## 3 Results
### Transmit Subsampling
#### 3.2.2 Verification of the Transmit Element Selection Scheme  
To verify the approach to subsampling we perform reconstructions of the simulated measurement data with wider or narrower transmit patterns. To generate these patterns, the vector \(\mathbf{c}\) is linearly interpolated or extrapolated to reflect a wider or narrower energy spread, the patterns are then chosen from that deviated distribution. This is done in order to verify that the chosen approach of taking the distribution from the model matrix is valid. In Figure 6, the interpolated distributions \(\mathbf{c}\) are related to the corresponding scale parameters. For larger values of the scaling factor, the curve becomes wider, yielding a more uniform transmitter distribution over the full aperture. In contrast, a small scale parameter leads to a narrower distribution that results in only picking the center elements of the array as transducers. Figure 4: Impact of different choices for the region of interest (ROI), given by the binary maps \(\mathbf{i}\), on the visibility by each transducer element, given by \(\mathbf{c}\). The ROI correspond to a defect at 13 mm for (\(\mathbf{a}\)), 15 mm for (\(\mathbf{b}\)), and the full weld for (\(\mathbf{c}\)). The ROI (\(\mathbf{c}\)) is used for selecting the transmit elements in the remainder of the paper. Figure 5: Illustration of the transmit patterns chosen for the scenario at hand based on the binary map displayed in Figure 4c. We evaluate the transmit element selection scheme in Figure 7 by choosing the transmit pattern from a scaled \(\mathbf{c}\) (the scaled \(\mathbf{c}\) are displayed in Figure 6 and performing a reconstruction both with FISTA as well as TFM on the given data. The reconstructions are then compared by the ratio of the energy in a square region around the defect to the energy in the whole image. The evaluation has been run both for a defect at \(z~{}=~{}13\) mm in (a) as well as for a defect at \(z~{}=~{}15\) mm averaging over 10 noise realizations for each. All plots have been scaled to maximum 1 for better comparison. The curves show that the TFM reconstruction performs best for narrow transmit patterns, except for a slight increase for 8 and 16 elements from scaling factor 0.1 to 0.25. The performance decreases for wider aperture choices. The original distribution at scaling factor 1.0 is approximately at half of the peak to minimum performance value. For FISTA, the reconstruction of the defect at \(z~{}=~{}13\) mm benefits from using a pattern with scaling factor 0.5 to 1, even failing to reconstruct when picking only the center elements. For the other defect, the algorithm performs in a similar fashion as TFM. The results for FISTA  
Figure 6: Plot of the energy distributions over the elements indices gained by linearly interpolating \(\mathbf{c}\) to a scaled axis. The original distribution, which reflects the \(\mathbf{c}\) obtained for the region of interest in Figure 4c, is given by the scaling factor 1.0. The distribution is scaled to emulate the misalignment of \(\mathbf{c}\) and investigate the impact on the reconstruction. Figure 7: Performance of the reconstruction for the proposed transmit pattern selection scheme if the underlying model deviates. The deviation is simulated by interpolation of the original \(\mathbf{c}\), the scaling factor refers to the distributions shown in Figure 6. The row (**a**) displays the results for a circular defect at \(z~{}=~{}13\) mm, the row (**b**) for a defect at \(z~{}=~{}15\) mm. The proposed transmit subsampling scheme yields the measures at scaling factor 1. For each scaling factor 10, noise realizations are evaluated, the opaque error range of each graph indicates the according standard deviations. show relatively large jumps for small changes of the scaling factor. This illustrates that the algorithm reacts strongly to small changes in the presence of multiple scattering events, leading to variations of the proportion of the front wall energy to defect energy in the reconstruction.