Results 1  10
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53
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 1513 (20 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
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 176 (22 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.
 In IEEE the TwentyEighth Asilomar Conference onSignals, Systems and Computers,
, 1994
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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 77 (10 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 &QUOT;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 76 (15 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 45 (7 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 45 (12 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 40 (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 40 (8 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.
Continuous frames, function spaces, and the discretization problem
 J. Fourier Anal. Appl
"... A continuous frame is a family of vectors in a Hilbert space which allows reproductions of arbitrary elements by continuous superpositions. Associated to a given continuous frame we construct certain Banach spaces. Many classical function spaces can be identified as such spaces. We provide a general ..."
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Cited by 31 (6 self)
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A continuous frame is a family of vectors in a Hilbert space which allows reproductions of arbitrary elements by continuous superpositions. Associated to a given continuous frame we construct certain Banach spaces. Many classical function spaces can be identified as such spaces. We provide a general method to derive Banach frames and atomic decompositions for these Banach spaces by sampling the continuous frame. This is done by generalizing the coorbit space theory developed by Feichtinger and Gröchenig. As an important tool the concept of localization of frames is extended to continuous frames. As a byproduct we give a partial answer to the question raised by Ali, Antoine and Gazeau whether any continuous frame admits a corresponding discrete realization generated by sampling.