📄 an introduction to wavelets what do some wavelets look like.htm
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<H3><FONT size=8>W</FONT>hat do <FONT size=6>S</FONT>ome <FONT
size=6>W</FONT>avelets <FONT size=6>L</FONT>ook <FONT size=6>L</FONT>ike?</H3>
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<P>Wavelet transforms comprise an infinite set. The different wavelet families
make different trade-offs between how compactly the basis functions are
localized in space and how smooth they are.
<P>Some of the wavelet bases have fractal structure. The Daubechies wavelet
family is one example (see Figure 3).
<P>
<UL><IMG alt=Fig3
src="An Introduction to Wavelets What Do Some Wavelets Look Like.files/IW_fig3.gif"
align=top></UL>
<P><B>Fig. 3. The fractal self-similiarity of the Daubechies mother
wavelet.</B> This figure was generated using the <A
href="http://playfair.stanford.edu/~wavelab">WaveLab</A> command:
<TT>wave=MakeWavelet(2, -4, 'Daubechies', 4, 'Mother', 2048).</TT> The inset
figure was created by zooming into the region <EM>x</EM>=1200 to 1500.
<P>Within each family of wavelets (such as the Daubechies family) are wavelet
subclasses distinguished by the number of coefficients and by the level of
iteration. Wavelets are classified within a family most often by the
<EM>number of vanishing moments.</EM> This is an extra set of mathematical
relationships for the coefficients that must be satisfied, and is directly
related to the number of coefficients <A
href="http://www.amara.com/IEEEwave/IW_ref.html#one">(1)</A>. For example,
within the Coiflet wavelet family are Coiflets with two vanishing moments, and
Coiflets with three vanishing moments. In Figure 4, I illustrate several
different wavelet families.
<P>
<UL><IMG alt=Fig4
src="An Introduction to Wavelets What Do Some Wavelets Look Like.files/IW_fig4.gif"
align=top></UL>
<P><B>Fig. 4. Several different families of wavelets.</B> The number next to
the wavelet name represents the number of vanishing moments (A stringent
mathematical definition related to the number of wavelet coefficients) for the
subclass of wavelet. Note: These figures were created using <A
href="http://playfair.stanford.edu/~wavelab">WaveLab,</A> by typing:
<P><TT>wave = MakeWavelet(2,-4,'Daubechies',6,'Mother', 2048);<BR>wave =
MakeWavelet(2,-4,'Coiflet',3,'Mother', 2048);<BR>wave =
MakeWavelet(0,0,'Haar',4,'Mother', 512);<BR>wave =
MakeWavelet(2,-4,'Symmlet',6,'Mother', 2048); <BR></TT>
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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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