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Wavelets
Theory
The research on this
topic has once again reminded us of the tremendous mathematical complexity
behind data compression and transmission. Developed independently in
the fields of mathematics, quantum physics, electrical engineering,
and seismic geology, the theory of wavelets has already found applications
in image compression, vision analysis, and earthquake prediction. Understanding
how it works in lay terms, however, is quite difficult.
Mathematical Transformation
Mathematical transformation
functions, the most famous of which is the Fourier Transform (FT),
translate a function from one domain, to another domain. The FT,
for example, transforms a signal that exists in the time domain (analyzing
what the signal is doing at a point in time) to the frequency domain
(analyzing what the signal is doing at a specific frequency). The
FT accompishes this by breaking down a complex signal into more basic
component sine and cosine waveforms, called basis functions. If the
basis functions were summed together, the result would be the original
signal. For complex signals the calculations for this are extensive.
A Fast Fourier Transform reduces the amount of calculation required
by using a regular sampling procedure of the signal, to produce a
close approximation. The reverse calculation, to tranform a signal
back to the time domain, is called an inverse FT.
There are currently
many applications for transformation functions. Two examples are:
Frequency Filtering
Because an FT allows a signal to be analyzed in the frequency domain, it
is possible to block certain frequencies within a signal, while leaving
others. As a signal comes in, it is transformed via an FT, certain
component frequencies are removed, and then it is inverse FT'd back
to the time domain. Industry applications include filtering data in
geologic seismic analysis, and filtering known frequencies of noise
from wireless signals. However, due to a phenomenon known as aliasing,
the sample rate of this procedure is limited, limiting its precision.
Image Compression
Current JPEG image compression is based on a transformation known as a
Discrete Cosine Transform (DCT). In the JPEG algorithm, the image is
sampled in 8 by 8 pixel blocks, and each sample is transformed via
DCT into component basis cosine functions. By comparing the coefficients
of these functions to a set coefficient, the algorithm identifies frequencies
in the samples that are undetectable to the human eye--frequencies
that will not affect the image quality if missing. By setting these
values to zero, the amount of data that needs to be stored to recreate
the image is greatly reduced. The base coefficients (there are varying
levels of compression) were determined by experimentation--trying numbers
and checking to see if the result is acceptable to a set of human eyes.
The near-universal use of JPEG for image compression is a testament
to its success.
Wavelets
Wavelet theory is also a form of mathematical transformation, similar to
the FT in that it takes a signal in time domain, and represents it in frequency
domain. Wavelet functions are distinguished from other transformations in
that they not only dissect signals into their component frequencies, they
also vary the scale at which the component frequencies are analyzed. Therefore
wavelets, as component pieces used to analyze a signal, are limited in space.
In other words, they have definite stopping points along the axis of a graph--they
do not repeat to infinity like a sine or cosine wave does. As a result, working
with wavelets produces functions and operators that are "sparse" (small),
which makes wavelets excellently suited for applications such as data compression
and noise reduction in signals. The ability to vary the scale of the function
as it addresses different frequencies also makes wavelets better suited to
signals with spikes or discontinuities than traditional transformations such
as the FT.
Applications of
wavelet theory are just beginning to show up in the marketplace:
JPEG2000
The Joint Photographic Experts Group (JPEG) has approved the next standard
for image compression, known as JPEG2000, based on wavelet compression
algorithms. By setting the "mother wave" for image compression
and decompression ahead of time as a part of the standard, JPEG2000
will be able to provide resolution at a compression of 200 to 1, equivalent
to current JPEG at 5 to 1.
Rainmaker Technologies
Wavelets are allowing network communications provider Rainmaker to develop
more efficient signal filtering and noise reduction algorithms. They
claim they will be able to equal fiber bandwidth on existing last mile
cable infrastructure with the technology, and greatly increase bandwidth
on other mediums such as phone or power lines.
More info:*
Introduction
to Wavelets
Rainmaker
Technologies
JPEG2000
EE
Times story about JPEG2000
*The WAVE Report
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