# Phase-Sensitive 2D Motion Estimators Using Frequency Spectra of Ultrasonic Echoes
## 2 Materials and Methods
### 2D Motion Estimator  
Let us define the shifted cross spectrum \(\gamma_{l}(i,j;\omega_{x},\omega_{z})\) as follows:  
\[\gamma_{l}(i,j;\omega_{x},\omega_{z})=S_{n}^{*}(x,z;\omega_{x},\omega_{z})\cdot S _{n+l}(x+i\cdot\delta_{x},z+j\cdot\delta_{z};\omega_{x},\omega_{z}), \tag{11}\]  
where \(l\) is 0 or 1, and \(i\) and \(j\) are integers determining the lateral and axial spatial shifts, respectively. Equation (11) becomes the commonly-used cross spectrum between echoes (without spatial shifts) in two frames when \(l=1\) and \(i=j=0\).  
[PERSON] et al. proposed a 2D motion estimator using the phase of the cross spectrum of ultrasonic echo signals without spatial shifts [30; 36]. The cross spectrum without spatial shifts can be modeled based on Equations (1) and (2) as follows:  
\[\gamma_{1}(0,0;\omega_{x},\omega_{z}) =S_{n}^{*}(x,z;\omega_{x},\omega_{z})\cdot S_{n+1}(x,z;\omega_{x}, \omega_{z}) \tag{12}\] \[=A_{n}(x,z;\omega_{x},\omega_{z})\cdot A_{n+1}(x,z;\omega_{x}, \omega_{z})\cdot e^{-j(\omega_{x}u_{x}+\omega_{z}u_{z})}.\]  
The mean difference \(\alpha\) between the phase of the cross spectrum \(\angle\gamma_{1}(0,0;\omega_{x},\omega_{z})\) without spatial shifts and its model \(-(\omega_{x}u_{x}+\omega_{z}u_{z})\) is defined as follows:  
\[\alpha=\sum\limits_{\omega_{x},\omega_{z}}w(\omega_{x},\omega_{z})\left| \angle\gamma_{1}(0,0;\omega_{x},\omega_{z})+\left(\omega_{x}u_{x}+\omega_{z} u_{z}\right)\right|^{2}, \tag{13}\]  
where \(w(\omega_{x},\omega_{z})\) is a weight function defined as follows [30]:  
\[w(\omega_{x},\omega_{z})=\left|\gamma_{1}(0,0;\omega_{x},\omega_{z})\right|^{ 2}. \tag{14}\]  
The least-square solution of Equation (13) is obtained by setting the partial derivatives of \(\alpha\) with respect to \(u_{x}\) and \(u_{z}\) at zero to determine \(\hat{u}_{x}\) and \(\hat{u}_{z}\), which minimize the mean squared difference \(\alpha\), as follows:  
\[\left(\begin{array}{c}\hat{u}_{x}\\ \hat{u}_{z}\end{array}\right)= -\frac{1}{\left\{\sum\limits_{\omega_{x},\omega_{z}}w(\omega_{x}, \omega_{z})\cdot\omega_{x}^{2}\right\}\left\{\sum\limits_{\omega_{x},\omega_{ z}}w(\omega_{x},\omega_{z})\cdot\omega_{z}^{2}\right\}-\left\{\sum\limits_{\omega_{x}, \omega_{z}}w(\omega_{x},\omega_{z})\cdot\omega_{x}\cdot\omega_{z}\right\}^{2}\right. \tag{15}\] \[\times\left(\begin{array}{c}\sum\limits_{\omega_{x},\omega_{z}} w(\omega_{x},\omega_{z})\cdot\omega_{x}\cdot\omega_{z}&\sum\limits_{\omega_{z}, \omega_{z}}w(\omega_{x},\omega_{z})\cdot\omega_{x}&\sum\limits_{\omega_{z}, \omega_{z}}w(\omega_{x},\omega_{z})\cdot\omega_{x}^{2}\\ \omega_{x}\omega_{z}\end{array}\right)\cdot\omega_{x}\cdot\omega_{z}\] \[\times\left(\begin{array}{c}\sum\limits_{\omega_{x},\omega_{z}} w(\omega_{x},\omega_{z})\cdot\omega_{x}\cdot\angle\gamma_{1}(0,0;\omega_{x}, \omega_{z})\\ \omega_{x}\omega_{z}\end{array}\right).\]In the estimations of \(\hat{u}_{x}\) and \(\hat{u}_{z}\), frequency components in the half plane of \((\omega_{x},\omega_{z})\) are used because the 2D frequency spectrum is Hermitian-symmetric \(S_{n}(x,z;\omega_{x},\omega_{z})=S_{n}^{*}(x,z;-\omega_{x},-\omega_{z})\).  
The lateral and axial velocity estimates \(\hat{v}_{x}\) and \(\hat{v}_{z}\) are obtained by dividing the estimated displacements \(\hat{u}_{x}\) and \(\hat{u}_{z}\) by the frame interval.