Stable Phase Retrieval in Infinite Dimensions

@article{Alaifari2019StablePR,
  title={Stable Phase Retrieval in Infinite Dimensions},
  author={Rima Alaifari and Ingrid Daubechies and Philipp Grohs and Rujie Yin},
  journal={Foundations of Computational Mathematics},
  year={2019},
  pages={1-32}
}
The problem of phase retrieval is to determine a signal $$f\in \mathcal {H}$$f∈H, with $$ \mathcal {H}$$H a Hilbert space, from intensity measurements $$|F(\omega )|$$|F(ω)|, where $$F(\omega ):=\langle f, \varphi _\omega \rangle $$F(ω):=⟨f,φω⟩ are measurements of f with respect to a measurement system $$(\varphi _\omega )_{\omega \in \Omega }\subset \mathcal {H}$$(φω)ω∈Ω⊂H. Although phase retrieval is always stable in the finite-dimensional setting whenever it is possible (i.e. injectivity… Expand
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References

SHOWING 1-10 OF 52 REFERENCES
On Lipschitz Analysis and Lipschitz Synthesis for the Phase Retrieval Problem
In this paper we prove two results regarding reconstruction from magnitudes of frame coefficients (the so called "phase retrieval problem"). First we show that phase retrievability as an algebraicExpand
Reconstructing Real-Valued Functions from Unsigned Coefficients with Respect to Wavelet and Other Frames
In this paper we consider the following problem of phase retrieval: given a collection of real-valued band-limited functions $$\{\psi _{\lambda }\}_{\lambda \in \Lambda }\subset L^2(\mathbbExpand
Stable Gabor Phase Retrieval and Spectral Clustering
We consider the problem of reconstructing a signal $f$ from its spectrogram, i.e., the magnitudes $|V_\varphi f|$ of its Gabor transform $$V_\varphi f (x,y):=\int_{\mathbb{R}}f(t)e^{-\piExpand
Phase recovery, MaxCut and complex semidefinite programming
TLDR
This work casts the phase retrieval problem as a non-convex quadratic program over a complex phase vector and formulates a tractable relaxation similar to the classical MaxCut semidefinite program. Expand
Phase Retrieval In The General Setting Of Continuous Frames For Banach Spaces
TLDR
It is shown that the problem of phase retrieval is never uniformly stable in infinite dimensions and the stability properties cannot be improved by oversampling the underlying discrete frame. Expand
Gabor phase retrieval is severely ill-posed
The problem of reconstructing a function from the magnitudes of its frame coefficients has recently been shown to be never uniformly stable in infinite-dimensional spaces [5]. This result also holdsExpand
Phase Retrieval of Real-Valued Signals in a Shift-Invariant Space
TLDR
An algorithm to reconstruct nonseparable signals in a shift-invariant space generated by a compactly supported continuous function is proposed and is robust against bounded sampling noise and it could be implemented in a distributed manner. Expand
Phase Retrieval via Wirtinger Flow: Theory and Algorithms
TLDR
This paper develops a nonconvex formulation of the phase retrieval problem as well as a concrete solution algorithm that is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements. Expand
Saving phase: Injectivity and stability for phase retrieval
Abstract Recent advances in convex optimization have led to new strides in the phase retrieval problem over finite-dimensional vector spaces. However, certain fundamental questions remain: What sortsExpand
Uniqueness results for the phase retrieval problem of fractional Fourier transforms of variable order
In this paper, we investigate the uniqueness of the phase retrieval problem for the fractional Fourier transform (FrFT) of variable order. This problem occurs naturally in optics and quantum physics.Expand
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