gray value gray value gray value 
M 
| Fe B i 
| | position / position ~ | position 
| A ^ ge [45^ cdd 
| » l/e diameter b | : 
  
Figure 8: Radial cross section through the gray value distribution 
of bubbles at different distance from the focal plane. The distance 
increases from left to right. 
The fact that the shape of the point-spread-function is inde- 
pendent of the distance z allows the PSF to be expressed in 
terms of a general shape function B(z) and a scaling factor 
Ve(2): 
PSF,(Z) - kB (+257) (5) 
with the normalization factor k=! = f dz B(z). At the focal 
plane (z — 0) V, is zero, resulting in a delta function for the 
PSF. This can be used to analyze the behavior of g,, in more 
detail: 
All bubbles with the same ratio between the radius r' and 
the size V, of the point spread function have the same mean 
gray value gm, because their images differ only in size and 
are of the same shape. Thereby r' — V;(z)r is the radius of 
the well focus object of radius r on the image plane. Thus 
  
= const & gm = const. (6) 
Denoting the constant in the above equation by y(gm) 
and resolving the mean gray value is given by gm (z,7) = 
y (e and therefore 
gm (2, T) — gm (72) (7) 
If we use a telecentric path of rays the magnification V, be- 
comes independ from z and with the use of q — 1/2 for 
segmentation the 1/q-area represents the true size of the 
particles. Furthermore, V5(z) is then a symmetric and linear 
function of z. Therefore, gm depends only on the normalized 
distance z/r: 
Im) SV (4) E Um (=) (8) 
5 CALIBRATION 
Calculating bubble size distributions from the image se- 
quences with the depth from focus approach requires that the 
instrument is carefully calibrated with a focus series of cali- 
bration targets of known size. Patterson reticles were used as 
calibration targets. This standard target for microscopic cali- 
bration consists of a series of black circles in a diameter range 
from 18 um up to 450 um on a glass plate (Fig. 9). Because 
Patterson globes are not absolutely accurate, the true size 
of each circle has to be measured independently, e.g. using 
a calibrated microscope. A black circle is a very good ap- 
proximation of the focused image of a bubble, since with the 
optical setup used in the experiments more than 99.6 % of 
the incident light is scattered away from the receiver. Nev. 
ertheless, the bright dot which appears in the center of well 
focused bubbles can be easily removed by applying a median 
filter to the normalized image. Depth series centered at the 
  
  
Figure 9: Partial view of the calibration target. 
focal plane are taken with a step size of 1 mm. Fig. 10 shows 
the radii measured from different circles of the Patterson tar- 
get. Within the measuring volume, the difference between 
the measured and the true radius in in the order of 10 to 15 
pm, which is about the size of one pixel. The variation of the 
mean gray value with increasing depth is shown in Fig. 11. A 
linear model g(z, r) — go — a(r)|z| fits well to the data. Be- 
cause a small axis offset and slight tilt of the target can often 
not be avoided, the axis offset for each circle is corrected by 
finding the center of symmetry in its depth series. 
= 160 
= 
8 140 
= 120 
ba 
T 10 
R 
2 —li— Q 450 um 
2 80 
s —9— 6 350 um 
= 60 —A—@ 270 pm 
—— 0 225 um 
40 —$— 9180 um 
—+— 6 144 um 
20 —— D 108 um 
—X— 9 72 um 
-8 -6 -4 2 0 2 4 6 8 
      
distance z from focal plane [mm] 
Figure 10: Independence of the size of blurred bubbles with the 
distance from the focal plane. The thin lines indicate the maximum 
size of the virtual measuring volume. 
6 CALCULATION OF PARTICLE CONCENTRATION 
6.1 Determination of the measuring volume 
The decrease of the mean gray value with increasing distance 
from the focal plane can now be used to define the measuring 
volume by a lower limit for g,». Only bubbles with mean gray 
values above this limit are taken into account for the calcu- 
lation of the size distribution. Thus the linear dependence of 
gm on the normalized distance 
glz,r) = go - alr)lz| = go as (©) 
gives the volume boundary: 
1 T 
MAT ET = 7 — 9min) = — — OUmin)- 10 
z m (90 — min.) = (go — Gmin) (10) 
The measuring volume is then given by 
Vir) = 225»az(T)49 (11) 
with Ag being the area imaged by the CCD at the focal plane. 
The depth of the volume is controlled by the parameter grin. 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996 
normalized mean gray value 
04 
o 
© 
  
  
  
  
  
distance z fron 
  
Figure 11: lef 
terson globes; rig 
z/R. This valida 
It is important 
size and increas: 
cle becomes les: 
image is convoh 
6.2 Calculatic 
As mentioned a 
particle size cai 
of the images. 
cated at the sar 
plane, the intrii 
lence in the dey 
completely. Th 
where the apert 
of the lens syste 
size does not n 
there is a uniqu 
z of an object 
the segmented 
[Geißler and Jäl 
rameters are ob 
values of the ot 
ular grid in the 
fast look-up tal 
6.3 Size distr 
Segmentation a 
of position and 
The data from 
calculate size d 
of an image se 
segmentation a 
  
Figure 12: R 
The dark gray li