-Yp (2.11b) 
with: 
A11>A12;-<-A33 : elements of the rotation matrix of the gimbal 
angles defining the orientation between the survey and photo 
coordinate systems. 
X, Y,Z : Object point coordinates in the survey system. 
X, Y,,Z, : Exposure station coordinates in the survey system. 
X,y : Observed image coordinates in the fiducial system. 
Xp>Yp Principal point coordinates in the fiducial system. 
c : Camera constant. 
K,,Ky,K3: Polynomial coefficients of symmetric radial 
distortion. 
P;,P,,P; : Polynomial coefficients of decentring distortion. 
In this model , parameters of interior orientation are to be 
recovered in a simultaneous least squares adjustment along 
with exterior orientation parameters and survey coordinates. 
the problem of highly correlated parameters has been 
demonstrated by many authors (Pogorelov and Popova, 1975: 
Mrchant, 1974; Salmanpera, 1974; Kupfer, 1985; Brown 
1969). Most often, the identification of correlated parameters 
has been based on the correlation matrix or the analysis of 
partial derivatives of the function with respect to each of the 
parameters. 
In this paper the procedure of variance decomposition is applied 
to data pertaining to in-flight camera calibration. 
3.1 Description of the Data 
To allow for extensive testing of the method a synthetic data 
was generated mathematically. Since in camera calibrartion the 
resulting matrix is large, to keep the computations within 
reasonable limits, only four (4) photos were considered. Object 
space coordinates were generated for 25' ground control points. 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
Exposure station coordinates and attitudes were assumed, 
allowing for the computation of the image coordinates with a 
focal length of 152.25 mm (Ettarid, 1992). Several factors such 
as elevation differences on ground control, use of convergent 
photos and a priori information on parameters were considered 
in the testing. 
To confirm the validity of the procedure and the conclusion 
drawn from simulations, the testing of the method was 
conducted also with real data. The configuration consisted of 4 
photos, two of which were vertical and the two others were 
convergent. The convergent photos were taken by modification 
of the flight scheme, using "the standard coordinted turn at 45 
degrees" as described in Tudhope (1988). 
3.2 Discussion of the Results 
The variance decomposition method was applied to the 
resulting design matrix. Different cases were considered such 
as the use of convergent photos, the influence of elevation 
differences on ground control and a priori information on 
parameters. As the resulting variance decomposition 
proportions matrices are very large, only extracts of the 
condition indices and parameters associated with are presented 
here: the reader interested in the complete set of results may 
refer to Ettarid (1992). Correlated parameters are identified as 
those having more than 50% of their variances associated with 
the same high condition index. 
* The variance decomposition applied to the design matrix 
resulting from calibration over a flat terrain showed that the 
high condition index is 58x10!? (Table 3.1), induced by a 
multiple correlation involving the parameters Pandw rotations, 
with 100% of the variance of P, associated with this index. 
Parameters K;,K;andK; are involved in a second corrlation 
associated with a condition index of magnitude 11x10” . The 
other correlated parameters identified are respectively the 
camera constant c and Z,'s, x, and Xo's, and y, and Y. vs. 
The involvement of P,in a stronger correlation with Omega 
rotations has masked its involvement with y, in a weaker 
correlation as we will see later. Similarly the involvement of 
xp with X's has masked the correlation between x, and P; . 
Table 3.1. Correlated Parameters in the Case of Calibration 
Over Flat Terrain 
  
Condition Correlated Prameters 
Indices (Variance-Decomposition Proportions) 
  
010 | Py (1.00) 91 C74) 02 C78) 03 (72) 04 (60) 
  
  
  
  
5.8x1 
RT) K,(87) K3(.96) K3(99) 
8.8x10° C (91) Z,(91) Z2(91) Z3(91) Z4(9N) 
2.8x10° xp(.96) Xo (96) Xo, (95) Xo, (97) Xo, (94) 
agudo | NC Yo C7) Y. C12 Yo 070) Y, (6 
  
  
  
  
The introduction of orthogonal Kappa rotations on successive 
exposures has no effect on correlated parameters in the case of 
calibration over flat terrain. 
183 
  
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