$α$-Molecules

Keiper, Sandra; Kutyniok, Gitta; Schäfer, Martin

doi:10.48550/arxiv.1407.4424

Cited by 2 publications

(20 citation statements)

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“…As mentioned before, the goal of this paper is to generalize the concept of α-molecules to arbitrary dimensions. The multivariate formulation extends the earlier results from [27,28,20] and gives valuable insights, e.g. on how they scale with the dimension.…”

Section: Our Contributionssupporting

confidence: 59%

“…The scope of parabolic molecules is limited to parabolically scaled systems, wherefore a major generalization was pursued in [28], namely the extension to α-molecules. These incorporate more general α-scaling (3) and can thus bridge the gap between wavelets and ridgelets, as well as curvelets and shearlets in between.…”

Section: α-Moleculesmentioning

confidence: 99%

“…Recalling the definition in [28], a system of bivariate α-molecules consists of functions in L 2 (R 2 ) obtained by applying α-scaling, rotations, and translations to a set of generating functions, which need to be sufficiently localized in time and frequency. Due to this construction, every α-molecule is naturally associated with a certain scale, orientation and spatial position, which -in the 2-dimensional case -is conveniently represented by a point in the corresponding parameter space…”

Section: α-Molecules In D Dimensionsmentioning

confidence: 99%

“…Each function m λ ∈ L 2 (R d ) of a system of d-dimensional α-molecules (m λ ) λ∈Λ shall by definition then be associated with a unique point (s λ , e λ , x λ ) ∈ P d , where the variable s λ ∈ R + shall represent its scale, the vector e λ ∈ S d−1 its orientation in R d , and x λ ∈ R d the spatial location. The relation between the index λ of a molecule m λ and its position (s λ , e λ , x λ ) in P d is described by a so-called parametrization, analogue to [28].…”

Section: α-Molecules In D Dimensionsmentioning

confidence: 99%

“…In case α = 1 this matrix scales R d isotropically, in the range α ∈ [0, 1) it scales uniformly in all directions except for the e d -direction. Note that here we choose e d as the distinguished direction in which the scaling is stronger -in contrast to the 2-dimensional case [28], where e 1 was chosen.…”

Section: α-Molecules In D Dimensionsmentioning

confidence: 99%

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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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Section: Our Contributionssupporting

confidence: 59%

Section: α-Moleculesmentioning

confidence: 99%

Section: α-Molecules In D Dimensionsmentioning

confidence: 99%

Section: α-Molecules In D Dimensionsmentioning

confidence: 99%

Section: α-Molecules In D Dimensionsmentioning

confidence: 99%

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Multivariate α-molecules

Flinth

Schäfer

2016

Journal of Approximation Theory

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Reconstructing Real-Valued Functions from Unsigned Coefficients with Respect to Wavelet and Other Frames

Alaifari

Daubechies

Grohs

et al. 2016

J Fourier Anal Appl

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Abstract. In this paper we consider the following problem of phase retrieval: Given a collection of real-valued band-limited functions {ψ λ } λ∈Λ ⊂ L 2 (R d ) that constitutes a semi-discrete frame, we ask whether any real-valued function f ∈ L 2 (R d ) can be uniquely recovered from its unsigned convolutions {|f * ψ λ |} λ∈Λ .We find that under some mild assumptions on the semi-discrete frame and if f has exponential decay at ∞, it suffices to know |f * ψ λ | on suitably fine lattices to uniquely determine f (up to a global sign factor).We further establish a local stability property of our reconstruction problem. Finally, for two concrete examples of a (discrete) frame of L 2 (R d ), d = 1, 2, we show that through sufficient oversampling one obtains a frame such that any real-valued function with exponential decay can be uniquely recovered from its unsigned frame coefficients.

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$α$-Molecules

Cited by 2 publications

References 0 publications

Multivariate α-molecules

Multivariate α-molecules

Reconstructing Real-Valued Functions from Unsigned Coefficients with Respect to Wavelet and Other Frames

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