text: We propose a new algorithm to learn a dictionary for reconstructing and sparsely encoding signals from measurements without phase. Specifically, we consider the task of estimating a two-dimensional image from squared-magnitude measurements of a complex-valued linear transformation of the original image. Several recent phase retrieval algorithms exploit underlying sparsity of the unknown signal in order to improve recovery performance. In this work, we consider such a sparse signal prior in the context of phase retrieval, when the sparsifying dictionary is not known in advance. Our algorithm jointly reconstructs the unknown signal—possibly corrupted by noise—and learns a dictionary such that each patch of the estimated image can be sparsely represented. Numerical experiments demonstrate that our approach can obtain significantly better reconstructions for phase retrieval problems with noise than methods that cannot exploit such “hidden” sparsity. Moreover, on the theoretical side, we provide a convergence result for our method.
  @cite_1: The theory of compressive sensing enables accurate and robust signal reconstruction from a number of measurements dictated by the signal's structure rather than its Fourier bandwidth. A key element of the theory is the role played by randomization. In particular, signals that are compressible in the time or space domain can be recovered from just a few randomly chosen Fourier coefficients. However, in some scenarios we can only observe the magnitude of the Fourier coefficients and not their phase. In this paper, we study the magnitude-only compressive sensing problem and in parallel with the existing theory derive sufficient conditions for accurate recovery. We also propose a new iterative recovery algorithm and study its performance. In the process, we develop a new algorithm for the phase retrieval problem that exploits a signal's compressibility rather than its support to recover it from Fourier transform magnitude measurements.
  @cite_6: 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.
  @cite_31: This paper considers the noisy sparse phase retrieval problem: recovering a sparse signal x ∈ R from noisy quadratic measurements yj = (a ′ jx) 2 + j , j = 1, . . . ,m, with independent sub-exponential noise j . The goals are to understand the effect of the sparsity of x on the estimation precision and to construct a computationally feasible estimator to achieve the optimal rates adaptively. Inspired by the Wirtinger Flow [12] proposed for non-sparse and noiseless phase retrieval, a novel thresholded gradient descent algorithm is proposed and it is shown to adaptively achieve the minimax optimal rates of convergence over a wide range of sparsity levels when the aj ’s are independent standard Gaussian random vectors, provided that the sample size is sufficiently large compared to the sparsity of x.
  @cite_34: The problem of signal recovery from the autocorrelation, or equivalently, the magnitudes of the Fourier transform, is of paramount importance in various fields of engineering. In this work, for one-dimensional signals, we give conditions, which when satisfied, allow unique recovery from the autocorrelation with very high probability. In particular, for sparse signals, we develop two non-iterative recovery algorithms. One of them is based on combinatorial analysis, which we prove can recover signals upto sparsity o(n1 3) with very high probability, and the other is developed using a convex optimization based framework, which numerical simulations suggest can recover signals upto sparsity o(n1 2) with very high probability.
  @cite_25: We demonstrate that sub-wavelength optical images borne on partially-spatially-incoherent light can be recovered, from their far-field or from the blurred image, given the prior knowledge that the image is sparse, and only that. The reconstruction method relies on the recently demonstrated sparsity-based sub-wavelength imaging. However, for partially-spatially-incoherent light, the relation between the measurements and the image is quadratic, yielding non-convex measurement equations that do not conform to previously used techniques. Consequently, we demonstrate new algorithmic methodology, referred to as quadratic compressed sensing, which can be applied to a range of other problems involving information recovery from partial correlation measurements, including when the correlation function has local dependencies. Specifically for microscopy, this method can be readily extended to white light microscopes with the additional knowledge of the light source spectrum.
summary: Signal sparsity (or compressability) can also be beneficially exploited in phase retrieval methods, cf. @cite_25 @cite_34 @cite_31 @cite_6 @cite_1 . However, to the best of our knowledge, existing methods assume that the signal is sparse itself or sparse with respect to a fixed pre-defined dictionary. This motivates the development of new algorithms and formulations to learn suitable dictionaries and reconstruct input signals from nonlinear measurements.