📄 an introduction to wavelets wavelet analysis.htm
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<H2><FONT size=8>W</FONT>avelet <FONT size=6>A</FONT>nalysis</H2>
<P>
<HR align=center noShade SIZE=2>
<P>Now we begin our tour of wavelet theory, when we analyze our signal in time
for its frequency content. Unlike Fourier analysis, in which we analyze
signals using sines and cosines, now we use wavelet functions.
<H3>The Discrete Wavelet Transform</H3>Dilations and translations of the
"Mother function," or "analyzing wavelet" <IMG alt=Phi(x)
src="An Introduction to Wavelets Wavelet Analysis.files/IW_eqphix.gif"
align=top> define an orthogonal basis, our wavelet basis:
<P>
<UL><IMG alt=eq4
src="An Introduction to Wavelets Wavelet Analysis.files/IW_eq4.gif"
align=top></UL>
<P>The variables <EM>s</EM> and <EM>l</EM> are integers that scale and dilate
the mother function <IMG alt=Phi(x)
src="An Introduction to Wavelets Wavelet Analysis.files/IW_eqphix.gif"
align=top> to generate wavelets, such as a Daubechies wavelet family. The
scale index <EM>s</EM> indicates the wavelet's width, and the location index
<EM>l</EM> gives its position. Notice that the mother functions are rescaled,
or "dilated" by powers of two, and translated by integers. What makes wavelet
bases especially interesting is the self-similarity caused by the scales and
dilations. Once we know about the mother functions, we know everything about
the basis.
<P>To span our data domain at different resolutions, the analyzing wavelet is
used in a scaling equation:
<P>
<UL><IMG alt=eq5
src="An Introduction to Wavelets Wavelet Analysis.files/IW_eq5.gif"
align=top></UL>
<P>where <EM>W(x)</EM> is the scaling function for the mother function <IMG
alt=Phi(x)
src="An Introduction to Wavelets Wavelet Analysis.files/IW_eqphix.gif"
align=top>, and <IMG alt=c_k
src="An Introduction to Wavelets Wavelet Analysis.files/IW_eqck.gif"
align=top> are the <EM>wavelet coefficients.</EM> The wavelet coefficients
must satisfy linear and quadratic constraints of the form
<P>
<UL><IMG alt=eq6
src="An Introduction to Wavelets Wavelet Analysis.files/IW_eq6.gif"
align=top></UL>
<P>where <IMG alt=delta
src="An Introduction to Wavelets Wavelet Analysis.files/IW_eqdelta.gif"
align=top> is the delta function and <EM>l</EM> is the location index.
<P>One of the most useful features of wavelets is the ease with which a
scientist can choose the defining coefficients for a given wavelet system to
be adapted for a given problem. In Daubechies' original paper <A
href="http://www.amara.com/IEEEwave/IW_ref.html#six">(6)</A>, she developed
specific families of wavelet systems that were very good for representing
polynomial behavior. The Haar wavelet is even simpler, and it is often used
for educational purposes.
<P>It is helpful to think of the coefficients <IMG alt="c coeffs"
src="An Introduction to Wavelets Wavelet Analysis.files/IW_eqcs.gif"
align=top> as a filter. The filter or coefficients are placed in a
transformation matrix, which is applied to a raw data vector. The coefficients
are ordered using two dominant patterns, one that works as a smoothing filter
(like a moving average), and one pattern that works to bring out the data's
"detail" information. These two orderings of the coefficients are called a
<EM>quadrature mirror filter pair</EM> in signal processing parlance. A more
detailed description of the transformation matrix can be found elsewhere <A
href="http://www.amara.com/IEEEwave/IW_ref.html#four">(4)</A>.
<P>To complete our discussion of the DWT, let's look at how the wavelet
coefficient matrix is applied to the data vector. The matrix is applied in a
hierarchical algorithm, sometimes called a <EM>pyramidal algorithm.</EM> The
wavelet coefficients are arranged so that odd rows contain an ordering of
wavelet coefficients that act as the smoothing filter, and the even rows
contain an ordering of wavelet coefficient with different signs that act to
bring out the data's detail. The matrix is first applied to the original,
full-length vector. Then the vector is smoothed and decimated by half and the
matrix is applied again. Then the smoothed, halved vector is smoothed, and
halved again, and the matrix applied once more. This process continues until a
trivial number of "smooth-smooth-smooth..." data remain. That is, each matrix
application brings out a higher resolution of the data while at the same time
smoothing the remaining data. The output of the DWT consists of the remaining
"smooth (etc.)" components, and all of the accumulated "detail" components.
<H3>The Fast Wavelet Transform</H3>The DWT matrix is not sparse in general, so
we face the same complexity issues that we had previously faced for the
discrete Fourier transform <A
href="http://www.amara.com/IEEEwave/IW_ref.html#seven">(7)</A>. We solve it as
we did for the FFT, by factoring the DWT into a product of a few sparse
matrices using self-similarity properties. The result is an algorithm that
requires only order <EM>n</EM> operations to transform an <EM>n</EM>-sample
vector. This is the "fast" DWT of Mallat and Daubechies.
<H3>Wavelet Packets</H3>The wavelet transform is actually a subset of a far
more versatile transform, the wavelet packet transform <A
href="http://www.amara.com/IEEEwave/IW_ref.html#eight">(8)</A>.
<P>Wavelet packets are particular linear combinations of wavelets <A
href="http://www.amara.com/IEEEwave/IW_ref.html#seven">(7)</A>. They form
bases which retain many of the orthogonality, smoothness, and localization
properties of their parent wavelets. The coefficients in the linear
combinations are computed by a recursive algorithm making each newly computed
wavelet packet coefficient sequence the root of its own analysis tree.
<H3>Adapted Waveforms</H3>Because we have a choice among an infinite set of
basis functions, we may wish to find the best basis function for a given
representation of a signal <A
href="http://www.amara.com/IEEEwave/IW_ref.html#seven">(7)</A>. A <EM>basis of
adapted waveform</EM> is the best basis function for a given signal
representation. The chosen basis carries substantial information about the
signal, and if the basis description is efficient (that is, very few terms in
the expansion are needed to represent the signal), then that signal
information has been compressed.
<P>According to Wickerhauser <A
href="http://www.amara.com/IEEEwave/IW_ref.html#seven">(7)</A>, some desirable
properties for adapted wavelet bases are
<OL>
<LI>speedy computation of inner products with the other basis functions;
<LI>speedy superposition of the basis functions;
<LI>good spatial localization, so researchers can identify the position of a
signal that is contributing a large component;
<LI>good frequency localization, so researchers can identify signal
oscillations; and
<LI>independence, so that not too many basis elements match the same portion
of the signal. </LI></OL>
<P>For adapted waveform analysis, researchers seek a basis in which the
coefficients, when rearranged in decreasing order, decrease as rapidly as
possible. to measure rates of decrease, they use tools from classical harmonic
analysis including calculation of <EM>information cost functions.</EM> This is
defined as the expense of storing the chosen representation. Examples of such
functions include the number above a threshold, concentration, entropy,
logarithm of energy, Gauss-Markov calculations, and the theoretical dimension
of a sequence.
<P>
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<H5>You may <A
href="http://www.amara.com/ftpstuff/IEEEwavelet.ps.gz">download</A> this
paper: "Introduction to Wavelets" </H5>
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<ADDRESS>Last Modified by <A href="mailto:amara@amara.com">Amara
Graps</A> on 8 October 1997.<BR>© Copyright Amara Graps, 1995-1997.
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