Stable Phase Retrieval in Infinite Dimensions

Alaifari, Rima; Daubechies, Ingrid; Yin, Rujie

doi:10.48550/arxiv.1609.00034

Cited by 11 publications

(37 citation statements)

References 24 publications

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“…We further note that a partial remedy to the instability of phase retrieval in the infinite-dimensional setting is to relax the notion of phase retrieval as introduced in [3]. There, phase retrieval is studied for the case of magnitude measurements |V ϕ f | from the Gabor or Cauchy wavelet transform.…”

Section: 24mentioning

confidence: 99%

Phase Retrieval In The General Setting Of Continuous Frames For Banach Spaces

Alaifari¹

2017

SIAM J. Math. Anal.

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Abstract. We develop a novel and unifying setting for phase retrieval problems that works in Banach spaces and for continuous frames and consider the questions of uniqueness and stability of the reconstruction from phaseless measurements. Our main result states that also in this framework, the problem of phase retrieval is never uniformly stable in infinite dimensions. On the other hand, we show weak stability of the problem. This complements recent work [11], where it has been shown that phase retrieval is always unstable for the setting of discrete frames in Hilbert spaces. In particular, our result implies that the stability properties cannot be improved by oversampling the underlying discrete frame.We generalize the notion of complement property (CP) to the setting of continuous frames for Banach spaces (over K = R or K = C) and verify that it is a necessary condition for uniqueness of the phase retrieval problem; when K = R the CP is also sufficient for uniqueness. In our general setting, we also prove a conjecture posed by Bandeira et al. [7], which was originally formulated for finite-dimensional spaces: for the case K = C the strong complement property (SCP) is a necessary condition for stability. To prove our main result, we show that the SCP can never hold for frames of infinite-dimensional Banach spaces.

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Section: 24mentioning

confidence: 99%

Phase Retrieval In The General Setting Of Continuous Frames For Banach Spaces

Alaifari¹

2017

SIAM J. Math. Anal.

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“…Given a concrete phase retrieval problem defined by a measurement system Φ it is notoriously difficult to study whether Φ does phase retrieval and there are only a few concrete instances where this is known. In the ∞-dimensional setting, examples of such instances include phase retrieval from Poisson wavelet measurements [47], from Gabor measurements [1] or from masked Fourier measurements [48], while it is known that the reconstruction of a compactly supported function from its Fourier magnitude is in general not uniquely possible [32].…”

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confidence: 99%

“…Phase Retrieval is Severely Ill-Posed. Despite its formidable relevance, the study of stability properties of phase retrieval problems has seen little progress until in recent work [1,13] a striking instability phenomenon has been identified by showing that sup f ∈B c(f ) = ∞, whenever dimB = ∞ 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 [3,13].…”

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confidence: 99%

“…A well-known source of instability (e.g., a very large constant c(f )), coined 'multicomponenttype instability' in [1] arises whenever the measurements Φ(f ) are separated in the sense that f = u + v with Φ(u) and Φ(v) concentrated in disjoint subsets of Ω. Intiutively, in this case the function g = u − v will produce measurements |Φ(g)| very close to the original measurements |Φ(f )|, while the distance d B (f, g) is not small at all, resulting in an instability (see also Figure 1 for an illustration). If B is a finite-dimensional Hilbert space over R (i.e., the real-valued case where only a sign and not the full phase needs to be determined) the correctness of this intuition has been proved in [8] and generalized in [2] to the setting of ∞-dimensional real or complex Banach spaces:…”

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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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“…The phaseless sampling and reconstruction of signals in a shift-invariant space is an infinite-dimensional phase retrieval problem, which has received considerable attention in recent years [1,2,3,12,20,34,36,37,40,44]. Phase retrieval plays important roles in signal/speech/image processing and it is a highly nonlinear mathematical problem.…”

Section: Introductionmentioning

confidence: 99%

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

Cheng

Jiang

Sun

2018

J Fourier Anal Appl

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A spatial signal is defined by its evaluations on the whole domain. In this paper, we consider stable reconstruction of real-valued signals with finite rate of innovations (FRI), up to a sign, from their magnitude measurements on the whole domain or their phaseless samples on a discrete subset. FRI signals appear in many engineering applications such as magnetic resonance spectrum, ultra wide-band communication and electrocardiogram. For an FRI signal, we introduce an undirected graph to describe its topological structure. We establish the equivalence between the graph connectivity and phase retrievability of FRI signals, and we apply the graph connected component decomposition to find all FRI signals that have the same magnitude measurements as the original FRI signal has. We construct discrete sets with finite density explicitly so that magnitude measurements of FRI signals on the whole domain are determined by their samples taken on those discrete subsets. In this paper, we also propose a stable algorithm with linear complexity to reconstruct FRI signals from their phaseless samples on the above phaseless sampling set. The proposed algorithm is demonstrated theoretically and numerically to provide a suboptimal approximation to the original FRI signal in magnitude measurements.

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Stable Phase Retrieval in Infinite Dimensions

Cited by 11 publications

References 24 publications

Phase Retrieval In The General Setting Of Continuous Frames For Banach Spaces

Phase Retrieval In The General Setting Of Continuous Frames For Banach Spaces

Stable Gabor Phase Retrieval and Spectral Clustering

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

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