# Phase-Sensitive 2D Motion Estimators Using Frequency Spectra of Ultrasonic Echoes
## 3 Experimental Results
For a graphical representation of the results, means and standard deviations of the lateral and axial velocity estimates summarized in Tables 1-4 are plotted in Figure 6a,b, respectively, as functions of the lateral window size. In Figure 6, the velocity estimates obtained with an axial window size of \(\pm 0.2464\) mm are shown as a typical case because the velocity estimates are more dependent on the lateral window size. As can be seen in Figure 6a, the accuracy in estimation of the lateral velocity was improved significantly, particularly in cases of smaller lateral window sizes. On the other hand, the accuracy in estimation of the axial velocity was much less dependent on the window size. To obtain the results shown in Tables 1-4, rectangular apodization was used in receive beamforming to obtain the beamformed RF signals, which were analyzed by the conventional and proposed motion estimators to obtain the lateral and axial velocity estimates. Figure 7(1-b,2-b) show B-mode images of the phantom obtained with Hanning apodization and apodization consisting of two Hanning functions (lateral modulation), respectively. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{6}{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) & \(-0.23\pm 0.82\) & \(-0.16\pm 0.48\) & \(0.10\pm 0.44\) & \(0.10\pm 0.31\) & \(0.04\pm 0.26\) \\ \hline \(\pm 0.2464\) (10) & \(-0.05\pm 0.50\) & \(-0.08\pm 0.34\) & \(0.03\pm 0.25\) & \(0.05\pm 0.22\) & \(0.00\pm 0.18\) \\ \hline \(\pm 0.3696\) (15) & \(-0.03\pm 0.34\) & \(-0.06\pm 0.29\) & \(0.00\pm 0.20\) & \(0.02\pm 0.18\) & \(0.01\pm 0.16\) \\ \hline \(\pm 0.4928\) (20) & \(0.00\pm 0.30\) & \(-0.01\pm 0.22\) & \(0.03\pm 0.18\) & \(0.04\pm 0.18\) & \(0.02\pm 0.17\) \\ \hline \multicolumn{6}{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) & \(0.00\pm 0.21\) & \(-0.01\pm 0.18\) & \(-0.01\pm 0.18\) & \(-0.02\pm 0.19\) & \(-0.02\pm 0.19\) \\ \hline \(\pm 0.2464\) (10) & \(-0.02\pm 0.20\) & \(-0.02\pm 0.19\) & \(-0.01\pm 0.19\) & \(-0.02\pm 0.19\) & \(-0.02\pm 0.18\) \\ \hline \(\pm 0.3696\) (15) & \(-0.02\pm 0.19\) & \(-0.01\pm 0.19\) & \(-0.01\pm 0.19\) & \(-0.02\pm 0.19\) & \(-0.02\pm 0.19\) \\ \hline \(\pm 0.4928\) (20) & \(-0.01\pm 0.19\) & \(-0.01\pm 0.19\) & \(-0.01\pm 0.19\) & \(-0.02\pm 0.19\) & \(-0.02\pm 0.19\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Bias errors \(e_{\text{bias}}\) and standard deviations \(\sigma\) of lateral and axial displacements at nine red points in Figure 4a estimated by the proposed paired 1D motion estimators with frame averaging. Beamformed echo signals were obtained with rectangular apodization. Actual lateral and axial velocities were 2 mm/s and -1 mm/s, respectively.