# Phase-Sensitive 2D Motion Estimators Using Frequency Spectra of Ultrasonic Echoes
## 2 Materials and Methods
### Paired 1D Motion Estimators
When an object moves by lateral and axial displacements \((u_{x},u_{z})\) between the \(n\)-th and \((n+1)\)-th frames, the 2D frequency spectrum \(S_{n+1}(x,z;\omega_{x},\omega_{z})\) of the ultrasonic echo in the \((n+1)\)-th frame can be modeled as follows:  
\[S_{n+1}(x,z;\omega_{x},\omega_{z})=A_{n+1}(x,z;\omega_{x},\omega_{z})\cdot e^{ j[\omega_{x}(x-u_{x})+\omega_{z}(z-u_{z})\}. \tag{2}\]  
Let us define the products of the 2D frequency spectra \(S_{n}(x,z;\omega_{x},\omega_{z})\) and \(S_{n+1}(x,z;\omega_{x},\omega_{z})\) as follows:  
\[\beta_{x,0}(\omega_{x},\omega_{z}) =S_{n}(x,z;\omega_{x},\omega_{z})\cdot S_{n}(x,z;\omega_{x},- \omega_{z}) \tag{3}\] \[=A_{n}(x,z;\omega_{x},\omega_{z})\cdot A_{n}(x,z;\omega_{x},- \omega_{z})\cdot e^{j2\omega_{x}x},\] (4) \[\beta_{x,1}(\omega_{x},\omega_{z}) =S_{n+1}(x,z;\omega_{x},\omega_{z})\cdot S_{n+1}(x,z;\omega_{x},- \omega_{z})\] \[=A_{n+1}(x,z;\omega_{x},\omega_{z})\cdot A_{n+1}(x,z;\omega_{x},- \omega_{z})\cdot e^{j2\omega_{x}(x-u_{x})},\] (5) \[\beta_{z,0}(\omega_{x},\omega_{z}) =S_{n}(x,z;\omega_{x},\omega_{z})\cdot S_{n}(x,z;-\omega_{x}, \omega_{z})\] \[=A_{n}(x,z;\omega_{x},\omega_{z})\cdot A_{n}(x,z;-\omega_{x}, \omega_{z})\cdot e^{j2\omega_{z}z},\] \[\beta_{z,1}(\omega_{x},\omega_{z}) =S_{n+1}(x,z;\omega_{x},\omega_{z})\cdot S_{n+1}(x,z;-\omega_{x}, \omega_{z})\] (6) \[=A_{n+1}(x,z;\omega_{x},\omega_{z})\cdot A_{n+1}(x,z;-\omega_{x}, \omega_{z})\cdot e^{j2\omega_{z}(z-u_{z})}.\]  
From Equations (3)-(6), the phase of those complex signals can be modeled as follows:  
\[S_{n+1}(x,z;\omega_{x},\omega_{z})=A_{n+1}(x,z;\omega_{x},\omega_{z})\cdot e^{ j[\omega_{x}(x-u_{x})+\omega_{z}(z-u_{z})\}. \tag{2}\]0 \[S_{n+1}(x,z;\omega_{x},\omega_{z})=A_{n+1}(x,z;\omega_{x},\omega_{z})\cdot e^{ j[\omega_{x}(x-u_{x})+\omega_{z}(z-u_{z})\}. \tag{2}\]1  
where \(\measuredleft\{\cdot\right\}\) denotes the phase angle of a complex number. Mean squared differences \(\alpha_{x}\) and \(\alpha_{z}\) between the measured phases \((\phi_{x}(\omega_{x}),\phi_{z}(\omega_{z}))\) and their models \((-2\omega_{x}u_{x},-2\omega_{z}u_{z})\) are defined as follows:  
\[S_{n+1}(x,z;\omega_{x},\omega_{z})=A_{n+1}(x,z;\omega_{x},\omega_{z})\cdot e^{ j[\omega_{x}(x-u_{x})+\omega_{z}(z-u_{z})\}. \tag{2}\]2 \[S_{n+1}(x,z;\omega_{x},\omega_{z})=A_{n+1}(x,z;\omega_{x},\omega_{z})\cdot e^{ j[\omega_{x}(x-u_{x})+\omega_{z}(z-u_{z})\}. \tag{2}\]3  
where \(w_{a}(\omega_{a})\) is the weight function defined as follows:  
\[S_{n+1}(x,z;\omega_{x},\omega_{z})=A_{n+1}(x,z;\omega_{x},\omega_{z})\cdot e^{ j[\omega_{x}(x-u_{x})+\omega_{z}(z-u_{z})\}. \tag{2}\]4 \[S_{n+1}(x,z;\omega_{x},\omega_{z})=A_{n+1}(x,z;\omega_{x},\omega_{z})\cdot e^{ j[\omega_{x}(x-u_{x})+\omega_{z}(z-u_{z})\}. \tag{2}\]5  
The least-square solutions of Equation (8) are obtained by setting the partial derivatives of \(\alpha_{x}\) and \(\alpha_{z}\) with respect to \(u_{x}\) and \(u_{z}\) at zero, respectively, to determine \(\hat{u}_{x}\) and \(\hat{u}_{z}\), which minimize the mean squared differences \(\alpha_{x}\) and \(\alpha_{z}\), as follows:\[S_{n+1}(x,z;\omega_{x},\omega_{z})=A_{n+1}(x,z;\omega_{x},\omega_{z})\cdot e^{ j[\omega_{x}(x-u_{x})+\omega_{z}(z-u_{z})\}. \tag{2}\]6 \[S_{n+1}(x,z;\omega_{x},\omega_{z})=A_{n+1}(x,z;\omega_{x},\omega_{z})\cdot e^{ j[\omega_{x}(x-u_{x})+\omega_{z}(z-u_{z})\}. \tag{2}\]7  
In the estimation of \(\hat{u}_{a}\), frequency components in the range of \(\omega_{a}>0\) 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.