Biography: David J. Brady is the Addy Family Professor of Electrical and Computer Engineering in the Pratt School of Engineering at Duke University. Brady's research focuses on computational optical sensors. He leads the DISP group (www.disp.duke.edu), which is pursing projects in biometric sensor networks, spectroscopic telescopy, multimodal spectroscopy for biomedical and compressive sampling for digital imaging. DISP is supported by grants from the Defense Advanced Research Projects Agency, the Air Force Office of Scientific Research, the Army Research Office, the National Institute on Alcohol Abuse and Alcoholism and the National Science Foudation. Brady holds a B.A. in physics and mathematics from Macalester College and M.S. and Ph.D. degrees in applied physics from the California Institute of Technology. He was on the faculty of electrical and computer engineering at the University of Illinois in Urbana-Champaign from 1990 until joining the Duke faculty in 2001. He was a David and Lucile Packard Foundation Fellow from 1990 until 1995.

Abstract: The history of artificial optical sensing is punctuated by three major innovations. The first innovation was the development, beginning approximately 700 years ago, of optical elements. These include lenses, prisms, gratings and mirrors. Optical elements enabled humans to see things that could not otherwise be seen. The second innovation was the development, beginning approximately 200 years ago, of photochemical recording processes. Photochemistry enabled humans to capture and store information produced by optical elements. The third innovation is the development, beginning approximately 50 years ago, of electronic recording processes and digital data analysis.

Just as 50-100 years passed between the first observations of photochemical behavior and its widespread use in photography, the implications of the electronic imaging remain unclear and in rapid development. From a mathematical perspective the most fundamental difference between electronic and photochemical photography lies in the association between the focal plane image and the display image. The photochemical display image is directly derived, through chemical processing, from the recorded focal plane image. The electronic image is created from recorded data by digital processing and need not be isomorphic in any sense to the focal plane field distribution.

As part of the disassociation of the focal plane distribution and the reconstructed image, discrete analysis plays a much greater role in digital imaging than in conventional systems. In view of this, the mathematical language of imaging is slowly evolving away from continuous transformations and toward discrete and multiscale analysis. While the instructor is not a professional mathematician, this tutorial will attempt to explain for a mathematical audience the current issues in the sampling theory of digital imaging and spectroscopy systems. In anticipation of an IMA workshop later in the fall on integrated sensing and processing, the tutorial will focus on integrated computational imaging system design, meaning joint analysis of physical layer filtering and processing and digital analysis and reconstruction algorithms.

Outline:

1. Geometric analysis of optical fields
   • Simple imaging systems
   • Coded aperture imaging
   • Tomography
   • Reference structure tomography
2. Wave analysis of optical fields
   • Fourier analysis of imaging and filtering
   • Coded wavefront imaging
   • Sampling and representation of wave fields
3. Correlation fields and interferometric imaging
   • The van Cittert Zernike theorem
   • Coherence and spectra
4. Compressive sampling and components of modern digital imaging systems
5. Spectroscopy and imaging
   • multiplex sensing in spectroscopy and imaging
   • Coded transformations in spectroscopy and imaging

Biography: Margaret Cheney is a Professor of Mathematics at Rensselaer Polytechnic Institute. Her Ph.D. in mathematics is from Indiana University; after a postdoc at Stanford University, she spent 3 years as assistant professor at Duke University before moving to RPI. She has held visiting appointments at NYU's Courant Institute (1987-1988) at the Minnesota Institue for Mathematics and Its Applications (1994-1995 and 1997), the Berkeley Mathematical Sciences Research Institute (2001), the Naval Air Warfare Center Weapons Division (2002), and the UCLA Institute for Pure and Applied Mathematics (2003). Most of her work has been on the inverse problems that arise in quantum mechanics, acoustics, and electromagnetic theory. Cheney has received several awards, including the Office of Naval Research Young Investigator Award in 1986, a National Science Foundation Faculty Award for Women in Science and Engineering in 1990, and the Lise Meitner Visiting Professorship at Lund Institute of Technology in 2000. She was a member of the Rensselaer Impedance Imaging team that received the 1993 ComputerWorld Smithsonian award in the Medicine category. She is a member of the SIAM board of Trustees, of the Electromagnetics Academy, and is a Fellow of the Institute of Physics. From 1994 to 2003, she served on the editorial board for the SIAM Journal of Applied Mathematics and was Editor-in-Chief 1995-97. She currently serves on the editorial board of Inverse Problems. She has 4 patents and roughly 90 publications, and she has given over 100 research lectures in the U.S. and Europe.

Abstract: Radar imaging is a technology that has been developed, very successfully, within the engineering community during the last 50 years. The key component that makes radar imaging possible, however, is mathematics, and many of the open problems are mathematical ones.

This tutorial will explain, in terms suitable for a mathematical audience, the basics of radar and the mathematics involved in producing high-resolution radar images.

This tutorial will help prepare participants for the upcoming workshop, and will provide them with a foundation that will enable them to read some of the theoretical engineering literature and begin research in the area.

Outline:

1. Nonimaging radar
   radar system architecture
   waveforms, correlation receiver, ambiguity function
   
   
2. Introduction to scattering
   Lippmann-Schwinger equation, Born approximation
3. Introduction to antennas
   electromagnetic vector potentials
   radiation from a current distribution
   far-field form of electric field
   examples: sinc pattern, phased arrays
   
   
4. Synthetic-aperture radar imaging
   model for the received signal
   interpretation as a Fourier Integral Operator
   approximate inversion
   point spread function and resolution
   
   
5. Properties of the image (time permitting)
   introduction to microlocal analysis
   microlocal analysis of the imaging operator
   

Recommended prerequisites are knowledge of vector calculus and the Fourier transform. Some familiarity with the wave equation would be helpful but not strictly necessary. The approach will be similar to that of the paper "A mathematical tutorial on Synthetic Aperture Radar", SIAM Review 43 (2001) 301-312.