Sparse phase retrieval of structured signals by Prony's method

Beinert, Robert; Plonka, Gerlind

doi:10.1002/pamm.201710382

Cited by 7 publications

(4 citation statements)

References 7 publications

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“…where δ is the Dirac delta function, c i ∈ C and t i ∈ R. In this setting, the uniqueness can be guaranteed by O(k 2 ) samples of the Fourier magnitude [17].…”

Section: Sparse Signalsmentioning

confidence: 99%

Fourier Phase Retrieval: Uniqueness and Algorithms

Bendory

Beinert

Eldar

2017

Compressed Sensing and Its Applications

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The problem of recovering a signal from its phaseless Fourier transform measurements, called Fourier phase retrieval, arises in many applications in engineering and science. Fourier phase retrieval poses fundamental theoretical and algorithmic challenges. In general, there is no unique mapping between a one-dimensional signal and its Fourier magnitude and therefore the problem is ill-posed. Additionally, while almost all multidimensional signals are uniquely mapped to their Fourier magnitude, the performance of existing algorithms is generally not well-understood. In this chapter we survey methods to guarantee uniqueness in Fourier phase retrieval. We then present different algorithmic approaches to retrieve the signal in practice. We conclude by outlining some of the main open questions in this field.

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“…where δ is the Dirac delta function, c i ∈ C and t i ∈ R. In this setting, the uniqueness can be guaranteed by O(k 2 ) samples of the Fourier magnitude [17].…”

Section: Sparse Signalsmentioning

confidence: 99%

Fourier Phase Retrieval: Uniqueness and Algorithms

Bendory

Beinert

Eldar

2017

Compressed Sensing and Its Applications

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“…Aiming at this problem, scholars have expanded the research field of phase retrieval. In [30], Prony's method is first used to realize phase retrieval of a specific type of continuous signals. Subsequently, the concept of continuous domain phase retrieval is first proposed in [31].…”

Section: Introductionmentioning

confidence: 99%

A Compact Super Resolution Phase Retrieval Method for Fourier Measurement

Zheng

et al. 2022

IEEE Trans. Instrum. Meas.

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“…In [4], the authors apply the Prony method to this problem to provide a sufficient condition for the recovery of a complex linear combination of s Dirac measures, µ = s j=1 c j δ tj using a number of measured quantities that is O(s 2 ). Here, µ is determined up to an overall unimodular multiplicative constant, so instead of the signal, we get [µ] = {cµ : c ∈ C, |c| = 1}.…”

Section: Introductionmentioning

confidence: 99%

Spikes, Roots, and Modulations: Phase Retrieval for Finitely-Supported Complex Measures

Bodmann¹,

Abouserie²

2022

Preprint

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We study the recovery of a finitely supported distribution, a complex linear combination of Dirac measures, from intensity measurements. The distribution µ = s j=1 c j δt j is given by a coefficient vector c ∈ C s and its support {t 1 , t 2 , . . . , ts} is contained in [0, Λ] for some Λ > 0. The intensity measurements evaluate (squared) magnitudes of a set of linear functionals applied to µ, obtained by sampling μ, the Fourier transform of µ, or by evaluating differences between modulated samples. Following a strategy by Alexeev et al., the structure of the linear functionals, and hence of the non-linear magnitude measurement, is encoded with a graph, where the vertices represent point evaluations of μ at {v 1 , v 2 , . . . , vn} ⊂ [−Ω, Ω] and each edge represents a (modulated) difference between vertices incident with it. We show that a Ramanujan graph with degree d ≥ 3 and n > 6(1+6/ ln(s/ΛΩ))svertices provides M = (d + 1)n magnitudes that are sufficient for identifying the complex measure up to an overall unimodular multiplicative constant. At the cost of including an additional oversampling step and with an additional requirement that n − 1 is prime, we construct an explicit recovery algorithm that is based on the Prony method.

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Sparse phase retrieval of structured signals by Prony's method

Cited by 7 publications

References 7 publications

Fourier Phase Retrieval: Uniqueness and Algorithms

Fourier Phase Retrieval: Uniqueness and Algorithms

A Compact Super Resolution Phase Retrieval Method for Fourier Measurement

Spikes, Roots, and Modulations: Phase Retrieval for Finitely-Supported Complex Measures

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