# Phase-Sensitive 2D Motion Estimators Using Frequency Spectra of Ultrasonic Echoes
## 1 Introduction  
Functional ultrasound imaging, such as vascular elastography [1; 2; 3; 4; 5] and myocardial contractility imaging [6; 7; 8; 9], has shown to be useful for diagnosis of the cardiovascular system. Recently, high-frame-rate ultrasound has gained attention because it enhances such functional imaging by measurements of tissue dynamics at a very high frame rate of over one hundred Hz.  
High-frame-rate ultrasound was first developed in 1980s [10]. After the development of shear wave elastography by [PERSON], et al. [11], which was the first practical application of high-frame-rate ultrasound, various studies have been conducted for cardiovascular applications, such as vascular strain and blood flow imaging [12; 13; 14], pulse wave imaging [15; 16], and cardiac strain and blood flow imaging [17; 18; 19; 20; 21]. In addition, various studies for improvement of the image quality in high-frame-rate ultrasound have been conducted [22; 23; 24; 25; 26].  
For such measurements of tissue dynamics, motion estimators are crucial. The autocorrelation method was introduced by [PERSON], et al. as a phase-sensitive motion estimator for blood flow imaging [27]. This method estimates the axial velocity of a target using the phase shift of the ultrasonic echo, but it suffers from the aliasing effect when the tissue displacement between consecutive ultrasound emissions is greater than half of the ultrasonic wavelength. In conventionalultrasonography, the imaging frame rate is typically tens of Hz. Under such a temporal resolution, the tissue displacement between frames can become larger than half of the ultrasonic wavelength, and phase-sensitive motion estimators possibly suffer from the aliasing effect. The speckle tracking method is suitable in such situations because it does not suffer from the aliasing effect. Therefore, the speckle tracking method is now widely used for estimation of tissue motion, such as in myocardial strain imaging [9].  
In high-frame-rate ultrasound, the tissue displacement between frames is significantly small owing to the high temporal resolution of over one hundred Hz. The speckle tracking method requires interpolation of the function, which is used for evaluation of the similarity of ultrasonic echoes in two frames, to estimate a sub-sample small displacement [28]. Such an interpolation process could be a heavy computational cost in high-frame-rate ultrasound because a number of frames needs to be processed. In such situations, a phase-sensitive motion estimator is feasible because it can estimate a sub-sample small displacement without interpolation and does not suffer from the aliasing effect owing to the high temporal resolution. In the present study, phase-sensitive motion estimators were newly developed and examined for motion estimation with high-frame-rate ultrasound.  
The autocorrelation approach, one of the common phase-sensitive motion estimators, was first used for estimation of the axial motion [27; 29]. Then, the phase-sensitive method was expanded to the 2D motion estimators using 2D frequency spectra of ultrasonic echoes [30] and the complex analytical signal obtained by modulating the ultrasonic field [31] or the Hilbert transform [32; 33; 34; 35]. In the present study, 2D motion estimators using 2D frequency spectra of ultrasonic echoes were developed, and the accuracies of the proposed methods were compared with that of the conventional method by the phantom experiments.