126 
4.3 Determination of the Radius-dependent Measuring Volume 
Because of the decreasing of the mean gray value with increasing distance from the focal plane, the measuring 
volume can now be defined by a lower limit for ge. Only bubbles with gray values above this limit are 
processed. All bubbles with the same ratio between the geometrically magnified radius and the size of the 
point spread function have the same mean gray value ge, because of the similarity of their images. Thus 
rVg(z) 
————- = const & g. = const. 7 
Denoting the constant in the above equation by *(ge) , the volume boundaries zy for the lower limit 9° 
become 
Vp(z) 
Va(z) 
In general, the solutions for positive and negative values of z are different. However, with the optical setup 
used in our device, v(z) ^ |z| and therefore |z;| = y(g3)r, yielding symmetric volume boundaries (Fig. 4 
right). 
  
zy -(g)v (r) with v(r)= (8) 
4.4 Implementation of Image Processing Algorithms 
An effective implementation of the depth-from-focus technique to measure the size distribution consists of 
the following five steps: 
e The measured point spread function and the calculated magnification factors V;(z) and V;(z) are used 
to compute the true bubble radius and the distance from the focal plane as a function of the 1 /e radius 
and the mean gray value. These computations are quite time consuming but only have to be done 
once for a given optical setup. The results are stored in 2-D look-up tables for further quick reference 
(Fig. 4a). 
e Gray-scale normalization and segmentation. 
e Computation of the 1/e radius and the mean gray value. 
e Computation of the true radius and distance from the focal plane using the previously computed 2-D 
lookup tables. 
e Computation of the size distribution of the bubbles. 
Using a i860 RISC processor board with real-time image transfer to a frame grabber, it took only 1-2 s 
per image to perform all the steps except for the first one. The segmentation algorithm used consists of 
two main steps. In order to localize the object, the image is binarized with a global threshold. For each 
object, the pixel with maximum gray value is chosen as starting point for the following region growing. 
Using a modified version of the region growing algorithm developed by [6], the 1/e area of each object is 
then segmented. Therefore, the value of the threshold does not influence the result of the segmentation. It 
only must be higher than the minimum value for ge. 
5 RESULTS 
As an example, the results of the analysis of two series of 600 and 400 images respectively, are shown in 
Fig. 5. The images were taken in the large wind/wave flume at Delft Hydraulics (The Netherlands) to 
explore the potential of the new technique. Image sequences were recorded on a S-VHS video recorder and 
later processed at the Institute for Environmental Physics in Heidelberg. The instrument was mounted 5 
cm below the mean water surface. Measurements were conducted at wind speeds of 11m /s and 14 m/s. The 
results nicely show the steep decrease in the bubble density towards larger radii and are in good agreement 
with measurements taken by [1]. 
IAPRS, Vol. 30, Part.5W1, ISPRS Intercommission Workshop "From Pixels to Sequences", Zurich, March 22-24 1995