eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2009-02-24
8492
1
18
10.4230/DagSemProc.08492.1
article
08492 Abstracts Collection – Structured Decompositions and Efficient Algorithms
Dahlke, Stephan
Daubechies, Ingrid
Elad, Michael
Kutyniok, Gitta
Teschke, Gerd
From 30.11. to 05.12.2008, the Dagstuhl Seminar 08492 ``Structured Decompositions and Efficient Algorithms '' was held in Schloss Dagstuhl~--~Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol08492/DagSemProc.08492.1/DagSemProc.08492.1.pdf
Sparse signal representation
optimal signal reconstruction
approximation
compression
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2009-02-24
8492
1
5
10.4230/DagSemProc.08492.2
article
08492 Executive Summary – Structured Decompositions and Efficient Algorithms
Dahlke, Stephan
Daubechies, Ingrid
Elad, Michael
Kutyniok, Gitta
Teschke, Gerd
New emerging technologies such as high-precision sensors or new MRI machines drive us towards a challenging
quest for new, more effective, and more daring mathematical models and algorithms. Therefore, in the last
few years researchers have started to investigate different methods to efficiently represent or extract relevant
information from complex, high dimensional and/or multimodal data. Efficiently in this context means a representation that is linked to the features or characteristics of interest, thereby typically providing a sparse expansion of such.
Besides the construction of new and advanced ansatz systems the central question is how to design algorithms that are able to treat complex and high dimensional data and that efficiently perform a suitable approximation of the signal. One of the main challenges is to design new sparse approximation algorithms that would ideally combine, with an adjustable tradeoff, two properties: a provably good `quality' of the resulting decomposition under mild assumptions on the analyzed sparse signal, and numerically efficient design.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol08492/DagSemProc.08492.2/DagSemProc.08492.2.pdf
Sparse signal representation
optimal signal reconstruction
approximation
compression
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2009-02-24
8492
1
35
10.4230/DagSemProc.08492.3
article
A Weighted Average of Sparse Representations is Better than the Sparsest One Alone
Elad, Michael
Yavneh, Irad
Cleaning of noise from signals is a classical and long-studied problem in signal
processing. Algorithms for this task necessarily rely on an a-priori knowledge about the signal characteristics, along with information about the noise properties. For signals that admit sparse representations over a known dictionary, a commonly used denoising technique is to seek the sparsest representation that synthesizes a signal close enough to the corrupted one. As this problem is too complex in general, approximation methods, such as greedy pursuit algorithms, are often employed.
In this line of reasoning, we are led to believe that detection of the sparsest representation is key in the success of the denoising goal. Does this mean that other competitive and slightly inferior sparse representations are meaningless? Suppose we are served with a group of competing sparse representations, each claiming to explain the signal differently. Can those be fused somehow to lead to a better result? Surprisingly, the answer to this question is positive; merging these representations can form a more accurate, yet dense, estimate of the original signal even when the latter is known to be sparse.
In this paper we demonstrate this behavior, propose a practical way to generate
such a collection of representations by randomizing the Orthogonal Matching Pursuit (OMP) algorithm, and produce a clear analytical justification for the superiority of the associated Randomized OMP (RandOMP) algorithm. We show that while the Maximum a-posterior Probability (MAP) estimator aims to find and use the sparsest representation, the Minimum Mean-Squared-Error (MMSE) estimator leads to a fusion of representations to form its result. Thus, working with an appropriate mixture of candidate representations, we are surpassing the MAP and tending towards the MMSE estimate, and thereby getting a far more accurate estimation, especially at medium and low SNR.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol08492/DagSemProc.08492.3/DagSemProc.08492.3.pdf
Sparse representations
MMSE
MAP
mathcing pursuit
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2009-02-24
8492
1
7
10.4230/DagSemProc.08492.4
article
An open question on the existence of Gabor frames in general linear position
Krahmer, Felix
Pfander, Götz E.
Rashkov, Peter
Uncertainty principles for functions defined on finite Abelian groups generally relate the cardinality of a function to the cardinality of its Fourier transform. We examine how the cardinality of a function is related to the cardinality of its short--time Fourier transform. We illustrate that for some cyclic groups of small order, both, the Fourier and the short--time Fourier case, show a remarkable resemblance. We pose the question whether this correspondence holds for all cyclic groups.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol08492/DagSemProc.08492.4/DagSemProc.08492.4.pdf
Gabor systems
erasure channels
time--frequency dictionaries
short--time Fourier transform
uncertainty principle.
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2009-02-24
8492
1
12
10.4230/DagSemProc.08492.5
article
Arbitrary Shrinkage Rules for Approximation Schemes with Sparsity Constraints
Ehler, Martin
Geisel, Simone
Finding a sparse representation of a possibly noisy signal is a common problem in signal representation and processing. It can be modeled as a variational minimization with $ell_ au$-sparsity constraints for $ au<1$. Applications whose computation time is crucial require fast algorithms for this minimization. However, there are no fast methods for finding the exact minimizer, and to circumvent this limitation, we consider minimization up to a constant factor. We verify that arbitrary shrinkage rules provide closed formulas for such minimizers, and we introduce a new shrinkage strategy, which is adapted to $ au<1$.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol08492/DagSemProc.08492.5/DagSemProc.08492.5.pdf
Frames
shrinkage
variational problems
sparse approximation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2009-02-24
8492
1
3
10.4230/DagSemProc.08492.6
article
Pseudospectral Fourier reconstruction with IPRM
Gröchenig, Karlheinz
Hrycak, Tomasz
We generalize the Inverse Polynomial Reconstruction Method (IPRM) for
mitigation of the Gibbs phenomenon by reconstructing a function as an
algebraic polynomial of degree $n-1$ from the function's $m$ lowest
Fourier coefficients ($m ge n$). We compute approximate Legendre
coefficients of the function by solving a linear least squares
problem, and we show that the condition number of the problem does not
exceed $sqrtfrac{m}{{m-alpha_0 n^2}}$, where $alpha_0 =
frac{4sqrt{2}}{pi^2} = 0.573 ldots$. Consequently, whenever
mbox{$m ge n^2$,} the convergence rate of the modified IPRM for an
analytic function is root exponential on the whole interval of
definition. Stability and accuracy of the proposed algorithm are
validated with numerical experiments.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol08492/DagSemProc.08492.6/DagSemProc.08492.6.pdf
IPRM
Fourier series
inverse methods
pseudospectral methods
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2009-02-24
8492
1
28
10.4230/DagSemProc.08492.7
article
Representation of Operators in the Time-Frequency Domain and Generalzed Gabor Multipliers
Dörfler, Monika
Torrésani, Bruno
Starting from a general operator representation in the time-frequency domain, this paper addresses the problem of approximating linear operators by operators that are diagonal or band-diagonal with respect to Gabor frames. A haracterization of operators that can be realized as Gabor multipliers is given and necessary conditions for the existence of (Hilbert-Schmidt) optimal Gabor multiplier approximations are discussed and an efficient method for the calculation of an operator's best approximation by a Gabor multiplier is derived. The spreading function of Gabor multipliers yields new error estimates for these approximations. Generalizations (multiple Gabor multipliers)
are introduced for better approximation of overspread operators. The Riesz property of the projection operators involved in generalized Gabor multipliers is characterized, and a method for obtaining an operator's best approximation by a multiple Gabor multiplier is suggested. Finally, it is shown that in certain situations, generalized Gabor multipliers reduce to a finite sum of regular Gabor multipliers with adapted windows.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol08492/DagSemProc.08492.7/DagSemProc.08492.7.pdf
Operator approximation
generalized Gabor multipliers
spreading function
twisted convolution
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2009-02-24
8492
1
8
10.4230/DagSemProc.08492.8
article
Sparse Reconstructions for Inverse PDE Problems
Raasch, Thorsten
We are concerned with the numerical solution of linear parameter identification problems for parabolic PDE, written as an operator equation $Ku=f$.
The target object $u$ is assumed to have a sparse expansion with respect to a wavelet system $Psi={psi_lambda}$ in space-time, being equivalent to a priori information on the regularity of $u=mathbf u^ opPsi$ in a certain scale
of Besov spaces $B^s_{p,p}$. For the recovery of the unknown coefficient array $mathbf u$, we miminize a Tikhonov-type functional
begin{equation*}
min_{mathbf u}|Kmathbf u^ opPsi-f^delta|^2+alphasum_{lambda}omega_lambda|u_lambda|^p
end{equation*}
by an associated thresholded Landweber algorithm, $f^delta$ being a noisy version of $f$.
Since any application of the forward operator $K$ and its adjoint
involves the numerical solution of a PDE, perturbed versions of the iteration
have to be studied. In particular, for reasons of efficiency,
adaptive applications of $K$ and $K^*$ are indispensable cite{Ra07}.
By a suitable choice of the respective tolerances and stopping criteria,
also the adaptive iteration could recently be shown to have regularizing properties cite{BoMa08a} for $p>1$. Moreover, the sequence of iterates linearly converges to the minimizer of the functional, a result which can also be proved
for the special case $p=1$, see [DaFoRa08]. We illustrate the performance of the resulting method by numerical computations for one- and two-dimensional inverse heat conduction problems.
References:
[BoMa08a] T. Bonesky and P. Maass,
Iterated soft shrinkage with adaptive operator evaluations, Preprint, 2008
[DaFoRa08] S. Dahlke, M. Fornasier, and T. Raasch,
Multiscale Preconditioning for Adaptive Sparse Optimization,
in preparation, 2008
[Ra07] T.~Raasch,
Adaptive wavelet and frame schemes for elliptic and parabolic equations,
Dissertation, Philipps-Universit"at Marburg, 2007
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol08492/DagSemProc.08492.8/DagSemProc.08492.8.pdf
Adaptivity
sparse reconstructions
l1 minimization
parameter identification
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2009-02-24
8492
1
7
10.4230/DagSemProc.08492.9
article
The Continuous Shearlet Transform in Arbitrary Space Dimensions
Dahlke, Stephan
Steidl, Gabriele
Teschke, Gerd
This note is concerned with the generalization of the continuous
shearlet transform to higher dimensions. Similar to the
two-dimensional case, our approach is based on translations,
anisotropic dilations and specific shear matrices. We show that the
associated integral transform again originates from a square-integrable
representation of a specific group, the full $n$-variate shearlet
group. Moreover, we verify that
by applying the coorbit theory, canonical scales of smoothness spaces
and associated Banach frames can be
derived. We also indicate how our transform can be used to
characterize singularities in signals.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol08492/DagSemProc.08492.9/DagSemProc.08492.9.pdf
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2009-02-24
8492
1
4
10.4230/DagSemProc.08492.10
article
Time-Frequency Analysis and PDE's
Tabacco, Anita
We study the action on modulation spaces of Fourier multipliers with symbols
$e^{imu(xi)}$, for real-valued functions $mu$ having unbounded second
derivatives. We show that if $mu$ satisfies the usual symbol estimates of order
$alphageq2$, or if $mu$ is a positively homogeneous function of degree $alpha$,
the corresponding Fourier multiplier is bounded as an operator between the weighted modulation spaces $mathcal{M}^{p,q}_delta$ and $mathcal{M}^{p,q}$,
for every $1leq p,qleqinfty$ and $deltageq d(alpha-2)|frac{1}{p}-frac{1}{2}|$.
Here $delta$ represents the loss of derivatives. The above threshold is shown to
be sharp for {it all} homogeneous functions $mu$ whose Hessian matrix is
non-degenerate at some point.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol08492/DagSemProc.08492.10/DagSemProc.08492.10.pdf
Fourier multipliers
modulation spaces
short-time Fourier transform