Results 1  10
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35
Near Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?
, 2004
"... Suppose we are given a vector f in RN. How many linear measurements do we need to make about f to be able to recover f to within precision ɛ in the Euclidean (ℓ2) metric? Or more exactly, suppose we are interested in a class F of such objects— discrete digital signals, images, etc; how many linear m ..."
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Cited by 832 (16 self)
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Suppose we are given a vector f in RN. How many linear measurements do we need to make about f to be able to recover f to within precision ɛ in the Euclidean (ℓ2) metric? Or more exactly, suppose we are interested in a class F of such objects— discrete digital signals, images, etc; how many linear measurements do we need to recover objects from this class to within accuracy ɛ? This paper shows that if the objects of interest are sparse or compressible in the sense that the reordered entries of a signal f ∈ F decay like a powerlaw (or if the coefficient sequence of f in a fixed basis decays like a powerlaw), then it is possible to reconstruct f to within very high accuracy from a small number of random measurements. typical result is as follows: we rearrange the entries of f (or its coefficients in a fixed basis) in decreasing order of magnitude f  (1) ≥ f  (2) ≥... ≥ f  (N), and define the weakℓp ball as the class F of those elements whose entries obey the power decay law f  (n) ≤ C · n −1/p. We take measurements 〈f, Xk〉, k = 1,..., K, where the Xk are Ndimensional Gaussian
Data compression and harmonic analysis
 IEEE Trans. Inform. Theory
, 1998
"... In this paper we review some recent interactions between harmonic analysis and data compression. The story goes back of course to Shannon’s R(D) theory... ..."
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Cited by 140 (24 self)
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In this paper we review some recent interactions between harmonic analysis and data compression. The story goes back of course to Shannon’s R(D) theory...
Unconditional bases are optimal bases for data compression and for statistical estimation
 Applied and Computational Harmonic Analysis
, 1993
"... An orthogonal basis of L 2 which is also an unconditional basis of a functional space F is a kind of optimal basis for compressing, estimating, and recovering functions in F. Simple thresholding operations, applied in the unconditional basis, work essentially better for compressing, estimating, and ..."
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Cited by 140 (23 self)
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An orthogonal basis of L 2 which is also an unconditional basis of a functional space F is a kind of optimal basis for compressing, estimating, and recovering functions in F. Simple thresholding operations, applied in the unconditional basis, work essentially better for compressing, estimating, and recovering than they do in any other orthogonal basis. In fact, simple thresholding in an unconditional basis works essentially better for recovery and estimation than other methods, period. (Performance is measured in an asymptotic minimax sense.) As an application, we formalize and prove Mallat's Heuristic, which says that wavelet bases are optimal for representing functions containing singularities, when there may be an arbitrary number of singularities, arbitrarily distributed.
Basis Pursuit
, 1994
"... The TimeFrequency and TimeScale communities have recently developed an enormous number of overcomplete signal dictionaries  wavelets, wavelet packets, cosine packets, wilson bases, chirplets, warped bases, and hyperbolic cross bases being a few examples. Basis Pursuit is a technique for decompos ..."
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Cited by 119 (15 self)
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The TimeFrequency and TimeScale communities have recently developed an enormous number of overcomplete signal dictionaries  wavelets, wavelet packets, cosine packets, wilson bases, chirplets, warped bases, and hyperbolic cross bases being a few examples. Basis Pursuit is a technique for decomposing a signal into an "optimal" superposition of dictionary elements. The optimization criterion is the l 1 norm of coefficients. The method has several advantages over Matching Pursuit and Best Ortho Basis, including superresolution and stability. 1 Introduction Over the last five years or so, there has been an explosion of awareness of alternatives to traditional signal representations. Instead of just representing objects as superpositions of sinusoids (the traditional Fourier representation) we now have available alternate dictionaries  signal representation schemes  of which the Wavelets dictionary is only the most wellknown. Wavelet dictionaries, Gabor dictionaries, Multiscale...
Localization of Frames, Banach Frames, and the Invertibility of the Frame Operator
"... We introduce a new concept to describe the localization of frames. In our main result we shown that the frame operator preserves this localization and that the dual frame possesses the same localization property. As an application we show that certain frames for Hilbert spaces extend automatically t ..."
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Cited by 49 (9 self)
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We introduce a new concept to describe the localization of frames. In our main result we shown that the frame operator preserves this localization and that the dual frame possesses the same localization property. As an application we show that certain frames for Hilbert spaces extend automatically to Banach frames. Using this abstract theory, we derive new results on the construction of nonuniform Gabor frames and solve a problem about nonuniform sampling in shiftinvariant spaces. 1.
A Banach space of test functions for Gabor analysis
 IN "GABOR ANALYSIS AND ALGORITHMS: THEORY AND APPLICATIONS
, 1998
"... We introduce the Banach space S 0 # L which has a variety of properties making it a useful tool in Gabor analysis. S 0 can be characterized as the smallest timefrequency homogeneous Banach space of (continuous) functions. We also present other characterizations of S 0 turning it into a very ..."
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Cited by 38 (9 self)
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We introduce the Banach space S 0 # L which has a variety of properties making it a useful tool in Gabor analysis. S 0 can be characterized as the smallest timefrequency homogeneous Banach space of (continuous) functions. We also present other characterizations of S 0 turning it into a very flexible tool for Gabor analysis and allowing for simplifications of various proofs. A careful
Localization of frames
 II, APPL. COMPUT. HARMONIC ANAL
, 2004
"... Several concepts for the localization of a frame are studied. The intrinsic localization of a frame is defined by the decay properties of its Gramian matrix. Our main result asserts that the canonical dual frame possesses the same intrinsic localization as the original frame. The proof relies heavi ..."
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Cited by 28 (6 self)
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Several concepts for the localization of a frame are studied. The intrinsic localization of a frame is defined by the decay properties of its Gramian matrix. Our main result asserts that the canonical dual frame possesses the same intrinsic localization as the original frame. The proof relies heavily on Banach algebra techniques, in particular on recent spectral invariance properties for certain Banach algebras of infinite matrices. Intrinsically localized frames extend in a natural way to Banach frames for a class of associated Banach spaces which are defined by weighted ℓ pcoefficients of their frame expansions. As an example the timefrequency concentration of distributions is characterized by means of localized (nonuniform) Gabor frames.
Gabor wavelets and the Heisenberg group: Gabor Expansions and Short Time Fourier Transform from the Group Theoretical Point of View
, 1992
"... We study series expansions of signals with respect to Gabor wavelets and the equivalent problem of (irregular) sampling of the short time Fourier tranform. Using Heisenberg group techniques rather than traditional Fourier analysis allows to design stable iterative algorithms for signal analysis ..."
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Cited by 24 (10 self)
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We study series expansions of signals with respect to Gabor wavelets and the equivalent problem of (irregular) sampling of the short time Fourier tranform. Using Heisenberg group techniques rather than traditional Fourier analysis allows to design stable iterative algorithms for signal analysis and synthesis. These algorithms converge for a variety of norms and are compatible with the timefrequency localization of signals.
History and evolution of the Density Theorem for Gabor frames
, 2007
"... The Density Theorem for Gabor Frames is one of the fundamental results of timefrequency analysis. This expository survey attempts to reconstruct the long and very involved history of this theorem and to present its context and evolution, from the onedimensional rectangular lattice setting, to arb ..."
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Cited by 17 (6 self)
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The Density Theorem for Gabor Frames is one of the fundamental results of timefrequency analysis. This expository survey attempts to reconstruct the long and very involved history of this theorem and to present its context and evolution, from the onedimensional rectangular lattice setting, to arbitrary lattices in higher dimensions, to irregular Gabor frames, and most recently beyond the setting of Gabor frames to abstract localized frames. Related fundamental principles in Gabor analysis are also surveyed, including the Wexler–Raz biorthogonality relations, the Duality Principle, the Balian–Low Theorem, the Walnut and Janssen representations, and the Homogeneous Approximation Property. An extended bibliography is included.
A First Survey of Gabor Multipliers
 Advances in Gabor Analysis. Birkhauser
, 2002
"... We describe various basic facts about Gabor multipliers and their continuous analogue which we will call STFTmultipliers. These operators are obtained by going from the signal domain to some transform domain, and applying a pointwise multiplication operator before resynthesis. Although such ope ..."
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Cited by 17 (2 self)
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We describe various basic facts about Gabor multipliers and their continuous analogue which we will call STFTmultipliers. These operators are obtained by going from the signal domain to some transform domain, and applying a pointwise multiplication operator before resynthesis. Although such operators have been in use implicitly for quite some time this paper appears to be the rst systematic mathematical treatment of Gabor multipliers.