Fusion Frames

Casazza, Peter G.; Kutyniok, Gitta

doi:10.1007/978-0-8176-8373-3_13

Cited by 12 publications

(13 citation statements)

References 37 publications

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“…We refer to [10] for more information on fusion frames. Although our work, as well as [2,6], is strongly motivated by fusion frames, we emphasize that our results below do not assume that the subspaces W j satisfy the fusion frame property (6).…”

Section: Block Sparsity and Fusion Framesmentioning

confidence: 99%

“…The collection of these subspaces may form a fusion frame (although this is not strictly required for our theory). Fusion frames generalize frames [11] and were first introduced in [9] under the name of 'frames of subspaces' (see also the survey [10]). They allow to analyze signals by projecting them onto multidimensional subspaces and for stable reconstruction from these projections.…”

Section: Introductionmentioning

confidence: 99%

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Uniform recovery of fusion frame structured sparse signals

Ayaz

Dirksen

Rauhut

2016

Applied and Computational Harmonic Analysis

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We consider the problem of recovering fusion frame sparse signals from incomplete measurements. These signals are composed of a small number of nonzero blocks taken from a family of subspaces. First, we show that, by using a-priori knowledge of a coherence parameter associated with the angles between the subspaces, one can uniformly recover fusion frame sparse signals with a significantly reduced number of vector-valued (sub-)Gaussian measurements via mixed ℓ 1 /ℓ 2 -minimization. We prove this by establishing an appropriate version of the restricted isometry property. Our result complements previous nonuniform recovery results in this context, and provides stronger stability guarantees for noisy measurements and approximately sparse signals. Second, we determine the minimal number of scalar-valued measurements needed to uniformly recover all fusion frame sparse signals via mixed ℓ 1 /ℓ 2 -minimization. This bound is achieved by scalar-valued subgaussian measurements. In particular, our result shows that the number of scalar-valued subgaussian measurements cannot be further reduced using knowledge of the coherence parameter. As a special case it implies that the best known uniform recovery result for block sparse signals using subgaussian measurements is optimal.

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Section: Block Sparsity and Fusion Framesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Uniform recovery of fusion frame structured sparse signals

Ayaz

Dirksen

Rauhut

2016

Applied and Computational Harmonic Analysis

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“…Thus the Gabor-Steiner ETF generated from any finite, abelian p-group is roux. We note that an immediate porism of this result is that the so-called Naimark complement [1], [2] of any G(p, . .…”

Section: Equiangular Tight Frames and Group Covariancementioning

confidence: 76%

“…Frames are generalizations of orthonormal bases which have applications in signal processing, quantization, coding theory, and more [1], [2]. Equiangular tight frames are the closest analog to orthonormal bases in a redundant setting and are known to give representations of data that are optimally robust to erasures and noise [3].…”

Section: Introductionmentioning

confidence: 99%

2- and 3-Covariant Equiangular Tight Frames

King

2019

2019 13th International Conference on Sampling Theory and Applications (SampTA)

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Equiangular tight frames (ETFs) are configurations of vectors which are optimally geometrically spread apart and provide resolutions of the identity. Many known constructions of ETFs are group covariant, meaning they result from the action of a group on a vector, like all known constructions of symmetric, informationally complete, positive operator-valued measures. In this short article, some results characterizing the transitivity of the symmetry groups of ETFs will be presented as well as a proof that an infinite class of so-called Gabor-Steiner ETFs are roux lines, where roux lines are a generalization of doubly transitive lines.

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“…Later, in 1986, Daubechies, Grossmann and Meyer [10] found new applications to wavelet and Gabor transforms in which frames played an important role. The basic theory of frames can be found in [3,7,18]. Let {f k } be a frame for H and let {α j,k } be a sequence of scalars.…”

Section: Introductionmentioning

confidence: 99%

Irregular Weyl–Heisenberg wave packet frames in L2(R)

Sah

Vashisht

2015

Bulletin des Sciences Mathématiques

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Let {f k } be a frame for a Hilbert H and let {α j,k } be a sequence of scalars. Aldroubi (1995) [1] considered a linear combination of the form g j = ∞ k=1 α j,k f k (j ∈ N) and proved sufficient conditions on {α j,k } such that {g j } constitutes a frame for H. In this paper we discuss linear combination of the irregular Weyl-Heisenberg wave packet frames for L 2 (R). Necessary and sufficient conditions for a finite sum of irregular Weyl-Heisenberg wave packet frames to be a frame for L 2 (R) are given.

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Fusion Frames

Cited by 12 publications

References 37 publications

Uniform recovery of fusion frame structured sparse signals

Uniform recovery of fusion frame structured sparse signals

2- and 3-Covariant Equiangular Tight Frames

Irregular Weyl–Heisenberg wave packet frames in L2(R)

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