Phase retrieval from local correlation measurements with fixed shift length

Melnyk, Oleh; Filbir, Frank; Krahmer, Felix

doi:10.1364/math.2019.mtu4d.3

Cited by 4 publications

(6 citation statements)

References 4 publications

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“…We also note the very recent and excellent work of Rayan Saab and Brian Preskitt [39] as well as that of Melnyk, Filbir, and Krahmer [35] which both prove results similar to Theorem 2. As in Theorem 2, the results of [35,39] can guarantee recovery with shift sizes a > 1. Their results primarily differ from Theorem 2 in that they don't used Wigner Distribution Deconvolution (WDD) based methods.…”

Section: Introductionmentioning

confidence: 53%

“…However, we make additional assumptions about either the support of m or the supports of m and x. We also note the very recent and excellent work of Rayan Saab and Brian Preskitt [39] as well as that of Melnyk, Filbir, and Krahmer [35] which both prove results similar to Theorem 2. As in Theorem 2, the results of [35,39] can guarantee recovery with shift sizes a > 1.…”

Section: Introductionmentioning

confidence: 92%

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Inverting Spectrogram Measurements via Aliased Wigner Distribution Deconvolution and Angular Synchronization

Perlmutter¹,

Merhi²,

Viswanathan³

et al. 2019

Preprint

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We propose a two-step approach for reconstructing a signal x ∈ C d from subsampled short-time Fourier transform magnitude (spectogram) measurements: First, we use an aliased Wigner distribution deconvolution approach to solve for a portion of the rank-one matrix x x * . Second, we use angular syncrhonization to solve for x (and then for x by Fourier inversion). Using this method, we produce two new efficient phase retrieval algorithms that perform well numerically in comparison to standard approaches and also prove two theorems, one which guarantees the recovery of discrete, bandlimited signals x ∈ C d from fewer than d STFT magnitude measurements and another which establishes a new class of deterministic coded diffraction pattern measurements which are guaranteed to allow efficient and noise robust recovery.

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Section: Introductionmentioning

confidence: 53%

Section: Introductionmentioning

confidence: 92%

Inverting Spectrogram Measurements via Aliased Wigner Distribution Deconvolution and Angular Synchronization

Perlmutter¹,

Merhi²,

Viswanathan³

et al. 2019

Preprint

View full text Add to dashboard Cite

show abstract

“…Very recently, excellent work by Pearlmutter et al [46] studied the related problem of inverting spectrogram measurements when the mask is locally supported and the signal is bandlimited. Finally, [43] studied the ptychographic setup we consider in the second part of the paper, and proved an analogous (but slightly different) result to our Theorem 4 using the magnitude estimation techniques of [32], rather than the more sophisticated techniques we consider herein.…”

Section: Related Workmentioning

confidence: 71%

“…as desired. The first two inequalities of (43) are immediate by observing that µ j = m when the covering is {[m] i } i∈ [d] . The third comes from setting i = X ii − (X 0 ) ii and writing…”

Section: Algorithm 3 Blockwise Magnitude Estimationmentioning

confidence: 99%

“…2 , and obviously max j µ j min j µ j ≥ 1, so the bound for BlockMag(X, {J i }) in (42) can never be better than that for BlockMag(X, δ) in (43). We also remark that two easy ways to ensure max j µ j min j µ j = 1 is minimized are to take J 0 ⊆ [δ] 0 and let J = J 0 + be a "cyclic" covering, or to let {J i } be a partition of [d].…”

Section: Algorithm 3 Blockwise Magnitude Estimationmentioning

confidence: 99%

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Admissible Measurements and Robust Algorithms for Ptychography

Preskitt¹,

Saab²

2019

Preprint

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We study an approach to solving the phase retrieval problem as it arises in a phase-less imaging modality known as ptychography. In ptychography, small overlapping sections of an unknown sample (or signal, say x 0 ∈ C d ) are illuminated one at a time, often with a physical mask between the sample and light source. The corresponding measurements are the noisy magnitudes of the Fourier transform coefficients resulting from the pointwise product of the mask and the sample. The goal is to recover the original signal from such measurements.The algorithmic framework we study herein relies on first inverting a linear system of equations to recover a fraction of the entries in x 0 x * 0 and then using non-linear techniques to recover the magnitudes and phases of the entries of x 0 . Thus, this paper's contributions are three-fold. First, focusing on the linear part, it expands the theory studying which measurement schemes (i.e., masks, shifts of the sample) yield invertible linear systems, including an analysis of the conditioning of the resulting systems. Second, it analyzes a class of improved magnitude recovery algorithms and, third, it proposes and analyzes algorithms for phase recovery in the ptychographic setting where large shifts -up to 50% the size of the mask -are permitted.

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Admissible Measurements and Robust Algorithms for Ptychography

Preskitt

Saab

2021

J Fourier Anal Appl

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Phase retrieval from local correlation measurements with fixed shift length

Cited by 4 publications

References 4 publications

Inverting Spectrogram Measurements via Aliased Wigner Distribution Deconvolution and Angular Synchronization

Inverting Spectrogram Measurements via Aliased Wigner Distribution Deconvolution and Angular Synchronization

Admissible Measurements and Robust Algorithms for Ptychography

Admissible Measurements and Robust Algorithms for Ptychography

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