GESPAR: Efficient Phase Retrieval of Sparse Signals

By Yoav Shechtman, Amir Beck, Yonina C. Eldar. IEEE Transactions on Signal Processing, 2013

362 citations recorded by the source corpus.

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 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.

Subfields: Computer Vision and Pattern Recognition, Radiation, Structural Biology

View Paper: https://www.semanticscholar.org/paper/d9c8f3aad0d505b2088dffb55bebe96f97a0169f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computer Vision and Pattern Recognition

1. Tom Goldstein
2. Yuxin Chen
3. Yonina C. Eldar
4. Christoph Studer
5. Yuejie Chi
6. E.J. Candes
7. Sundeep Rangan
8. G. Wang

In Radiation

1. Yuxin Chen
2. Tom Goldstein
3. Christoph Studer
4. Yonina C. Eldar
5. Lei Tian
6. E.J. Candes
7. Sundeep Rangan
8. Philip Schniter

In Structural Biology

1. Yuxin Chen
2. E.J. Candes
3. Philip Schniter
4. Sundeep Rangan
5. Yonina C. Eldar
6. Yuejie Chi
7. Christoph Studer
8. Tom Goldstein

Citations in the corpus

• Phase Retrieval with Application to Optical Imaging: A contemporary overview (2015, methodology): “more extensive comparisons, we refer the reader to [51].”
• Compressive phase retrieval via generalized approximate message passing (2012, methodology): “Again, we repeat an experiment from [28], where the measurement and signal lengths were fixed at and the signal sparsity was varied.”
• Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems (2015, methodology): “It is worth noting that apart from WF, various other nonconvex procedures have been proposed as well for phase retrieval, including the error reduction schemes dating back to Gerchberg-Saxton and Fienup [23,24], iterated projections [22], alternating minimization [36], generalized approximate messag”
• Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow (2016, background): “One possibility is to assume additional constraints on the unknown signal su ch as sparsity [24], [25], [17], [26].”
• Computational illumination for high-speed in vitro Fourier ptychographic microscopy (2015, background): “Reconstruction algorithms that explicitly take a priori information into account, such as sparsity based methods [29,30], could further improve capture speed; here, for generality, we choose not to make any assumptions on the sample.”
• PhaseMax: Convex Phase Retrieval via Basis Pursuit (2016, background): “The convexity of PhaseMax enables a natural extension to sparse phase retrieval [55], [56] or other signal priors (e.”
• Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolution (2017, background): “A small sample of other nonconvexphase retrievalmethods include [6,10,22,36,43,47,92,98,100,109,122], which are beyond the scope of this paper.”
• Phase Retrieval Using Alternating Minimization (2013, neutral): “[21, 38] develop algorithms for phase retrieval from Fourier magnitude measurements.”