September 19 - 23, 2005
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:
- Geometric analysis of optical fields
- Simple imaging systems
- Coded aperture imaging
- Tomography
- Reference structure tomography
- Wave analysis of optical fields
- Fourier analysis of imaging and filtering
- Coded wavefront imaging
- Sampling and representation of wave fields
- Correlation fields and interferometric imaging
- The van Cittert Zernike theorem
- Coherence and spectra
- Compressive sampling and components of modern digital imaging systems
- 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:
- Nonimaging radar
radar system architecture
waveforms, correlation receiver, ambiguity function
- Introduction to scattering
Lippmann-Schwinger equation, Born approximation
- Introduction to antennas
electromagnetic vector potentials
radiation from a current distribution
far-field form of electric field
examples: sinc pattern, phased arrays
- Synthetic-aperture radar imaging
model for the received signal
interpretation as a Fourier Integral Operator
approximate inversion
point spread function and resolution
- 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.