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By David F. Salisbury
January 7, 2002
A teenager turns on her portable CD
player, attaches the earphones and tunes out the rest of the world.
A tourist with a digital camera snaps a picture of his wife standing
in front of a famous monument. A neurosurgeon examines half-a-dozen
MRI images of a patient's brain as he plans the surgery needed to
remove a tumor.
These three scenes have something in
common. The people they portray are all dealing with digital representations
of one kind or another. Such representations are finding increasing
use in modern society because they are so easy to store, manipulate,
display and transmit.
In the foreseeable future, digital
music recordings may be audibly clearer, digital pictures may be
much sharper and MRI scans more precise due to a new mathematical
theory developed by mathematics professors Akram Aldroubi from Vanderbilt
University and Karlheinz Gröchenig from the University of Connecticut.
Writing in the Dec. 5 issue of the Society of Industrial and Applied
Mathematics Review, Aldroubi and Gröchenig describe a new mathematical
theory for producing digital representations of complex signals
that overcomes a number of the limitations of current methods.
Not only does the theory have application
in areas such music, photography and medical imaging, but it also
promises improvements in areas as disparate as astronomy, geophysics
and communications.
Most real-world signals, such as voices,
are "analog" in nature. That is, they vary continuously over time
or space. As a result, they contain a tremendous amount of information.
Before these signals can be saved and manipulated in computers,
however, they must be converted to digital form. This is done by
a process called sampling. The strength of the analog signal, say
the sound of a clarinet, is measured, or sampled, at regular intervals.
As a result, digital depictions are
not perfect. But the shorter the sampling interval and the more
accurate the measurements, the more realistic the resulting digital
representation tends to be. Still, the string of ones and zeros
burned onto the surface of a music CD does not completely capture
the sound of a live band. Nor can a digital picture reproduce all
the features of a landscape that the human eye perceives. Similarly,
an MRI image of the brain provides only an approximate representation
of the contours of gray matter hidden beneath the skull.
"Our theory - which is based on a lot
of beautiful new mathematics - can produce more accurate digital
representations of all kinds of samples, including those that classical
methods handle poorly or cannot handle at all," says Aldroubi. "It
generates algorithms [sets of mathematical procedures] that are
fast, efficient, stable and robust."
Traditional sampling procedures date
back half a century to the work of Claude Shannon, the Bell Labs
mathematician who laid the foundation of modern information theory.
These methods generally require that the sampling be done at regular
intervals. Classical techniques also require that the original signal
be "band limited" - a technical term meaning that the signal must
stay within certain, defined limits. Take the case of music. Because
human hearing does not extend above 20,000 hertz (cycles per second),
extremely accurate digital representations can be made with digital
representations that totally ignore sounds above this frequency.
The new theory, however, handles situations
where the sampling is non-uniform and the signal is not band-limited.
A number of important applications
stand to benefit. Oil and gas exploration, for example, makes heavy
use of geophysical data collected at locations that frequently deviate
from a regular grid pattern due to factors like variations in surface
terrain. This is analogous to the intermittent nature of astronomical
observations when light from a given star is blocked by clouds passing
overhead.
Much of the impetus for developing
the new sampling theory comes from the need for improving the quality
of medical imaging. Aldroubi worked on sampling problems at the
National Institutes of Health before coming to Vanderbilt. One of
the figures in the paper shows three MRI images. The first is the
original image. The second is the same image peppered with black
splotches where 50 percent of the data has been removed. The third
image was produced by one of the new reconstruction algorithms,
which accurately restored all the major features in the original
image.
In an attempt to make the new theory
apply to the widest range of possible applications, Aldroubi and
Gröchenig also designed it to take into account two other important
factors that are not covered by the classical theory. It deals explicitly
with the real-world problem of noise. Similarly, it takes the actual
characteristics of the sampling device into account, unlike current
methods that assume the measurements are perfect.
The new approach can produce some
startling effects. In one exercise, for example, Aldroubi takes
a large MRI image, shrinks it down to a small size and then expands
it back to the same size as the original. The original and reconstructed
images look almost identical, and a pixel-by-pixel comparison shows
that only a minute amount of information has been lost.
The research was funded by the National
Science Foundation.
Prof. Aldroubi's
home page:
http://atlas.math.vanderbilt.edu/~aldroubi/
Preprint of SIAM Review paper:
http://atlas.math.vanderbilt.edu/~aldroubi/preprints.html
SIAM Review paper:
http://epubs.siam.org/sam-bin/dbq/article/38698
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