Yale Computational Mathematics

<BODY> <P> </P> <P ALIGN=CENTER> </P> <P ALIGN=CENTER><IMG SRC="IMAGES/GIFS/YCM.GIF" WIDTH="368" HEIGHT="135" ALIGN="BOTTOM" NATURALSIZEFLAG="3"></P> <P ALIGN=CENTER> </P> <P ALIGN=CENTER><IMG SRC="IMAGES/GIFS/YALE.GIF" WIDTH="110" HEIGHT="119" ALIGN="BOTTOM" NATURALSIZEFLAG="3"></P> <P ALIGN=CENTER> </P> <P ALIGN=CENTER>To view this site, you will need a frames & tables capable browser.</P> <P ALIGN=CENTER>Both Netscape Navigator and Microsoft Explorer will work.</P> <P ALIGN=CENTER>To view the movies contained in this site, you will need Quicktime.</P> <P ALIGN=CENTER><A HREF="http://www.netscape.com/"><IMG SRC="IMAGES/GIFS/NETNOW3.GIF" WIDTH="88" HEIGHT="31" ALIGN="MIDDLE" NATURALSIZEFLAG="3" BORDER="1"></A><IMG SRC= "IMAGES/GIFS/SPACING.GIF" WIDTH="10" HEIGHT="10" ALIGN="BOTTOM" NATURALSIZEFLAG= "3"><A HREF="http://www.quicktime.apple.com/"><IMG SRC="IMAGES/GIFS/QT.GIF" WIDTH="36" HEIGHT="44" ALIGN="MIDDLE" NATURALSIZEFLAG="3" BORDER="1"></A><IMG SRC= "IMAGES/GIFS/SPACING.GIF" WIDTH="10" HEIGHT="10" ALIGN="BOTTOM" NATURALSIZEFLAG= "3"><A HREF="http://www.microsoft.com/"><IMG SRC="IMAGES/GIFS/IE.GIF" WIDTH= "88" HEIGHT="31" ALIGN="MIDDLE" NATURALSIZEFLAG="3" BORDER="1"></A></P> <P> </P> <P><FONT SIZE=+1>Yale University computational mathematics program in areas related to fast computation and processing.</FONT></P> <P><FONT SIZE=+1>Over the last few years we have tried at <A HREF="http://www.yale.edu/">Yale</A> (<A HREF="http://www.math.yale.edu/">Mathematics</A> and <A HREF="http://www.cs.yale.edu/">Computer Science</A>) 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.</FONT></P> <P><FONT SIZE=+1>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.</FONT></P> <P><FONT SIZE=+1>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. </FONT></P> <P><FONT SIZE=+1>As an outgrowth of extensive collaborative efforts in efficient algorithmic developments by <A HREF="mailto:roklhin@cs.yale.edu">V. Rokhlin</A> and his team who have introduced Fast Multipole Methods for computational compression and <A HREF="mailto:coifman@math.yale.edu">R .Coifman</A> and his collaborators at <A HREF="http://www.yale.edu/">Yale</A> and elsewhere (Y. Meyer In Paris, V. Wickerhauser In St Louis) we see emerging a general collection of methods for addressing these tasks.</FONT></P> <P><FONT SIZE=+1>Adapted Waveform Analysis is a toolkit for efficient transcription of large sets of measured data; <A HREF="AUDIO/AUDIODEN.HTM">sounds</A>, 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.</FONT></P> <P><FONT SIZE=+1>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.</FONT></P> <P><FONT SIZE=+1>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.)</FONT></P> <P><FONT SIZE=+1>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.)</FONT></P> <P><FONT SIZE=+1>For one dimensional data such as <A HREF="AUDIO/AUDIODEN.HTM">audio</A> we should think of waveforms as musical notes having three characteristics, pitch (or oscillation number), duration and intensity (or amplitude). </FONT></P> <P><FONT SIZE=+1>An efficient description of a recorded piece of music can be given by writing a musical score. </FONT></P> <P><FONT SIZE=+1>Adapted waveform analysis provides a mathematical alphabet (of notes) and a method for transcribing a recorded signal into a most efficient notation. </FONT></P> <P><FONT SIZE=+1>This is achieved by matching the patterns in the recording to a library of waveforms and selecting transcriptions optimised for the task at hand (<A HREF="HTML/NCOMPRES.HTM">compression</A>, classification, discrimination, <A HREF="HTML/NDENOISE.HTM">enhancement</A>, fast processing, etc...)</FONT></P> <P><FONT SIZE=+1>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.</FONT></P> <P><FONT SIZE=+1>Other applications for medical diagnostic and medical image enahancing as well as feature extraction have been developed. </FONT></P> <P><FONT SIZE=+1>In a joint project with<A HREF="mailto:jonathan.berger@yale.edu"> J. Berger</A> at the <A HREF="http://www.music.yale.edu/">Yale Music School</A>, 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 Brahms. </FONT></P> <P><FONT SIZE=+1>As an application of this method various remarkable enhancement methods for <A HREF="HTML/NDENOISE.HTM">Echoplanar Magnetic resonance</A> fast imaging have been obtained by <A HREF="mailto:woog-lionel@cs.yale.edu">L. Woog</A> and <A HREF="mailto:kevin.johnson@yale.edu">K. Johnson.</A></FONT></P> <P><FONT SIZE=+1>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 <A HREF="http://www.yale.edu/">Yale</A> is being pursued extensively by the <A HREF="http://www.cs.yale.edu/">Computer Science</A> and <A HREF= "http://www.math.yale.edu/">Mathematics</A> departments both for the purpose of scientific and industrial simulations as well as large scale digitized real data processing.</FONT></P> <P><A HREF="http://www.hughes.com/"><FONT SIZE=+1>Hughes Research Laboratories</FONT></A><FONT SIZE=+1> is converting various fast algorithms for Electromagnetic scattering computation into engineering software for such tasks.</FONT></P> <P><FONT SIZE=+1>Wavelet based Signal Analysis and image processing software, as well as a variety of fast numerical algorithms have been developed. </FONT></P> <P><FONT SIZE=+1>Specific application software toolkits including</FONT></P> <OL> <LI><FONT SIZE=+1>Audio denoising software for music restoration and enhancement.</FONT> <LI><FONT SIZE=+1>High quality image <A HREF="HTML/NCOMPRES.HTM">compression</A> suite of algorithms, including Fingerprint standard, enhanced JPEG, Wavelet and wavelet-packet adapted coding.</FONT> <LI><FONT SIZE=+1>Image and video denoising and enhancement code, currently being used on Echoplanar NMR images as well as radiation and synthetic aperture radar.</FONT> <LI><FONT SIZE=+1>Classification and discrimination algorithms for diagnostic and feature extraction.</FONT> <LI><FONT SIZE=+1>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 .</FONT> </OL> </BODY>