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Illinois/Missouri Applied Harmonic Analysis Seminar


April 24, 2010

Radu Balan - University of Maryland, College Park
Signal Reconstruction From its Spectrogram
This paper presents a framework for discrete-time signal reconstruction from absolute values of its short-time Fourier coefficients. Our approach has two steps. In step one we reconstruct a band-diagonal matrix associated to the rank-one operator $K_x=xx^*$. In step two we recover the signal $x$ by solving an optimization problem. The two steps are somewhat independent, and one purpose of this talk is to present a framework that decouples the two problems. The solution to the first step is connected to the problem of constructing frames for spaces of Hilbert-Schmidt operators. The second step is somewhat more elusive. Due to inherent redundancy in recovering $x$ from its associated rank-one operator $K_x$, the reconstruction problem allows for imposing supplemental conditions. In this paper we make one such choice that yields a fast and robust reconstruction. However this choice may not necessarily be optimal in other situations. It is worth mentioning that this second step is related to the problem of finding a rank-one approximation to a matrix with missing data.

Chris Heil - Georgia Institute of Technology, Atlanta
A Selection of Problems in Time-Frequency Analysis and Wavelet Theory
This talk will survey a range of open problems in time-frequency analysis and wavelet theory, ranging from esoteric to practical. Some of the problems include the HRT conjecture on linear independence of time-frequency shifts, the Olson-Zalik conjecture on Schauder bases of translates, and frame bounds of finite wavelet and Gabor systems.

Emily King - National Institutes of Health, Washington DC
Representation theory of generalized shearlets
A number of new transformations have sprung up in an attempt to detect directional trends in $2$-$d$ data, including shearlets, for which nice algorithms and theory exist. There is a need to develop higher dimensional tools for various applications, like biomedical imaging. We exploit the representation theory of the extended metaplectic group in order to construct isotropic and anisotropic analogs of the shearlet transform over $L^2(R^d)$ for $d \geq 2$. The new representations are then analyzed and co-orbit space theory is applied in order to create discrete versions of these systems.

Maria Skopina - St. Petersburg State University, Russia
Multivariate wavelet frames and frame-like systems
In order to construct a wavelet frame with a desirable approximation order, it is necessary to provide vanishing moments for the generating wavelet functions.In the multi-dimensional case this problem is much more complicated than its one-dimensional version. In particular, two open algebraic problems are obstacles for the construction of compactly supported multivariate tight wavelet frames. Namely, first, it is not known if any appropriate row can be extended to a unitary matrix whose entries are trigonometric polynomials. Second, it is not known if any non-negative trigonometric polynomial can be represented as a finite sum of squared magnitude of trigonometric polynomials. We suggest a way to get around these obstacles and give a constructive method for the improvement an arbitrary appropriate mask to obtain a scaling mask
generating a compactly supported tight wavelet frame with a required approximation order. A method for the construction of dual wavelet frames is also developed. It appears that frame-type decompositions hold for some MRA-based wavelet systems which are not frames in L_2. Due to resent results by B.Han and Z.Shen, it may happen that such a system is a frame in the Sobolev space. We study frame-type decompositions and their approximation order in a more general situation. 

Ahmed Zayed - DePaul University, Chicago
Shift-invariant Spaces in the Fractional Fourier Transform Domain
In this talk we first introduce the factional Fourier transform and some of its applications in optics, and then proceed to discuss properties of shift-invariant spaces in the fractional Fourier transform domain.