Compressed sensing with preconditioning for sparse recovery with subsampled matrices of Slepian prolate functions

Gosse, Laurent

doi:10.1007/s11565-012-0159-3

Cited by 8 publications

(9 citation statements)

References 58 publications

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“…Izu and Lakey [40] have drawn an analogy between sampling bounds for multiband signals and classical results in CS, but not specifically for the purpose of using the finite-dimensional CS framework for sparse recovery of sample vectors from multiband analog signals. Gosse [37] has considered the recovery of smooth functions from random samples; however, this work focuses on a very different setting, employing a PSWF (not DPSS) dictionary, considering only baseband signals, and exploiting sparsity in a different way than our work. Senay et al [60,61] have also considered a PSWF dictionary for reconstruction of signals from nonuniform samples; however, this work also focuses on baseband signals and lacks formal approximation and CS recovery guarantees.…”

Section: Relation To Existing Workmentioning

confidence: 99%

“…As a basis for efficiently representing many such vectors x, we propose the following. First, let W = B band Ts 2 , and as in (37), let S N,W denote the N × N matrix containing the N DPSS vectors (constructed with parameters N and W ) as columns. Next, define f c = F c T s and let E fc denote an N × N diagonal matrix with entries…”

Section: A Bandpass Modulated Dpss Basismentioning

confidence: 99%

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Compressive sensing of analog signals using Discrete Prolate Spheroidal Sequences

Davenport

Wakin

2012

Applied and Computational Harmonic Analysis

116

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Compressive sensing (CS) has recently emerged as a framework for efficiently capturing signals that are sparse or compressible in an appropriate basis. While often motivated as an alternative to Nyquist-rate sampling, there remains a gap between the discrete, finite-dimensional CS framework and the problem of acquiring a continuous-time signal. In this paper, we attempt to bridge this gap by exploiting the Discrete Prolate Spheroidal Sequences (DPSS's), a collection of functions that trace back to the seminal work by Slepian, Landau, and Pollack on the effects of time-limiting and bandlimiting operations. DPSS's form a highly efficient basis for sampled bandlimited functions; by modulating and merging DPSS bases, we obtain a dictionary that offers high-quality sparse approximations for most sampled multiband signals. This multiband modulated DPSS dictionary can be readily incorporated into the CS framework. We provide theoretical guarantees and practical insight into the use of this dictionary for recovery of sampled multiband signals from compressive measurements.

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Section: Relation To Existing Workmentioning

confidence: 99%

Section: A Bandpass Modulated Dpss Basismentioning

confidence: 99%

Compressive sensing of analog signals using Discrete Prolate Spheroidal Sequences

Davenport

Wakin

2012

Applied and Computational Harmonic Analysis

116

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“…To the best of our knowledge, C. Niven was the first, in 1880, to give a remarkably detailed theoretical, as well as computational studies of the eigenfunctions and the eigenvalues of L c , see [24]. Nowadays work on PSWFs is mainly connected with possible applications in signal processing [6,12,17,18] and other scientific issues. In geophysics, for instance, they provide good approximations of the Rossby waves that constitute the planetary scale waves in the atmosphere and ocean, see [7,20,21,22,25].…”

Section: Introductionmentioning

confidence: 99%

Uniform Approximation and Explicit Estimates for the Prolate Spheroidal Wave Functions

Bonami

Karoui

2015

Constr Approx

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For fixed c, Prolate Spheroidal Wave Functions (PSWFs), denoted by ψ n,c , form an orthogonal basis with remarkable properties for the space of band-limited functions with bandwith c. They have been largely studied and used after the seminal work of D. Slepian and his co-authors. In several applications, uniform estimates of the ψ n,c in n and c, are needed. To progress in this direction, we push forward the uniform approximation error bounds and give an explicit approximation of their values at 1 in terms of the Legendre complete elliptic integral of the first kind. Also, we give an explicit formula for the accurate approximation the eigenvalues of the Sturm-Liouville operator associated with the PSWFs.2010 Mathematics Subject Classification. Primary 42C10, 65L70. Secondary 41A60, 65L15. Key words and phrases. Prolate spheroidal wave functions, asymptotic and uniform estimates, eigenvalues and eigenfunctions, Sturm-Liouville operator.

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“…To describe the errors, we introduce the broken Sobolev space: 25) equipped with the norm and semi-norm 26) then for any u ∈ H σ (a,b) with σ ≥ 1, we have 27) where D and δ are positive constants independent of u, N and c.…”

Section: Prolate-element Methods and Nonconvergence Of H-refinementmentioning

confidence: 99%

“…(iv) Diverse applications in sampling, signal processing, time series analysis and image processing (see, e.g., [12,17,26,27,35,38,46,57,74]).…”

Section: Introductionmentioning

confidence: 99%

A Review of Prolate Spheroidal Wave Functions from the Perspective of Spectral Methods

Wang¹

2017

JMS

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Compressed sensing with preconditioning for sparse recovery with subsampled matrices of Slepian prolate functions

Cited by 8 publications

References 58 publications

Compressive sensing of analog signals using Discrete Prolate Spheroidal Sequences

Compressive sensing of analog signals using Discrete Prolate Spheroidal Sequences

Uniform Approximation and Explicit Estimates for the Prolate Spheroidal Wave Functions

A Review of Prolate Spheroidal Wave Functions from the Perspective of Spectral Methods

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