Discrete and Continuous Sparse Recovery Methods and Their Applications

Low dimensional signal processing has drawn an increasingly broad amount of attention in the past decade, because prior information about a low-dimensional space can be exploited to aid in the recovery of the signal of interest. Among all the different forms of low di- mensionality, in this dissertation we focus on the synthesis and analysis models of sparse recovery. This dissertation comprises two major topics. For the first topic, we discuss the synthesis model of sparse recovery and consider the dictionary mismatches in the model. We further introduce a continuous sparse recovery to eliminate the existing off-grid mismatches for DOA estimation. In the second topic, we focus on the analysis model, with an emphasis on efficient algorithms and performance analysis. In considering the sparse recovery method with structured dictionary mismatches for the synthesis model, we exploit the joint sparsity between the mismatch parameters and original sparse signal. We demonstrate that by exploiting this information, we can obtain a robust reconstruction under mild conditions on the sensing matrix. This model is very useful for radar and passive array applications. We propose several efficient algorithms to solve the joint sparse recovery problem. Using numerical examples, we demonstrate that our proposed algorithms outperform several methods in the literature. We further extend the mismatch model to a continuous sparse model, using the mathematical theory of super resolution. Statistical analysis shows the robustness of the proposed algorithm. A number-detection algorithm is also proposed for the co-prime arrays. By using numerical examples, we show that continuous sparse recovery further improves the DOA estimation accuracy, over both the joint sparse method and also MUSIC with spatial smoothing. In the second topic, we visit the corresponding analysis model of sparse recovery. Instead of assuming a sparse decomposition of the original signal, the analysis model focuses on the existence of a linear transformation which can make the original signal sparse. In this work we use a monotone version of the fast iterative shrinkage- thresholding algorithm (MFISTA) to yield efficient algorithms to solve the sparse recovery. We examine two widely used relaxation techniques, namely smoothing and decomposition, to relax the optimization. We show that although these two techniques are equivalent in their objective functions, the smoothing technique converges faster than the decomposition technique. We also compute the performance guarantee for the analysis model when a LASSO type of reconstruction is performed. By using numerical examples, we are able to show that the proposed algorithm is more efficient than other state of the art algorithms

Maximum Likelihood-based Gridless DoA Estimation Using Structured Covariance Matrix Recovery and SBL with Grid Refinement

We consider the parametric data model employed in applications such as line spectral estimation and direction-of-arrival estimation. We focus on the stochastic maximum likelihood estimation (MLE) framework and offer approaches to estimate the parameter of interest in a gridless manner, overcoming the model complexities of the past. This progress is enabled by the modern trend of reparameterization of the objective and exploiting the sparse Bayesian learning (SBL) approach. The latter is shown to be a correlation-aware method, and for the underlying problem it is identified as a grid-based technique for recovering a structured covariance matrix of the measurements. For the case when the structured matrix is expressible as a sampled Toeplitz matrix, such as when measurements are sampled in time or space at regular intervals, additional constraints and reparameterization of the SBL objective leads to the proposed structured matrix recovery technique based on MLE. The proposed optimization problem is non-convex, and we propose a majorization-minimization based iterative procedure to estimate the structured matrix; each iteration solves a semidefinite program. We recover the parameter of interest in a gridless manner by appealing to the Caratheodory-Fejer result on decomposition of PSD Toeplitz matrices. For the general case of irregularly spaced time or spatial samples, we propose an iterative SBL procedure that refines grid points to increase resolution near potential source locations, while maintaining a low per iteration complexity. We provide numerical results to evaluate and compare the performance of the proposed techniques with other gridless techniques, and the CRB. The proposed correlation-aware approach is more robust to environmental/system effects such as low number of snapshots, correlated sources, small separation between source locations and improves sources identifiability.Comment: Submitted to the IEEE Transactions on Signal Processing (Previous submission date: 29-Oct-2021

Structured Sub-Nyquist Sampling with Applications in Compressive Toeplitz Covariance Estimation, Super-Resolution and Phase Retrieval

Sub-Nyquist sampling has received a huge amount of interest in the past decade. In classical compressed sensing theory, if the measurement procedure satisfies a particular condition known as Restricted Isometry Property (RIP), we can achieve stable recovery of signals of low-dimensional intrinsic structures with an order-wise optimal sample size. Such low-dimensional structures include sparse and low rank for both vector and matrix cases. The main drawback of conventional compressed sensing theory is that random measurements are required to ensure the RIP property. However, in many applications such as imaging and array signal processing, applying independent random measurements may not be practical as the systems are deterministic. Moreover, random measurements based compressed sensing always exploits convex programs for signal recovery even in the noiseless case, and solving those programs is computationally intensive if the ambient dimension is large, especially in the matrix case. The main contribution of this dissertation is that we propose a deterministic sub-Nyquist sampling framework for compressing the structured signal and come up with computationally efficient algorithms. Besides widely studied sparse and low-rank structures, we particularly focus on the cases that the signals of interest are stationary or the measurements are of Fourier type. The key difference between our work from classical compressed sensing theory is that we explicitly exploit the second-order statistics of the signals, and study the equivalent quadratic measurement model in the correlation domain. The essential observation made in this dissertation is that a difference/sum coarray structure will arise from the quadratic model if the measurements are of Fourier type. With these observations, we are able to achieve a better compression rate for covariance estimation, identify more sources in array signal processing or recover the signals of larger sparsity. In this dissertation, we will first study the problem of Toeplitz covariance estimation. In particular, we will show how to achieve an order-wise optimal compression rate using the idea of sparse arrays in both general and low-rank cases. Then, an analysis framework of super-resolution with positivity constraint is established. We will present fundamental robustness guarantees, efficient algorithms and applications in practices. Next, we will study the problem of phase-retrieval for which we successfully apply the sparse array ideas by fully exploiting the quadratic measurement model. We achieve near-optimal sample complexity for both sparse and general cases with practical Fourier measurements and provide efficient and deterministic recovery algorithms. In the end, we will further elaborate on the essential role of non-negative constraint in underdetermined inverse problems. In particular, we will analyze the nonlinear co-array interpolation problem and develop a universal upper bound of the interpolation error. Bilinear problem with non-negative constraint will be considered next and the exact characterization of the ambiguous solutions will be established for the first time in literature. At last, we will show how to apply the nested array idea to solve real problems such as Kriging. Using spatial correlation information, we are able to have a stable estimate of the field of interest with fewer sensors than classic methodologies. Extensive numerical experiments are implemented to demonstrate our theoretical claims

Sparse Array Signal Processing

This dissertation details three approaches for direction-of-arrival (DOA) estimation or beamforming in array signal processing from the perspective of sparsity. In the first part of this dissertation, we consider sparse array beamformer design based on the alternating direction method of multipliers (ADMM); in the second part of this dissertation, the problem of joint DOA estimation and distorted sensor detection is investigated; and off-grid DOA estimation is studied in the last part of this dissertation. In the first part of this thesis, we devise a sparse array design algorithm for adaptive beamforming. Our strategy is based on finding a sparse beamformer weight to maximize the output signal-to-interference-plus-noise ratio (SINR). The proposed method utilizes ADMM, and admits closed-form solutions at each ADMM iteration. The algorithm convergence properties are analyzed by showing the monotonicity and boundedness of the augmented Lagrangian function. In addition, we prove that the proposed algorithm converges to the set of Karush-Kuhn-Tucker stationary points. Numerical results exhibit its excellent performance, which is comparable to that of the exhaustive search approach, slightly better than those of the state-of-the-art solvers, and significantly outperforms several other sparse array design strategies, in terms of output SINR. Moreover, the proposed ADMM algorithm outperforms its competitors, in terms of computational cost. Distorted sensors could occur randomly and may lead to the breakdown of a sensor array system. In the second part of this thesis, we consider an array model in which a small number of sensors are distorted by unknown sensor gain and phase errors. With such an array model, the problem of joint DOA estimation and distorted sensor detection is formulated under the framework of low-rank and row-sparse decomposition. We derive an iteratively reweighted least squares (IRLS) algorithm to solve the resulting problem. The convergence property of the IRLS algorithm is analyzed by means of the monotonicity and boundedness of the objective function. Extensive simulations are conducted in view of parameter selection, convergence speed, computational complexity, and performance of DOA estimation as well as distorted sensor detection. Even though the IRLS algorithm is slightly worse than the ADMM in detecting the distorted sensors, the results show that our approach outperforms several state-of-the-art techniques in terms of convergence speed, computational cost, and DOA estimation performance. In the last part of this thesis, the problem of off-grid DOA estimation is investigated. We develop a method to jointly estimate the closest spatial frequency (the sine of DOA) grids, and the gaps between the estimated grids and the corresponding frequencies. By using a second-order Taylor approximation, the data model under the framework of joint-sparse representation is formulated. We point out an important property of the signals of interest in the model, namely the proportionality relationship. The proportionality relationship is empirically demonstrated to be useful in the sense that it increases the probability of the mixing matrix satisfying the block restricted isometry property. Simulation examples demonstrate the effectiveness and superiority of the proposed method against several state-of-the-art grid-based approaches

Efficient algorithms and data structures for compressive sensing

Wegen der kontinuierlich anwachsenden Anzahl von Sensoren, und den stetig wachsenden Datenmengen, die jene produzieren, stößt die konventielle Art Signale zu verarbeiten, beruhend auf dem Nyquist-Kriterium, auf immer mehr Hindernisse und Probleme. Die kürzlich entwickelte Theorie des Compressive Sensing (CS) formuliert das Versprechen einige dieser Hindernisse zu beseitigen, indem hier allgemeinere Signalaufnahme und -rekonstruktionsverfahren zum Einsatz kommen können. Dies erlaubt, dass hierbei einzelne Abtastwerte komplexer strukturierte Informationen über das Signal enthalten können als dies bei konventiellem Nyquistsampling der Fall ist. Gleichzeitig verändert sich die Signalrekonstruktion notwendigerweise zu einem nicht-linearen Vorgang und ebenso müssen viele Hardwarekonzepte für praktische Anwendungen neu überdacht werden. Das heißt, dass man zwischen der Menge an Information, die man über Signale gewinnen kann, und dem Aufwand für das Design und Betreiben eines Signalverarbeitungssystems abwägen kann und muss. Die hier vorgestellte Arbeit trägt dazu bei, dass bei diesem Abwägen CS mehr begünstigt werden kann, indem neue Resultate vorgestellt werden, die es erlauben, dass CS einfacher in der Praxis Anwendung finden kann, wobei die zu erwartende Leistungsfähigkeit des Systems theoretisch fundiert ist. Beispielsweise spielt das Konzept der Sparsity eine zentrale Rolle, weshalb diese Arbeit eine Methode präsentiert, womit der Grad der Sparsity eines Vektors mittels einer einzelnen Beobachtung geschätzt werden kann. Wir zeigen auf, dass dieser Ansatz für Sparsity Order Estimation zu einem niedrigeren Rekonstruktionsfehler führt, wenn man diesen mit einer Rekonstruktion vergleicht, welcher die Sparsity des Vektors unbekannt ist. Um die Modellierung von Signalen und deren Rekonstruktion effizienter zu gestalten, stellen wir das Konzept von der matrixfreien Darstellung linearer Operatoren vor. Für die einfachere Anwendung dieser Darstellung präsentieren wir eine freie Softwarearchitektur und demonstrieren deren Vorzüge, wenn sie für die Rekonstruktion in einem CS-System genutzt wird. Konkret wird der Nutzen dieser Bibliothek, einerseits für das Ermitteln von Defektpositionen in Prüfkörpern mittels Ultraschall, und andererseits für das Schätzen von Streuern in einem Funkkanal aus Ultrabreitbanddaten, demonstriert. Darüber hinaus stellen wir für die Verarbeitung der Ultraschalldaten eine Rekonstruktionspipeline vor, welche Daten verarbeitet, die im Frequenzbereich Unterabtastung erfahren haben. Wir beschreiben effiziente Algorithmen, die bei der Modellierung und der Rekonstruktion zum Einsatz kommen und wir leiten asymptotische Resultate für die benötigte Anzahl von Messwerten, sowie die zu erwartenden Lokalisierungsgenauigkeiten der Defekte her. Wir zeigen auf, dass das vorgestellte System starke Kompression zulässt, ohne die Bildgebung und Defektlokalisierung maßgeblich zu beeinträchtigen. Für die Lokalisierung von Streuern mittels Ultrabreitbandradaren stellen wir ein CS-System vor, welches auf einem Random Demodulators basiert. Im Vergleich zu existierenden Messverfahren ist die hieraus resultierende Schätzung der Kanalimpulsantwort robuster gegen die Effekte von zeitvarianten Funkkanälen. Um den inhärenten Modellfehler, den gitterbasiertes CS begehen muss, zu beseitigen, zeigen wir auf wie Atomic Norm Minimierung es erlaubt ohne die Einschränkung auf ein endliches und diskretes Gitter R-dimensionale spektrale Komponenten aus komprimierten Beobachtungen zu schätzen. Hierzu leiten wir eine R-dimensionale Variante des ADMM her, welcher dazu in der Lage ist die Signalkovarianz in diesem allgemeinen Szenario zu schätzen. Weiterhin zeigen wir, wie dieser Ansatz zur Richtungsschätzung mit realistischen Antennenarraygeometrien genutzt werden kann. In diesem Zusammenhang präsentieren wir auch eine Methode, welche mittels Stochastic gradient descent Messmatrizen ermitteln kann, die sich gut für Parameterschätzung eignen. Die hieraus resultierenden Kompressionsverfahren haben die Eigenschaft, dass die Schätzgenauigkeit über den gesamten Parameterraum ein möglichst uniformes Verhalten zeigt. Zuletzt zeigen wir auf, dass die Kombination des ADMM und des Stochastic Gradient descent das Design eines CS-Systems ermöglicht, welches in diesem gitterfreien Szenario wünschenswerte Eigenschaften hat.Along with the ever increasing number of sensors, which are also generating rapidly growing amounts of data, the traditional paradigm of sampling adhering the Nyquist criterion is facing an equally increasing number of obstacles. The rather recent theory of Compressive Sensing (CS) promises to alleviate some of these drawbacks by proposing to generalize the sampling and reconstruction schemes such that the acquired samples can contain more complex information about the signal than Nyquist samples. The proposed measurement process is more complex and the reconstruction algorithms necessarily need to be nonlinear. Additionally, the hardware design process needs to be revisited as well in order to account for this new acquisition scheme. Hence, one can identify a trade-off between information that is contained in individual samples of a signal and effort during development and operation of the sensing system. This thesis addresses the necessary steps to shift the mentioned trade-off more to the favor of CS. We do so by providing new results that make CS easier to deploy in practice while also maintaining the performance indicated by theoretical results. The sparsity order of a signal plays a central role in any CS system. Hence, we present a method to estimate this crucial quantity prior to recovery from a single snapshot. As we show, this proposed Sparsity Order Estimation method allows to improve the reconstruction error compared to an unguided reconstruction. During the development of the theory we notice that the matrix-free view on the involved linear mappings offers a lot of possibilities to render the reconstruction and modeling stage much more efficient. Hence, we present an open source software architecture to construct these matrix-free representations and showcase its ease of use and performance when used for sparse recovery to detect defects from ultrasound data as well as estimating scatterers in a radio channel using ultra-wideband impulse responses. For the former of these two applications, we present a complete reconstruction pipeline when the ultrasound data is compressed by means of sub-sampling in the frequency domain. Here, we present the algorithms for the forward model, the reconstruction stage and we give asymptotic bounds for the number of measurements and the expected reconstruction error. We show that our proposed system allows significant compression levels without substantially deteriorating the imaging quality. For the second application, we develop a sampling scheme to acquire the channel Impulse Response (IR) based on a Random Demodulator that allows to capture enough information in the recorded samples to reliably estimate the IR when exploiting sparsity. Compared to the state of the art, this in turn allows to improve the robustness to the effects of time-variant radar channels while also outperforming state of the art methods based on Nyquist sampling in terms of reconstruction error. In order to circumvent the inherent model mismatch of early grid-based compressive sensing theory, we make use of the Atomic Norm Minimization framework and show how it can be used for the estimation of the signal covariance with R-dimensional parameters from multiple compressive snapshots. To this end, we derive a variant of the ADMM that can estimate this covariance in a very general setting and we show how to use this for direction finding with realistic antenna geometries. In this context we also present a method based on a Stochastic gradient descent iteration scheme to find compression schemes that are well suited for parameter estimation, since the resulting sub-sampling has a uniform effect on the whole parameter space. Finally, we show numerically that the combination of these two approaches yields a well performing grid-free CS pipeline