Phase Retrieval with Sparse Phase Constraint

16th IFAC Symposium on System Identification, 2012

Given a linear system in a real or complex domain, linear regression aims to recover the model parameters from a set of observations. Recent studies in compressive sensing have successfully shown that under certain conditions, a linear program, namely, 1 -minimization, guarantees recovery of sparse parameter signals even when the system is underdetermined. In this paper, we consider a more challenging problem: when the phase of the output measurements from a linear system is omitted. Using a lifting technique, we show that even though the phase information is missing, the sparse signal can be recovered exactly by solving a simple semidefinite program when the sampling rate is sufficiently high, albeit the exact solutions to both sparse signal recovery and phase retrieval are combinatorial. The results extend the type of applications that compressive sensing can be applied to those where only output magnitudes can be observed. We demonstrate the accuracy of the algorithms through theoretical analysis, extensive simulations and a practical experiment. arXiv:1111.6323v2 [math.ST]

2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2012

Signal recovery from the amplitudes of the Fourier transform, or equivalently from the autocorrelation function is a classical problem. Due to the absence of phase information, signal recovery requires some form of additional prior information. In this paper, the prior information we assume is sparsity. We develop a convex optimization based framework to retrieve the signal support from the support of the autocorrelation, and propose an iterative algorithm which terminates in a signal with the least sparsity satisfying the autocorrelation constraints. Numerical results suggest that unique recovery up to a global sign change, time shift and/or time reversal is possible with a very high probability for sufficiently sparse signals.

This paper presents a new iterative algorithm called constraint removal (CR) for the recovery of a sparse signal x from an incomplete number of linear measurements y such that y mÂ1 ¼ A mÂn x nÂ1 and m o n. It is empirically demonstrated that the CR algorithm has a recovery performance which is between basis pursuit linear programming (BP-LP) and subspace pursuit (SP) for both zero-one and Gaussian type signals.

2011

Sparse modeling is one of the efficient techniques for imaging that allows recovering lost information. In this paper, we present a novel iterative phase-retrieval algorithm using a sparse representation of the object amplitude and phase. The algorithm is derived in terms of a constrained maximum likelihood, where the wave field reconstruction is performed using a number of noisy intensity-only observations with a zero-mean additive Gaussian noise. The developed algorithm enables the optimal solution for the object wave field reconstruction. Our goal is an improvement of the reconstruction quality with respect to the conventional algorithms. Sparse regularization results in advanced reconstruction accuracy, and numerical simulations demonstrate significant enhancement of imaging.

Vietnam Journal of Mathematics, 2013

The Method of Alternating Projections (MAP), a classical algorithm for solving feasibility problems, has recently been intensely studied for nonconvex sets. However, intrinsically available are only local convergence results: convergence occurs if the starting point is not too far away from solutions to avoid getting trapped in certain regions. Instead of taking full projection steps, it can be advantageous to underrelax, i.e., to move only part way towards the constraint set, in order to enlarge the regions of convergence. In this paper, we thus systematically study the Method of Alternating Relaxed Projections (MARP) for two (possibly nonconvex) sets. Complementing our recent work on MAP, we establish local linear convergence results for the MARP. Several examples illustrate our analysis.

2015

Let A and B be nonempty, convex and closed subsets of a Hilbert spaceH. In the practical considerations we need to find an element of the intersection A ∩B or, more general, to solve the following problem: find a ∗ ∈ A and b ∗ ∈ B such that ‖a ∗ − b∗ ‖ = inf a∈A,b∈B ‖a − b‖. One of the important methods generating sequences which converge weakly to a solution of above problems is the von Neumann alternating projection method xk+1 = PAPBxk. The method has found application in different areas of mathematics. These include probability and statistics, image re-construction and intensity modulated radiation therapy, where the convex subsets are described by a large and sparse system of linear equations or inequalities. We deal with generalization of the von Neumann alternating projec-tion method of the form xk+1 = PA(xk + λk(xk)(PAPBxk − xk)), where FixPAPB 6 = ∅. We give sufficient conditions for the weak convergence of the sequence (xk) to FixPAPB in the general case and in the case A ...

SIAM Journal on Scientific Computing, 2010

We propose a fast algorithm for solving the 1-regularized minimization problem min x∈R n μ x 1 + Ax − b 2 2 for recovering sparse solutions to an undetermined system of linear equations Ax = b. The algorithm is divided into two stages that are performed repeatedly. In the first stage a first-order iterative "shrinkage" method yields an estimate of the subset of components of x likely to be nonzero in an optimal solution. Restricting the decision variables x to this subset and fixing their signs at their current values reduces the 1-norm x 1 to a linear function of x. The resulting subspace problem, which involves the minimization of a smaller and smooth quadratic function, is solved in the second phase. Our code FPC AS embeds this basic two-stage algorithm in a continuation (homotopy) approach by assigning a decreasing sequence of values to μ. This code exhibits state-of-the-art performance in terms of both its speed and its ability to recover sparse signals.

Physics in Medicine and Biology, 2012

In this work, we investigate optimization-based image reconstruction from few-view (i.e., <10 views) projections of sparse objects such as coronary-artery specimens. Using optimization programs as a guide, we formulate constraint programs as reconstruction programs and develop algorithms to reconstruct images through solving the reconstruction programs. Characterization studies are carried out for elucidating the algorithm properties of "convergence" (relative to designed solutions) and "utility" (relative to desired solutions) by using simulated few-view data calculated from a discrete FORBILD coronary-artery phantom, and real few-view data acquired from a human coronary-artery specimen. Study results suggest that carefully designed reconstruction programs and algorithms can yield accurate reconstructions of sparse images from few-view projections.

arXiv: Optimization and Control, 2017

In this paper, we study the convergence of Alternating Projection (AP) algorithm for the matrix completion and compressed sensing problems. We also present computational evidence for the excellent performance of the algorithm. Also, in the last section, we prove using algebraic-geometric techniques that, fixing the known positions, if a rank r matrix can be completed in finitely many ways using one given set of known entries, then, for almost all set of known entries, the matrix can be only completed into a rank r matrix in finitely many ways.

2018 6th International Conference on Wireless Networks and Mobile Communications (WINCOM), 2018

The emerging field of Compressive Sensing (CS) has shown that sparse signals can be acquired using a rate far less than the one required by the classical Shannon-Nyquist theorem. In that, CS acquires a sparse signal by correlating it with the rows of a sensing matrix to form a small set of measurements. And then these measurements are used to reconstruct the original sparse signal using an optimization algorithm. In this paper, we extend our recently published work, to more investigate the power of Constraint Programming (CP) solvers with the CS problem. We show that a statistical property of the sensing matrix can highly affects the performance of CP solvers in the case of CS problem. Which enable us to improve the CP solvers performance and even reach the optimal case. The effectiveness of our method is demonstrated via simulation results.