GESPAR: Efficient Phase Retrieval of Sparse Signals

We consider the problem of phase retrieval, namely, recovery of a signal from the magnitude of its Fourier transform, or of any other linear transform. Due to the loss of the Fourier phase information, this problem is ill-posed. Therefore, prior information on the signal is needed in order to enable its recovery. In this work we consider the case in which the signal is known to be sparse, i.e., it consists of a small number of nonzero elements in an appropriate basis. We propose a fast local search method for recovering a sparse signal from measurements of its Fourier transform (or other linear transform) magnitude which we refer to as GESPAR: GrEedy Sparse PhAse Retrieval. Our algorithm does not require matrix lifting, unlike previous approaches, and therefore is potentially suitable for large scale problems such as images. Simulation results indicate that GESPAR is fast and more accurate than existing techniques in a variety of settings.Comment: Generalized to non-Fourier measurements, added 2D simulations, and a theorem for convergence to stationary poin

An evaluation of the data space dimension in phase retrieval: results in Fresnel zone

In this paper, we address the problem of computing the dimension of data space in phase retrieval problem. Starting from the quadratic formulation of the phase retrieval, the analysis is performed in two steps. First, we exploit the lifting technique to obtain a linear representation of the data. Later, we evaluate the dimension of data space by computing analytically the number of relevant singular values of the linear operator that represents the data. The study is done with reference to a 2D scalar geometry consisting of an electric current strip whose square amplitude of the electric radiated field is observed on a twodimensional extended domain in Fresnel zone

Phasing Two-Dimensional Crystal Diffraction Pattern with Iterative Projection Algorithms

abstract: Phase problem has been long-standing in x-ray diffractive imaging. It is originated from the fact that only the amplitude of the scattered wave can be recorded by the detector, losing the phase information. The measurement of amplitude alone is insufficient to solve the structure. Therefore, phase retrieval is essential to structure determination with X-ray diffractive imaging. So far, many experimental as well as algorithmic approaches have been developed to address the phase problem. The experimental phasing methods, such as MAD, SAD etc, exploit the phase relation in vector space. They usually demand a lot of efforts to prepare the samples and require much more data. On the other hand, iterative phasing algorithms make use of the prior knowledge and various constraints in real and reciprocal space. In this thesis, new approaches to the problem of direct digital phasing of X-ray diffraction patterns from two-dimensional organic crystals were presented. The phase problem for Bragg diffraction from two-dimensional (2D) crystalline monolayer in transmission may be solved by imposing a compact support that sets the density to zero outside the monolayer. By iterating between the measured stucture factor magnitudes along reciprocal space rods (starting with random phases) and a density of the correct sign, the complex scattered amplitudes may be found (J. Struct Biol 144, 209 (2003)). However this one-dimensional support function fails to link the rod phases correctly unless a low-resolution real-space map is also available. Minimum prior information required for successful three-dimensional (3D) structure retrieval from a 2D crystal XFEL diffraction dataset were investigated, when using the HIO algorithm. This method provides an alternative way to phase 2D crystal dataset, with less dependence on the high quality model used in the molecular replacement method.Dissertation/ThesisDoctoral Dissertation Physics 201

Orthogonal Matrix Retrieval in Cryo-Electron Microscopy

In single particle reconstruction (SPR) from cryo-electron microscopy (cryo-EM), the 3D structure of a molecule needs to be determined from its 2D projection images taken at unknown viewing directions. Zvi Kam showed already in 1980 that the autocorrelation function of the 3D molecule over the rotation group SO(3) can be estimated from 2D projection images whose viewing directions are uniformly distributed over the sphere. The autocorrelation function determines the expansion coefficients of the 3D molecule in spherical harmonics up to an orthogonal matrix of size $(2l+1)\times (2l+1)$ for each $l=0,1,2,...$. In this paper we show how techniques for solving the phase retrieval problem in X-ray crystallography can be modified for the cryo-EM setup for retrieving the missing orthogonal matrices. Specifically, we present two new approaches that we term Orthogonal Extension and Orthogonal Replacement, in which the main algorithmic components are the singular value decomposition and semidefinite programming. We demonstrate the utility of these approaches through numerical experiments on simulated data.Comment: Modified introduction and summary. Accepted to the IEEE International Symposium on Biomedical Imagin

Frames and Phase Retrieval

Phase retrieval tackles the problem of recovering a signal after loss of phase. The phase problem shows up in many different settings such as X-ray crystallography, speech recognition, quantum information theory, and coherent diffraction imaging. In this dissertation we present some results relating to three topics on phase retrieval. Chapters 1 and 2 contain the relevant background materials. In chapter 3, we introduce the notion of exact phase-retrievable frames as a way of measuring a frame\u27s redundancy with respect to its phase retrieval property. We show that, in the d-dimensional real Hilbert space case, exact phase-retrievable frames can be of any lengths between 2d - 1 and d(d + 1)=2, inclusive. The complex Hilbert space case remains open. In chapter 4, we investigate phase-retrievability by studying maximal phase-retrievable subspaces with respect to a given frame. These maximal PR-subspaces can have different dimensions. We are able to identify the ones with the largest dimension and this can be considered as a generalization of the characterization of real phase-retrievable frames. In the basis case, we prove that if M is a k-dimensional PR-subspace then |supp(x)| ≥ k for every nonzero vector x 2 M. Moreover, if 1 ≤ k \u3c [(d + 1)=2], then a k-dimensional PR-subspace is maximal if and only if there exists a vector x ϵ M such that |supp(x)| = k|. Chapter 5 is devoted to investigating phase-retrievable operator-valued frames. We obtain some characterizations of phase-retrievable frames for general operator systems acting on both finite and infinite dimensional Hilbert spaces; thus generalizing known results for vector-valued frames, fusion frames, and frames of Hermitian matrices. Finally, in Chapter 6, we consider the problem of characterizing projective representations that admit frame vectors with the maximal span property, a property that allows for an algebraic recovering of the phase-retrieval problem. We prove that every irreducible projective representation of a finite abelian group admits a frame vector with the maximal span property. All such vectors can be explicitly characterized. These generalize some of the recent results about phase-retrieval with Gabor (or STFT) measurements