124 
    
  
  
   
ISSN 
SLR 
rm ORR SS 
e 0 ANN 
A 
    
      
NN 
        
Figure 2: Pseudo-3D plot of the irradiance distribution B(z,y) at the diaphragm of the receiver lens. The resulting 
point spread function shows two maxima. The larger one comes from the direct imaging of the filament, the smaller 
one from the image reflected by a spherical mirror. 
B(Z) on the filament of the halogen bulb. Thus, the normalized PSF can be written as 
PSF,(Z) = kB (757) (2) 
with the normalization factor k^! = [dZB(Z). At the focal plane (z — 0) V, is zero, resulting in a delta 
function for the PSF. The resulting brightness distribution on the CCD sensor plane is then 
  
T 
12) - 9) 1 -7( 2) «Psr.) 3 
@ =v(@ [1-7 (7 (z) (3) 
with v(Z) being the vignetting function. 
4 DETERMINATION OF THE CONCENTRATION BY 
DEPTH-FROM-FOCUS 
4.1 Low Level Preprocessing 
For the depth-from-focus technique it is critical that the gray values in the images are known absolutely and 
are independent of the actual brightness of the illumination and the actual settings of the camera and frame 
grabber. 
A linear illumination model is applied assuming that the image shows a background level b(Z) and that the 
measured gray values g(Z) are further proportional to the irradiance I(£). Then 
9(&) = a(2)I(Z) + b(Z). (4) 
D 
The unknown quantities b(Z) and a(Z) are obtained by taking a background image gs(Z) with illumination 
switched off (I(Z) = 0) and a zero image g,(Z) in which no bubbles are present (I(Z) = Io(Z)). Then the 
linear inhomogeneous point operation 
gz (Z) = Jo(F) 
results in a normalized gray value n(Z) in the range of 0 to 1. It is important to note that this procedure 
removes any type of inhomogeneities caused by uneven illumination or small dust particles on the CCD chip 
or the lenses. Applying Eq. 5 to Eq. 3, one yields 
nur d + PSF.(2). (6) 
for the normalized image. 
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop “From Pixels to Sequences”, Zurich, March 22-24 1995