Page 391 - Introduction to Information Optics

376 7. Pattern Recognition with Optics

7.3.1. DETECTION WITH TEMPORAL FOURIER-DOMAIN FILTERS

The technique is similar to exploiting the coherence content from an
incoherent source for partially coherent processing, as described in Sec, 2.8. Let
us consider that a conventional VLC is illuminated by a red, green, and blue
(RGB) coherent plane wave. A color image transparency in contact with a
sinusoidal grating is assumed at the input plane. The spectral distribution at
the Fourier domain can be written as

g.b g,b / /•; \
S(«, ft; A) = £ S H(o, /?; /„) + £ S n a ± Po, ft, (1.16)

where (a, //) denotes the spatial coordinate at the Fourier plane, / is the focal
1
length of the transform lens, and S r(a, p), S g(x, /?), and S b(tx, ft] are the
corresponding color Fourier spectra. It is trivial to see that the red, green, and
blue color spectra are scattering diffracted along the a axis. Since the transpar-
ency is spatial frequency limited, the RGB spectra are physically separated in
the Fourier domain by using a sufficiently high sampling frequency p 0.
A set of RGB matched filters (called temporal Fourier holograms) can be
synthesized by this interferometric technique of Vander Lugt, as written by

//„(«, p 0,
7.17)
:
•cos —-
[ 2n
where x 0 is an arbitrary carrier spatial frequency, Ks are appropriate propor-
1
tionality constants, and S n(a, /?) = \S n(x, f$}\ exp[z'0 n(a, p )], are the primary
color image spectra. If these temporal matched spatial filters are inserted in the
Fourier domain, as illustrated in Fig. 7.22, the complex light distribution at the
output plane of the VLC would be

A
A
"\A PYT^^/fl VI
/it \- lA — \
t/\ ! y} — /.; L^fiv-^? y/ CA FV'FO /
|~c / v
n — r
+ s n(x, y) exp(/p 0;c) * s a(x - x 0,3;) exp(/p 0x) (7.18)
+ S B(X, y) exp(/p 0x) * S B (-X + x 0, y) exp(ip 0x)],
in which the last term represents the RGB image correlations that will be
superimposely diffracted at x = x 0.
For experimental demonstration, a color image transparency of a campus