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Yale University computational mathematics program in areas
related to fast computation and processing.
Over the last few years we have tried at Yale
(Mathematics and Computer
Science) to come up with a broad collection of tools enabling a comprehensive
approach the problem of efficient computation and processing of large scale
digitized analog data sets.
The situation now resembles roughly the state of affairs
in arithmetic computation before the invention of a good notation for numbers,
where various computations were carried out in strange ways by the Greeks
and other cultures. The ability to have a good digital or binary representation
for numbers permitted to automate the process.
Currently the main obstacle is our ability to represent
functions (or digitized measured data, such as images or audio) efficiently.
A good notational scheme for both storage and computation is essential to
surmount the complexity hurdle.
As an outgrowth of extensive collaborative efforts in efficient
algorithmic developments by V. Rokhlin
and his team who have introduced Fast Multipole Methods for computational
compression and R .Coifman and
his collaborators at Yale and elsewhere
(Y. Meyer In Paris, V. Wickerhauser In St Louis) we see emerging a general
collection of methods for addressing these tasks.
Adapted Waveform Analysis is a toolkit for efficient transcription
of large sets of measured data; sounds,
images, seismic vibrations, atmospheric pressure maps, electrocardiograms,
etc. The main issue involves "feature extraction" for the analysis
of the data, for diagnostic recognition, storage purposes and transcription.
As a result, the storage requirements are lowered substantially.
Moreover, tasks involving extraction of features, elimination of noise,
diagnostic analysis are considerably simplified and accelerated.
For two dimensional data such as images or three dimensional
data such as video we should think of the waveform as simple patterns or
templates which can be used by superposition to resenthetise the data in
a way somewhat analoguous to overlaying various simple transparencies to
form a complex image. This "musical notation" for images and video
is a powerful processing tool enabling computational tasks which are otherwise
impossible to achieve even on the most elaborate hardware. (for example
in areas of acoustic and electromagnetic scattering, or even in elaborate
tasks as image rendering.)
This is achieved by describing the data in terms of carefully
selected collection of templates called waveforms. (wavelets, wavelet-packets
and Local Trigonometric Waveforms are used as our elementary "atoms"
for feature extraction or "alphabet" for data storage.)
For one dimensional data such as audio
we should think of waveforms as musical notes having three characteristics,
pitch (or oscillation number), duration and intensity (or amplitude).
An efficient description of a recorded piece of music can
be given by writing a musical score.
Adapted waveform analysis provides a mathematical alphabet
(of notes) and a method for transcribing a recorded signal into a most efficient
This is achieved by matching the patterns in the recording
to a library of waveforms and selecting transcriptions optimised for the
task at hand (compression, classification,
discrimination, enhancement, fast processing,
To date algorithms developed at Yale have been extensively
tested and adapted by the FBI and Scotland Yard as being the most efficient
method for storing digitally their library of fingerprint cards, permitting
a 20 fold saving in storage and transmission time.
Other applications for medical diagnostic and medical image
enahancing as well as feature extraction have been developed.
In a joint project with
J. Berger at the Yale Music School,
an old recording of Brahms, made by Thomas Edison, is being processed for
enhancement as a "musical archeological dig". A big problem in
this noisy recording involves finding all of the actual notes played by
As an application of this method various remarkable enhancement
methods for Echoplanar Magnetic resonance
fast imaging have been obtained by L.
Woog and K. Johnson.
This ability to represent complex physical functions with
few parameters is a remarkable tool for fast and efficient numerical simulations
of natural phenomena. This program of numerical compression which is unique
to Yale is being pursued extensively
by the Computer Science and Mathematics departments both for the purpose
of scientific and industrial simulations as well as large scale digitized
real data processing.
Hughes Research Laboratories is converting various fast algorithms for Electromagnetic scattering
computation into engineering software for such tasks.
Wavelet based Signal Analysis and image processing software,
as well as a variety of fast numerical algorithms have been developed.
Specific application software toolkits including
- Audio denoising software for music restoration and enhancement.
- High quality image compression
suite of algorithms, including Fingerprint standard, enhanced JPEG, Wavelet
and wavelet-packet adapted coding.
- Image and video denoising and enhancement code, currently
being used on Echoplanar NMR images as well as radiation and synthetic
- Classification and discrimination algorithms for diagnostic
and feature extraction.
- Fast numerical codes for Scientific and engineering tasks,
including various fast multipole based algorithms for a variety of elliptic
solvers and for acoustic and electromagnetic scattering computations .