{"@context":"https:\/\/schema.org\/","@type":"PublicationVolume","@id":"#volume729","volumeNumber":8492,"name":"Dagstuhl Seminar Proceedings, Volume 8492","dateCreated":"2009-02-24","datePublished":"2009-02-24","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"Periodical","@id":"#series119","name":"Dagstuhl Seminar Proceedings","issn":"1862-4405","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume729"},"hasPart":[{"@type":"ScholarlyArticle","@id":"#article2314","name":"08492 Abstracts Collection \u2013 Structured Decompositions and Efficient Algorithms","abstract":"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.\r\nDuring the seminar, several participants presented their current\r\nresearch, and ongoing work and open problems were discussed. Abstracts of\r\nthe presentations given during the seminar as well as abstracts of\r\nseminar results and ideas are put together in this paper. The first section\r\ndescribes the seminar topics and goals in general.\r\nLinks to extended abstracts or full papers are provided, if available.","keywords":["Sparse signal representation","optimal signal reconstruction","approximation","compression"],"author":[{"@type":"Person","name":"Dahlke, Stephan","givenName":"Stephan","familyName":"Dahlke"},{"@type":"Person","name":"Daubechies, Ingrid","givenName":"Ingrid","familyName":"Daubechies"},{"@type":"Person","name":"Elad, Michael","givenName":"Michael","familyName":"Elad"},{"@type":"Person","name":"Kutyniok, Gitta","givenName":"Gitta","familyName":"Kutyniok"},{"@type":"Person","name":"Teschke, Gerd","givenName":"Gerd","familyName":"Teschke"}],"position":1,"pageStart":1,"pageEnd":18,"dateCreated":"2009-02-26","datePublished":"2009-02-26","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Dahlke, Stephan","givenName":"Stephan","familyName":"Dahlke"},{"@type":"Person","name":"Daubechies, Ingrid","givenName":"Ingrid","familyName":"Daubechies"},{"@type":"Person","name":"Elad, Michael","givenName":"Michael","familyName":"Elad"},{"@type":"Person","name":"Kutyniok, Gitta","givenName":"Gitta","familyName":"Kutyniok"},{"@type":"Person","name":"Teschke, Gerd","givenName":"Gerd","familyName":"Teschke"}],"copyrightYear":"2009","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.08492.1","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume729"},{"@type":"ScholarlyArticle","@id":"#article2315","name":"08492 Executive Summary \u2013 Structured Decompositions and Efficient Algorithms","abstract":"New emerging technologies such as high-precision sensors or new MRI machines drive us towards a challenging\r\nquest for new, more effective, and more daring mathematical models and algorithms. Therefore, in the last\r\nfew years researchers have started to investigate different methods to efficiently represent or extract relevant\r\ninformation 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.\r\n 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.","keywords":["Sparse signal representation","optimal signal reconstruction","approximation","compression"],"author":[{"@type":"Person","name":"Dahlke, Stephan","givenName":"Stephan","familyName":"Dahlke"},{"@type":"Person","name":"Daubechies, Ingrid","givenName":"Ingrid","familyName":"Daubechies"},{"@type":"Person","name":"Elad, Michael","givenName":"Michael","familyName":"Elad"},{"@type":"Person","name":"Kutyniok, Gitta","givenName":"Gitta","familyName":"Kutyniok"},{"@type":"Person","name":"Teschke, Gerd","givenName":"Gerd","familyName":"Teschke"}],"position":2,"pageStart":1,"pageEnd":5,"dateCreated":"2009-02-24","datePublished":"2009-02-24","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Dahlke, Stephan","givenName":"Stephan","familyName":"Dahlke"},{"@type":"Person","name":"Daubechies, Ingrid","givenName":"Ingrid","familyName":"Daubechies"},{"@type":"Person","name":"Elad, Michael","givenName":"Michael","familyName":"Elad"},{"@type":"Person","name":"Kutyniok, Gitta","givenName":"Gitta","familyName":"Kutyniok"},{"@type":"Person","name":"Teschke, Gerd","givenName":"Gerd","familyName":"Teschke"}],"copyrightYear":"2009","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.08492.2","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume729"},{"@type":"ScholarlyArticle","@id":"#article2316","name":"A Weighted Average of Sparse Representations is Better than the Sparsest One Alone","abstract":"Cleaning of noise from signals is a classical and long-studied problem in signal\r\nprocessing. 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.\r\nIn 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.\r\nIn this paper we demonstrate this behavior, propose a practical way to generate\r\nsuch 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.","keywords":["Sparse representations","MMSE","MAP","mathcing pursuit"],"author":[{"@type":"Person","name":"Elad, Michael","givenName":"Michael","familyName":"Elad"},{"@type":"Person","name":"Yavneh, Irad","givenName":"Irad","familyName":"Yavneh"}],"position":3,"pageStart":1,"pageEnd":35,"dateCreated":"2009-02-24","datePublished":"2009-02-24","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Elad, Michael","givenName":"Michael","familyName":"Elad"},{"@type":"Person","name":"Yavneh, Irad","givenName":"Irad","familyName":"Yavneh"}],"copyrightYear":"2009","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.08492.3","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume729"},{"@type":"ScholarlyArticle","@id":"#article2317","name":"An open question on the existence of Gabor frames in general linear position","abstract":"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.","keywords":["Gabor systems","erasure channels","time--frequency dictionaries","short--time Fourier transform","uncertainty principle."],"author":[{"@type":"Person","name":"Krahmer, Felix","givenName":"Felix","familyName":"Krahmer"},{"@type":"Person","name":"Pfander, G\u00f6tz E.","givenName":"G\u00f6tz E.","familyName":"Pfander"},{"@type":"Person","name":"Rashkov, Peter","givenName":"Peter","familyName":"Rashkov"}],"position":4,"pageStart":1,"pageEnd":7,"dateCreated":"2009-02-24","datePublished":"2009-02-24","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Krahmer, Felix","givenName":"Felix","familyName":"Krahmer"},{"@type":"Person","name":"Pfander, G\u00f6tz E.","givenName":"G\u00f6tz E.","familyName":"Pfander"},{"@type":"Person","name":"Rashkov, Peter","givenName":"Peter","familyName":"Rashkov"}],"copyrightYear":"2009","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.08492.4","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume729"},{"@type":"ScholarlyArticle","@id":"#article2318","name":"Arbitrary Shrinkage Rules for Approximation Schemes with Sparsity Constraints","abstract":"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 $ell_\tau$-sparsity constraints for $\tau<1$. 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 $\tau<1$.","keywords":["Frames","shrinkage","variational problems","sparse approximation"],"author":[{"@type":"Person","name":"Ehler, Martin","givenName":"Martin","familyName":"Ehler"},{"@type":"Person","name":"Geisel, Simone","givenName":"Simone","familyName":"Geisel"}],"position":5,"pageStart":1,"pageEnd":12,"dateCreated":"2009-02-24","datePublished":"2009-02-24","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Ehler, Martin","givenName":"Martin","familyName":"Ehler"},{"@type":"Person","name":"Geisel, Simone","givenName":"Simone","familyName":"Geisel"}],"copyrightYear":"2009","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.08492.5","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume729"},{"@type":"ScholarlyArticle","@id":"#article2319","name":"Pseudospectral Fourier reconstruction with IPRM","abstract":"We generalize the Inverse Polynomial Reconstruction Method (IPRM) for\r\nmitigation of the Gibbs phenomenon by reconstructing a function as an\r\nalgebraic polynomial of degree $n-1$ from the function's $m$ lowest\r\nFourier coefficients ($m ge n$). We compute approximate Legendre\r\ncoefficients of the function by solving a linear least squares\r\nproblem, and we show that the condition number of the problem does not\r\nexceed $sqrtfrac{m}{{m-alpha_0 n^2}}$, where $alpha_0 =\r\nfrac{4sqrt{2}}{pi^2} = 0.573 ldots$. Consequently, whenever \r\nmbox{$m ge n^2$,} the convergence rate of the modified IPRM for an\r\nanalytic function is root exponential on the whole interval of\r\ndefinition. Stability and accuracy of the proposed algorithm are\r\nvalidated with numerical experiments.","keywords":["IPRM","Fourier series","inverse methods","pseudospectral methods"],"author":[{"@type":"Person","name":"Gr\u00f6chenig, Karlheinz","givenName":"Karlheinz","familyName":"Gr\u00f6chenig"},{"@type":"Person","name":"Hrycak, Tomasz","givenName":"Tomasz","familyName":"Hrycak"}],"position":6,"pageStart":1,"pageEnd":3,"dateCreated":"2009-02-24","datePublished":"2009-02-24","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Gr\u00f6chenig, Karlheinz","givenName":"Karlheinz","familyName":"Gr\u00f6chenig"},{"@type":"Person","name":"Hrycak, Tomasz","givenName":"Tomasz","familyName":"Hrycak"}],"copyrightYear":"2009","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.08492.6","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume729"},{"@type":"ScholarlyArticle","@id":"#article2320","name":"Representation of Operators in the Time-Frequency Domain and Generalzed Gabor Multipliers","abstract":"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) \r\nare 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.","keywords":["Operator approximation","generalized Gabor multipliers","spreading function","twisted convolution"],"author":[{"@type":"Person","name":"D\u00f6rfler, Monika","givenName":"Monika","familyName":"D\u00f6rfler"},{"@type":"Person","name":"Torr\u00e9sani, Bruno","givenName":"Bruno","familyName":"Torr\u00e9sani"}],"position":7,"pageStart":1,"pageEnd":28,"dateCreated":"2009-02-24","datePublished":"2009-02-24","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"D\u00f6rfler, Monika","givenName":"Monika","familyName":"D\u00f6rfler"},{"@type":"Person","name":"Torr\u00e9sani, Bruno","givenName":"Bruno","familyName":"Torr\u00e9sani"}],"copyrightYear":"2009","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.08492.7","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume729"},{"@type":"ScholarlyArticle","@id":"#article2321","name":"Sparse Reconstructions for Inverse PDE Problems","abstract":"We are concerned with the numerical solution of linear parameter identification problems for parabolic PDE, written as an operator equation $Ku=f$.\r\nThe 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 $u=mathbf u^\topPsi$ in a certain scale\r\nof Besov spaces $B^s_{p,p}$. For the recovery of the unknown coefficient array $mathbf u$, we miminize a Tikhonov-type functional\r\nbegin{equation*}\r\n min_{mathbf u}|Kmathbf u^\topPsi-f^delta|^2+alphasum_{lambda}omega_lambda|u_lambda|^p\r\nend{equation*}\r\nby an associated thresholded Landweber algorithm, $f^delta$ being a noisy version of $f$.\r\nSince any application of the forward operator $K$ and its adjoint\r\ninvolves the numerical solution of a PDE, perturbed versions of the iteration\r\nhave to be studied. In particular, for reasons of efficiency,\r\nadaptive applications of $K$ and $K^*$ are indispensable cite{Ra07}.\r\nBy a suitable choice of the respective tolerances and stopping criteria,\r\nalso 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\r\nfor 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.\r\n\r\n\r\nReferences:\r\n\r\n[BoMa08a] T. Bonesky and P. Maass,\r\n Iterated soft shrinkage with adaptive operator evaluations, Preprint, 2008\r\n\r\n[DaFoRa08] S. Dahlke, M. Fornasier, and T. Raasch,\r\n Multiscale Preconditioning for Adaptive Sparse Optimization,\r\n in preparation, 2008\r\n\r\n[Ra07] T.~Raasch,\r\n Adaptive wavelet and frame schemes for elliptic and parabolic equations,\r\n Dissertation, Philipps-Universit\"at Marburg, 2007","keywords":["Adaptivity","sparse reconstructions","l1 minimization","parameter identification"],"author":{"@type":"Person","name":"Raasch, Thorsten","givenName":"Thorsten","familyName":"Raasch"},"position":8,"pageStart":1,"pageEnd":8,"dateCreated":"2009-02-24","datePublished":"2009-02-24","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Raasch, Thorsten","givenName":"Thorsten","familyName":"Raasch"},"copyrightYear":"2009","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.08492.8","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume729"},{"@type":"ScholarlyArticle","@id":"#article2322","name":"The Continuous Shearlet Transform in Arbitrary Space Dimensions","abstract":"This note is concerned with the generalization of the continuous\r\nshearlet transform to higher dimensions. Similar to the\r\ntwo-dimensional case, our approach is based on translations,\r\nanisotropic dilations and specific shear matrices. We show that the\r\nassociated integral transform again originates from a square-integrable\r\nrepresentation of a specific group, the full $n$-variate shearlet\r\ngroup. Moreover, we verify that\r\nby applying the coorbit theory, canonical scales of smoothness spaces\r\nand associated Banach frames can be\r\nderived. We also indicate how our transform can be used to\r\ncharacterize singularities in signals.","author":[{"@type":"Person","name":"Dahlke, Stephan","givenName":"Stephan","familyName":"Dahlke"},{"@type":"Person","name":"Steidl, Gabriele","givenName":"Gabriele","familyName":"Steidl"},{"@type":"Person","name":"Teschke, Gerd","givenName":"Gerd","familyName":"Teschke"}],"position":9,"pageStart":1,"pageEnd":7,"dateCreated":"2009-03-10","datePublished":"2009-03-10","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Dahlke, Stephan","givenName":"Stephan","familyName":"Dahlke"},{"@type":"Person","name":"Steidl, Gabriele","givenName":"Gabriele","familyName":"Steidl"},{"@type":"Person","name":"Teschke, Gerd","givenName":"Gerd","familyName":"Teschke"}],"copyrightYear":"2009","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.08492.9","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume729"},{"@type":"ScholarlyArticle","@id":"#article2323","name":"Time-Frequency Analysis and PDE's","abstract":"We study the action on modulation spaces of Fourier multipliers with symbols\r\n$e^{imu(xi)}$, for real-valued functions $mu$ having unbounded second\r\nderivatives. We show that if $mu$ satisfies the usual symbol estimates of order\r\n$alphageq2$, or if $mu$ is a positively homogeneous function of degree $alpha$,\r\nthe corresponding Fourier multiplier is bounded as an operator between the weighted modulation spaces $mathcal{M}^{p,q}_delta$ and $mathcal{M}^{p,q}$,\r\nfor every $1leq p,qleqinfty$ and $deltageq d(alpha-2)|frac{1}{p}-frac{1}{2}|$.\r\nHere $delta$ represents the loss of derivatives. The above threshold is shown to\r\nbe sharp for {it all} homogeneous functions $mu$ whose Hessian matrix is\r\nnon-degenerate at some point.","keywords":["Fourier multipliers","modulation spaces","short-time Fourier transform"],"author":{"@type":"Person","name":"Tabacco, Anita","givenName":"Anita","familyName":"Tabacco"},"position":10,"pageStart":1,"pageEnd":4,"dateCreated":"2009-02-24","datePublished":"2009-02-24","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Tabacco, Anita","givenName":"Anita","familyName":"Tabacco"},"copyrightYear":"2009","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.08492.10","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume729"}]}