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10
Adaptive frame methods for elliptic operator equations: the steepest descent approach
, 2007
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Kronecker Compressive Sensing
"... Compressive sensing (CS) is an emerging approach for acquisition of signals having a sparse or compressible representation in some basis. While the CS literature has mostly focused on problems involving 1-D signals and 2-D images, many important applications involve signals that are multidimensional ..."
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Cited by 38 (2 self)
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Compressive sensing (CS) is an emerging approach for acquisition of signals having a sparse or compressible representation in some basis. While the CS literature has mostly focused on problems involving 1-D signals and 2-D images, many important applications involve signals that are multidimensional; in this case, CS works best with representations that encapsulate the structure of such signals in every dimension. We propose the use of Kronecker product matrices in CS for two purposes. First, we can use such matrices as sparsifying bases that jointly model the different types of structure present in the signal. Second, the measurement matrices used in distributed settings can be easily expressed as Kronecker product matrices. The Kronecker product formulation in these two settings enables the derivation of analytical bounds for sparse approximation of multidimensional signals and CS recovery performance as well as a means to evaluate novel distributed measurement schemes. I.
Modulation Spaces: Looking Back and Ahead
- Sampl. Theory Signal Image Process
"... This note provides historical perspectives and background on the moti-vations which led to the invention of the modulation spaces by the author almost 25 years ago, as well as comments about their role for ongoing re-search efforts within time-frequency analysis. We will also describe the role of mo ..."
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Cited by 27 (2 self)
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This note provides historical perspectives and background on the moti-vations which led to the invention of the modulation spaces by the author almost 25 years ago, as well as comments about their role for ongoing re-search efforts within time-frequency analysis. We will also describe the role of modulation spaces within the more general coorbit theory developed jointly with Karlheinz Gröchenig, and which eventually led to the develop-ment of the concept of Banach frames and more recently to the so-called localization theory for frames. A comprehensive bibliography is included. Key words and phrases: modulation spaces, Wiener amalgam spaces, Ga-bor analysis, time-frequency analysis, Heisenberg group, time-frequency
Generalized coorbit theory, Banach frames, and the relation to α-modulation spaces
- Proceedings of the London Mathematical Society
, 2008
"... This paper is concerned with generalizations and specific applications of the coorbit space theory based on group representations modulo quotients that has been developed quite recently. We show that the general theory applied to the affine Weyl–Heisenberg group gives rise to families of smoothness ..."
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Cited by 19 (8 self)
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This paper is concerned with generalizations and specific applications of the coorbit space theory based on group representations modulo quotients that has been developed quite recently. We show that the general theory applied to the affine Weyl–Heisenberg group gives rise to families of smoothness spaces that can be identified with α-modulation spaces. Key Words: Square integrable group representations, time–frequency analysis, atomic decompositions, (Banach) frames, homogeneous spaces, weighted coorbit
Model-Based Compressive Sensing for Signal Ensembles
"... Abstract—Compressive sensing (CS) is an alternative to Shannon/Nyquist sampling for acquiring sparse or compressible signals. Instead of taking N periodic samples, we measure M ≪ N inner products with random vectors and then recover the signal via a sparsity-seeking optimization or greedy algorithm. ..."
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Cited by 14 (3 self)
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Abstract—Compressive sensing (CS) is an alternative to Shannon/Nyquist sampling for acquiring sparse or compressible signals. Instead of taking N periodic samples, we measure M ≪ N inner products with random vectors and then recover the signal via a sparsity-seeking optimization or greedy algorithm. A new framework for CS based on unions of subspaces can improve signal recovery by including dependencies between values and locations of the signal’s significant coefficients. In this paper, we extend this framework to the acquisition of signal ensembles under a common sparse supports model. The new framework provides recovery algorithms with theoretical performance guarantees. Additionally, the framework scales naturally to large sensor networks: the number of measurements needed for each signal does not increase as the network becomes larger. Furthermore, the complexity of the recovery algorithm is only linear in the size of the network. We provide experimental results using synthetic and real-world signals that confirm these benefits. I.
Shearlet smoothness spaces
- J. Fourier Anal. Appl
, 2013
"... The shearlet representation has gained increasingly more prominence in recent years as a flexible mathematical framework which enables the efficient analysis of anisotropic phenomena by combining multiscale analysis with the ability to handle directional information. In this paper, we introduce a cl ..."
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Cited by 6 (0 self)
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The shearlet representation has gained increasingly more prominence in recent years as a flexible mathematical framework which enables the efficient analysis of anisotropic phenomena by combining multiscale analysis with the ability to handle directional information. In this paper, we introduce a class of shearlet smoothness spaces which is derived from the theory of decomposition spaces recently developed by L. Borup and M. Nielsen. The introduction of these spaces is motivated by recent results in image processing showing the advantage of using smoothness spaces associated with directional multiscale representations for the design and performance analysis of improved image restoration algorithms. In particular, we examine the relationship of the shearlet smoothness spaces with respect to Besov spaces, curvelet-type decomposition spaces and shearlet coorbit spaces. With respect to the theory of shearlet coorbit space, the construction of shearlet smoothness spaces presented in this paper does not require the use of a group structure. Key words and phrases: atomic decompositions, Banach frames, Besov spaces, decomposition spaces, shearlets.
ON ANISOTROPIC TRIEBEL-LIZORKIN TYPE SPACES, WITH APPLICATIONS TO THE STUDY OF PSEUDO-DIFFERENTIAL OPERATORS
, 2006
"... On anisotropic Triebel-Lizorkin type spaces with applications to the study of pseudo-differential operators by ..."
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Cited by 1 (0 self)
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On anisotropic Triebel-Lizorkin type spaces with applications to the study of pseudo-differential operators by
Citation for pulished version (APA):
, 2011
"... Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download a ..."
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Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal? Take down policy If you believe that this document breaches copyright please contact us at vbn@aub.aau.dk providing details, and we will remove access to the work immediately and investigate your claim.
ISSN 1399–2503 On-line version ISSN 1601–7811 ON A TRANSFERENCE RESULT FOR MATRIX WEIGHTS
, 2009
"... On a transference result for matrix weights by ..."
