## 4 Discussion  
In this study, the effect of beam-forming strategies and line density on the accuracy of normalized cross-correlation-based displacement estimation was evaluated. Therefore, we designed a method to evaluate the accuracy by phantom rotation, which had two main advantages. First, displacement fields were induced with axial and lateral displacements separately varying within \(\pm\)0.5 mm (Figure 1a-d, i-l) instead of pure axial or lateral displacements. Second, the local displacement gradients are expected to be zero independent of the rotation center and angle in this case where the rotation angles was small (\(<\)\(10^{\circ}\)). Consequently, the error of the gradient (RMSE) was used as measure for the accuracy of displacement estimates. Displacements were calculated by normalized cross-correlation and block-matching. Although other methods are available, only this method was used in the study since it is widely used in (shear) strain imaging. The effect of different methods on the accuracy of displacements was not the scope  
Figure 10: Lateral displacement and strain fields using Lu\({}^{\prime}\)s-fk for the L12-5 transducer: (**a,c**) displacement field using one (**a**) or two (**c**) lines per pitch. Please refer to Figure 60 for the color bar; (**b,d**) strain field using one (**b**) or two (**d**) lines per pitch. Please refer to Figure 6j and Figure 6t for the corresponding color bars. Figure 9: The root-mean-squared error (RMSE) without the top left artifact of the (**a**) axial strain (\(s_{zz}\)) and (**b**) lateral strain (\(s_{xx}\)) of the 12L4 VF transducer; \(s_{zz}\) and \(s_{xx}\) values in the top left corner (axial and lateral position smaller than 28 and 0 cm respectively) were neglected in these RMSE calculations. of this study and so not evaluated. Furthermore, a one dimensional, three-point least-square-strain estimator (LSQSE) was implemented to calculate the gradients for evaluation. Although not presented here, equal results were obtained for LSQSE sizes of 7 and 11 points. The accuracy of displacement estimates, especially lateral, improved significantly using a line-density of multiple lines per pitch for all transducers except the high frequency transducer MS250 in which this improvement was minimal or the accuracy even slightly decreased for some beam-forming strategies. Improvement by higher line-densities was also observed by [PERSON] et al. [20] in which conventional focused ultrasound was used and RF-lines were interpolated (up to 64 lines per pitch) to gain higher line-densities to improve results. The accuracy of displacement estimates already improved by two or three lines per pitch in this study. However, interpolation of the cross-correlation peak after the final iteration was implemented in this study which also contributes to improved sub-sample displacement estimates [2]. As can be seen in Figure 10a,b, the performance of cross-correlation peak interpolation seemed to be decreased in displacements at half-pitch distances resulting in increased gradient bands (Figure 10b). This effect seemed to disappear at a line density of two lines per pitch (Figure 10c,d). The increased performance of sub-sample interpolation might be explained by the point spread function (PSF). According to [PERSON] et al. [17], beam-forming to a PSF-based USGrid results in a circular cross-correlation peak which improves the performance of two-dimensional spline peak interpolation. The optimal number lines should be between two or three lines according to the point spread function for all transducers. This line-density matched the optimal density found in this study. For all transducers except the 12L4 VF transducer, apodization or angular weighting seemed to decrease or minimally affect the accuracy of axial and lateral displacement estimates (Figures 7 and 8). Weighting and apodization increased the lateral resolution of the beam-formed data (Appendix A), which may explain the decreased accuracy of lateral estimates.