Frequency Sub-Sampling of Ultrasound Non-Destructive Measurements: Acquisition, Reconstruction and Performance

In ultrasound nondestructive testing, a widespread approach is to take synthetic aperture measurements from the surface of a specimen to detect and locate defects within it. Based on these measurements, imaging is usually performed using the Synthetic Aperture Focusing Technique (SAFT). However, SAFT is sub-optimal in terms of resolution and requires oversampling in time domain to obtain a fine grid for the Delay-and-Sum (DAS). On the other hand, parametric reconstruction algorithms give better resolution, but their usage for imaging becomes computationally expensive due to the size of the parameter space and the large amount of measurement data in realistic 3-D scenarios. In the literature, the remedies to this are twofold: First, the amount of measurement data can be reduced using state of the art sub-Nyquist sampling approaches to measure Fourier coefficients instead of time domain samples. Second, parametric reconstruction algorithms mostly rely on matrix-vector operations that can be implemented efficiently by exploiting the underlying model structure. In this paper, we propose and compare different strategies to choose the Fourier coefficients to be measured. Their asymptotic performance is compared by numerically evaluating the Cram\'er-Rao-Bound for the localizability of the defect coordinates. These subsampling strategies are then combined with an $\ell_1$-minimization scheme to compute 3-D reconstructions from the low-rate measurements. Compared to conventional DAS, this allows us to formulate a fully physically motivated forward model. To enable this, the projection operations of the forward model matrix are implemented matrix-free by exploiting the underlying 2-level Toeplitz structure. Finally, we show that high resolution reconstructions from as low as a single Fourier coefficient per scan are possible based on simulated data as well as on measurements.