Page 105 - Introduction to Information Optics

90 2. Signal Processing with Optics

where * denotes the convolution operation, K is a proportionality constant,
and h(x, y) is the spatial impulse response of the Fourier domain filter, which
can be generated on SLM2. We note that a Fourier domain filter can be
described by a complex amplitude transmittance such as

H(p,q)= H(p,q)\expti(t>(p,q)l
Let us further assume that a holographic type matched filter (as described in
Sec. 2.4.1) is generated at SLM2, as given by

2
H(p,q) = K{\ + \F(p,q)\ + 2 F(p,q)\ cos[a 0p + <j>(p, q)~]}, (2.47)
where a 0 //l is the spatial carrier frequency. It is straightforward to show that
the output complex light distribution can be written as

0(a, ft) = K[/(.x, y) + /(x, >') * /(x, y) * /*( - x, - y) + /(x, y) * /(x + a 0, y)
+ /(x, y) * /*( -x + « 0, -y)]. (2.48)

We see that third and fourth terms are the convolution and cross-correlation
terms, which are diffracted in the neighborhood of a = a 0 and a = a 0, respect-
ively.
If we assume the input object is embedded in an additive white Gaussian
noise n\ that is,

/'(.x,y) = /(*, v) + «(x,y), (2.49)

then the correlation term would be

J?(a, p) = /C[/(x, y) + n(x, y)] * /*(-x + a 0, -y).

Since the cross-correlation between n(x, y) and f*( — x + « 0, — y) can be shown
to be approximately equal to zero, the preceding equation reduces to

K(a, ft) = /(x, y) * /*(-x + a 0, -y), (2.50)
which, in fact, is the autocorrelation detection of /(x, y).
Notice that to ensure that the zero-order and the first-order diffraction
terms will not overlap, a 0 is required that

«<>>// + Ik (2.51)

where l f and / s are the spatial lengths in the x direction of the input scene (or
frame) and the detecting signal /(x, y), respectively.