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17
Phase Retrieval via Wirtinger Flow: Theory and Algorithms
, 2014
"... We study the problem of recovering the phase from magnitude measurements; specifically, we wish to reconstruct a complexvalued signal x ∈ Cn about which we have phaseless samples of the form yr = ∣⟨ar,x⟩∣2, r = 1,...,m (knowledge of the phase of these samples would yield a linear system). This pape ..."
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We study the problem of recovering the phase from magnitude measurements; specifically, we wish to reconstruct a complexvalued signal x ∈ Cn about which we have phaseless samples of the form yr = ∣⟨ar,x⟩∣2, r = 1,...,m (knowledge of the phase of these samples would yield a linear system). This paper develops a nonconvex formulation of the phase retrieval problem as well as a concrete solution algorithm. In a nutshell, this algorithm starts with a careful initialization obtained by means of a spectral method, and then refines this initial estimate by iteratively applying novel update rules, which have low computational complexity, much like in a gradient descent scheme. The main contribution is that this algorithm is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements. Indeed, the sequence of successive iterates provably converges to the solution at a geometric rate so that the proposed scheme is efficient both in terms of computational and data resources. In theory, a variation on this scheme leads to a nearlinear time algorithm for a physically realizable model based on coded diffraction patterns. We illustrate the effectiveness of our methods with various experiments on image data. Underlying our analysis are insights for the analysis of nonconvex optimization schemes that may have implications for computational problems beyond phase retrieval.
Proximal heterogeneous block inputoutput method and application to blind ptychographic diffraction imaging, arXiv:1408.1887v1
"... We propose a general alternating minimization algorithm for nonconvex optimization problems with separable structure and nonconvex coupling between blocks of variables. To fix our ideas, we apply the methodology to the problem of blind ptychographic imaging. Compared to other schemes in the literatu ..."
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We propose a general alternating minimization algorithm for nonconvex optimization problems with separable structure and nonconvex coupling between blocks of variables. To fix our ideas, we apply the methodology to the problem of blind ptychographic imaging. Compared to other schemes in the literature, our approach differs in two ways: (i) it is posed within a clear mathematical framework with practically verifiable assumptions, and (ii) under the given assumptions, it is provably convergent to critical points. A numerical comparison of our proposed algorithm with the current stateoftheart on simulated and experimental data validates our approach and points toward directions for further improvement.
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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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
ADVANCED IMAGING WITH SCATTERING LENSES
"... The work described in this thesis is financially supported by the stichting FOM which is financially supported by the ‘Nederlandse Organisatie voor Wetenschappelijk Onderzoek ’ (NWO). Additional funding is provided by ERC, ANP, and MESA+. It was carried out at the ..."
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The work described in this thesis is financially supported by the stichting FOM which is financially supported by the ‘Nederlandse Organisatie voor Wetenschappelijk Onderzoek ’ (NWO). Additional funding is provided by ERC, ANP, and MESA+. It was carried out at the
Speckle correlation resolution enhancement of widefield fluorescence imaging
"... Highresolution fluorescence imaging is essential in nanoscience and biological sciences. Due to the diffraction limit, conventional imaging systems can only resolve structures larger than 200 nm. Here, we introduce a new fluorescence imaging method that enhances the resolution by using a highindex ..."
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Highresolution fluorescence imaging is essential in nanoscience and biological sciences. Due to the diffraction limit, conventional imaging systems can only resolve structures larger than 200 nm. Here, we introduce a new fluorescence imaging method that enhances the resolution by using a highindex scattering medium as an imaging lens. Simultaneously, we achieve a wide field of view. We develop a new image reconstruction algorithm that converges even for complex object structures. We collect twodimensional fluorescence images of a collection of 100 nm diameter dyedoped nanospheres, and demonstrate a deconvolved Abbe resolution of 116 nm with a field of view of 10μm × 10 μm. Our method is robust against optical aberrations and stage drifts, and therefore is well suited to image nanostructures with high resolution under ambient conditions. © 2015 Optical Society of America
Phase Retrieval Using Feasible Point Pursuit: Algorithms and CramérRao Bound
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NonCoherent Direction of Arrival Estimation from MagnitudeOnly Measurements
"... Abstract—We consider the classical Direction of arrival (DOA) estimation problem in the presence of random sensor phase errors are present at each sensor. To eliminate the effect of these phase errors, we propose a DOA recovery technique that relies only on magnitude measurements. This approach is i ..."
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Abstract—We consider the classical Direction of arrival (DOA) estimation problem in the presence of random sensor phase errors are present at each sensor. To eliminate the effect of these phase errors, we propose a DOA recovery technique that relies only on magnitude measurements. This approach is inspired by phase retrieval for applications in other fields. Ambiguities typically associated with phase retrieval methods are resolved by introducing reference targets with known DOA. The DOA estimation problem is formulated as a nonlinear optimization in a sparse framework, and is solved by the recently proposed GESPAR algorithm modified to accommodatemultiple snapshots. Numerical results demonstrate good DOA estimation performance. For example, the probability of error in locating a single target within 2 degrees is less than 0.1 for dB and one snapshot, and negligible for dB and five snapshots. Index Terms—Direction of arrival, phase retrieval, sparsity. I.
Sparse Phase Retrieval from ShortTime Fourier Measurements
"... Abstract—We consider the classical 1D phase retrieval problem. In order to overcome the difficulties associated with phase retrieval from measurements of the Fourier magnitude, we treat recovery from the magnitude of the shorttime Fourier transform (STFT). We first show that the redundancy offere ..."
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Abstract—We consider the classical 1D phase retrieval problem. In order to overcome the difficulties associated with phase retrieval from measurements of the Fourier magnitude, we treat recovery from the magnitude of the shorttime Fourier transform (STFT). We first show that the redundancy offered by the STFT enables unique recovery for arbitrary nonvanishing inputs, under mild conditions. An efficient algorithm for recovery of a sparse input from the STFT magnitude is then suggested, based on an adaptation of the recently proposed GESPAR algorithm. We demonstrate through simulations that using the STFT leads to improved performance over recovery from the oversampled Fourier magnitude with the same number of measurements. Index Terms—GESPAR, phase retrieval, shorttime Fourier transform, sparsity.
Proximal Heterogeneous Block ImplicitExplicit Method and Application to Blind
"... Abstract. We propose a general alternating minimization algorithm for nonconvex optimization problems with separable structure and nonconvex coupling between blocks of variables. To fix our ideas, we apply the methodology to the problem of blind ptychographic imaging. Compared to other schemes in th ..."
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Abstract. We propose a general alternating minimization algorithm for nonconvex optimization problems with separable structure and nonconvex coupling between blocks of variables. To fix our ideas, we apply the methodology to the problem of blind ptychographic imaging. Compared to other schemes in the literature, our approach differs in two ways: (i) it is posed within a clear mathematical framework with practical verifiable assumptions, and (ii) under the given assumptions, it is provably convergent to critical points. A numerical comparison of our proposed algorithm with the current state of the art on simulated and experimental data validates our approach and points toward directions for further improvement.