2. Optical Image Processing 
Optical image processing is subdivided into incoherent and coherent 
optical processing. In this chapter it is only intended to give a 
short introduction to coherent optical systems, because those sy- 
stems are more commonly used than incoherent optical systems. A 
detailed description of the problems related with optical proces- 
sing can be found in /1,2,3,4/. 
2.1 Principal mode of operation 
In principle the optical processor consists of a coherent, mono- 
chromatic light source and several lenses (figure 6). Laser beams 
fit the conditions for the light source in an ideal way and there- 
fore the development of the laser technique and of the coherent 
optical image processor was strongly correlated. 
The input plane P4 (left hand focal plane of lens L4) is assumed 
to have the local transparency distribution S(x,y) and is illumi- 
nated by the coherent light source. This transparency distribu- 
tion is transformed by lens L, into the frequency plane P, (right 
1 
hand focal plane of lens L,) and represents in this frequency 
plane a fairly good approximation of the Fourier transform 
s(ju,, Jay) of the input signal (equ. 1). 
+œ 
s(u,, Joy) - ff S(x,y)* exp {73 (w, X + e Ux)! dxdy (1) 
- 00 
x y 
spatial frequencies in x- and y-direction. A second lens L, is 
s (Ju, Ju) is a complex function and uo, and vw, represent the 
added in such a way that its left hand focal plane is identical 
with the frequency plane E This lens L, performs the inverse 
Fourier transformation so that the modulation of the light in the 
output plane is equal to the transparency distribution in the input 
plane (equ. 2). 
To 
S(x,y) = 3 SIs (Ju. jo 
"E ) - exp {J (wx + uyY) d». du 
y y 
In equ. 3 the Fourier and the inverse Fourier transformations 
according to equ. 1 and 2 are described in a symbolic matter 
S (Gu, Ju y) #6—D S(x,y) 
S(x,y) CÓ» s (Ju, Ju) 
Many problems in image processing can at least partially be solved 
by applying convolution and correlation techniques (equ. 4). 
convolution: 
+ co 
B,(x',y') = JIB (x,y) G(x" = x, y' = y)dudy 
correlation: 
+o 
Bo(x',y!) s T/B.(x,y)* Gix - x', y - y')dxdy B, ®G 
B, (x,y) is the original image function (e.g. transparency distri- 
bution of a film), Gix,y) is a weighting function and B,(x',y") 
is the processed image. The computation according to equ. 4 can 
be done digitally in the image domain but in many applications 
the computations are very time consuming. 
In the frequency domain correlation and convolution are reduced 
to a point by point matrix-multiplication (equ. 5). 
convolution: 
p, 
correlation: 
B. * 
2 By 96 TW bh. .=5h +g 
2 1 
* ; 
(g complex conjugate of g) 
where the small letters represent the Fourier transform of the 
corresponding capital letters.