# A PSF-Shape-Based Beamforming Strategy for Robust 2D Motion Estimation in Ultrafast Data
## 4 Discussion  
In this paper, a new method has been proposed to determine the beamforming grid for ultrafast data based on the dimensions of the imaging system's PSF. As a result of this data sampling, the CCFs used in the ST algorithm will contain circular-shaped peaks. It was hypothesized that once combined with 2D cubic interpolation, this would increase the accuracy and precision of subsample displacement estimation and thereby enlarge the range of detectable displacements and velocities. Both simulations and experiments were conducted to characterize the performance of the method. A comparison was made with conventionally spaced beamformed data and the very commonly used 1D parabolic subsample displacement estimation method. The results confirmed our hypothesis: the PSF-shape-based beamforming grid combined with 2D cubic interpolation (PSF_2 Dcub) showed the most accurate and stable performance with respect to the range of inter-frame displacements, both for the assessment of blood flow and vessel wall dynamics. Besides, the performance of the method was least affected by motion direction or beam-to-flow angle. An important finding was that largest performance differences between the four methods were found for the lateral displacement component. This can be appreciated for the vessel wall (Figures 6, 7 and 10), but also for the flow results (Figure 8), in the upper and lower part of the disk, where velocities are predominantly in lateral direction. Furthermore, the accuracy and precision of the axial displacement estimates is consistently higher as compared to the lateral estimates, due to the higher resolution and presence of phase information. This also explains why the performance difference is less obvious in the axial direction and very clearly visible in the lateral direction. No phase information is available and the frequency content is lower in the lateral direction. Consequently, more benefit is gained from the proposed methodology. Several functions can be used to interpolate the CCF for subsample displacement estimation. In this work, we used 1D parabolic interpolation, one of the most widely used functions, to compare to our proposed methodology. The results however, show that 1D parabolic is less optimal for subsample  
Figure 10: Median and interquartile ranges of the absolute bias and SD for the axial (**a**, **b**) and lateral (**c**, **d**) inter-frame displacements obtained from the rotating disk setup processed as rotational vessel wall displacements. The performance of the methods is visualized as function of the PRF, which corresponds to a range of inter-frame displacement (Table 2). displacement estimation. Large bias, SD and RMSE values were found when 1D parabolic interpolation was used for CCF interpolation. More specifically, large bias and SD values on the lateral axis were shown when subsample displacements existed on the axial axis. This effect can be appreciated in Figure 8, where vertically stripped patterns are visible in the magnitude and angle estimates for the PSF\(\_\)1 Dpar en conv\(\_\)1 Dpar methods. These patterns originate from the inaccurate lateral displacement estimates in regions of subsample axial displacements, as also reported earlier [33]. The benefit of using 1D parabolic interpolation is the low computational cost, whereas 2D cubic interpolation increases computational time. Therefore, the trade-off between computational time and performance needs to be considered. One limitation of this study is that a single 2D interpolator was investigated. Many other studies have investigated the use of other shapes and functions, such as cosine, parabolic or spline [25; 33; 51; 52], most often in 1D. However, none of these interpolators have been studied before in the context of the proposed methodology, i.e., in combination with the freedom of choosing the beamforming grid and thereby influencing the CCF shape. Further investigation should be performed to compare different 2D interpolators in combination with the PSF-shape-based grid.