[Paper Review] GESPAR: Efficient Phase Retrieval of Sparse Signals

Yoav Shechtman, Amir Beck+1 more|arXiv (Cornell University)|Jan 6, 2013

Advanced X-ray Imaging Techniques15 references8 citations

TL;DR

GESPAR is a fast, greedy local search algorithm for phase retrieval of sparse signals that avoids matrix lifting, enabling efficient recovery from Fourier magnitude measurements. It iteratively optimizes signal support and phase using a damped Gauss-Newton method, achieving higher accuracy and scalability than existing methods like Fienup-type algorithms and SDP-based approaches, especially for large-scale 1D and 2D problems.

ABSTRACT

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.

Motivation & Objective

• Address the ill-posed nature of phase retrieval by exploiting signal sparsity as prior information.
• Overcome limitations of existing methods such as slow convergence in Fienup-type algorithms and high computational cost of SDP-based approaches.
• Develop a scalable, efficient algorithm suitable for large-scale problems like 2D image recovery.
• Enable robust recovery from noisy or incomplete measurements while maintaining high success probability.
• Provide a practical alternative to matrix-lifting techniques that are infeasible for large-scale problems.

Proposed method

• Formulates sparse phase retrieval as a non-convex, non-linear optimization problem with sparsity constraints.
• Employs a greedy local search strategy that iteratively updates the signal's support set.
• Uses the damped Gauss-Newton (DGN) algorithm to minimize the data-fidelity term for a fixed support set.
• Alternates between support selection and DGN refinement to converge toward a stationary point.
• Avoids matrix lifting by directly optimizing over the signal vector, reducing computational complexity.
• Applies a thresholding rule to select new support indices based on gradient and Hessian information.

Experimental results

Research questions

• RQ1 Can a fast, non-SDP-based algorithm achieve high recovery accuracy for sparse phase retrieval in 1D and 2D settings?
• RQ2 How does GESPAR compare in performance and scalability to Fienup-type and SDP-based methods?
• RQ3 What is the required number of measurements for reliable recovery as a function of sparsity?
• RQ4 How robust is GESPAR to noise and inexact prior knowledge?
• RQ5 Can GESPAR effectively recover large-scale signals (e.g., 2D images) without matrix lifting?

Key findings

• GESPAR recovers signals with sparsity level $ s $ from $ N = 2n $ measurements with high probability, and the required number of measurements scales approximately as $ s^3 $.
• For $ n = 512, s = 35 $, GESPAR recovers signals in 33.5 seconds with $ ITER = 6400 $, significantly outperforming SDP-based methods in scalability.
• In 2D phase retrieval, GESPAR successfully recovers signals up to $ s = 57 $ for $ n = 6400 $, while sparse-Fienup only recovers up to $ s = 42 $ under the same conditions.
• GESPAR’s computation time scales approximately linearly with the number of support swaps, and the mean number of swaps grows logarithmically with $ s $ and $ n $.
• GESPAR demonstrates robustness to noise and performs well even without autocorrelation-derived support information.
• The algorithm achieves higher recovery probability than sparse-Fienup across various signal sizes and sparsity levels in both 1D and 2D settings.