📄 an introduction to wavelets historical perspective.htm
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<H2><FONT size=8>H</FONT>istorical <FONT size=6>P</FONT>erspective</H2>
<P>
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<P>In the history of mathematics, wavelet analysis shows many different
origins <A href="http://www.amara.com/IEEEwave/IW_ref.html#two">(2)</A>. Much
of the work was performed in the 1930s, and, at the time, the separate efforts
did not appear to be parts of a coherent theory.
<H3>Pre-1930</H3>Before 1930, the main branch of mathematics leading to
wavelets began with Joseph Fourier (1807) with his theories of frequency
analysis, now often referred to as Fourier synthesis. He asserted that any
<IMG alt=2pi
src="An Introduction to Wavelets Historical Perspective.files/IW_eq2pi.gif"
align=top> -periodic function <EM>f(x)</EM> is the sum
<P>
<UL>
<UL><IMG alt=eq1
src="An Introduction to Wavelets Historical Perspective.files/IW_eq1.gif"
align=middle></UL></UL>
<P>of its Fourier series. The coefficients <IMG alt=a_0
src="An Introduction to Wavelets Historical Perspective.files/IW_eqa0.gif"
align=top>, <IMG alt=a_k
src="An Introduction to Wavelets Historical Perspective.files/IW_eqak.gif"
align=top>, and <IMG alt=b_k
src="An Introduction to Wavelets Historical Perspective.files/IW_eqbk.gif"
align=top> are calculated by
<P><IMG alt=eq2
src="An Introduction to Wavelets Historical Perspective.files/IW_eq2.gif"
align=middle>
<P>Fourier's assertion played an essential role in the evolution of the ideas
mathematicians had about the functions. He opened up the door to a new
functional universe.
<P>After 1807, by exploring the meaning of functions, Fourier series
convergence, and orthogonal systems, mathematicians gradually were led from
their previous notion of <EM>frequency analysis</EM> to the notion of
<EM>scale analysis.</EM> That is, analyzing <EM>f(x)</EM> by creating
mathematical structures that vary in scale. How? Construct a function, shift
it by some amount, and change its scale. Apply that structure in approximating
a signal. Now repeat the procedure. Take that basic structure, shift it, and
scale it again. Apply it to the same signal to get a new approximation. And so
on. It turns out that this sort of scale analysis is less sensitive to noise
because it measures the average fluctuations of the signal at different
scales.
<P>The first mention of wavelets appeared in an appendix to the thesis of A.
Haar (1909). One property of the Haar wavelet is that it has <EM>compact
support,</EM> which means that it vanishes outside of a finite interval.
Unfortunately, Haar wavelets are not continuously differentiable which
somewhat limits their applications.
<H3>The 1930s</H3>In the 1930s, several groups working independently
researched the representation of functions using <EM>scale-varying basis
functions.</EM> Understanding the concepts of basis functions and
scale-varying basis functions is key to understanding wavelets; the sidebar <A
href="http://www.amara.com/IEEEwave/IW_basis.html">next</A> provides a short
detour lesson for those interested.
<P>By using a scale-varying basis function called the Haar basis function
(more on this later) Paul Levy, a 1930s physicist, investigated Brownian
motion, a type of random signal <A
href="http://www.amara.com/IEEEwave/IW_ref.html#two">(2)</A>. He found the
Haar basis function superior to the Fourier basis functions for studying small
complicated details in the Brownian motion.
<P>Another 1930s research effort by Littlewood, Paley, and Stein involved
computing the energy of a function <EM>f(x)</EM>:
<P>
<UL>
<UL><IMG alt=eq3
src="An Introduction to Wavelets Historical Perspective.files/IW_eq3.gif"
align=top></UL></UL>
<P>The computation produced different results if the energy was concentrated
around a few points or distributed over a larger interval. This result
disturbed the scientists because it indicated that energy might not be
conserved. The researchers discovered a function that can vary in scale
<EM>and</EM> can conserve energy when computing the functional energy. Their
work provided David Marr with an effective algorithm for numerical image
processing using wavelets in the early 1980s.
<P>
<H3>1960-1980</H3>Between 1960 and 1980, the mathematicians Guido Weiss and
Ronald R. Coifman studied the simplest elements of a function space, called
<EM>atoms,</EM> with the goal of finding the atoms for a common function and
finding the "assembly rules" that allow the reconstruction of all the elements
of the function space using these atoms. In 1980, Grossman and Morlet, a
physicist and an engineer, broadly defined wavelets in the context of quantum
physics. These two researchers provided a way of thinking for wavelets based
on physical intuition.
<H3>Post-1980</H3>In 1985, Stephane Mallat gave wavelets an additional
jump-start through his work in digital signal processing. He discovered some
relationships between quadrature mirror filters, pyramid algorithms, and
orthonormal wavelet bases (more on these later). Inspired in part by these
results, Y. Meyer constructed the first non-trivial wavelets. Unlike the Haar
wavelets, the Meyer wavelets are continuously differentiable; however they do
not have compact support. A couple of years later, Ingrid Daubechies used
Mallat's work to construct a set of wavelet orthonormal basis functions that
are perhaps the most elegant, and have become the cornerstone of wavelet
applications today.
<P>
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<P><B><A href="http://www.amara.com/index.html">[Home]</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_overview.html">[Previous]</A> <A
href="http://www.amara.com/IEEEwave/IW_basis.html">[Next]</A> </B>
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<P>
<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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<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>
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