No previous knowledge of wavelet analysis will be assumed.

Twelve sessions, about 90 minutes each, with extra time for discussions:

1. The Haar multiresolution analysis
2. Subband coding
3. Orthogonal and biorthogonal filter banks
4. From filter banks to wavelets
5. Sample applications using MATLAB
6. Continuous time-scale analysis
7. Continuous time-frequency analysis
8. Bases and frames
9. Sampling frames
10. The lifting method
11. Integer transforms and lossless coding
12. Software demonstrations
    
    Note: The new JPEG2000 and MPEG-4 standards are based on wavelet compression technology.

COURSE OUTLINE

1. Haar analysis, discovered in 1910, already displays the essential elements of modern wavelet theory in their simplest form. The fast Haar transform (FHT) is a good alternative to the fast Fourier transform (FFT).
2. We use the FHT as a paradigm to develop general subband coding schemes based on filter banks.
3. Orthogonal filters generalize the FHT. Symmetric biorthogonal filters are preferable for compression. Biorthogonal spline filters are an example.
4. Multiresolution analysis} embeds subband coding in continuous time. This leads from filter banks to continuous-time wavelets and reveals the stability of subband coding under recursions.
5. Subband coding is extended to 2-D and applied to image analysis and compression, with many MATLAB examples. Questions addressed include: What makes an algorithm good for compression?
6. We extend the wavelet transform to continuous time and scale and show how to filter in scale.
7. Continuous time-frequency analysis is viewed as the narrowband limit of continuous time-scale analysis.
8. Frames unify several ideas: orthogonal bases, biorthogonal bases, and continuous coherent-state representations as known in physics. This will be the basis for constructing physical wavelets.
9. Sampling is possible in domains other than time and frequency. The ensuing sampling theorems include discrete wavelet and time-frequency representations.
10. The lifting method is a very flexible way to construct fast wavelet transforms by breaking them into constituent ``atoms.'' The resulting algorithms are simpler, faster, and require less memory since they can be implemented "in-place." They are also more general, being able to deal naturally with boundaries and irregular sampling.
11. Lifting can also be used to modify every FWT to an integer-valued algorithm, giving efficient and lossless coding by avoiding roundoff errors. We derive the two simplest examples, the S-transform and the ST transform, which are integer-valued versions of the Haar transform and the biorthogonal transform of order (3,1).
12. Demonstrations will be given of wavelet software.

Short Courses taught 1994 - 2000

• 11/13-16/00: A Detailed Introduction to Wavelets
  with Applications to Image Analysis and Lossless Coding
  The Virginia Center for Signals and Waves
  
  
• 5/29/00: Radar Analysis with Causal Pulsed-Beam Wavelets
  EuroElectromagnetics (EUROEM) 2000 Conference, Edinburgh, Scotland
  
  
• 5/1-4/00: A Detailed Introduction to Mathematical and Physical Wavelets
  Applied Technology Institute, Newport, RI
  
  
• 11/29-12/3/99: A Detailed Introduction to Mathematical and Physical Wavelets
  Applied Technology Institute, Arlington, VA
  
  
• 9/20-23/99: A Detailed Introduction to Wavelets with Applications to Image Analysis and Lossles Coding
  Given at Eastman Kodak Company, Rochester, NY
  
  
• 3/8-11/99: A Detailed Introduction to Mathematical and Physical Wavelets
  Applied Technology Institute, Boston, MA
  
  
• 5/11-14/98: A Detailed Introduction to Mathematical and Physical Wavelets
  Given at the European Space Agency (ESA/ESTEC), Noordwijk, The Netherlands
  
  
• 12/16-18/97: Mathematics and Physics of Wavelets
  Applied Technology Institute, Brookline, MA
  
  
• 7/30/97: Introduction to Radar via Physical Wavelets
  Internat. Soc. for Optical Engineering (SPIE), San Diego, CA
  
  
• 4/21/97: Remote Sensing via Physical Wavelets
  Internat. Soc. for Opt. Eng. (SPIE) AeroSense, Orlando, FL
  
  
• 3/21/97: Radar via Physical Wavelets
  IEEE Appl. Comp. Electromagnetics Soc. (ACES) Conference, Monterey, CA
  
  
• 2/10/97: Wavelet Fundamentals
  Internat. Soc. for Opt. Eng. (SPIE) Photonics West, San Jose, CA
  
  
• 12/17-19/96: Mathematics and Physics of Wavelets
  The Applied Technology Institute, College Park, MD
  
  
• 8/4/96: Wavelet Transforms: Theory and Applications
  Internat. Soc. for Opt. Eng. (SPIE) 1996, Denver, CO
  
  
• 7/21/96: Electromagnetic Wavelets in Radar and Sonar
  IEEE Antennas & Prop. Soc. Internat. Symp. and URSI Radio Science Meeting, Baltimore, MD
  
  
• 6/4-6/96: Mathematics and Physics of Wavelets
  The Applied Technology Institute, College Park, MD
  
  
• 5/16/96: Applications of Electromagnetic Wavelets to Radar and Sonar
  IEEE 1996 National Radar Conference, Ann Arbor, MI
  
  
• 4/8/96: Introduction to Wavelets
  Internat. Soc. for Opt. Eng. (SPIE) AeroSense, Orlando, FL
  
  
• 6/6-8/95: Mathematics and Physics of Wavelets
  The Applied Technology Institute, College Park, MD
  
  
• 5/9/95: Physical Wavelets with Applications to Radar and Sonar
  IEEE International Radar Conference, Alexandria, VA
  
  
• 3/20/95: Physical Wavelets, with Applications to Remote Sensing
  IEEE Appl. Comp. Electromagnetics Soc. (ACES) Conference, Monterey, CA
  
  
• 9/2/94: Physical Wavelets
  Ohio State University, Columbus, OH
  
  
• 9/1/94: Physical Wavelets in Electromagnetics, Radar and Acoustics
  Air Force Institute of Technology, Dayton, OH
  
  
• 6/94-8/94: Wavelet Analysis, with Applications to Geophysics
  Phillips Laboratory, Hanscom Air Force Base, MA
  
  
• 3/21/94: Wavelet Electrodynamics
  IEEE Appl. Comp. Electromagnetics Soc. (ACES) Conference, Monterey, CA

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