Page 97 - Introduction to Information Optics

82 2. Signal Processing with Optics

The corresponding intensity distribution over the recording medium is

2
I(p, q) - 1 + |F(p, q)\ + 2|F(p, q)\ cos[« 0p + </;(/;, $)]. (130)
We assume that if the amplitude transmittance of the recording is linear, the
corresponding amplitude transmittance function of the spatial filter is given by
2
H(p, q) = K{1 + |F(p, q)\ + 2|F(p, q)\ cos[« 0p + </>(/>, <?)]}, (2.31)

which is, in fact, a real positive function.
Remember also that, in principle, a complex Fourier domain filter can be
synthesized with a spatial light modulator (SLM) using computer-generating
techniques [1]. (This will be discussed in more detail when we reach the
discussion of hybrid optical processing further on.)

2,4.1. SPATIAL DOMAIN FILTER

In a JTP we see that the (input) spatial function and the spatial domain
filter are physically separated, as can be seen in Fig. 2.10. In other words a
spatial domain filter can also be synthesized using the impulse response of the
Fourier domain filter; that is

Note that h(x, y) can be a complex function and is also limited by the similar
physical realizable conditions of a Fourier domain filter, such as

0 <

Needless to say, such a filter can be synthesized by the combination of an
amplitude and a phase filter. In fact, such a filter can also be synthesized by
computer-generation technique and then displayed on a spatial light modula-
tor. A matched Fourier domain filter is given by

H(p, q) - KS*(p< q\ (2.34)

where 5(p, q) is the signal (or target) spectrum. The corresponding impulse
response is given by
/i(x,y) = S(-.Y, -.v), (2.35)