文档介绍：March 18, 2004 8:53 WSPC/Trim Size: 9in x 6in for Proceedings jpa˙wavelets
WAVELET ANALYSIS AND SOME OF ITS APPLICATIONS
IN PHYSICS
J.-P. ANTOINE
Institut de Physique Th´eorique, Universit´eCatholique de Louvain
B-1348 Louvain-la-Neuve, Belgium
E-mail: ******@
We review the general properties of the wavelet transform, both in its continuous
and its discrete versions, in one or two dimensions, and we describe some of its
applications in signal and image processing. We also consider its extension to
higher dimensions.
1. What is wavelet analysis?
Wavelet analysis is a particular time- or space-scale representation of signals
which has found a wide range of applications in physics, signal processing
and applied mathematics in the last few years. In order to get a feeling
for it and to understand its ess, let us consider first the case of one-
dimensional signals.
It is a fact that most real life signals are nonstationary. They often
contain ponents, sometimes very significant physically, and
mostly cover a wide range of frequencies. In addition, there is frequently a
direct correlation between the characteristic frequency of a given segment
of the signal and the time duration of that segment. Low frequency pieces
tend to last a long interval, whereas high frequencies occur in general for
a short moment only. Human speech signals are typical in this respect.
Vowels have a relatively low mean frequency and last quite long, whereas
consonants contain a wide spectrum, up to very high frequencies, especially
in the attack, but they are very short.
Clearly standard Fourier analysis is inadequate for treating such signals,
since it loses all information about the time localization of a given frequency
component. In addition, it is very uneconomical. If a segment of the
signal is almost flat, thus carries little information, one still has to sum an
(alternating) infinite series for reproducing it. Worse y