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UPCOMING TALK
May 16, 2007
Joint NWC
- CSCAMM seminar.
Title: The Surprising Structure of Gaussian Point Clouds and its
Implications for Signal Processing
Speaker:
Jared Tanner (University of Utah),
Contact:
tanner@math.utah.edu
Please notice the time change for
this talk 12:15PM
Abstract
We will explore connections between the
structure of high-dimensional convex polytopes and information
acquisition for compressible signals. A classical result in the
field of convex polytopes is that if N points are distributed
Gaussian iid at random in dimension n<<N, then only order (log N)^n
of the points are vertices of their convex hull. Recent results show
that provided n grows slowly with N, then with high probability all
of the points are vertices of its convex hull. More surprisingly, a
rich "neighborliness" structure emerges in the faces of the convex
hull. One implication of this phenomenon is that an N-vector with k
non-zeros can be recovered computationally efficiently from only n
random projections with n=2e k log(N/n). Alternatively, the best
k-term approximation of a signal in any basis can be recovered from
2e k log(N/n) non-adaptive measurements, which is within a log
factor of the optimal rate achievable for adaptive sampling.
Additional implications for randomized error correcting codes will
be presented.
This work was joint with David L. Donoho.
 Back
Joint NWC
- CSCAMM seminar.
Title: The Surprising Structure of Gaussian Point Clouds and its
Implications for Signal Processing
Speaker:
Jared Tanner (University of Utah),
Contact:
tanner@math.utah.edu
Please notice the time change for
this talk 12:15PM
Abstract
We will explore connections between the
structure of high-dimensional convex polytopes and information
acquisition for compressible signals. A classical result in the
field of convex polytopes is that if N points are distributed
Gaussian iid at random in dimension n<<N, then only order (log N)^n
of the points are vertices of their convex hull. Recent results show
that provided n grows slowly with N, then with high probability all
of the points are vertices of its convex hull. More surprisingly, a
rich "neighborliness" structure emerges in the faces of the convex
hull. One implication of this phenomenon is that an N-vector with k
non-zeros can be recovered computationally efficiently from only n
random projections with n=2e k log(N/n). Alternatively, the best
k-term approximation of a signal in any basis can be recovered from
2e k log(N/n) non-adaptive measurements, which is within a log
factor of the optimal rate achievable for adaptive sampling.
Additional implications for randomized error correcting codes will
be presented.
This work was joint with David L. Donoho.
Back
Title: A critical-exponent Balian-Low theorem
Speaker:
Zubin Sushrut Gautam (UCLA),
Contact:
sgautam@math.ucla.edu
Please notice the time change for
this talk
3:00PM
Abstract
The classical Balian-Low Theorem
is a manifestation of the uncertainty principle, presenting high
time-frequency regularity as an obstruction to a function's
generating a Gabor frame for L2(IR). The Zak transform
allows us to interpret this result as a consequence of the endpoint
Sobolev embedding into VMO. Using a variant of this Sobolev
embedding theorem, we can prove a version of the Balian-Low Theorem
for modified time-frequency regularity conditions; this answers a
question raised in a paper of Benedetto, Czaja, Powell, and Sterbenz.
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Title: Not Quite a Canonical Form
Speaker:
Chandler Davis (University of Toronto),
Contact:
davis@math.toronto.edu
Abstract
There is a canonical form for a contractive
operator from one Hilbert space to another; it is
given by the polar resolution, together with the
spectral theorem. If we seek a canonical form for
a contractive operator together with a distinguished
subspace of the domain space, we are bound to be
disappointed. I will discuss the general form for
this problem, explain why it doesn't quite qualify
as "canonical" in the same sense, and show some
wonderful things it can nevertheless do for us.
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Joint
Statistics - NWC seminar.
Speaker:
Dennis Healy (UMCP)
Contact:
dhealy@math.umd.edu
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Title: Construction of sampling theorems for unions of shifted
lattices.
Speaker:
David Walnut (George Mason University),
Contact:
dwalnut@gmu.edu
Abstract
The classical sampling theorem permits
reconstruction of a bandlimited function from its values on a shifted
lattice. This work, which is joint with Hamid Behmard from Western
Oregon University and Adel Faridani from Oregon State, considers
sampling sets which are unions of possibly different shifted lattices.
The approach is based on a suitable decomposition of the domain of
support of the function's Fourier transform. Two methods to construct
such decompositions are given, and it is demonstrated that the
decompositions can be used to construct sampling theorems and recursive
reconstruction algorithms.
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Title: From Wavelets to Shearlets:
Introducing Directionality into Traditional Multiscale Analysis
Speaker:
Gitta Kutyniok (Princeton University)
Contact:
kutyniok@math.princeton.edu
Abstract
In data analysis, one main focus of
current research is on the development of directional representation
systems which precisely detect orientations of singularities like
edges in a 2-D image while providing optimally sparse
representations. The shearlet systems are the first directional
representation systems, which not only possess those properties, but
are moreover equipped with a rich mathematical structure similar to
wavelets.
In this talk we will first give an introduction into the theory of
shearlets. We will present the main properties of both the
continuous and discrete shearlet transform, thereby in particular
illustrating the detection of orientations and emphasizing the
usefulness of the structural similarity to wavelets. We will then
discuss some very recent results on improving the accuracy of the
shearlet transform, on Banach spaces associated with the decay of
the shearlet coefficients, and on shearlet subdivision schemes
aiming at a fast algorithm for shearlet decomposition using
FIR-filters.
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Title: A unified
approach to iterative thresholding algorithms for sparse recovery
Speaker:
Massimo Fornaiser (Princeton University)
Contact:
massimo.fornasier@oeaw.ac.at ,
mfornasi@math.princeton.edu
Since the work of Donoho and
Johnstone, soft and hard thresholding operators have been
extensively studied for denoising of digital signals, mainly in a
statistical framework. Usually associated to wavelet or curvelet
expansions, thresholding allows to eliminate those coefficients
which encode noise components. The assumption is that signals are
sparse in origin with respect to such expansions and that the
effect of noise is essentially to perturb the sparsity structure
by introducing non zero coefficients with relatively small
magnitude. While a simple and direct thresholding is used for
statistical estimation of the relevant components of an explicitly
given signal, and to discard those considered disturbance, the
computation of the sparse representation of a signal implicitly
given as the solution of an operator equation or of an inverse
problem requires more sophistication. We refer, for instance, to
deconvolution and superresolution problems, image recovery and
enhancing, and problems arising in geophysics and brain imaging.
In these cases, thresholding has been combined with classical
Landweber iterations to compute the solutions. In this talk we
present a general theory of iterative thresholding algorithms
which includes soft, hard, and the so-called firm thresholding
operators. In particular, we develop a unified variational
approach of such algorithms which allows for a complete
characterization of their convergence properties. As a matter of
fact, despite of their simplicity which makes them very appealing
to users and their enormous impact for applications, iterative
thresholding algorithms converges very slowly and might be
impracticable in certain situations. By analyzing their typical
convergence dynamics we propose acceleration methods based 1. on
projected gradient iterations, 2. on alternating subspace
corrections (domain decompositions.) Also for both these latter
families of algorithms, a variational approach is fundamental in
order to correctly analyse the convergence.
The talk partially summarizes recent joint results with Ingrid
Daubechies, Ron DeVore, Sinan Gunturk, and Holger Rauhut.
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Title: A Semiparametric Approach
to Time Series Prediction.
Speaker: Benjamin Kedem (UMCP)
Contact:
bnk@math.umd.edu
Abstract
Given m time series regression models, linear
or not, with additive noise components, it is shown how to
estimate the predictive probability distribution of all the time
series conditional on the
observed and covariate data at the time of prediction. This
is done by a certain synergy argument, assuming that the distributions
of the residual components associated with the regression models are
tilted versions of a reference distribution. Point predictors are
obtained from the predictive distribution as a byproduct. Applications
to US mortality rates prediction and to value at risk (VaR) estimation
will be discussed. A connection with harmonic analysis will be pointed
to.
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Title: Analysis
and Improvements of the Total Variation Method for Wavelet-based
Denoising.
Speaker: Glenn Easley (System Planning
Corporation)
Contact:
geasley@sysplan.com
Back
Title: Denoising
Natural Color Images
Speaker:
Yang Wang (NSF / Georgia Institute of Technology),
Contact:
wang@math.gatech.edu
Abstract
One of the big challenges in digital
photography is denoising. With the ever increasing demand for higher
pixel count in a digital camera noise becomes increasingly an issue that
needs to be addressed. Even with high end digital cameras the photos can
be very noisy under low lighting and artificial lighting. Although image
denosing has been studied extensively, many of these schemes are
evaluated using artificial pixelwise independent Gausssian noise. For
natural color images this may not be entirely realistic.
In this talk we will give a brief overview on color imaging and image
denoising. We'll also introduce several techniques for denoising natural
color images, including a new color space specifically for this purpose
and the Multiscale Total Variation denoising scheme, a wavelet based PDE
technique. We show that combined together they lead to very effective
denoising of natural color images.
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Title: A new
effective multidimensional spectral factorization algorithm
Speaker:
Lasha Ephremidze (ISR / A. Razmadze Mathematical Institute)
Contact:
lephremi@umd.edu
,
05k054@scc.u-tokai.ac.jp
Abstract
Spectral Factorization appeared first in
N.Wiener's and A.N.Kolmogorov's papers as a mathematical method for
solving certain estimation and prediction problems for stochastic
processes. Namely Wiener in 1941, in an effort to make his own
contribution to the war effort, was led to formulate the antiaircraft
fire control problem as a linear prediction problem the solution of
which required spectral factorization. Wiener also studied the problem
of filtering signals out of noise and noted that similar approaches
could be used for other problems in control theory and circuit theory.
In fact, although Wiener's solution was not effective for the
anti-aircraft problem, his emphasis on the statistical nature of
communication problems and on seeking solutions that met specified
optimization criteria influenced heavily the development of mathematical
system theory. Spectral factorization has since found further
applications in various branches of engineering and economics. Due to
its importance, it is not surprising that numerous methods were
developed for its solution.
Spectral factorization can be explicitly written, and is relatively easy
to calculate, for scalar valued processes. However, even in the case of
rational power spectrum, whenever the order of the polynomials involved
is large, the factorization requires some non-trivial special methods.
In addition to a few traditional methods of scalar spectral
factorization named after Kolmogorov, Bauer, Levinson-Durbin, Wilson,
etc. there exist also some relatively new methods as well.
A significantly more difficult task is to actually compute the spectral
factor of a matrix-valued function, which arises in the case of
vector-valued observations. The lack of efficient methods for such
calculations is considered to be a major bottleneck that makes many
theoretical developments in multidimensional signals and systems
infeasible. Since Wiener's original efforts to create a computable
method of multidimensional spectral factorization, dozens of papers
addressed the development of appropriate algorithms. Nevertheless, the
methods derived so far, even those that are algorithmic and very
general, are far from being numerically sound computational procedures
when the dimension of the factorized matrix is high. Specifically, it
was assumed so far that the most computationally satisfactory way of
computing the spectral factors is by solving a discrete-time algebraic
Riccati equation, which is unfortunately a nonlinear equation with
non-unique solution.
I will present a completely new approach to the multi-dimensional
spectral factorization problem which was originated by my former adviser
Prof. G.Janashia together with his students. Speaking about the
advantages of the proposed method I will mention the following: 1)
Methods of Complex Analysis are used for the solution of the problem;
this turns out to be very effective since the problem itself is set in
this branch of mathematics (other methods mostly use Functional Analysis
or some System-theoretic approaches); 2) A dense class of matrix
functions is identified for which an explicit spectral factorization is
constructed; 3) Using flexible manipulations, the hard technical
difficulties of the problem are completely absorbed by the mathematics
leaving very few and simple procedures for computation. Namely, the
multi-dimensional spectral factorization is reduced to the scalar
factorization problem by solving in addition only one system of linear
algebraic equations; 4) Although this system of linear equations might
sometimes be of high dimension in order to achieve good accuracy, it is
always positive definite and enjoys the so called displacement structure
which further reduces the computational burden. In summary the
longstanding mathematical problem, initiated by Wiener, of finding a
computationally reliable matrix spectral factorization algorithm is
solved.
During my talk I will describe the details of the proposed algorithm. A
corresponding software implementation which proves the efficiency of the
method will be demonstrated as well. I will mention also some
connections of the established method of spectral factorization with the
Corona Problem (solved by Carleson).
Back
Joint
Statistics - NWC seminar.
Title: A Multivariate Statistical Approach to Performance
Analysis of Wireless Communication Systems.
Speaker: Siamak Sorooshyari (Lucent
Technologies - Bell Laboratories)
Contact:
sorooshyari@alcatel-lucent.com
Time and room change
3:30PM, Math1313
Abstract
The explosive growth of
wireless communication technologies has placed paramount importance on
accurate performance analysis of the fidelity of a service offered by a
system to a user. Unlike the channels of wireline systems, a wireless
medium subjects a user to time-varying detriments such as multipath
fading, cochannel interference, and thermal receiver noise. As a
countermeasure, structured redundancy in the form of diversity has been
instrumental in ensuring reliable wireless communication characterized
by a low bit error probability (BEP). In the performance analysis of
diversity systems the common assumption of uncorrelated fading among
distinct branches of system diversity tends to exaggerate diversity gain
resulting in an overly optimistic view of performance. A limited number
of works take into account the problem of statistical dependence. This
is primarily due to the mathematical complication brought on by relaxing
the unrealistic assumption of independent fading among degrees of system
diversity.
We present a multivariate statistical approach to the performance
analysis of wireless communication systems employing diversity. We show
how such a framework allows for the statistical modeling of the
correlated fading among the diversity branches of the system users.
Analytical results are derived for the performance of maximal-ratio
combining (MRC) over correlated Gaussian vector channels. Generality is
maintained by assuming arbitrary power users and no specific form for
the covariance matrices of the received faded signals. The analysis and
results are applicable to binary signaling over a multiuser single-input
multiple-output (SIMO) channel. In the second half of the presentation,
attention is given to the performance analysis of a frequency diversity
system known as multicarrier code-division multiple-access (MC-CDMA).
With the promising prospects of MC-CDMA as a predominant wireless
technology, analytical results are presented for the performance of MC-CDMA
in the presence of correlated Rayleigh fading. In general, the empirical
results presented in our work show the effects of correlated fading to
be non-negligible, and most pronounced for lightly-loaded communication
systems.
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February Fourier Talks at the Norbert Wiener Center
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Title: Wiener's
Generalized Harmonic Analysis and Waveform Design
Speaker: Somantika
Datta (UMCP),
Contact:
soma@math.umd.edu
Room change
Math1310
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Title: Density
of Gabor Frames
Speaker:
Emily King (UMCP)
Contact:
eking@math.umd.edu
Back
Title: Riesz
sequences of exponentials and the Feichtinger conjecture.
Speaker:
Darrin Speegle (Saint Louise University)
Contact:
speegled@slu.edu
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Title: Uncertainty
principles for Gabor systems and the Zak transform.
Speaker:
Wojciech Czaja (UMCP)
Contact:
wojtek@math.umd.edu
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Title: Weak
uncertainty principles for fractals, graphs, and manifolds.
Speaker:
Kasso Okoudjou (UMCP)
Contact:
kasso@math.umd.edu
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Title: Certain
properties of convolution operators and Fourier transform in the affine
group.
Speaker:
Abdelkrim Bourouihiya (UMCP)
Contact:
karim@math.umd.edu
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Joint CSCAMM - Math - NWC seminar.
Title: Fast Algorithms for Sparse
Analysis.
Speaker:
Anna Gilbert (University of Michigan),
Contact:
annacg@umich.edu
Time change
3:30PM
Abstract
I will present several extremely fast
algorithms for recovering a compressible signal from a few linear
measurements. These examples span a variety of orthonormal bases,
including one large redundant dictionary. As part of the presentation of
these algorithms, I will give an explanation of the crucial role of
group testing in each algorithm.
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Title: Can we
improve JPEG2000 using shearlets?
Speaker: Glenn Easley (System Planning
Corporation)
Contact:
geasley@sysplan.com
Time change 2:00PM
Abstract
Sparse representations of data play an
increasingly important role in areas across applied mathematics,
science and engineering. Over the past few years, there has been a
rapidly increasing pressure to handle ever larger and higher dimensional
data sets, with the challenge of providing representations of these data
that are sparse and fast to compute. JPEG2000 is the latest compression
scheme that makes use of the wavelet transform. However, the wavelet
transform is known to be suboptimal when dealing with a certain class of
images, e.g. images described as C 2 functions away from piecewise C 2
curves. In this talk, we develop a transform called the shearlet
transform designed to handle such images optimally. We conclude with
many examples indicating that shearlets are the latest competitor to
beat for representing images sparsely.
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Title: The
concentration phenomenon for Nonlinear Schr�inger equation.
Speaker:
Svetlana Roudenko
(Arizona State University)
Contact:
svetlana@math.la.asu.edu
Abstract
I will first review the known results on
(mass) concentration phenomenon for solutions to NLS with finite time of
existence (blow up solutions), and then will focus on cubic NLS in 2D,
in particularly, establish the dependence between the window size of
concentration and the rate of explosion of the space-time (Strichartz)
norm by revisiting Bourgain's argument (98) in one direction and using
the frequency restriction in the opposite. Finally, I will discuss L3
norm concentration for cubic NLS in 3D (joint work with J. Colliander
and J. Holmer).
Back
Title: Alternative
duals for linearly reconstructing sigma-delta quantized frame
coefficients.
Speaker:
Alex Powell (Vanderbilt University),
Contact:
alexander.m.powell@vanderbilt.edu
Back
Title: Deconvolution
results from the summer workshop at the IMA.
Speaker:
David P. Widemann (UMCP),
Contact:
widemann@math.umd.edu
Abstract
IMA summer workshop results. A team of six
graduate students worked on the problem of image blur. Image blurring is
modeled by convolving the image with a point spread function and adding
noise. Direct and indirect deconvolution methods for solving this
problem and the team's motion blur results will be presented.
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Title: The Balian
- Low Theorem.
Speaker:
Chris Flake (UMCP)
Contact:
jcflake@math.umd.edu
Time and room change
3:30PM, Math1310
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Organizational Meeting for Fall 2006.
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Title: Operator sampling used for communication
channel measurements and radar target identification.
Speaker:
Goetz Pfander (International University Bremen)
Contact:
g.pfander@iu-bremen.de |
