# Phase-Sensitive 2D Motion Estimators Using Frequency Spectra of Ultrasonic Echoes
## 2 Materials and Methods
### Estimation of Mean Frequency  
In general, the 2D frequency spectrum \(S_{n}(x,z;\omega_{x},\omega_{z})\) is obtained by applying the discrete 2D Fourier transform to the target signal extracted with a 2D spatial window. When the lateral and axial sizes of the spatial window are \((2M+1)\cdot\delta_{x}\) and \((2K+1)\cdot\delta_{z}\), respectively, the angular frequencies \(\omega_{x}\) and \(\omega_{z}\) of each component of the spectrum \(S_{n}(x,z;\omega_{z},\omega_{z})\) are determined as follows:  
\[\omega_{x} = 2\pi\cdot\frac{m}{(2M+1)\cdot\delta_{x}}\qquad(m=-M,-M+1,...,-1, 0,1,...,M-1,M), \tag{16}\] \[\omega_{z} = 2\pi\cdot\frac{k}{(2K+1)\cdot\delta_{z}}\qquad(k=-K,-K+1,...,-1, 0,1,...,K-1,K). \tag{17}\]  
However, the frequency spectrum \(S_{n}(x,z;\omega_{x},\omega_{z})\) obtained by the discrete Fourier transform is the convolution of the true frequency spectrum of the ultrasonic RF signal and the frequency spectrum of the spatial window. Equations (16) and (17) are the case when the size of the spatial window is infinitesimally large (the main lobe of the frequency spectrum of the spatial window is infinitesimally narrow). However, a small spatial window is desired because it realizes a better spatial resolution in motion estimation. In such cases, the main lobe of the frequency spectrum of the spatial window is broad, and the mean frequency of each component of the spectrum, which is obtained as the convolution of the true spectrum and the spectrum of the spatial window, is ambiguous. In the present study, a method using the shifted cross spectrum \(\gamma_{I}(i,j;\omega_{x},\omega_{z})\) was introduced to estimate the mean frequency of each component of the spectrum. By referring to the relationships in Equations (1) and (2), the shifted cross spectra \(\gamma_{0}(i,  
Microsonic, Tokyo, Japan), which acquired ultrasonic echo signals from a sponge phantom received by individual transducer elements at a sampling frequency of 31.25 MHz, resulting in the axial sampling interval of 0.02464 mm (for both element and beamformed RF signals) by assuming the speed of sound as 1540 m/s. The element pitch of the linear array was 0.1 mm, which corresponds to the lateral sampling interval of the beamformed RF signals. As illustrated in Figure 1, the ultrasonic probe was moved by an automatic stage at constant lateral and axial velocities of 2 mm/s and -1 mm/s, respectively, which were the maximum velocities available by the automatic stage, to make the relative movement between the ultrasonic probe and the sponge phantom. The transmit-receive sequence is described in [12]. In the present study, the number of emissions of plane waves for creation of one image frame was set at 4, and each plane wave was emitted using 96 transducer elements. In each transmission, 24 focused receiving beams were created at intervals of 0.1 mm, and the aperture used to create one focused receiving beam consisted of 72 elements. In the present study, the beamformed RF signals were obtained with rectangular apodization, Hanning apodization, and apodization consisting of two Hanning functions, which realized the lateral modulation of the ultrasonic field [31]. Consequently, one image frame consisting of 24 \(\times\) 4 = 96 focused receiving beams was obtained by 4 emissions of plane waves. The frame rate achieved under such a transmit-receive condition was 1302 Hz at the pulse repetition frequency of 5208 Hz. The beamforming procedure was performed off-line on the ultrasound echo signals received by the individual elements using an in-house software. In the subsequent experimental validation, the point spread function obtained by the imaging method described above was evaluated by measuring an ultrasonic echo from a fine stainless wire of 15 \(\upmu\)m in diameter. Owing to the high frame rate, as illustrated in Figure 2, temporal averaging was performed to obtain a set of velocity estimates \((\hat{v}_{x},\hat{v}_{z})\) for reduction of the variance in the estimates.