Research Programs

The Norbert Wiener Center is built upon a strong foundation of innovative research in the theory and application of harmonic analysis, signal processing, and multi-resolution analysis. The group pursues fundamental discoveries in these research areas both for their own sake and for their further development into sophisticated mathematical tools enabling critical advances in key technology areas. Our group has successfully applied state-of-the-art mathematics to an extensive set of modern technologies including remote sensing, medical imaging, communications, and "smart" sensor systems.

In its applied work, our research group works by directly engaging domain experts in specific technology areas in order to help us quickly grasp the major underlying issues and to be prepared to respond quickly to new opportunities. In the last year we have pursued this strategy by collaborating with engineers and scientists at several Maryland institutions including Northrop Grumman, the NIH, the Naval Research Lab, and the Army Research Lab, as well as with faculty members in several departments at the University of Maryland and the Center for Imaging Science at the Johns Hopkins University. We also work closely with technologists outside the state at institutions including the Systems Planning Corporation and Naval Surface Warfare Center in Virginia, Raytheon Missile Systems in Arizona, the SPAWARS System Center and Computational Sensors Corporation in California, Fast Mathematical Algorithms and Hardware in Connecticut, as well as engaging researchers in many disciplines at several universities.

Our group's work is recognized for its innovation, and our efforts have been recognized and supported in the last year by DARPA, NSF, and ONR, as well as industry subcontracts. Expositions of the work have been presented in a wide variety of journals, workshops, and meetings sponsored by the leading mathematics and engineering professional societies.

The experience of our researchers suggests that the exciting and challenging mathematical questions arising in technological problems rival those provided by more traditional sources of mathematical inspiration. In our efforts to deal with these challenges we have developed a wide variety of mathematical tools including methodologies for device modeling and simulation, for the analysis of analog and mixed-signal electronic components, for the optimization and adaptation of communication and sensing waveforms, as well as algorithms for robust and efficient pattern analysis and control in adaptive sensor systems.

Our efforts are aimed at solving real problems in practical technological systems in order to produce tangible benefits. A few current examples of these include the following:

With bioengineers at Louisiana Tech we are pursuing theoretical and experimental approaches in ultrasound velocimetry aimed at imaging arterial flow for noninvasive diagnosis of thrombosis and related health problems in the vasculature.

With guidance from the Navy we are developing new families of radar and communication waveforms and their associated signal processing to support adaptive multifunctional Radio Frequency systems of interest to both Defense Department and commercial entities We are applying new results from harmonic analysis and ergodic theory to the problems of fading, multipath, and interference problems which are familiar to every cell phone caller who has posed the question: "Can you hear me now?"

With our partners in industry and with guidance from the Missile Defense Agency we are working on a next generation of infrared staring array video imaging technologies capable of dealing with the tremendous real-time throughput requirements of missile defense. Our work here includes analysis and control strategies for adaptive analog image processing hardware integrated into these architectures.

Further examples of our group's efforts follow:

Algorithm for fast data acquisition in magnetic resonance imaging (MRI)

We have applied an arsenal of techniques to the difficult problem of long imaging times in standard MRI scanners. These approaches range from sophisticated classical harmonic analysis dealing with Beurling densities to state of the art linear algebra, auto focus techniques, and computer simulations. Several viable algorithms for this important problem are being constructed, and one of these has been tested in an actual scanner to image joint motion.

Concurrent signal processing algorithms

Fast implementation of concurrent signal processing algorithms is essential in multifunctional sensor and communication arrays. We have documented and evaluated the interaction of various useful signal-processing utilities e.g., the FFT, for several computing important platforms. This paves the way towards optimal real time performance for signal processing applications comprised of many concurrent signal-processing primitives.

Waveform design

This research is fundamental in current radar and communications theory problems, e.g., in cell phones and other CDMA-type technology. Our results use discrepancy theory (from number theoretic uniform distribution theory), Wiener's generalized harmonic analysis, and ergodic theory. Our constructions and implementations are for generalized versions of the Shapiro codes and for constant amplitude zero autocorrelation (CAZAC) codes, where we have developed a user-friendly toolbox evaluating the impact of Doppler shift and various noises in radar problems.

Sigma-Delta quantization

This nonlinear process is an essential component of analog to digital conversion in signal processing. Effective high resolution, high bandwidth Sigma-Delta quantization algorithms are required in a host of applications, and we are constructing such algorithms in the context of the theory of frames, where we have an expertise. Basic Fourier analysis, tiling constructions, and algebraic number theory are tools.

Spectral wavelet sets

The recursion theory we have developed to construct single dyadic wavelets for multidimensional spaces is exciting because of applicable problems, e.g., in medical imaging, because its generality is comparable to the qualitative representation theoretic approach, and because the geometric structure of these sets leads to a host of fascinating problems.

Uncertainty principles and signal decomposition

Quantitative forms of the Heisenberg inequality and Balian-Low uncertainty principle play a major role in evaluation the effective of various Weyl-Heisenberg or wavelet decompositions. Our theory has been developed for the Weyl-Heisenberg case and we have proved fine quantitative estimates as well as the developing the theory for the case of the Euclidean symplectic form for simplifying Hamiltonian systems.

p-adic wavelet theory

Wavelet theory is usually formulated in the Euclidean setting, because of applications in signal processing, including those from our own group. Now we have a wavelet theory for locally compact abelian groups with compact open subgroups, so that we can address p-adic and adelic number-theoretic problems.