Page 101 - Introduction to Information Optics

86 2. Signal Processing with Optics
in which we see that the object function f(x , v) convolves with the spatial
impulse response h(x, y) and are diffracted around ( —2a 0 ,0) and (2a 0,0),
respectively, where ® and * represent the correlation and convolution oper-
ations, respectively.
It is therefore apparent that the JTP can, in principle, perform all the
processing operations that a conventional FDP can. The inherent advantages
of using JTP are: (1) avoidance of complex spatial filter synthesis, (2) higher
input space-bandwidth product, (3) lower spatial carrier frequency require-
ment, and (4) higher output diffraction efficiency, particularly by using the
hybrid JTP, as will be seen later.
Matched filtering can also be performed with a JTP, which we shall call a
joint transform correlator (JTC), as opposed to the Vander-Lugt correlator
(VLC) described earlier. We assume that the two identical object functions
f( x — oc 0, y) and f(x -f « 0, y) are inserted in the input plane of a JTP. The
complex light distribution arriving at the square-law detector in Fourier plane
P z will then be

E(p, q) = F(p, q) exp(-/a 0 p) + F(p, q) exp(za 0p), (2.42)

where F(p, q) is the Fourier spectrum of the input object. The corresponding
irradiance at the input end of the square-law detector is given by

2
I(p 1q) = 2\F(p,q)\ [l +cos2a 0 p].
By coherent readout, the complex light distribution at the output plane can be
shown as

0(a, 0) - 2f(x, y) <g> /*(*, y) + f(x, y) ® f*(x - 2a 0, y)
(2.43)
+ /*(x, y) (x) f( x + 2a 0, v),

in which we see that two major autocorrelation terms are diffracted at a = 2a 0
and a = ~2a 0, respectively. Notice that the aforementioned square-law con-
verter or devices such as photographic plates, liquid crystal light valves, or
charge-coupled device cameras, can be used.
To illustrate the shift invariant property of a JTC, we assume a target "B"
is located at (« 0, j> 0) and a reference image is located at ( —a 0 ,0), as shown in
Fig. 2.16a. The set of input functions can be written as f(x + oe 0, y — >' 0) and
/(x + a 0, y), respectively. Thus the complex light distribution at the input end
of the square-law detector can be written as

E(p, q) = F(p, <?){exp[-i(a 0p + >' 0g)] + exp[(ia 0p)}.