Mathematics Awareness Week - April 1998

Mathematics and Imaging

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Description of Theme
Mathematics is an essential element of imaging in fields as diverse as
medicine, computer science, and space exploration, and many other
settings as well. For instance, medicine benefits from the techniques
of tomography, including cutting-edge ideas like electrical impedance
tomography that may greatly improve the detection of cancerous tumors,
and from level set and marching methods that can extract images of
beating hearts from MRI images.

Computer imaging depends upon a whole range of mathematical tools.
Wavelet transforms permit efficient illumination of two-dimensional
representations of three-dimensional scenes so that the images created
by computer graphics are realistic and visually pleasing. Familiar
references like Microsoft's Encarta encyclopedia and its 7,000 color
images fit on a single CD-ROM because of fractal image compression.

Image compression is also one of the essential tools of space
exploration. For example, the scientists monitoring transmissions from
the Mars Pathfinder will be receiving data at a mere 40 bytes per
second (about 700 times slower than a typical modem!) when the
spacecraft reaches Mars. Compression routinely increases the effective
transmission rate of image data by a factor of fifteen or twenty.

New techniques are being developed that will raise compression ratios
into the hundreds while still preserving essential features of the
image. One of these new ideas, wavelet image compression , has already
enabled electronic storage and retrieval of the FBI's vast archive of
fingerprint records.

Image restoration tools can extract additional detail otherwise hidden
in images from a wide range of sources, including satellites, medical
imaging devices, telescopes, and even amateur video pictures submitted
as courtroom evidence. Images from astronomical telescopes are cleaner
to begin with if they have benefited from the control strategies of
active and adaptive optics that minimize degradation from atmospheric
blur and mechanical tremors.
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