Page 116 - Introduction to Information Optics

2.5. Image Processing with Optics 101

where C is a complex constant, and

F(p) = f(x')e" lpx dx',
J
x' = x 4- (In — l)w/2 and sine X ^
X

For simplicity of illustration, we assume f(x') = exp(ip 0x) a complex
sinusoidal signal, w & h,b & a, and N » 1; then the corresponding intensity
distribution can be shown as

„ ,
2
7(p 5 q] = K sine (164)

in which the first sine factor represents a narrow spectral line located at p = p 0.
The second sine factor represents a relatively broad spectral band in the q
direction, which is due to the narrow channel width a. This last factor deserves
special mention; for large values of N, it approaches a sequence of narrow
pulses,

q — - (Inn — wp 0), n — 1,2,... (2.65)

which yields a fine spectral resolution in the q direction.
In view of the spectral intensity distribution Eq. (2.64), we see that the first
sine factor is confined within a very narrow region along the p direction, due
to large w. The half-width spread can be written as

O
Ap = —, (2.66)
w
which is equal to the resolution limit of the transform lens.
However, along the q direction, the intensity is first confined within a
relatively broad spectral band, due to narrow channel width a, and then
modulated by a sequence of narrow periodic pulses, as can be seen in Fig. 2.28.
Thus, along the q axis modulation we can envision a series of spectral points
which are located at p = p Q and q = i/b(2nn — wp 0). As the input signal
frequency changes, the position of the spectral points also changes by the
amounts

dq = ~dp (). (2.67)