Illinois/Missouri Applied Harmonic Analysis Seminar
April 24, 2010
- University of Maryland, College Park
From its Spectrogram
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
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.
- Georgia Institute of Technology, Atlanta
Time-Frequency Analysis and Wavelet Theory
will survey a range of open problems in time-frequency analysis
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 finite wavelet and Gabor systems.
- National Institutes of Health, Washington DC
of generalized shearlets
of new transformations have sprung up in an attempt to
directional trends in $2$-$d$ data, including shearlets, for
algorithms and theory exist. There is a need to develop higher
tools for various applications, like biomedical imaging.
exploit the representation theory of the extended metaplectic
in order to construct isotropic and anisotropic analogs of
shearlet transform over $L^2(R^d)$ for $d \geq 2$. The new representations
then analyzed and co-orbit space theory is applied in order
create discrete versions of these systems.
- St. Petersburg State University, Russia
Multivariate wavelet frames and frame-like
to construct a wavelet frame with a desirable approximation
it is necessary to provide vanishing moments for the generating wavelet
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
can be extended to a unitary matrix whose entries are trigonometric
Second, it is not known if any non-negative trigonometric polynomial
can be represented as a finite sum of squared magnitude of
We suggest a way to get around these obstacles and give a constructive
for the improvement an arbitrary appropriate mask to obtain a scaling
tight wavelet frame with a required approximation
construction of dual wavelet frames is also developed.
decompositions hold for some MRA-based wavelet
in L_2. Due to resent results by B.Han and Z.Shen,
such a system is a frame in the Sobolev space. We
and their approximation order in a
more general situation.
the Fractional Fourier Transform Domain
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.