📄 an introduction to wavelets overview.htm
字号:
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<!-- saved from url=(0046)http://www.amara.com/IEEEwave/IW_overview.html -->
<!-- Amara Graps' IEEE Paper: An Intro to Wavelets --><HTML><HEAD><TITLE>An Introduction to Wavelets: Overview</TITLE>
<META http-equiv=Content-Type content="text/html; charset=gb2312"><LINK rev=made
href="mailto:amara@amara.com"><!-- Make Background color GhostWhite and all links color DarkBlue-->
<META content="MSHTML 6.00.2600.0" name=GENERATOR></HEAD>
<BODY link=#00008b bgColor=#f8f8ff>
<BLOCKQUOTE>
<HR align=center noShade SIZE=2>
<P>
<H2><FONT size=8>O</FONT>verview</H2>
<P>
<HR align=center noShade SIZE=2>
<P>The fundamental idea behind wavelets is to analyze according to scale.
Indeed, some researchers in the wavelet field feel that, by using wavelets,
one is adopting a whole new mindset or perspective in processing data.
<P>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
1800's, when Joseph Fourier discovered that he could superpose sines and
cosines to represent other functions. However, in wavelet analysis, the
<EM>scale</EM> that we use to look at data plays a special role. Wavelet
algorithms process data at different <EM>scales</EM> or <EM>resolutions.</EM>
If we look at a signal with a large "window," we would notice gross features.
Similarly, if we look at a signal with a small "window," we would notice small
features. The result in wavelet analysis is to see both the forest
<EM>and</EM> the trees, so to speak.
<P>This makes wavelets interesting and useful. For many decades, scientists
have wanted more appropriate functions than the sines and cosines which
comprise the bases of Fourier analysis, to approximate choppy signals <A
href="http://www.amara.com/IEEEwave/IW_ref.html#one">(1)</A>. By their
definition, these functions are non-local (and stretch out to infinity). They
therefore do a very poor job in approximating sharp spikes. But with wavelet
analysis, we can use approximating functions that are contained neatly in
finite domains. Wavelets are well-suited for approximating data with sharp
discontinuities.
<P>The wavelet analysis procedure is to adopt a wavelet prototype function,
called an <EM>analyzing wavelet</EM> or <EM>mother wavelet.</EM> Temporal
analysis is performed with a contracted, high-frequency version of the
prototype wavelet, while frequency analysis is performed with a dilated,
low-frequency version of the same wavelet. Because the original signal or
function can be represented in terms of a wavelet expansion (using
coefficients in a linear combination of the wavelet functions), data
operations can be performed using just the corresponding wavelet coefficients.
And if you further choose the best wavelets adapted to your data, or truncate
the coefficients below a threshold, your data is sparsely represented. This
sparse coding makes wavelets an excellent tool in the field of data
compression.
<P>Other applied fields that are making use of wavelets include astronomy,
acoustics, nuclear engineering, sub-band coding, signal and image processing,
neurophysiology, music, magnetic resonance imaging, speech discrimination,
optics, fractals, turbulence, earthquake-prediction, radar, human vision, and
pure mathematics applications such as solving partial differential equations.
<P>
<HR align=center noShade SIZE=2>
<P><B><A href="http://www.amara.com/index.html">[Home Page]</A> <A
href="http://www.amara.com/current/wavelet.html">[Wavelet Page]</A> <A
href="http://www.amara.com/IEEEwave/IEEEwavelet.html#contents">[Contents]</A>
<A href="http://www.amara.com/IEEEwave/IW_history.html">[Next Section]</A>
</B>
<P>
<HR align=center noShade SIZE=2>
<P>
<H5>You may <A
href="http://www.amara.com/ftpstuff/IEEEwavelet.ps.gz">download</A> this
paper: "Introduction to Wavelets" </H5>
<P>
<HR align=center noShade SIZE=2>
<CENTER>
<TABLE>
<TBODY>
<TR><!-- Miscellaneous Contact Information -->
<TD><BASEFONT size=2>
<ADDRESS>Last Modified by <A href="mailto:amara@amara.com">Amara
Graps</A> on 8 October 1997.<BR>© Copyright Amara Graps, 1995-1997.
</ADDRESS></BASEFONT></TD></TR></TBODY></TABLE></CENTER></BLOCKQUOTE></BODY></HTML>
⌨️ 快捷键说明
复制代码
Ctrl + C
搜索代码
Ctrl + F
全屏模式
F11
切换主题
Ctrl + Shift + D
显示快捷键
?
增大字号
Ctrl + =
减小字号
Ctrl + -