## 4 Discussion  
We show that a significant reduction of transmit cycles is possible for the simulation scenario at hand. Using only 8 instead of 128 elements in transmit, the TFM and FISTA reconstructions still yield a clear image of the defects. Consequently, the measurement time for the acquisition of one dataset can be reduced by factor 16 with virtually no impairment of the imaging quality for the given region of interest. Furthermore, the computation time of the reconstruction algorithms is scaled down by approximately the same factor. For  
Figure 10: B-scan views of the reconstructions under presence of additive white Gaussian noise and structural noise normalized such that the maximum of the colormap of each image is the maximum value of the defect, where each column shows a different number of transmit elements. The rows indexed by (**a**), (**b**) have been calculated for a defect located at \(z~{}=~{}13\) mm and \(z~{}=~{}15\) mm, respectively. highly scattering media, a tuning of the number of transmit channels according to the noise level is advised. The transmit elements are chosen according to a deterministic scheme which, for a given region of interest, selects an as wide as suitable aperture while emphasizing those transmit elements that contribute more energy. The scheme is verified by evaluating the performance of the reconstruction algorithms for wider and narrower choices of the aperture. From these observations, it becomes apparent that using a slightly narrower aperture would benefit the imaging of the side-drilled holes used in the simulations. Still, as less well-behaving and more directional reflections are expected when imaging actual cracks or inclusions, the given aperture choice seems reasonable and in a good performance range over the full region of interest. Inspired by SART, we introduce a normalization of the model matrix, which improves the performance of FISTA in the given scenario. The algorithm does not perform optimally for problems that give varying weight to contributions in the imaging region. Though, this is natural for ultrasound imaging scenarios, as different areas in the imaging region are in most cases being observed by a varying number of transducers. We show empirically that the proposed normalization improves the results in the given scenario. The reconstructions, normalization for FISTA, as well as transmit element selection are based on a linear forward model which is derived using a ray casting scheme. It includes effects such as the directivity of the transducer elements as well as the refraction at the pipe boundary. The algorithms are expressed in terms of linear algebra, allowing easy and fast parallel computation and precomputation. As next step, the findings will be applied and verified on real measurement data. Furthermore, including other propagation modes than only longitudinal propagation seems beneficial, further influencing the aperture choice as well as the possibilities for imaging from a larger aperture. Including multiple propagation modes is inherently possible given the way the algorithms are expressed here, yet the fusion of the multimode images will require further investigation [11]. Furthermore, in order to reduce the processing time further, including subsampling on the receive side might be beneficial. Conceptualization, F.K., J.K., E.P., T.S., F.R., and A.O. ; methodology, F.K., J.K., E.P., and F.R. ; software, F.K. and J.K.; validation, F.K., J.K., E.P., and F.R. ; formal analysis, F.K. and J.K.; investigation, F.K. ; resources, F.K. and T.S. ; data curation, T.S. ; writing--original draft preparation, F.K. ; writing--review and editing, F.K., J.K., E.P., and F.R. ; visualization, F.K. ; supervision, F.R. and A.O. ; project administration, F.R. and A.O. ; funding acquisition, F.R. and A.O. All authors have read and agreed to the published version of the manuscript. This research was funded by DFG under the project \"CoSMaDU\" with grant GA 2062/5-1 and by the Fraunhofer Internal Programs under grant Attract 025-601128. Not applicable.