Page 278 - Introduction to Information Optics

5.4. Wavelet Transform 263

Regularity of the wavelet is not an obligated condition but is usually
required, because regularity leads to the localization of the wavelet in both time
and frequency domains. One measure of wavelet regularity requires the
wavelets to have the first n + 1 moments up to the order n, equal to zero as
[14]:

p
M_ = t h(t) dt = 0 for p = 0,1,2,..., n.

According to the admissibility condition given in the preceding equation the
wavelet must oscillate to have a zero mean. If at the same time the wavelet is
regularly satisfying the zero high-order moment condition, then the wavelet
also should have fast decay in the time domain. As a result, the wavelet must
be a "small" wave with a fast-decreasing amplitude, as described by the name
wavelet. The wavelet is localized in the time domain.
By using the Taylor expansion of the signal f(t) the wavelet transform
defined in Eq. (5.9) can be written as

(P)
W f(s, 0) = 4= (l / (0) f -. h(t/s)
s pl
\ p J
(p)
where / (0) denotes the pth derivative of f(t) at t = 0, and the integrals on
the right-hand side of the above equation are the moments of pth orders of the
wavelet. Hence, we have

0)
1

According to the regularity condition, the high-order moments of the wavelet
up to M n are all equal to zero. Therefore, the wavelet transform coefficient
+(3/2}
decays with decreasing of the scale s, or increasing of 1/s, as fast as s" for
a smooth signal f(t), whose derivatives of orders higher than the (n + l)th
order have finite values.
The regularity of the wavelet leads to localization of the wavelet transform
in frequency. The wavelet transform is therefore a local operator in the
frequency domain. According to the admissible condition, the wavelet already
must be zero at the zero frequency. According to the regularity condition, the
wavelet must have a fast decay in frequency. Hence, the wavelet transform is a
bandpass filter in the Fourier plane, as described in Eq. (5.10).