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> imaging 
system in a self-calibrating bundle adjustment in order to 
achieve high accuracy. Extended collinearity equations are 
used as the functional model in self-calibrating bundle 
adjustments. 
m,(X- X) «m,4(Y- Y) m4(Z-Zj) 
Mg, (X= Xo) + Mo (Y= Yo) + Myg(2- 20) 
  
X+AX-X,=~C 
My (X=Xo) + Mop (Y= Yo) + Myy(Z- 2) 
m (X- Xy) *my4(Y- Y5) my(Z-Z;) 
  
y*Ay-yo*-C 
(1) 
where, x, y are the image coordinates of the image points; X, 
Y, and Z denote the object coordinates of the corresponding 
object points, and Xy, y and c represent the basic interior 
orientation parameters, i.e., the location of the principal point 
and the calibrated principal distance; Ax and Ay stand for the 
correction terms applied to the image coordinate 
measurements, which are the functions of the selected APs; 
m, (i,j21, 3) are the elements of the rotation matrix between 
image space and object space based on three independent 
orientation angles, and X,, Y,, Z, are the object coordinates 
of the projection centre. 
Many versions of self-calibrating bundle adjustment 
algorithms exist. One of the main differences is the selection 
of the APs and the composition of Ax and Ay. Due to the 
non-linearity of eq.(1), linearization and iterations are 
necessary for its solution. 
2.2 Digital Camera and Its Calibration 
A digital camera is very much like a non-metric camera in 
many aspects, except for the substitution of the film with solid 
state sensors. Therefore, the images captured by a digital 
camera are also suffering from certain distortions due to 
systematic errors during imaging, which leads to inaccurate 
results. In order to exploit the potential of the 
photogrammetric techniques, camera calibration is necessary 
to determine the quality of the images and the imaging 
system and thus to be able to efficiently compensate for the 
systematic influences, which should lead to improved final 
accuracy. 
On the other hand, the interior geometric structure of a digital 
camera is also instable, which means that its geometric 
characteristics may alter to a certain extent over time. This 
suggests that the calibration has to be carried out 
simultaneously with the data reduction rather than as 
separate laboratory calibration with subsequent data 
evaluation. Consequently, self-calibration seems to be an 
ideal tool to combine the two steps. 
2.3 UNBASC2 Self-Calibration Method 
A self-calibrating bundle block adjustment program 
(UNBASC2) was developed for aerotriangulation and close- 
range photogrammetric applications (Moniwa, 1977), which 
can be successfully applied to a non-metric camera (Faig et 
al,1990). Compared to other methods, UNBASC2 possesses 
two advantages: 
. Itis a photo-variant procedure with no limitation on 
the number of photos, the type of camera used, 
e.g. metric camera, non-metric camera or even 
61 
digital camera. 
. No approximate initial values are needed for the 
adjustment, although linearization and iteration 
have to be dealt with. This is convenient for 
applications, where the approximations are difficult 
to obtain. 
In UNBASC2, the error model comprises of the following 
three independent functions which model systematic 
distortions of the camera system. 
2.3.1 Radial Lens Distortion. Odd power polynomials are 
used to describe radial symmetric lens distortion: 
= 3 5 7 
dra kr? «ker ekor^ s. 
(2) 
which leads to its components in the photo coordinate 
system: 
dr =X WE 72 Kr rk su 
dr,- (y- yo) (Kir? * or^ «kr? «...) (3) 
where dr is the radial distortion and dr,, dr, represent the 
components in x and y directions, respectively. r refers to the 
radial distance between principal point and image point; k; are 
the coefficients of radial distortion functions. 
2.3.2 Decentering Lens Distortion. The decentering lens 
distortion is defined by Conrady-Brown’s model (Brown, 
1966) in UNBASC2: 
dA Ap 2 prt) 
[p, U? +2 (x-X0) + 2p, (XX) (Y-¥o)] 
dp, = (1 «DIDI s.) 
[pur *2(y-yol * 2p, (x-X3) Q/-9)] (4) 
where p; are decentering distortion parameters. 
Usually, k,, k,, k, andp ,p, are sufficient to define the 
radial and decentering lens distortion for most commercial 
non-metric cameras (Faig, 1973). If not, higher order terms 
could be included. 
2.3.3 Film Deformation. Originally, an affine transformation 
is employed to model the film deformation, 
dq,- A(y-X,) 
dq, - B(y-y)) ©) 
where A and B are parameters defining scale change and 
non-perpendicularity of coordinate axes. Due to the fact that 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B1. Vienna 1996