# Phase-Sensitive 2D Motion Estimators Using Frequency Spectra of Ultrasonic Echoes
## 4 Discussion  
In the present study, two phase-sensitive motion estimators, namely, the proposed paired 1D motion estimators and the proposed 2D motion estimator, were developed and examined by the phantom experiment. Both proposed methods significantly reduced the bias errors in the lateral velocity estimates compared with the conventional 2D motion estimator particularly when the lateral window size was small. However, the performance of the proposed paired 1D motion estimators was  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{5}{c}{**Bias Errors and Standard Deviation of Estimated Lateral Velocities (mm/s)**} \\ \hline \multirow{2}{*}{axial window size (mm) (\(K\))} & \multicolumn{4}{c}{lateral window size (mm) (\(M\))} \\ \cline{2-6}  & \(\pm\)0.5 (5) & \(\pm\)1.0 (10) & \(\pm\)1.5 (15) & \(\pm\)2.0 (20) & \(\pm\)2.5 (25) \\ \hline \(\pm\)0.1232 (5) & 0.07 \(\pm\) 0.29 & 0.11 \(\pm\) 0.24 & 0.08 \(\pm\) 0.21 & 0.08 \(\pm\) 0.16 & 0.06 \(\pm\) 0.15 \\ \hline \(\pm\)0.2464 (10) & 0.02 \(\pm\) 0.23 & 0.08 \(\pm\) 0.19 & 0.06 \(\pm\) 0.17 & 0.06 \(\pm\) 0.15 & 0.06 \(\pm\) 0.15 \\ \hline \(\pm\)0.3696 (15) & -0.01 \(\pm\) 0.21 & 0.04 \(\pm\) 0.16 & 0.04 \(\pm\) 0.15 & 0.04 \(\pm\) 0.15 & 0.05 \(\pm\) 0.15 \\ \hline \(\pm\)0.4928 (20) & -0.02 \(\pm\) 0.19 & 0.01 \(\pm\) 0.15 & 0.02 \(\pm\) 0.14 & 0.02 \(\pm\) 0.13 & 0.03 \(\pm\) 0.14 \\ \hline \hline \multicolumn{5}{c}{**Bias Errors and Standard Deviation of Estimated Axial Velocities (mm/s)**} \\ \hline \multirow{2}{*}{axial window size (mm) (\(K\))} & \multicolumn{4}{c}{lateral window size (mm) (\(M\))} \\ \cline{2-6}  & \(\pm\)0.5 (5) & \(\pm\)1.0 (10) & \(\pm\)1.5 (15) & \(\pm\)2.0 (20) & \(\pm\)2.5 (25) \\ \hline \(\pm\)0.1232 (5) & 0.01 \(\pm\) 0.11 & 0.01 \(\pm\) 0.10  
worse than the proposed 2D motion estimator. The reason for this can be considered as follows: To estimate the displacement in the direction of interest (lateral or axial direction), [PERSON] et al. remove the phase shift of the ultrasonic echo caused by the displacement in the other direction from the total phase shift using the narrow band analytical signal obtained by the Hilbert transform [35]. In the proposed paired 1D motion estimators, similar procedures were done by Equation (7) by assuming that the absolute values of the mean frequencies of the components \(S_{n}(x,z;\omega_{x},\omega_{z})\) and \(S_{n}(x,z;\omega_{x},-\omega_{z})\) were same, i.e., \(\tilde{\omega}_{z}=-\tilde{\omega}_{z}\). However, the absolute values of the mean frequencies of the components \(S_{n}(x,z;\omega_{x},\omega_{z})\) and \(S_{n}(x,z;\omega_{x},-\omega_{z})\) were not always same due to the ambiguity in the mean frequency of each component of the spectrum, particularly when the window size was small, as described in Section 2.3. As a result, the velocity estimate in the direction of interest was degraded because the phase shift caused by the displacement in the other direction was not removed accurately from the total phase shift. The proposed 2D motion estimator realized the best performance among the three methods examined in the present study because it did not assume that the absolute values of the mean frequencies of the components \(S_{n}(x,z;\omega_{x},\omega_{z})\) and \(S_{n}(x,z;\omega_{x},-\omega_{z})\) were same and estimates the mean frequency of each component of spectrum. As shown by the results in Table 2, the conventional 2D motion estimator requires a window size, which is larger than or equal to \(\pm 1.5\) mm (\(M=15\)) in the lateral direction and \(\pm 0.2464\) mm (\(K=10\)) in the axial direction, to reduce the bias error in the lateral velocity estimates. The Gaussian spatial windows used in the discrete Fourier transform and their frequency spectra are shown in Figure 8. Figure 8(1-a) shows the Gaussian window with the lateral and axial sizes of (\(\pm 0.5\) mm (\(M=5\)), \(\pm 0.2464\) mm (\(K=10\))), respectively, and Figure 8(1-b) shows its power spectrum.