ISPRS Commission II, Vol.34, Part 3A „Photogrammetric Computer Vision‘, Graz, 2002 
  
coordinate systems to each other. It maps the coordinates from 
the object space into the image space (Mikhail, Bethel and 
McGlone, 2001). There are two types of parameters. First, the 
exterior parameters which include the location and orientation 
parameters. Lens distortion and focal length are examples of 
the second type, which are called the interior parameters. The 
model specifically covers and takes into account the lens 
distortion through some parameters that model radial, 
decentering, and affinity distortion. 
X-x mry'ey + Ax 
=—f FA AA "1, tn, 12 72.) 
LXE YL {Y =F +7. (2-7) 
(1) 
  
y=y.=2y=y +Ay 
ze EX-XerBOY- Y» (Z-—27) 
LX ~X +r (TF =Y Yr. (Z=2) 
  
Where: x, y and x', y: ideal and measured target coordinates 
in image space 
x,, y, : Principal point coordinates in image space 
? x, ? y: distortion corrections in x, y directions 
f: camera focal length 
r ij: the ith row and jth element of the orientation 
matrix R 
X,Y,Z: target coordinates in object space 
Xc,Yc,Zc: exposure station coordinate in object space 
The rotation matrix R expresses the orientation of the image 
coordinate system with respect to the object coordinate system. 
The distortion effects including radial, lens decentering, and 
affinity were computed through the equations below. 
Ax =X(kr® + kt - ky) p(r? +232) +2p,X y 
(2) 
Ay zv Tk +kr°)+2p,x y+ pr +2ÿ")+a,x+a,ÿ 
-— /, — / 2 —2 —2 
where: ur 1 dt CUS vy 
k, , P;» À; radial , decentering, affinity distortion 
coefficients 
The two condition equations for each target will be: 
F,=X+X(kr’ +k,r* + kr’) + p,(r’ + 2X) + 2p, Xp 
+f X ANZ ZZ.) x 
LX Xn -Y)tuuz-2Z) 
  
(3) 
F,-yty(kr * kr! - kr) -2pxy* pr 423^?) + 
LUX =X y+, (Y~Y +n ll~Z) =0 
axo yd = 
2 YET 
  
From the equations above, each target observation will 
generate two equations. Consequently, the number of equations 
will be twice the number of targets in the image for each 
camera. 
4.2 Solution method 
The unified least squares approach was used to solve this 
system since some a priori knowledge is available for a 
number of parameters (Mikhail and Ackerman, 1976). Using 
the a priori knowledge of the parameters is the distinction 
between ordinary least squares and unified least squares. This 
knowledge is utilized to give those parameters initial values 
and weights. In this sense, some of the parameters were treated 
as observations with low precision by assigning large variances 
to them. Since the system is non-linear, the parameter values 
will be updated iteratively by adding the correction to them. 
The system will converge when the correction vector values are 
negligible. Then the final correction will be added to the 
parameters to get the final estimated values. In our case here, 
the system converges with few iterations since the precision of 
the observations was very high. 
5. DISTORTION ANALYSIS 
5.1 Radial Distortion 
The term used for the displacement of an imaged object 
radially either towards or away from the principle point is 
radial distortion (Atkinson, 1996). The magnitude of this 
displacement is usually determined to micrometer precision 
and it varies with the lens focusing. Radial Distortion is 
included in the math model and its magnitude can be 
calculated as follows: 
Ar=kr°+kr+kr" 
(4) 
0, =Ar*x/r 0, =Arxylr 
The radial distortion curve was constructed based on the 
equation above as shown in figure 5. The resulting curves were 
obtained for all four cameras and the maximum radial 
distortion was around 30 micrometers. 
The following step was done to level or balance the curve 
based on equalizing the maximum and the minimum distortion 
values. This procedure is done only to balance the positive and 
negative excursions of the distortion function about zero. This 
step has no effect on the final results of the corrected 
coordinates; it is just cosmetic but accepted professional 
practice. Mathematically, balancing the curve leads to a change 
in the radial distortion parameters and consequently the focal 
length and other related camera parameters. The aim of this 
balancing procedure is to make Ka io as shown in 
figure 6 and the condition equation will be: 
tox — CELXtan(e )+r. —CFLxtan(e_ )=0 (5) 
So the new focal length is: 
ro +r. 
max min 
) + tan(e_. ) 
7 tan(o 
max 
(6) 
After getting the revised focal length, the calibration 
adjustment program is run again but with a fixed focal length 
(CF 
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