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Phase Retrieval from Coded Diffraction Patterns
, 2013
"... This paper considers the question of recovering the phase of an object from intensityonly measurements, a problem which naturally appears in Xray crystallography and related disciplines. We study a physically realistic setup where one can modulate the signal of interest and then collect the inten ..."
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Cited by 21 (5 self)
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This paper considers the question of recovering the phase of an object from intensityonly measurements, a problem which naturally appears in Xray crystallography and related disciplines. We study a physically realistic setup where one can modulate the signal of interest and then collect the intensity of its diffraction pattern, each modulation thereby producing a sort of coded diffraction pattern. We show that PhaseLift, a recent convex programming technique, recovers the phase information exactly from a number of random modulations, which is polylogarithmic in the number of unknowns. Numerical experiments with noiseless and noisy data complement our theoretical analysis and illustrate our approach.
Invertibility and robustness of phaseless reconstruction
 Appl. Comput. Harmon. Anal
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Phase Retrieval using Lipschitz Continuous Maps, available online arXiv:1403.2304v1
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Stability of Phase Retrievable Frames
 proceedings of SPIE 2013
"... In this paper we study the property of phase retrievability by redundant sysems of vectors under perturbations of the frame set. Specifically we show that if a set F of m vectors in the complex Hilbert space of dimension n allows for vector reconstruction from magnitudes of its coefficients, then th ..."
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In this paper we study the property of phase retrievability by redundant sysems of vectors under perturbations of the frame set. Specifically we show that if a set F of m vectors in the complex Hilbert space of dimension n allows for vector reconstruction from magnitudes of its coefficients, then there is a perturbation bound ρ so that any frame set within ρ from F has the same property. In particular this proves the recent construction in15 is stable under perturbations. By the same token we reduce the critical cardinality conjectured in11 to proving a stability result for non phaseretrievable frames.
FRAMES AND PHASELESS RECONSTRUCTION AMS SHORT COURSE: JOINT MATHEMATICS MEETINGS . . .
"... Frame design for phaseless reconstruction is now part of the broader problem of nonlinear recon struction and is an emerging topic in harmonic analysis. The problem of phaseless reconstruction can be simply stated as follows. Given the magnitudes of the coefficients of an output of a linear redunda ..."
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Frame design for phaseless reconstruction is now part of the broader problem of nonlinear recon struction and is an emerging topic in harmonic analysis. The problem of phaseless reconstruction can be simply stated as follows. Given the magnitudes of the coefficients of an output of a linear redundant system (frame), we want to reconstruct the unknown input. This problem has first occurred in Xray crystallography starting from the early 20th century. The same nonlinear reconstruction problem shows up in speech processing, particularly in speech recognition. In this lecture we shall cover existing analysis results as well as algorithms for signal recovery including: necessary and sufficient conditions for injectivity, Lipschitz bounds of the nonlinear map and its left inverses, stochastic performance bounds, convex relaxation algorithms for inversion, leastsquares inversion algorithms.
STABILITY OF FRAMES WHICH GIVE PHASE RETRIEVAL
"... In this paper we study the property of phase retrievability by redundant sysems of vectors under perturbations of the frame set. Specifically we show that if a set F of m vectors in the complex Hilbert space of dimension n allows for vector reconstruction from magnitudes of its coefficients, then th ..."
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In this paper we study the property of phase retrievability by redundant sysems of vectors under perturbations of the frame set. Specifically we show that if a set F of m vectors in the complex Hilbert space of dimension n allows for vector reconstruction from magnitudes of its coefficients, then there is a perturbation bound ρ so that any frame set within ρ from F has the same property. In particular this proves a recent construction for the case m = 4n − 4 s stable under perturbations. Additionally we provide estimates of the stability radius.
(Chair)
, 2014
"... or the United States Government. This material is declared a work of the U.S. ..."
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or the United States Government. This material is declared a work of the U.S.
Contemporary Mathematics On Lipschitz Inversion of Nonlinear Redundant Representations
"... Abstract. In this note we show that reconstruction from magnitudes of frame coefficients (the so called “phase retrieval problem”) can be performed using Lipschitz continuous maps. Specifically we show that when the nonlinear analysis map α: H → Rm is injective, with (α(x))k = 〈x, fk〉2, where {f ..."
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Abstract. In this note we show that reconstruction from magnitudes of frame coefficients (the so called “phase retrieval problem”) can be performed using Lipschitz continuous maps. Specifically we show that when the nonlinear analysis map α: H → Rm is injective, with (α(x))k = 〈x, fk〉2, where {f1, · · · , fm} is a frame for the Hilbert space H, then there exists a left inverse map ω: Rm → H that is Lipschitz continuous. Additionally we obtain that the Lipschitz constant of this inverse map is at most 12 divided by the lower Lipschitz constant of α. 1.