# A PSF-Shape-Based Beamforming Strategy for Robust 2D Motion Estimation in Ultrafast Data
## 4 Discussion
2D fitting of the CCF peak instead of interpolation is not considered in this work, due to the enormous increase of computational time, which makes its use less practical. Another method to improve subsample displacement estimation is by interpolation of the echo signal. This method was often used in combination with conventionally acquired data and reduced the error by the up sample factor. However, the computational cost increased significantly [25; 53]. The benefit of the use of ultrafast data is that instead of interpolating the echo signals, actual focusing of the signals can be performed in receive, as a post-processing step. This work shows that no excessive sampling of the RF data is required when using knowledge of the PSF dimensions to beamform the data combined with a suitable 2D subsample interpolation method. The proposed methodology is based upon the statement that the CCF shape is primarily determined by the shape of the PSF. While it is one of the most important factors influencing the appearance of the CCF, it is known that other factors will also be of influence. In general, the performance of motion estimation methods will degrade as a result of signal decorrelation. Factors such as out-of-plane motion, presence of strong gradients within the displacement field and large deformations can hamper the match of the signal template within the search window, thereby influencing the appearance of the CCF. Smartly choosing data type and window sizes is recommended in those situations [25; 54] and aligning and stretching methods are suggested to minimize signal decorrelation [55; 56]. Although these and other factors will also influence the CCF shape, the proposed methodology was successful under the circumstances tested in this work. To determine the spacing of the PSF-shape-based beamforming grid, a single average PSF\(\_\)ratio was used in this work. This was allowed since for carotid artery imaging, the PSF is nearly constant within the region of interest. However, for other applications, such as cardiac imaging, the size of the PSF will change with respect to the position in the imaging plane, amongst others due to the frequency dependent attenuation and location of the echo with respect to the ultrasonic beam [34]. These effects can be seen in the work of [PERSON] et al. [57]. However, application of the PSF-shape-based grid is still possible, when using variable grid spacing tuned to the PSF shape locally. [PERSON] et al. [58] showed first results using such a multi-zone, depth-dependent PSF-shape-based grid for cardiac motion imaging. The beamforming grid resolution was adjusted to the local PSF shape. This work also shows that the proposed beamforming strategy is not limited to linear array imaging. The PSF-shape-based beamforming grid can easily be used for data obtained with different transducer types, such as a phased array. The only requirement for using this method is the characterization of the imaging system by determining the (local) dimensions of the PSF. In this work, the dimensions of the PSFs were determined using the response of single scatterers. When processing data acquired in vivo, a more sophistic approach could be favorable, which includes other factors influencing the shape of the PSF, such as attenuation. A representative estimate of the PSF\({}_{\text{ratio}}\) could be obtained by using the spatial auto covariance method [59] to determine the speckle size in a B-mode image. Since speckle size is proportional to the dimensions of the PSF [60; 61], this will provide the required information to determine the PSF-shape based beamforming grid. In the present work, a large range of inter-frame displacements were studied by imaging with different PRFs. A stable and accurate performance of the proposed method was shown with respect to this large range of displacements.