Sparse signal representation based algorithms with application to ultrasonic array imaging

We address one- and two-layer ultrasonic array imaging. We use an array of transducers to inspect the internal structure of a given specimen. In the case of one-layer imaging we also address the problem of mode conversion. We propose a sparse signal representation based method for imaging solid materials in the presence of mode conversion phenomenon. In the case of two-layer imaging we model the signal propagation effect using Huygens principle and Rayleigh-Sommerfeld diffraction formula. We then use this model to develop a sparse signal representation based imaging technique for a test sample immersed in water. Moreover, we develop a new sparse Bayesian technique. In the model that we develop, the reflectivity coefficients of the desired reflectors are nonnegative real numbers and sparse in nature. Therefore, we use Weibull distribution function with two hyperparameters, namely the shape parameter and the scaling parameter, to model the prior distribution function of the reflectivity coefficients of the reflectors. As we show, the Weibull distribution, whose scale parameter obeys the inverse Gamma distribution, will enforce sparsity. We then propose a method for estimating the shape parameter of the Weibull distribution using Mellin transform

Compressive Sensing and Imaging of Guided Ultrasonic Wavefields

Structural health monitoring (SHM) and Nondestructive Evaluation (NDE) technologies can be used to predict the structural remaining useful life through appropriate diagnosis and prognosis methodologies. The main goal is the detection and characterization of defects that may compromise the integrity and the operability of a structure. The use of Lamb waves, which are ultrasonic guided waves (GW), have shown potential for detecting damage in specimens as a part of SHM or NDT systems. These methods can play a significant role in monitoring and tracking the integrity of structures by estimating the presence, location, severity, and type of damage. One of the advantages of GW is their capacity to propagate over large areas with excellent sensitivity to a variety of damage types while guaranteeing a short wavelength, such that the detectability of large structural damages is guaranteed. The Guided ultrasonic wavefield imaging (GWI) is an advanced technique for Damage localization and identification on a structure. GWI is generally referred to as the analysis of a series of images representing the time evolution of propagating waves and, possibly, their interaction with defects. This technique can provide useful insights into the structural conditions. Nowadays, high-resolution wavefield imaging has been widely studied and applied in damage identification. However, full wavefield imaging techniques have some limitations, including slow data acquisition and lack of accuracy. The objectives of this dissertation are to develop novel and high resolution Guided Wavefield Imaging techniques able to detect defects in metals and composite materials while reducing the acquisition time without losing in detection accuracy

Time of flight diffraction and imaging (TOFDI)

Time of flight diffraction and imaging (TOFDI) is based on time of flight diffraction (TOFD), adding cross-sectional imaging of the sample bulk by exploiting the scattering of ultrasonic waves from bulk defects in metals. Multiple wave modes are emitted by a pulsed laser ultrasound ablative source, and received by a sparse array of receiving electromagnetic acoustic transducers (EMATs), for non-contact (linear) scanning, with mode-conversions whenever waves are scattered. Standard signal processing techniques, such as band-pass filters, reduce noise. A B-scan is formed from multiple data captures (A-scans), with time and scan position axes, and colour representing amplitude or magnitude. B-scans may contain horizontal lines from surface waves propagating directly from emitter to receiver, or via a back-wall, and angled lines after reflection off a surface edge. A Hough transform (HT), modified to deal with the constraints of a B-scan, can remove such lines. A parabola matched filter has been developed that identifies the features in the B-scan caused by scattering from point-like defects, reducing them to peaks and minimising noise. Multiple B-scans are combined to reduce noise further. The B-scan is also processed to form a cross-sectional image, enabling detection and positioning of multiple defects. The standard phase correlation technique applied to camera images, has been used to track the relative position between transducer and sample. Movement has been determined to sub-pixel precision, with a median accuracy of 0.01mm of linear movement (0.06 of a pixel), despite uneven illumination and the use of a basic low resolution camera. The prototype application is testing rough steel products formed by continuous casting, but the techniques created to facilitate operation of TOFDI are applicable elsewhere

Fast image reconstruction for near-field terahertz imaging with multistatic non-uniform sparse arrays

Conference paper presented at Radar Sensor Technology XXVII, Orlando, FL, USA, 30 April - 5 May, 2023.In this paper, first, the structure of a linear sparse periodic array for two-dimensional scanning is described. Then, based on its characteristics, an algorithm is presented for fast image reconstruction of the scene in a near-field (NF) multistatic terahertz imaging scenario. Although the basis of this algorithm is developed in the Fourier domain, it is compatible with the non-uniform structure of the array and also takes into account the phase deviations caused by multistatic imaging in NF. The performance of the proposed approach is evaluated with numerical data obtained from electromagnetic simulations in FEKO as well as experimental data. The results are discussed in terms of computational time on the central processing unit and graphics processing unit as well as the quality of the reconstructed image.Leverhulme Trust under Research Leadership Award RL-2019-019. The work of Shaoqing Hu is funded by Brunel University London under Research Development Fund LBG194 and 2022/2023 Brunel Research Initiative and Enterprise Fund 12455

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