# A PSF-Shape-Based Beamforming Strategy for Robust 2D Motion Estimation in Ultrafast Data
## Materials and Methods
### Multi-Step Speckle Tracking Settings  
Inter-frame displacements were estimated using data beamformed on both the conventional grid and the PSF-shape-based grid. The size of the search windows was tuned to the maximum displacement expected in each situation. Spatial smoothing of the estimated displacements was performed using median filters. Inter-frame displacement estimates were multiplied by frame rate to calculate velocities. Subsample resolution was resolved by interpolation of the CCF in both steps of the algorithm. Two types of interpolation were used: 1D parabolic fitting in each direction of the CCF independently, since it is one of the most often used interpolators in ultrasound applications as described in the Introduction, and 2D cubic interpolation, where a joint estimation of subsample displacement in both  
Figure 3: (**a**) Simulated PSFs at spatial positions relevant for carotid artery imaging. (**b**) Intensity curves in axial and lateral direction. The red lines indicate the \(-6\) dB level. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Grid Type** & **Parameter** & **Simulations** & **Experiments** \\ \hline \multirow{4}{*}{**PSF-shape-based grid**} & PSFratio & 1.5 & 2.0 \\  & d\({}_{\text{ax,RF}}\) & 14.2 \(\upmu\)m & 14.4 \(\upmu\)m \\  & d\({}_{\text{lat,RF}}\) \& d\({}_{\text{lat,ENV}}\) & 107 \(\upmu\)m & 143 \(\upmu\)m \\  & d\({}_{\text{ax,ENV}}\) & 71 \(\upmu\)m & 72 \(\upmu\)m \\ \hline \multirow{2}{*}{**Conventional grid**} & d\({}_{\text{ax}}\) & 21.4 \(\upmu\)m & 21.6 \(\upmu\)m \\  & d\({}_{\text{lat}}\) & 198 \(\upmu\)m & 198 \(\upmu\)m \\ \hline \multirow{2}{*}{**Both grids**} & Apodization window & Hamming \\  & F-number & 0.875 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Beamforming parameters. PSF: point-spread-function. directions was obtained. The parabolic fitting is solved analytically using the commonly applied three-point parabola fitting [25; 33]. 2D cubic interpolation was performed with a resulting effective upsampling factor of 10 and 1000 in the first and second step of the algorithm, respectively. For the second step, a fast in-house built implementation was used, where the high final resolution was reached iteratively by zooming in onto the peak of the CCF in multiple iteration steps. Given the CCF with its maximum at position (i,j), the new sampling points (ix,iy) are defined by ix = ixm*5 iter + i and iy = iym*5 iter + j, with matrices ixm and iym representing the direct neighbors of (i,j) according to ixm = [-1 0 1;-1 0 1;-1 0 1] and iym = [-1 -1 -1; 0 0 0; 1 1 1]. Ten iteration steps (iter = 1-10) were used, resulting in the effective upsampling factor of 1000. A complete overview of the settings used in the ST algorithm can be found in Table 4.  
Summarized, four different combinations of beamforming grid and CCF interpolation type were studied in this work: (1) data beamformed on the conventional grid combined with 1D parabolic interpolation (conv_1 Dpar) and (2) combined with 2D cubic interpolation (conv_2 Dcub), (3) data beamformed on the PSF-shape-based grid combined with 1D parabolic interpolation (PSF_1 Dpar) and (4) combined with 2D cubic interpolation (PSF_2 Dcub). The performance of the four approaches was studied in terms of the accuracy and precision of the velocity magnitude, velocity angle and the displacement estimates. The accuracy of the velocity magnitude and the inter-frame displacement estimates was assessed by calculating the absolute error percentage between the mean estimates and the ground truth as follows:  
\[\text{Error (i) }=\frac{|EST_{mean}(i)-GT(i)|}{GT(i)}*100\% \tag{1}\]  
with \(i\) all estimated samples within the region-of-interest (ROI), \(EST_{mean}\) the mean of the estimates and \(GT\) the ground truth value. \(EST_{mean}\) was based on \(n\) = 10 realizations for the simulated parabolic flow, \(n\) = 100 ensemble averaged velocity estimates for the experimental rotational flow, and \(n\) = 100 inter-frame displacement estimates for the rotational displacement experiments.