£ 
" 
2 
ax ^ P4 (7^2 x, ^) 2p, x) yy.) d 
day ^ 2PB4(X7x9) (y7-y9)* p, (r^«2(y-y )?) 
In equations (5) and (4) there are 
X) Yo coordinates of the 
principal point, 
X. y measured comparator co- 
ordinates of a point, 
ky, k^; ks, P1» D5 distortion parameters, 
2. 2 an 
wo 'e(yey ^ 
Thus we obtain the resulting corrections for 
the measured image coordinates out of 
AX = (x-x3)d,, + dx 
(5) 
Ay 9 (y7y d, * dy 
The extended collinearity equations (1) 
become 
a,x + ayy + a2 + a) 
  
  
Xt AX = 
agX + aft act 1 (6) 
a-X +a-Y + a7 t à 
yt oy = 2 6 1 S 
agX + a4oY * a z+ 1 
and they can be used together with the 
conditions (2) for the determination of the 
11 transformation and the 5 distortion 
parameters out of given object space co- 
ordinates and measured image coordinates of 
control points. 
4, DETERMINATION OF THE PARAMETERS AND 
THE OBJECT SPACE COORDINATES 
The determination of the unknown parameters 
out of (6) and (2) leads to a non-linear 
least-squares adjustment if enough object 
space control is available. We presume small 
corrections Ax and Ay due to distortion and 
consider the measured image coordinates 
x and y as well as the coordinates x. and y 
of the principal point in (3) and (49 as © 
constants. 
Thus we obtain linear fractional observation 
equations out of (6) with the non-linear 
additional constraints (2). For n control 
points these 2n+2 equations contain besides 
the 11 transformation parameters up to 5 
distortion parameters. We take three cases 
for the determination of ax and ay into 
consideration: 
  
case |incorporated distortion parameters 
T k 
  
1 
IT ki» Kos K4 
III ky» ky, Kgs Pgs Py 
  
  
  
  
The initial values of the unknowns for the 
iterative least-squares adjustment are com- 
puted in two steps. In the first step we 
neglect Ax and Ay in (6) and solve the 
resulting system of 11 linear normal 
equations for the transformation parameters 
a. including all control points. With 
- 2 2 2 
X7 B (ag + 840 * 44 ) 
: 2 à 2 (7) 
Yo: C7 (ag + alg + 8414 ) 
and equations (5) we determine in the 
Second step the initial values of the 
distortion parameters again by solving a 
linear system of 1, 5 or 5 normal equations. 
Now we have the initial values for the 
12, 14 or 16 parameters in the non-linear 
least-squares adjustment problem which is 
solved iteratively by updating x, and y 
in each iteration step and by regarding 
them as constants as mentioned above. 
After this determination of the transfor- 
mation and distortion parameters the measured 
image coordinates x and y of all points are 
corrected by Ax and Ay out of (5). 
With these corrected image coordinates the 
object space coordinates of all points, 
imaged in at least two photos, are computed 
in à second non-linear least-squares adjust- 
ment derived from (1) as shown in /2/. 
5. PRACTICAL TEST 
During the photogrammetric control survey /1/ 
of a plastic membrane covering an open-air 
café we took several slides on a cheap 
24 mm x 36 mm film (19 DIN) from Quelle 
(DM 6.50) with a KODAK RETINETTE 1B 
(1:2,8/45 mm) for documentation purposes. 
In order to get a first idea of the 
accuracy which is obtainable with the 
described on-the-job calibration method and 
common slides taken with an inexpensive non- 
metric camera we restituted four of these 
slides, containing enough object space 
control, on a ZEISS PSK. 
The position of the ! camera stations and the 
control points is given in Figure 1. 
14 
1300 oO o15 
12 
o16 017 
control points 
o 10 
19 
26 © 
240 © 018 
021 o 22 
o 
23 27 
025 
y 
A A a’ A! 
camera stations 
E ———————À 
30m 
Figure 1: Situation of camera stations and 
control points