Sparsity-Aware Signal Processing for Multichannel Ultrasound NDT Data

Ultrasound Nondestructive Evaluation (UNDE) systems enjoy widespread usage thanks to their comparative ease of deployment, safety, and ability to characterize critical structures. However, this inspection modality is governed by a complex physical phenomenon and requires extensive post-processing before yielding useful results. Furthermore, when employed for localization and imaging, UNDE necessitates multichannel data, and benefits from an ever-increasing amount of spatially-rich measurements from which a high spatial and temporal resolution can be obtained.

Naively, this can be achieved by increasing the number of sensors and their bandwidth. However, this has the consequence of increasing the number of transmission events, and therefore the measurement time, as well as the sampling rate, data volume, and data rate. Accordingly, measurement hardware becomes more complex and expensive, and the computational and storage costs increase. Instead, the inherent characteristics of UNDE data, the measurement process, and post-processing can be considered holistically to develop novel acquisition and processing schemes that can a.) achieve the same performance as classical methods using less data, or b.) achieve a higher performance when applied to the same data volume.

In this dissertation, the inherent sparsity of multichannel UNDE data is exploited under the Sparse Signal Recovery (SSR) and Compressed Sensing (CS) framework. The dissertation and contributions are organized into three parts encompassing efficient forward modeling, sparse recovery, and compression matrix design. The corresponding main contributions are the efficient implementation of matrix-free Full Matrix Capture (FMC) forward models exploiting translational invariance and incorporating multiple scattering, the incorporation of the aforementioned models into an alternating optimization framework for SSR and CS using soft thresholding algorithms, and the usage of the Cramér–Rao Bound (CRB) both for the study of the models and for the design of compression matrices. The results are validated on real and simulated UNDE data. Despite the focus on UNDE localization and imaging, it should be highlighted that the contributions generalize to other multichannel/multidimensional wave-based measurement modalities and inverse problems.

In the first part, the modeling of scalar wave fields in piecewise isotropic and homogeneous media is studied with emphasis on FMC. Interrelated forward modeling concepts found in the mathematics, physics, and UNDE literature under different nomenclatures are curated and contextualized in terms of their suitability for signal processing. FMC data is shown to exhibit multiple forms of invariance, allowing both the reduction of memory requirements by several orders of magnitude, and the acceleration of computations with off-the-shelf software.

Motivated by the sparsity of the sought-after quantity in UNDE imaging and localization, the second part of the work focuses on CS with soft thresholding algorithms. The importance of the choice of sparse recovery algorithm is highlighted by studying the behavior of two representative algorithms in practical conditions. An optimization framework that alternates between a linear forward modeling problem and a linear inverse problem is presented. The framework is then applied to measurement data and simulations with and without compression, culminating with tomographic reconstructions incorporating multiple scattering and using single digit percentages of the original, uncompressed data volume.

The final part addresses the design of compression matrices for the aforementioned optimization framework. The CRB is chosen as the core quantity to be optimized, choice justified by relating the CRB to the sparse recovery identifiability conditions. First, the CRB is shown to be susceptible to assumptions regarding the presence of multiple scattering, requiring a judicious choice of model parameters. Next, the design of compression matrices, and more specifically, subsampling matrices, is addressed. A collection of original greedy and gradient-based optimization methods for CRB-based structured subsampling matrix design is presented, enabling the design of matrices for realistically-sized problems that are easily implementable in hardware.

The contributions of this dissertation enable the design of subsampling matrices which result in faster acquisition times, data volumes as low as 1 % of what is used in traditional methods, and low data rates. Importantly, the subsampling matrices are easily implementable in the real world. The forward models and sparse recovery algorithms presented in this work compose a computationally efficient framework that can be applied to large-scale problems. The localization and imaging quality attained on real and simulated data is on par with, or superior to, classical methods, depending on the desired amount of compression. The results in this work lend themselves well to model-based Deep Learning (DL) extensions as a line of future research.