Stable Phase Retrieval in Infinite Dimensions

Alaifari, Rima; Daubechies, Ingrid; Grohs, Philipp; Yin, Rujie

doi:10.1007/s10208-018-9399-7

Cited by 58 publications

(98 citation statements)

References 44 publications

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“…In this subsection we briefly relate our results to the stable solution of the backwards heat equation and our previous work [1].…”

Section: Putting Our Results In Perspectivementioning

confidence: 86%

“…While this conjecture seems to be folklore in the phase retrieval community 1 and ensuring connectedness of the essential support of the measurements is a common empirical regularization strategy [8,26,35,47], we are not aware of any mathematical result that resolves Conjecture 1.1 for any concrete phase retrieval problem.…”

Section: What Are the Sources For Instability?mentioning

confidence: 96%

“…In the 1-dimensional setting, examples of such instances include phase retrieval from Poisson wavelet measurements [48], from Gabor measurements [1], and from masked Fourier measurements [49], while it is known that the reconstruction of a compactly supported function from its Fourier magnitude is in general not uniquely possible [33]. In the 1-dimensional setting, examples of such instances include phase retrieval from Poisson wavelet measurements [48], from Gabor measurements [1], and from masked Fourier measurements [49], while it is known that the reconstruction of a compactly supported function from its Fourier magnitude is in general not uniquely possible [33].…”

Section: Motivationmentioning

confidence: 99%

“…Despite its formidable relevance, the study of stability properties of phase retrieval problems has seen little progress until recently [1,13]; a striking instability phenomenon has been identified by showing that sup f 2B c.f / D 1 whenever dim B D 1 and some natural conditions on B and D are satisfied. Even worse, the stability of finite-dimensional approximations to such problems in general degenerates exponentially in a power of the dimension [2,13].…”

Section: Phase Retrieval Is Severely Ill-posedmentioning

confidence: 99%

“…A well-known source of instability (e.g., a very large constant c.f /), coined "multicomponent-type instability" in [1] arises whenever the measurementsˆ.f / are separated in the sense that f D u C v withˆ.u/ andˆ.v/ concentrated in disjoint subsets of . A well-known source of instability (e.g., a very large constant c.f /), coined "multicomponent-type instability" in [1] arises whenever the measurementsˆ.f / are separated in the sense that f D u C v withˆ.u/ andˆ.v/ concentrated in disjoint subsets of .…”

Section: What Are the Sources For Instability?mentioning

confidence: 99%

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Stable Gabor Phase Retrieval and Spectral Clustering

Rathmair

2018

Comm Pure Appl Math

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We consider the problem of reconstructing a signal f from its spectrogram, i.e., the magnitudes jV ' f j of its Gabor transformSuch problems occur in a wide range of applications, from optical imaging of nanoscale structures to audio processing and classification. While it is well-known that the solution of the above Gabor phase retrieval problem is unique up to natural identifications, the stability of the reconstruction has remained wide open. The present paper discovers a deep and surprising connection between phase retrieval, spectral clustering, and spectral geometry. We show that the stability of the Gabor phase reconstruction is bounded by the reciprocal of the Cheeger constant of the flat metric on R 2 , conformally multiplied with jV ' f j. The Cheeger constant, in turn, plays a prominent role in the field of spectral clustering, and it precisely quantifies the "disconnectedness" of the measurements V ' f .It has long been known that a disconnected support of the measurements results in an instability-our result for the first time provides a converse in the sense that there are no other sources of instabilities.Due to the fundamental importance of Gabor phase retrieval in coherent diffraction imaging, we also provide a new understanding of the stability properties of these imaging techniques: Contrary to most classical problems in imaging science whose regularization requires the promotion of smoothness or sparsity, the correct regularization of the phase retrieval problem promotes the "connectedness" of the measurements in terms of bounding the Cheeger constant from below. Our work thus, for the first time, opens the door to the development of efficient regularization strategies.

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“…In this subsection we briefly relate our results to the stable solution of the backwards heat equation and our previous work [1].…”

Section: Putting Our Results In Perspectivementioning

confidence: 86%

Section: What Are the Sources For Instability?mentioning

confidence: 96%

Section: Motivationmentioning

confidence: 99%

Section: Phase Retrieval Is Severely Ill-posedmentioning

confidence: 99%

Section: What Are the Sources For Instability?mentioning

confidence: 99%

See 3 more Smart Citations

Stable Gabor Phase Retrieval and Spectral Clustering

Rathmair

2018

Comm Pure Appl Math

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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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Ill-Posed Problems: From Linear to Nonlinear and Beyond

Alaifari

2021

Harmonic and Applied Analysis

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Stable Phase Retrieval in Infinite Dimensions

Abstract: The problem of phase retrieval

Cited by 58 publications

References 44 publications

Stable Gabor Phase Retrieval and Spectral Clustering

Stable Gabor Phase Retrieval and Spectral Clustering

Generalized phase retrieval in quaternion Euclidean spaces

Ill-Posed Problems: From Linear to Nonlinear and Beyond

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