Jan Kirchhof | Searcharxiv

arXiv2021

Fast Linear Transformations in Python

Scientific computing requires handling large linear models, which are often composed of structured matrices. With increasing model size, dense representations quickly become infeasible to compute or store. Matrix-free implementations are suited to mitigate this problem but usually complicate research and development effort by months, when applied to practical research problems. Fastmat is a framework for handling large composed or structured matrices by offering an easy-to-use abstraction model. It allows expressing and using linear operators in a mathematically intuitive way, while maintaining a strong focus on efficient computation and memory storage. The implemented user interface allows for very readable code implementation with very close relationship to the actual mathematical notation of a given problem. Further it provides means for quickly testing new implementations and also allows for run-time execution path optimization. Summarizing, fastmat provides a flexible and extensible framework for handling matrix-free linear structured operators efficiently, while being intuitive and generating easy-to-reuse results.

cs.MS↗

arXiv2020

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 Cramér-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.

eess.SP↗