123 
is assumed [8]. This approach cannot be followed in our application. In fact, the point spread function of 
our device does not even have rotational symmetry. On the other hand, like in many other applications, 
the type of objects that occur are precisely known. This paper describes a novel depth-from-focus technique 
based on the precise knowledge of the 3-D point spread function and the properties of the observed objects. 
No assumptions about the shape of the point spread functions are necessary. 
2 OPTICAL SETUP 
The air bubbles are visualized using a light blocking technique (Fig. 1). A CCD camera looks directly into 
the light source, which is a Halogen bulb or a pulsed LED array. A bubble entering the light beam refracts 
and reflects light in such a way that it is imaged as a black disk on a uniformly bright background. A short 
illumination time of 20 us avoids motion blur. The images are stored on video tape or laser video disc for 
later processing. An area of typically 6 x8 mm? is imaged on 240 x 512 pixel images with a spatial resolution 
of 27 um x 15 um. The measuring volume is at most several cm deep. The visualization system has been 
described in more detail by [5]. 
  
water surface 
       
  
OA Glas Window : 
av > Adiustable Minor \/ 
= SCC Telecentric Stop — 
[Scattering Plate 
LED Array 
| — water tight housing 
   
  
  
     
   
  
  
  
  
  
  
  
  
of flume bottom 
ion E NENNEN 
nce 
the Figure 1: left: Sketch of the optical bubble measuring device used at linear flume in Delft. The light source and 
les the optical receiver with CCD camera are mounted 5-20 cm below the water surface. The free optical path between 
hly source and receiver is about 50 cm. The measuring volume is located in the middle of this path. right: Air bubbles 
at several cm beneath the mean water surface. 
3 MODEL OF IMAGE FORMATION 
to 
an An air bubble can be modelled by its light absorbing coefficient 7(Z) in the object plane. A circular bubble at 
ns the position Xo and with the radius r is then described as a circular box function 7(X) = II (IX — Xol/ 2r) 
us where II(x) is the unit step function. The capital letters indicate object plane coordinates. The Z axis of the 
system coincides with the optical axis of the system. The image of the bubble is given by the convolution of 
n the well focused image with the point spread function PSF(Z) of the optical system: 
ia. 
hic en E— | dj 
th- HE) = (1 —II ( 27 )) * PSF(£). (1) 
To 
not In this equation, the object coordinates have been replaced by the image coordinates. The radius of the 
the bubble at the image plane is given by r — V;(z)r, where V, is the geometrical magnification factor. It is 
ect not possible to use a simple model (like a box or Gaussian function ) for the point spread function of our 
ics. optical system. Since the filament of the halogen bulb is imaged on the receiver lens diaphragm, its image 
om forms the effective diaphragm of the system. Therefore, the shape of the point spread function is given by 
nd the irradiance at the lens diaphragm which is, except for a size factor, equal to the brightness distribution 
ion 
5 IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop "From Pixels to Sequences", Zurich, March 22-24 1995