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S/CNAf 6J /MAG€ PROCESSING
An
Introduction
to
Wavelets
Wavelets were developed
independently by
mathematicians, quantum
physicists, electrical
engineers, and geologists,
but collaborations among
these fields during the
last decade have led to
new and varied
applications. What are
wavelets, and why might
they be useful to you?
.:
Amara Graps
analyze according to scale. Indeed, some researchers feel that using
wavelets means adopting a whole new mindset or perspective in process
ing data.
Wavelets are functions that satisfy certain mathematical requirements
and are used in representing data or other functions. This idea is not new.
Approximation using superposition of functions has existed since the
early 18OOs, when Joseph Fourier discovered that he could superpose
sines and cosines to represent other functions. However, in wavelet analy
sis, the scale that we use to look at data plays a special role. Wavelet algo
rithms process data at different scales or resolutions. If we look at a signal
(or a function) through a large “window,” we would notice gross features.
Similarly, if we
look
at a signal through a small “window,” we would no
tice small features. The result in wavelet analysis is to see both the forest
and the trees, so to speak.
This makes wavelets interesting and useful. For many decades scientists
have wanted more appropriate functions than the sines and cosines,
which are the basis of Fourier analysis, to approximate choppy signals.’
By their definition, these functions are nonlocal (and stretch out to infin
ity). They therefore do a very poor job in approximating sharp spikes. But
with wavelet analysis, we can use approximating functions that are con
tained neatly in finite domains. Wavelets are wtllsuited for approximat
ing data with sharp discontinuities.
The wavelet analysis procedure is to adopt a wavelet prototype function,
called an
analyzing wavelet
or
mother wavelet.
Temporal analysis is per
formed with a contracted, highfrequency version of the prototype wavelet,
while frequency analysis is performed with a dilated, lowfrequency version
of the same wavelet. Because the original signal or function can be repre
sented in terms of a wavelet expansion (using coefficients in a linear combi
An
earlier
version of this article will appear in the
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Sczewtzfic
d?
Enginerring Applications
of
the
:Ilizcintush
1991
Cuiference
CDROiI.1, MacSciTech, Worcester, Mass., 1995.
50
10709924/95/$4.00
0 1995 IEEE
IEEE COMPUTATIONAL
SCIENCE & ENGINEERING
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View Full Document nation of the wavelet functions), data operations
can be performed using just the corresponding
wavelet coefficients. And if you further choose
the wavelets best adapted to your data, or
trun
cate the coefficients below a threshold, your data
are sparsely represented. This sparse coding
makes wavelets an excellent tool in the field of
data compression.
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This note was uploaded on 05/28/2010 for the course EE EE564 taught by Professor Runyiyu during the Spring '10 term at Eastern Mediterranean University.
 Spring '10
 RunyiYu

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