4.2 Definition of the search space 
It makes sense to adapt the search space for the fiducial marks 
in the i-th pyramid level to the quality of the localization on 
the previous 7 + 1-th pyramid level. We use the co of the 
transformation estimation between pixel and plate system as 
a quality measure and the result of the consistency check to 
define the search space. On the highest level (1 = imac), if 
there are no approximate values available, the search space 
is set to a multiple of the size of the fiducial template M(?. 
The search space A(? at the i-th pyramid level is defined as: 
(i) M® 4 Vig opt Dj e hus j 
AY = (i) (i+1) ; (1) 
MUI os otherwise 
where f is set to 3, which corresponds to a 99.7% confidence 
region. 
The consistency check (ref. 4.5) may not be solvable, if all 
the found positions are inconsistent. This indicates that at 
least 5096 of the fiducials are not correctly located. In this 
case the search space is opened again to a multiple of the size 
of the fiducial template. Thus a very efficient definition of the 
search space is possible. If the quality of the localization on 
the previous level is high, the search space is small, normally 
only about 30 by 30 pixels. 
4.3 Binarization 
Corresponding to the fiducial marks i.e. in a positive im- 
age where the fiducials are black, the binarization divides the 
image in dark and homogeneous versus non-dark and inho- 
mogeneous areas. The binary image B is derived from: 
i ] : I1<T : (VI € T2 
B(z,y) = { 0 otherwise 
with the absolute value of the squared gradient of the image 
li 
(2) 
: 2 2 
IVI 5 = Bosna] + Mas) — Ianue] 
(3) 
The thresholds for the binarization are adaptively derived 
from the corresponding subsections of the image. Ti is 
found using a histogram analysis and T? can be derived us- 
ing an estimation of the noise variance o, [BRUGELMANN, 
FORSTNER]. From the expectation value E(||VI||?) = 407 
To = 9° (40) — 36 o2 can be found. 
  
Figure 3: Subsection of an Figure 4: Binarization using 
image with fiducial mark only T; — 8 [gr] 
Figures 3 to 6 demonstrate the efficiency of using both crite- 
ria for the binarization. The over- and under-segmentation in 
Fig. 4 and Fig. 5 respectively show how sensitive the bina- 
rization is if only the grey level is used. The thresholds only 
748 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
  
  
      
  
    
    
    
    
   
    
  
    
   
   
    
    
   
   
   
   
   
  
    
    
   
  
   
    
  
  
  
  
    
   
     
   
   
   
   
   
  
Figure 5: Binarization using Figure 6: Binarization using 
only Tı = 12 [gr] T, = 20 [gr] and T2 = 5 [gr”] 
differ in 4 gray levels. Whereas Fig. 6 shows a clear seg- 
mentation when both criteria are used, independent of small 
changes in the thresholds. 
4.4 Binary correlation 
The cross correlation [HARALICK/SHAPIRO 1993], following 
called grey level correlation, is based on the following model: 
Two corresponding images only differ 
1. In geometry by a simple translation T'(u, v), and 
2. In radiometry by a linear transformation in contrast 
and brightness. 
As the binary correlation only deals with binary images, only 
point one is valid here. 
Using a binary correlation to locate a fiducial mark, the tem- 
plate £ is translated by T(u,v). A first estimation for the 
position P(gj(&, 0) of the template in the image b can be 
found by: 
MAL (vv) Pot > (4,9)? (4) 
with 
ove (u,v 
pyu(u,v) , xn (5) 
On(U,V)Ot 
  
  
  
oun) = zz |[#6n) EF] © 
ou) = #5 wj)-Æ (7) 
« - sp-EZ (8) 
where m. is the number of pixels in the template £, 3b and 
#t the sum of all black pixels in the corresponding area of b 
and the template £ respectively. #(b Nt) is the size of the 
intersection of all the black pixels in b and ¢. 
The estimated position P(j(à, 9) is integer valued, with a 
rounding error of 1/3 of a pixel. A sub-pixel estimation can be 
achieved by approximating the surface of the two dimensional 
correlation function ps; by a second order polynomial in a 
neighborhood of P(g(à,0$). The final sub-pixel position is 
defined as the local maximum of the second order polynomial 
which leads to 
A T n -1 
(à, à) = (BD = [Hp ka] Vp l(à,6)(0) (9) 
with the Hesse matrix H and the gradient V von p(u, v) 
Puu Puv 
22 nays Us ^ ) [4,00 (10) 
    
   
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