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Register a new userPhase Retrieval for Sparse Signals: Uniqueness Conditions
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Added by
Juri Ranieri
Publication date
2013
fields
Informatics Engineering
and research's language is
English
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In a variety of fields, in particular those involving imaging and optics, we often measure signals whose phase is missing or has been irremediably distorted. Phase retrieval attempts the recovery of the phase information of a signal from the magnitude of its Fourier transform to enable the reconstruction of the original signal. A fundamental question then is: Under which conditions can we uniquely recover the signal of interest from its measured magnitudes? In this paper, we assume the measured signal to be sparse. This is a natural assumption in many applications, such as X-ray crystallography, speckle imaging and blind channel estimation. In this work, we derive a sufficient condition for the uniqueness of the solution of the phase retrieval (PR) problem for both discrete and continuous domains, and for one and multi-dimensional domains. More precisely, we show that there is a strong connection between PR and the turnpike problem, a classic combinatorial problem. We also prove that the existence of collisions in the autocorrelation function of the signal may preclude the uniqueness of the solution of PR. Then, assuming the absence of collisions, we prove that the solution is almost surely unique on 1-dimensional domains. Finally, we extend this result to multi-dimensional signals by solving a set of 1-dimensional problems. We show that the solution of the multi-dimensional problem is unique when the autocorrelation function has no collisions, significantly improving upon a previously known result.
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In a variety of fields, in particular those involving imaging and optics, we often measure signals whose phase is missing or has been irremediably distorted. Phase retrieval attempts to recover the phase information of a signal from the magnitude of its Fourier transform to enable the reconstruction of the original signal. Solving the phase retrieval problem is equivalent to recovering a signal from its auto-correlation function. In this paper, we assume the original signal to be sparse; this is a natural assumption in many applications, such as X-ray crystallography, speckle imaging and blind channel estimation. We propose an algorithm that resolves the phase retrieval problem in three stages: i) we leverage the finite rate of innovation sampling theory to super-resolve the auto-correlation function from a limited number of samples, ii) we design a greedy algorithm that identifies the locations of a sparse solution given the super-resolved auto-correlation function, iii) we recover the amplitudes of the atoms given their locations and the measured auto-correlation function. Unlike traditional approaches that recover a discrete approximation of the underlying signal, our algorithm estimates the signal on a continuous domain, which makes it the first of its kind. Along with the algorithm, we derive its performance bound with a theoretical analysis and propose a set of enhancements to improve its computational complexity and noise resilience. Finally, we demonstrate the benefits of the proposed method via a comparison against Charge Flipping, a notable algorithm in crystallography.
We consider the phase retrieval problem for signals that belong to a union of subspaces. We assume that amplitude measurements of the signal of length $n$ are observed after passing it through a random $m times n$ measurement matrix. We also assume that the signal belongs to the span of a single $d$-dimensional subspace out of $R$ subspaces, where $dll n$. We assume the knowledge of all possible subspaces, but the true subspace of the signal is unknown. We present an algorithm that jointly estimates the phase of the measurements and the subspace support of the signal. We discuss theoretical guarantees on the recovery of signals and present simulation results to demonstrate the empirical performance of our proposed algorithm. Our main result suggests that if properly initialized, then $O(d+log R)$ random measurements are sufficient for phase retrieval if the unknown signal belongs to the union of $R$ low-dimensional subspaces.
We consider the problem of testing for the presence (or detection) of an unknown sparse signal in additive white noise. Given a fixed measurement budget, much smaller than the dimension of the signal, we consider the general problem of designing compressive measurements to maximize the measurement signal-to-noise ratio (SNR), as increasing SNR improves the detection performance in a large class of detectors. We use a lexicographic optimization approach, where the optimal measurement design for sparsity level $k$ is sought only among the set of measurement matrices that satisfy the optimality conditions for sparsity level k-1. We consider optimizing two different SNR criteria, namely a worst-case SNR measure, over all possible realizations of a k-sparse signal, and an average SNR measure with respect to a uniform distribution on the locations of the up to k nonzero entries in the signal. We establish connections between these two criteria and certain classes of tight frames. We constrain our measurement matrices to the class of tight frames to avoid coloring the noise covariance matrix. For the worst-case problem, we show that the optimal measurement matrix is a Grassmannian line packing for most---and a uniform tight frame for all---sparse signals. For the average SNR problem, we prove that the optimal measurement matrix is a uniform tight frame with minimum sum-coherence for most---and a tight frame for all---sparse signals.
In recent years, the mathematical and algorithmic aspects of the phase retrieval problem have received considerable attention. Many papers in this area mention crystallography as a principal application. In crystallography, the signal to be recovered is periodic and comprised of atomic distributions arranged homogeneously in the unit cell of the crystal. The crystallographic problem is both the leading application and one of the hardest forms of phase retrieval. We have constructed a graded set of benchmark problems for evaluating algorithms that perform this type of phase retrieval. The data, publicly available online, is provided in an easily interpretable format. We also propose a simple and unambiguous success/failure criterion based on the actual needs in crystallography. Baseline runtimes were obtained with an iterative algorithm that is similar but more transparent than those used in crystallography. Empirically, the runtimes grow exponentially with respect to a new hardness parameter: the sparsity of the signal autocorrelation. We also review the algorithms used by the leading software packages. This set of benchmark problems, we hope, will encourage the development of new algorithms for the phase retrieval problem in general, and crystallography in particular.
The Random Demodulator (RD) and the Modulated Wideband Converter (MWC) are two recently proposed compressed sensing (CS) techniques for the acquisition of continuous-time spectrally-sparse signals. They extend the standard CS paradigm from sampling discrete, finite dimensional signals to sampling continuous and possibly infinite dimensional ones, and thus establish the ability to capture these signals at sub-Nyquist sampling rates. The RD and the MWC have remarkably similar structures (similar block diagrams), but their reconstruction algorithms and signal models strongly differ. To date, few results exist that compare these systems, and owing to the potential impacts they could have on spectral estimation in applications like electromagnetic scanning and cognitive radio, we more fully investigate their relationship in this paper. We show that the RD and the MWC are both based on the general concept of random filtering, but employ significantly different sampling functions. We also investigate system sensitivities (or robustness) to sparse signal model assumptions. Lastly, we show that block convolution is a fundamental aspect of the MWC, allowing it to successfully sample and reconstruct block-sparse (multiband) signals. Based on this concept, we propose a new acquisition system for continuous-time signals whose amplitudes are block sparse. The paper includes detailed time and frequency domain analyses of the RD and the MWC that differ, sometimes substantially, from published results.
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