Intrinsic Localization of Anisotropic Frames II: $$\alpha $$ α -Molecules

Vigogna, Stefano

doi:10.1007/s00041-014-9365-y

Cited by 6 publications

(11 citation statements)

References 33 publications

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“…As mentioned previously, many of the popular frames such as curvelets, shearlets and wavelet frames are intrinsically localized so that their Gram matrices are near diagonal. This property has been studied for wavelet frames in [26,10,18] and more recently for anisotropic systems such as shearlets and curvelets in [23,24]. In this section, we will show how the property of intrinsic localization can yield estimates on the localized sparsity term, κ, and the localized level sparsity terms, κ j 's.…”

Section: Intrinsic Localizationmentioning

confidence: 97%

“…Remark 5.1 Under this definition, wavelet frames have been shown to be intrinsically localized [10] with the parameter L being dependent on the regularity of the generating wavelets. For the anisotropic systems studied in [23] and [24], the definition of intrinsic localization used is more complex than the definition presented above. However, the key idea of how to exploit this property to obtain bounds on the localized sparsity values should still be applicable.…”

Section: Intrinsic Localizationmentioning

confidence: 99%

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Structure dependent sampling in compressed sensing: Theoretical guarantees for tight frames

Poon

2017

Applied and Computational Harmonic Analysis

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Many of the applications of compressed sensing have been based on variable density sampling, where certain sections of the sampling coefficients are sampled more densely. Furthermore, it has been observed that these sampling schemes are dependent not only on sparsity but also on the sparsity structure of the underlying signal. This paper extends the result of (Adcock, Hansen, Poon and Roman, arXiv:1302.0561, 2013) to the case where the sparsifying system forms a tight frame. By dividing the sampling coefficients into levels, our main result will describe how the amount of subsampling in each level is determined by the local coherences between the sampling and sparsifying operators and the localized level sparsities -the sparsity in each level under the sparsifying operator.

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Section: Intrinsic Localizationmentioning

confidence: 97%

Section: Intrinsic Localizationmentioning

confidence: 99%

Structure dependent sampling in compressed sensing: Theoretical guarantees for tight frames

Poon

2017

Applied and Computational Harmonic Analysis

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“…Hence the function ω α can be viewed as a kind of multiplicative pseudo-metric. A proof of these properties for the 2-dimensional case can be found in [29], which translates very well to higher dimensions. Now we are in a position to formulate the main theorem of this paper.…”

Section: Index Distance In D Dimensionsmentioning

confidence: 68%

“…Altogether, this leads to the following definition. It directly generalizes the metric introduced in [29], which is a simplified version of the original metric from [28].…”

Section: Index Distance In D Dimensionsmentioning

confidence: 99%

Multivariate α-molecules

Flinth

Schäfer

2016

Journal of Approximation Theory

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The suboptimal performance of wavelets with regard to the approximation of multivariate data gave rise to new representation systems, specifically designed for data with anisotropic features. Some prominent examples of these are given by ridgelets, curvelets, and shearlets, to name a few.The great variety of such so-called directional systems motivated the search for a common framework, which unites many under one roof and enables a simultaneous analysis, for example with respect to approximation properties. Building on the concept of parabolic molecules, the recently introduced framework of α-molecules does in fact include the previous mentioned systems. Until now however it is confined to the bivariate setting, whereas nowadays one often deals with higher dimensional data. This motivates the extension of this unifying theory to dimensions larger than 2, put forward in this work. In particular, we generalize the central result that the cross-Gramian of any two systems of α-molecules will to some extent be localized.As an exemplary application, we investigate the sparse approximation of video signals, which are instances of 3D data. The multivariate theory allows us to derive almost optimal approximation rates for a large class of representation systems.Suitable representation systems are provided e.g. by so-called frame systems [6], which ensure stable measurement of the coefficients and also stable reconstruction. A system (m λ ) λ∈Λ in H forms a frame if there exist constants A, B > 0, called the frame bounds, such that A f 2 ≤ λ∈Λ | f, m λ | 2 ≤ B f 2 for all f ∈ H.

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“…[CD04;GK14]). Since many of the proofs (for example of approximation rates) often resemble each other between constructions, some effort has been made recently to unify them by discovering and clarifying the underlying concepts -curvelets, shearlets and contourlets fall into the framework of so-called "parabolic molecules" [GK14], while all of the mentioned systems (including wavelets and ridgelets) are encompassed by the even broader framework of α-molecules [GKKS14].…”

Section: Higher-dimensional Singularitiesmentioning

confidence: 99%

On the approximation of functions with line singularities by ridgelets

Obermeier

2019

Journal of Approximation Theory

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In [GO15], the authors discussed the existence of numerically feasible solvers for advection equations that run in optimal computational complexity. In this paper, we complete the last remaining requirement to achieve this goal -by showing that ridgelets, on which the solver is based, approximate functions with line singularities (which may appear as solutions to the advection equation) with the best possible approximation rate.Structurally, the proof resembles [Can01], where a similar result was proved for a different ridgelet construction, which is however not well-suited for use in a PDE solver (and in particular, not suitable for the CDD-schemes [CDD01] we are interested in). Due to the differences between the two ridgelet constructions, we have to deal with quite a different set of issues, but are also able to relax the (support) conditions on the function being approximated. Finally, the proof employs a new convolution-type estimate that could be of independent interest due to its sharpness.

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Intrinsic Localization of Anisotropic Frames II: $$\alpha $$ α -Molecules

Cited by 6 publications

References 33 publications

Structure dependent sampling in compressed sensing: Theoretical guarantees for tight frames

Structure dependent sampling in compressed sensing: Theoretical guarantees for tight frames

Multivariate α-molecules

On the approximation of functions with line singularities by ridgelets

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