Characterization of coorbit spaces with phase-space covers

Romero, José Luis

doi:10.1016/j.jfa.2011.09.005

Cited by 22 publications

(29 citation statements)

References 49 publications

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“…Indeed, when we identify a signal with its canonical frame coefficients, it turns out that truncating a frame expansion is a Toeplitz-like operation: it sets some coefficients to zero and then projects the result onto the space of coefficients that are compatible with the restrictions imposed by redundancy. This perspective has been exploited in different contexts in [32,33,19] and we use some technical insights from that work.…”

Section: Technical Overviewmentioning

confidence: 99%

Computing reconstructions from nonuniform Fourier samples: Universality of stability barriers and stable sampling rates

Adcock

Gatarić

Romero

2019

Applied and Computational Harmonic Analysis

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We study the problem of recovering an unknown compactly-supported multivariate function from samples of its Fourier transform that are acquired nonuniformly, i.e. not necessarily on a uniform Cartesian grid. Reconstruction problems of this kind arise in various imaging applications, where Fourier samples are taken along radial lines or spirals for example.Specifically, we consider finite-dimensional reconstructions, where a limited number of samples is available, and investigate the rate of convergence of such approximate solutions and their numerical stability. We show that the proportion of Fourier samples that allow for stable approximations of a given numerical accuracy is independent of the specific sampling geometry and is therefore universal for different sampling scenarios. This allows us to relate both sufficient and necessary conditions for different sampling setups and to exploit several results that were previously available only for very specific sampling geometries.The results are obtained by developing: (i) a transference argument for different measures of the concentration of the Fourier transform and Fourier samples; (ii) frame bounds valid up to the critical sampling density, which depend explicitly on the sampling set and the spectrum.As an application, we identify sufficient and necessary conditions for stable and accurate reconstruction of algebraic polynomials or wavelet coefficients from nonuniform Fourier data.

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Section: Technical Overviewmentioning

confidence: 99%

Computing reconstructions from nonuniform Fourier samples: Universality of stability barriers and stable sampling rates

Adcock

Gatarić

Romero

2019

Applied and Computational Harmonic Analysis

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“…Filters optimizing classical window quality measures are constructed in [32,33], without frame theoretic considerations. Recent results on phase space covers [34,35], based on a collection of Gabor systems, require certain decay of the dual windows, not necessarily provided by the canonical dual. Thus, the study of alternate dual windows is necessary.…”

Section: Innovations Of This Contributionmentioning

confidence: 99%

Designing Gabor windows using convex optimization

Perraudin

Holighaus

Søndergaard

et al. 2018

Applied Mathematics and Computation

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Redundant Gabor frames admit an infinite number of dual frames, yet only the canonical dual Gabor system, constructed from the minimal 2 -norm dual window, is widely used. This window function however, might lack desirable properties, e.g. good timefrequency concentration, small support or smoothness. We employ convex optimization methods to design dual windows satisfying the Wexler-Raz equations and optimizing various constraints. Numerical experiments suggest that alternate dual windows with considerably improved features can be found.

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“…If Ω ⊆ R 2d is compact, then H Ω is a compact and positive operator on L 2 (R d ) [9,10,16,39]. Hence H Ω can be diagonalized as…”

Section: Localization Operatorsmentioning

confidence: 99%

Sharp rates of convergence for accumulated spectrograms

2017

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Abstract. We investigate an inverse problem in time-frequency localization: the approximation of the symbol of a time-frequency localization operator from partial spectral information by the method of accumulated spectrograms (the sum of the spectrograms corresponding to large eigenvalues). We derive a sharp bound for the rate of convergence of the accumulated spectrogram, improving on recent results.

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Characterization of coorbit spaces with phase-space covers

Cited by 22 publications

References 49 publications

Computing reconstructions from nonuniform Fourier samples: Universality of stability barriers and stable sampling rates

Computing reconstructions from nonuniform Fourier samples: Universality of stability barriers and stable sampling rates

Designing Gabor windows using convex optimization

Sharp rates of convergence for accumulated spectrograms

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