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Nonlinear approximation in α-modulation spaces (2003)

by L Borup, M Nielsen
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Kronecker Compressive Sensing

by Marco F. Duarte, Richard G. Baraniuk
"... 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 ..."
Abstract - Cited by 38 (2 self) - Add to MetaCart
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.
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... multidimensional structures provide an advantage during compression. Some promising candidates include modulation spaces, which contain signals that can be compressed using Wilson and brushlet bases =-=[41, 42]-=-. Our KCS also motivates the formulation of novel structured representations using sparsifying bases in applications where transform coding compression schemes have not been developed. While we focuse...

Adaptive frame methods for elliptic operator equations: the steepest descent approach

by Stephan Dahlke, Thorsten Raasch, Manuel Werner, Massimo Fornasier, Rob Stevenson , 2007
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Abstract - Cited by 37 (16 self) - Add to MetaCart
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Modulation Spaces: Looking Back and Ahead

by Hans G. Feichtinger - SAMPL. THEORY SIGNAL IMAGE PROCESS , 2006
"... 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 ..."
Abstract - Cited by 26 (2 self) - Add to MetaCart
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.
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...theory of decomposition spaces (cf. [73, 63]). In the last few years the alpha-modulation spaces have received a lot of attention through the work of Massimo Fornasier, Lasse Borup and Morten Nielsen =-=[73, 63, 161, 22, 99, 101, 23, 24, 25, 41]-=-. Among other results, the existence of Banach frames for these spaces has been established. The developments around modulation spaces and their generalizations — i.e., the general coorbit theory in i...

Generalized coorbit theory, Banach frames, and the relation to α-modulation spaces

by Stephan Dahlke, Massimo Fornasier, Holger Rauhut, Gabriele Steidl, Gerd Teschke - 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 ..."
Abstract - Cited by 19 (8 self) - Add to MetaCart
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

by Marco F. Duarte, Volkan Cevher, Richard G. Baraniuk
"... 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. ..."
Abstract - Cited by 14 (3 self) - Add to MetaCart
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.
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... multidimensional structures provide an advantage during compression. Some promising candidates include modulation spaces, which contain signals that can be compressed using Wilson and brushlet bases =-=[140, 141]-=-. This framework also motivates the formulation of novel structured representations using sparsifying bases in applications where transform coding compression schemes have not been developed, such as ...

Banach frames for α-modulation spaces

by Massimo Fornasier , 2006
"... ..."
Abstract - Cited by 6 (2 self) - Add to MetaCart
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Shearlet smoothness spaces

by Demetrio Labate, Lucia Mantovani, Pooran Negi - 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 ..."
Abstract - Cited by 6 (0 self) - Add to MetaCart
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.
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...t to an appropriate smoothness measure in Besov spaces [25]. Similarly, the Gabor systems, which are widely used in time-frequency analysis, are naturally associated to the class of modulation spaces =-=[1, 16]-=-. In the case of shearlets, a sequence of papers by Dahlke, Kutyniok, Steidl and Teschke [8, 9, 10] have recently introduced a class of shearlet spaces within the framework of the coorbit space theory...

ON ANISOTROPIC TRIEBEL-LIZORKIN TYPE SPACES, WITH APPLICATIONS TO THE STUDY OF PSEUDO-DIFFERENTIAL OPERATORS

by Lasse Borup, Morten Nielsen, Lasse Borup, Morten Nielsen , 2006
"... On anisotropic Triebel-Lizorkin type spaces with applications to the study of pseudo-differential operators by ..."
Abstract - Cited by 1 (0 self) - Add to MetaCart
On anisotropic Triebel-Lizorkin type spaces with applications to the study of pseudo-differential operators by
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... to define α-modulation spaces. It is not difficult to verify that M βp,q(hα, 〈·〉) = Mβ,αp,q (R d), and consequently F βp,q(hα, 〈·〉) can be considered the T-L equivalent of the αmodulation spaces. In =-=[5]-=-, the authors introduced so-called α-Triebel-Lizorkin space in the one dimensional case. One can verify that the α-Triebel-Lizorkin scale equals the T-L type space F βp,q(hα, 〈·〉). However, the author...

Citation for pulished version (APA):

by Aalborg Universitet, Christian H. T, Christian Tollestrup , 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.
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...at has been successfully applied to compress sound signals and images with smoothness in some Besov space, see e.g. [13, 12]. Other interesting examples include sparse expansions in modulation spaces =-=[33, 3]-=-, and sparse curvelet expansions [43, 7]. In this paper we consider a general construction of smoothness spaces, a subclass of so-called decomposition spaces, defined on Rd for which it is possible to...

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