125 
4.2 Determination of Size and Distance from Focal Plane 
  
With increasing blurring, the edges of the bubble become less steep, the radius (defined at the gray values 
| that are 1/e of the maximum gray value) increases, and finally also the maximum gray value decreases. Figure 
3 illustrates these effects by showing the 1/e area and its mean gray value as a function of the distance z 
from the focal plane. In order to describe the blurring, two parameters of the bubbles are measured: 
l/e radius r, : For small blurring, this is a good approximation for the true radius of the bubble. It 
increases with increasing blurring. In case of elliptically shaped bubbles, an equivalent radius was 
computed from the area. 
Mean gray value of the 1/e area (ge) : This parameter is a good integral measure of the blurring. It 
decreases with increasing blurring. 
  
    
  
  
  
  
  
  
  
  
  
  
  
  
  
ing 
ller 
gray value gray value gray value 
| e S. 
o © 
| | position AN : position S position : > 
e c > 
[d d LATE eet © 9 
(2) a ||1/e diameter b C - z [em] “z [em] 
  
Figure 3: left: Radial cross section through the gray value distribution of bubbles at different distances from the 
Ita focal plane. The distance increases from a) to c). right: 1/e area (a) and its mean gray value (b) as a function of the 
distance z from the focal plane for a bubble with 0.25 mm radius. 
(3) The basic idea of our approach is to use these two parameters re and ge to infer the two quantities of interest: 
the true radius, r, and and distance from the focal plane, z. For a given point spread function, there is a 
unique and monotonic relation for r and z as a function of r, and ge. These relations can be computed 
from the known point spread function and the magnification factors V;(z) and V,(z). Both were calculated 
from the setup of the optical system using geometric optics. Using Eq. 6, the normalized images n(Z) of 
bubbles of different sizes and distances from the focal plane were computed. These images were segmented 
to obtain both re and ge. In a second step, the values for r(re, ge) and z(re, Je) were mapped on a regular 
grid in (re, ge) coordinates and stored in look-up tables (Fig. 4 left). There is an intrinsic ambivalence in 
this measurement: it is impossible to distinguish if a bubble is located in front or behind the focal plane. 
This results in an ambivalence in the bubble size measurement, for the distance information is needed to 
nd reconstruct the true size of blurred bubbles. The effects are small and not significant if only size distributions 
me are computed. Since the probability for a bubble to occur in front and behind the focal plane is the same, 
we can simply take the mean value of the relations for positive and negative z. To overcome this problem, 
the we now use a second generation device with a telecentric path of rays. The LUTS for positive and negative 
distances from the focal plane become then identical if further the 1/2 area is used as blurred size instead of 
the 1/e area. 
    
  
    
    
(4) 
ion 2 
he m 4 
2 ns Imin = 0.4 
E 3 
true 2 
i E 
(5) radius z [mm]. 2 2.5 Imin = 0-7 
[mm]..: A > 2 
2 5 
ire 39^ ixel 1.8 | ns 3 i Imin = 0.9 
hip i re[pixel] e. relpixel] & 
l/e gray value Sm 1/e gray valu E à 
0.0 s 200 400 600 800 1000 
a) b) bubble radius [um] 
(6) 
Figure 4: left: True bubble radius, r, (a) and distance from the focal plane, z, (b) as a function of the mean 1/e 
gray value ge and the 1/e radius r.. right: Measuring volume as a function of bubble radius for three different values 
of the lower gray value limit gmin 
> IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop “From Pixels to Sequences”, Zurich, March 22-24 1995