# A PSF-Shape-Based Beamforming Strategy for Robust 2D Motion Estimation in Ultrafast Data
## 1 Introduction  
Ultrasound imaging is often used to visualize and estimate motion in our body [1; 2; 3; 4; 5; 6; 7; 8; 9; 10]. Tissue motion and deformation can provide information about the structure, composition and functioning of the tissue and changes in the dynamic behavior of tissue can be used as indicator of disease. For example, stiff regions in breasts are often related to cancers [11; 12], the assessment of local arterial deformation provides information about the atherosclerotic progression and vulnerability of lesions [13; 14; 15; 16], increased blood velocities indicate the presence of a stenosis in the vessel [17; 18; 19], and infarcted myocardial tissue shows changed deformation patterns as compared to healthy tissue [20]. Speckle tracking (ST), first introduced by [PERSON] et al. [21; 22], is one of the motion estimation methods which is frequently utilized. Speckle patterns are tracked to find the displacement of the underlying tissue. Pattern matching techniques are used, where a kernel region in the first acquisition is matched within a surrounding search region in the following acquisition. The location of the best match defines the displacement of the kernel region and thus the displacement of the underlyingtissue. ST can be performed using radio frequency (RF) signals [23], envelope signals [24], B-mode images [20; 22], or a combination thereof [25]. [PERSON] et al. [26] wrote a review on the development of this technique, focusing on multi-dimensional flow estimation. They concluded that decorrelation of speckle patterns is one of the major challenges in ST. Strong displacement gradients and velocities with low beam-to-flow angles were described as major sources of speckle decorrelation. Increasing the imaging frame rate can minimize speckle decorrelation [26]. The first published approach for increasing imaging frame rate was by reconstructing multiple image lines simultaneously from each transmit, so-called parallel beamforming [27]. Nowadays, plane wave and diverging wave imaging are used more often, where the transmitted pulse fully illuminates the entire region of interest, enabling full image reconstruction for every transmit event. This allows for imaging at ultrafast frame rates in the kHz range. Hereby, decorrelation of speckle can be minimized and fast moving, sometimes short-lived, motion patterns can be tracked [28; 29; 30; 31; 32]. However, frame-to-frame tissue displacements become smaller, since they scale with the imaging frame rate. Consequently, acquiring accurate subsample displacement estimates is of great importance. The estimation of subsample displacements in ST was already a challenge before the introduction of ultrafast data acquisition, since echo signals are sampled. For conventionally acquired, line-by-line data, interpolation of the pattern matching function is often applied to acquire subsample displacement estimates [25; 33; 34]. The sampling grid for conventional data is predefined, with normally a line distance equal to the pitch and an axial grid spacing with typically four samples per wavelength. Consequently, the sampling of the pattern matching function is also set. A predefined shape or curve, often parabolic or cosine [34; 35], is assumed and fitted to the peak of the function, either in each direction of the 2D function independently [25; 35], or by fitting in 2D to obtain a joint estimate of subsample displacement in both directions. [PERSON] et al. [33] investigated 2D polynomial fitting to get a joint estimation of subsample displacement and showed significant improvements in terms of bias and standard deviation (SD) compared to the commonly used 1D parabolic and cosine fitting. Interpolating in between echo signals is also often applied to reduce the error of the subsample displacement estimates [25; 36; 37]. With the use of ultrafast data, subsample displacements estimation remains a challenge.