Page 423 - Introduction to Information Optics

408 7. Pattern Recognition with Optics

where the superasterisk represents the complex conjugate of the wavelet, which
is defined as

(7.61)

This function is obtained by translating and dilating the analyzing wavelet
function h(x, y). Thus, the WT can be written as

b y)h* (—, — dx dy
'
(7,62)
x y'
• s(x, y) (x)
a ra v

which is essentially the cross-correlation between the signal and the dilated
analyzing wavelet. Furthermore, h(x/a x, y/a y) can be interpreted as a bandpass
filter governed by the (a x, a y) dilation. Thus, both the dominant frequency and
the bandwidth of the filter can be adjusted by changing the dilation of
h(x/a x, y/a y). In other words, the WT is essentially a filtered version of the
input signal s {(x, y). The correlation of two WTs with respect to the input
signal is an estimation of the similarity between the two signals. Due to the
inherent local feature selection characteristic of the WT, the wavelet matched
filter (WMF) generally offers a higher discriminability than the conventional
matched filter (CMF). Although implementing the WMF in conventional VLC
and JTC has been reported, the WMF can be synthesized with a PR-based
Fourier-domain correlator.
We first illustrate WMF construction. If we let the WT target signal s(x, y)
be

w(a x, a.,, b x, b v) = ~- 7= s(x, y) <g) h* (—, — ), (7.63)
\<* x «,/
^/a xa y

the corresponding Fourier-domain representation is

W(p, q) = -^= S(p, q)H*(p, q). (7.64)
fl fl
V x y
Similarly, the Fourier-domain representation of the WT reference signal