Phaseless Sampling and Reconstruction of Real-Valued Signals in Shift-Invariant Spaces

Cheng, Cheng; Jiang, Junzheng; Sun, Qiyu

doi:10.1007/s00041-018-9639-x

Cited by 28 publications

(30 citation statements)

References 75 publications

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“…Our second problem concerns the determination of signals in the shift‐invariant space

V false(scriptH ϕ false) : = ({}, \sum_{l \in double-struckZ} c_{l} scriptH ϕ false(\cdot - l false) : false{c_{l} : l \in double-struckZ false} \in ℓ^{2} false(double-struckZ false))

by phaseless sampling, where ϕ is compactly supported. The problem is closely related with both the phaseless sampling in shift‐invariant spaces (cf Cheng et al and Sun) and analytic signal theory (cf previous studies()).…”

Section: Introductionmentioning

confidence: 97%

“…On the other hand, the Hilbert transform of a compactly supported function is commonly not compactly supported. Consequently, if ϕ is compactly supported, then the phaseless sampling results in Cheng et al and Sun do not hold for

V false(scriptH ϕ false)

in . Unlike the methods in previous studies,() we do not intend to directly determine the target signal in

V false(scriptH ϕ false)

by its phaseless sampling.…”

Section: Introductionmentioning

confidence: 98%

“…Please refer to Cheng et al, Sun et al, and Sun for the related developments of phaseless sampling in a shift‐invariant space

V false(φ false) : = true{\sum_{l \in double-struckZ} c_{l} φ false(\cdot - l false) : false{c_{l} : l \in double-struckZ false} \in ℓ^{2} false(double-struckZ false) true}

. Recall that in previous studies,() the generator φ for V ( φ ) is required to be compactly supported. On the other hand, the Hilbert transform of a compactly supported function is commonly not compactly supported.…”

Section: Introductionmentioning

confidence: 99%

“…by phaseless sampling, where is compactly supported. The problem is closely related with both the phaseless sampling in shift-invariant spaces (cf Cheng et al 16 and Sun 17 ) and analytic signal theory (cf previous studies 11,[18][19][20]. Phaseless sampling is a special phase-retrieval problem.…”

Section: Introductionmentioning

confidence: 99%

“…It aims at reconstructing a real-valued signal f from samples of |f|. Please refer to Cheng et al, 16 Sun et al, 21 and Sun 17 for the related developments of phaseless sampling in a…”

Section: Introductionmentioning

confidence: 99%

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Phaseless inverse discrete Hilbert transform and determination of signals in shift‐invariant space

Yuan

2019

Math Methods in App Sciences

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In this paper, we first address the uniqueness of phaseless inverse discrete Hilbert transform (phaseless IDHT for short) from the magnitude of discrete Hilbert transform (DHT for short). The measurement vectors of phaseless IDHT do not satisfy the complement property, a traditional requirement for ensuring the uniqueness of phase retrieval. Consequently, the uniqueness problem of phaseless IDHT is essentially different from that of the traditional phase retrieval. For the phaseless IDHT related to compactly supported functions, conditions are given in our first main theorem to ensure the uniqueness. A condition on the step size of DHT is crucial for the insurance. The second main theorem concerns the uniqueness related to noncompactly supported functions. It is not the trivial generalization of the first main theorem. Our third main result is on the determination of signals in shift‐invariant spaces by phaseless IDHT. Note that the measurements used for the determination are the DHT magnitudes. They are the approximations to the Hilbert transform magnitudes. Recall that for the existing methods of phaseless sampling (a special phase‐retrieval problem), to determine a signal depends on the exact measurements but not the approximative ones. Therefore, our determination method is essentially different from the traditional phaseless sampling. Numerical simulations are conducted to examine the efficiency of phaseless IDHT and its application in determining signals in spline Hilbert shift‐invariant space.

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“…Our second problem concerns the determination of signals in the shift‐invariant space

V false(scriptH ϕ false) : = ({}, \sum_{l \in double-struckZ} c_{l} scriptH ϕ false(\cdot - l false) : false{c_{l} : l \in double-struckZ false} \in ℓ^{2} false(double-struckZ false))

Section: Introductionmentioning

confidence: 97%

V false(scriptH ϕ false)

in . Unlike the methods in previous studies,() we do not intend to directly determine the target signal in

V false(scriptH ϕ false)

by its phaseless sampling.…”

Section: Introductionmentioning

confidence: 98%

“…Please refer to Cheng et al, Sun et al, and Sun for the related developments of phaseless sampling in a shift‐invariant space

V false(φ false) : = true{\sum_{l \in double-struckZ} c_{l} φ false(\cdot - l false) : false{c_{l} : l \in double-struckZ false} \in ℓ^{2} false(double-struckZ false) true}

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…It aims at reconstructing a real-valued signal f from samples of |f|. Please refer to Cheng et al, 16 Sun et al, 21 and Sun 17 for the related developments of phaseless sampling in a…”

Section: Introductionmentioning

confidence: 99%

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Phaseless inverse discrete Hilbert transform and determination of signals in shift‐invariant space

Yuan

2019

Math Methods in App Sciences

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Generalized phase retrieval in quaternion Euclidean spaces

Yang,

2024

Math Methods in App Sciences

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Recently, quaternionic Fourier analysis has received increasing attention due to its applications in signal analysis and image processing. This paper addresses quaternionic generalized phase retrieval (QGPR) problem in quaternion Euclidean spaces . We introduce the concept of QGPR which aims to reconstruct a signal in from the quadratic measurements , where each is an self‐adjoint quaternion matrix. We characterize QGPR sequences in terms of their real Jacobian matrices, prove that the set of QGPR sequences is an open set in some sense, and present some phaselift‐based sufficient conditions on QGPR which gives a method to construct QGPR sequences.

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Shift-Invariant Spaces of Entire Functions

Gröchenig¹

2021

Trends in Mathematics

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Phaseless Sampling and Reconstruction of Real-Valued Signals in Shift-Invariant Spaces

Cited by 28 publications

References 75 publications

Phaseless inverse discrete Hilbert transform and determination of signals in shift‐invariant space

Phaseless inverse discrete Hilbert transform and determination of signals in shift‐invariant space

Generalized phase retrieval in quaternion Euclidean spaces

Shift-Invariant Spaces of Entire Functions

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