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29
Relaxed averaged alternating reflections for diffraction imaging
 Inverse Problems 21
, 2005
"... We report on progress in algorithms for iterative phase retrieval. The theory of convex optimization is used to develop and to gain insight into counterparts for the nonconvex problem of phase retrieval. We propose a relaxation of averaged alternating reflectors and determine the fixed point set of ..."
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We report on progress in algorithms for iterative phase retrieval. The theory of convex optimization is used to develop and to gain insight into counterparts for the nonconvex problem of phase retrieval. We propose a relaxation of averaged alternating reflectors and determine the fixed point set of the related operator in the convex case. A numerical study supports our theoretical observations and demonstrates the effectiveness of the algorithm compared to the current state of the art. 1
Highresolution ab initio threedimensional xray diffraction microscopy
 Journal of the Optical Society of America A
, 2006
"... Coherent Xray diffraction microscopy is a method of imaging nonperiodic isolated objects at resolutions only limited, in principle, by the largest scattering angles recorded. We demonstrate Xray diffraction imaging with high resolution in all three dimensions, as determined by a quantitative anal ..."
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Cited by 24 (4 self)
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Coherent Xray diffraction microscopy is a method of imaging nonperiodic isolated objects at resolutions only limited, in principle, by the largest scattering angles recorded. We demonstrate Xray diffraction imaging with high resolution in all three dimensions, as determined by a quantitative analysis of the reconstructed volume images. These images are retrieved from the 3D diffraction data using no a priori knowledge about the shape or composition of the object, which has never before been demonstrated on a nonperiodic object. We also construct 2D images of thick objects with infinite depth of focus (without loss of transverse spatial resolution). These methods can be used to image biological and materials science samples at high resolution using Xray undulator radiation, and establishes the techniques to be used in atomicresolution ultrafast imaging at Xray freeelectron laser sources. OCIS codes: 340.7460, 110.1650, 110.6880, 100.5070, 100.6890, 070.2590, 180.6900 1.
Phase retrieval using alternating minimization
 In NIPS
, 2013
"... Phase retrieval problems involve solving linear equations, but with missing sign (or phase, for complex numbers) information. Over the last two decades, a popular generic empirical approach to the many variants of this problem has been one of alternating minimization; i.e. alternating between estima ..."
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Phase retrieval problems involve solving linear equations, but with missing sign (or phase, for complex numbers) information. Over the last two decades, a popular generic empirical approach to the many variants of this problem has been one of alternating minimization; i.e. alternating between estimating the missing phase information, and the candidate solution. In this paper, we show that a simple alternating minimization algorithm geometrically converges to the solution of one such problem – finding a vector x from y,A, where y = ATx  and z  denotes a vector of elementwise magnitudes of z – under the assumption that A is Gaussian. Empirically, our algorithm performs similar to recently proposed convex techniques for this variant (which are based on “lifting ” to a convex matrix problem) in sample complexity and robustness to noise. However, our algorithm is much more efficient and can scale to large problems. Analytically, we show geometric convergence to the solution, and sample complexity that is off by log factors from obvious lower bounds. We also establish close to optimal scaling for the case when the unknown vector is sparse. Our work represents the only known theoretical guarantee for alternating minimization for any variant of phase retrieval problems in the nonconvex setting. 1
Ultrafast singleshot diffraction imaging of nanoscale dynamics
 Nature Photonics
, 2008
"... nanoscale dynamics ..."
Biological physics
, 2002
"... Xray diffraction microscopy on frozen hydrated specimens ..."
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Xray diffraction microscopy on frozen hydrated specimens
XXXXXXXXXX Ab initio compressive phase retrieval
, 809
"... Any object on earth has two fundamental properties: it is finite, and it is made of atoms. Structural information about an object can be obtained from diffraction amplitude measurements that account for either one of these traits. Nyquistsampling of the Fourier amplitudes is sufficient to image sin ..."
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Any object on earth has two fundamental properties: it is finite, and it is made of atoms. Structural information about an object can be obtained from diffraction amplitude measurements that account for either one of these traits. Nyquistsampling of the Fourier amplitudes is sufficient to image single particles of finite size at any resolution. Atomic resolution data is routinely used to image molecules replicated in a crystal structure. Here we report an algorithm that requires neither information, but uses the fact that an image of a natural object is compressible. Intended applications include tomographic diffractive imaging, crystallography, powder diffraction, small angle xray scattering and random Fourier amplitude measurements.
Solving Random Quadratic Systems of Equations is nearly as easy as . . .
, 2015
"... We consider the fundamental problem of solving quadratic systems of equations in n variables, where yi = 〈ai,x〉2, i = 1,...,m and x ∈ Rn is unknown. We propose a novel method, which starting with an initial guess computed by means of a spectral method, proceeds by minimizing a nonconvex functional ..."
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We consider the fundamental problem of solving quadratic systems of equations in n variables, where yi = 〈ai,x〉2, i = 1,...,m and x ∈ Rn is unknown. We propose a novel method, which starting with an initial guess computed by means of a spectral method, proceeds by minimizing a nonconvex functional as in the Wirtinger flow approach [11]. There are several key distinguishing features, most notably, a distinct objective functional and novel update rules, which operate in an adaptive fashion and drop terms bearing too much influence on the search direction. These careful selection rules provide a tighter initial guess, better descent directions, and thus enhanced practical performance. On the theoretical side, we prove that for certain unstructured models of quadratic systems, our algorithms return the correct solution in linear time, i.e. in time proportional to reading the data {ai} and {yi} as soon as the ratio m/n between the number of equations and unknowns exceeds a fixed numerical constant. We extend the theory to deal with noisy systems in which we only have yi ≈ 〈ai,x〉2 and prove that our algorithms achieve a statistical accuracy, which is nearly unimprovable. We complement our theoretical study with numerical examples showing that solving random quadratic systems is both computationally and statistically not much harder than solving linear systems of the same size—hence the title of this paper. For instance, we
Bit retrieval: intractability and application to digital watermarking
"... Bit retrieval, the problem of determining a binary sequence from its cyclic autocorrelation, is a special case of the phase retrieval problem. Algorithms for phase retrieval are extensively used in several scientific disciplines, and yet, very little is known about the complexity of these algorithms ..."
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Bit retrieval, the problem of determining a binary sequence from its cyclic autocorrelation, is a special case of the phase retrieval problem. Algorithms for phase retrieval are extensively used in several scientific disciplines, and yet, very little is known about the complexity of these algorithms or phase retrieval in general. Here we show that bit retrieval, in particular, is closely related to computations that arise in algebraic number theory and can also be formulated as an integer program. We find that general purpose algorithms from these fields, when applied to bit retrieval, are outperformed by a particular iterative phase retrieval algorithm. This algorithm still has exponential complexity and motivates us to propose a new public key signature scheme based on the intractability of bit retrieval, and image watermarking as a possible application.
Random projections and the optimization of an algorithm for phase retrieval
"... Abstract. Iterative phase retrieval algorithms typically employ projections onto constraint subspaces to recover the unknown phases in the Fourier transform of an image, or, in the case of xray crystallography, the electron density of a molecule. For a general class of algorithms, where the basic i ..."
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Abstract. Iterative phase retrieval algorithms typically employ projections onto constraint subspaces to recover the unknown phases in the Fourier transform of an image, or, in the case of xray crystallography, the electron density of a molecule. For a general class of algorithms, where the basic iteration is specified by the difference map, solutions are associated with fixed points of the map, the attractive character of which determines the effectiveness of the algorithm. The behavior of the difference map near fixed points is controlled by the relative orientation of the tangent spaces of the two constraint subspaces employed by the map. Since the dimensionalities involved are always large in practical applications, it is appropriate to use random matrix theory ideas to analyze the averagecase convergence at fixed points. Optimal values of the γ parameters of the difference map are found which differ somewhat from the values previously obtained on the assumption of orthogonal tangent spaces. PACS numbers: Submitted to: J. Phys. A: Math. Gen. Random projections and the optimization of an algorithm for phase retrieval 2 1.
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