Page 121 - Introduction to Information Optics

106 2. Signal Processing with Optics

x
y
y
x
y
x
f(x,y) Nonlinear f(e ,e ) Linear f(e ,e }®f(e ,e )
i_^ i _ p
Coordination Optical
^
Transformers Correlators
Fig. 2.30. A block diagram representation of Mellin transform.
2
. -J_ f " • -,-'«». (2.76)
271 J 0 '
im
F m(r, 0) = F m(r)e ° is called the mth-order circular harmonic.
If the object is rotated by an angle a, it can be written by

imt im9
/(r, 0 + a) = F m(r)e e . (2,77)

Let us denote /(x, y) and f a(x, y) as object functions of /(r, 0) and /(r, 0 + a),
respectively. By referring to matched filtering, when a rotated object
/ a(r, 0 + a) is applied to the input end of a Fourier domain process, the output
light field can be evaluated, as given by

(2.78)

It is apparent that if a = 0, the autocorrelation peak appears at x = y = 0. By
transforming the preceding convolution integral into a polar coordinate
system, the center of correlation can be shown as

C2K
C(a) = rdr\ f(r, 9 + a)/*(r, 0) dB. (2.79)
' Jo

In view of the definition of circular harmonic expansion of Eq. (2.75), we see
that

C(a) = (2.80)

where

2
2n\ |F M(r)| r dr.
!o