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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.
Reconstruction of signals from magnitudes of redundant representations
"... Abstract. This paper is concerned with the question of reconstructing a vector in a finitedimensional real or complex Hilbert space when only the magnitudes of the coefficients of the vector under a redundant linear map are known. We present new invertibility results as well an iterative algorithm ..."
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Cited by 12 (6 self)
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Abstract. This paper is concerned with the question of reconstructing a vector in a finitedimensional real or complex Hilbert space when only the magnitudes of the coefficients of the vector under a redundant linear map are known. We present new invertibility results as well an iterative algorithm that finds the leastsquare solution and is robust in the presence of noise. We analyze its numerical performance by comparing it to two versions of the CramerRao lower bound. 1.
Alternating Projection, Ptychographic Imaging and Phase Synchronization. ArXiv eprints
, 2014
"... Abstract. We demonstrate necessary and sufficient conditions of the global convergence of the alternating projection algorithm to a unique solution up to a global phase factor. Additionally, for the ptychographic imaging problem, we discuss phase synchronization and connection graph Laplacian, and ..."
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Cited by 5 (1 self)
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Abstract. We demonstrate necessary and sufficient conditions of the global convergence of the alternating projection algorithm to a unique solution up to a global phase factor. Additionally, for the ptychographic imaging problem, we discuss phase synchronization and connection graph Laplacian, and show how to construct an accurate initial guess to accelerate convergence speed to handle the big imaging data in the coming new light source era. 1.
Phase Retrieval using Lipschitz Continuous Maps, available online arXiv:1403.2304v1
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Algorithms and theory for clustering . . .
, 2014
"... In this dissertation we discuss three problems characterized by hidden structure or information. The first part of this thesis focuses on extracting subspace structures from data. Subspace Clustering is the problem of finding a multisubspace representation that best fits a collection of points tak ..."
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Cited by 1 (0 self)
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In this dissertation we discuss three problems characterized by hidden structure or information. The first part of this thesis focuses on extracting subspace structures from data. Subspace Clustering is the problem of finding a multisubspace representation that best fits a collection of points taken from a highdimensional space. As with most clustering problems, popular techniques for subspace clustering are often difficult to analyze theoretically as they are often nonconvex in nature. Theoretical analysis of these algorithms becomes even more challenging in the presence of noise and missing data. We introduce a collection of subspace clustering algorithms, which are tractable and provably robust to various forms of data imperfections. We further illustrate our methods with numerical experiments on a wide variety of data segmentation problems. In the second part of the thesis, we consider the problem of recovering the seemingly hidden phase of an object from intensityonly measurements, a problem which naturally appears in Xray crystallography and related disciplines. We formulate the
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.
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.
Phase retrieval for the Cauchy wavelet transform
"... We consider the phase retrieval problem in which one tries to reconstruct a function from the modulus of its wavelet transform. We study the unicity and stability of the reconstruction. In the case where the wavelets are Cauchy wavelets, we prove that the modulus of the wavelet transform uniquely de ..."
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We consider the phase retrieval problem in which one tries to reconstruct a function from the modulus of its wavelet transform. We study the unicity and stability of the reconstruction. In the case where the wavelets are Cauchy wavelets, we prove that the modulus of the wavelet transform uniquely determines the function up to a global phase. We show that the reconstruction operator is continuous but not uniformly continuous. We describe how to construct pairs of functions which are far away in L2norm but whose wavelet transforms are very close, in modulus. The principle is to modulate the wavelet transform of a fixed initial function by a phase which varies slowly in both time and frequency. This construction seems to cover all the instabilities that we observe in practice; we give a partial formal justification to this fact. Finally, we describe an exact reconstruction algorithm and use it to numerically confirm our analysis of the stability question. 1