# Phase-Sensitive 2D Motion Estimators Using Frequency Spectra of Ultrasonic Echoes
## 3 Experimental Results
Furthermore, the proposed method was evaluated with a different motion velocity of the automatic stage, i.e., lateral and axial velocities of 1 mm/s and 0.5 mm/s, respectively. Tables 9 and 10 show the lateral and axial velocities estimated at the same points shown in Figure 4a obtained by the conventional and proposed 2D motion estimators, respectively. As in the results with higher lateral and axial velocities shown in Tables 2 and 4, respectively, bias errors \(e_{\text{bias}}\) in the lateral velocity estimates were significantly reduced by the proposed method at a small lateral window size of \(\pm\)0.5 mm. Figure 7: B-mode images of phantom obtained with **(1)** Hanning apodization and **(2)** lateral modulation [31]. **(a)** Receive apodization. (b) B-mode image. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{6}{c}{**Bias Errors and Standard Deviation of Estimated Lateral Velocities (mm/s)**} \\ \hline \hline \multirow{2}{*}{axial window size (mm) (\(K\))} & \multicolumn{6}{c}{lateral window size (mm) (\(M\))} \\ \cline{2-7}  & \(\pm 0.5\) (5) & \(\pm 1.0\) (10) & \(\pm 1.5\) (15) & \(\pm 2.0\) (20) & \(\pm 2.5\) (25) \\ \hline \(\pm 0.1232\) (5) & \(-0.06\pm 0.41\) & \(0.09\pm 0.33\) & \(0.01\pm 0.25\) & \(0.01\pm 0.24\) & \(0.04\pm 0.24\) \\ \hline \(\pm 0.2464\) (10) & \(-0.09\pm 0.28\) & \(-0.02\pm 0.23\) & \(0.01\pm 0.21\) & \(0.01\pm 0.19\) & \(0.03\pm 0.18\) \\ \hline \(\pm 0.3696\) (15) & \(0.04\pm 0.22\) & \(0.02\pm 0.19\) & \(0.01\pm 0.18\) & \(0.01\pm 0.17\) & \(0.02\pm 0.17\) \\ \hline \(\pm 0.4928\) (20) & \(0.04\pm 0.21\) & \(0.03\pm 0.18\) & \(0.02\pm 0.17\) & \(0.01\pm 0.17\) & \(0.02\pm 0.17\) \\ \hline \multicolumn{6}{c}{**Bias Errors and Standard Deviation of Estimated Axial Velocities (mm/s)**} \\ \hline \hline \multicolumn{6}{c}{lateral window size (mm) (\(M\))} \\ \hline \hline \multirow{2}{*}{axial window size (mm) (\(K\))} & \multicolumn{6}{c}{lateral window size (mm) (\(M\))} \\ \cline{2-7}  & \(\pm 0.5\) (5) & \(\pm 1.0\) (10) & \(\pm 1.5\) (15) & \(\pm 2.0\) (20) & \(\pm 2.5\) (25) \\ \hline \(\pm 0.1232\) (5) & \(0.00\pm 0.19\) & \(0.00\pm 0.19\) & \(-  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \multicolumn{6}{c}{**Bias Errors and Standard Deviation of Estimated Lateral Velocities (mm/s)**} \\ \hline \hline \multirow{2}{*}{axial window size (mm) (K)} & \multicolumn{6}{c}{lateral window size (mm) (\(M\))} \\ \cline{2-7}  & \(\pm\)0.5 (5) & \(\pm\)1.0 (10) & \(\pm\)1.5 (15) & \(\pm\)2.0 (20) & \(\pm\)2.5 (25) \\ \hline \(\pm\)0.1232 (5) & –0.76 \(\pm\) 0.50 & –0.30 \(\pm\) 0.80 & –0.10 \(\pm\) 0.98 & –0.27 \(\pm\) 0.63 & –0.33 \(\pm\) 0.61 \\ \hline \(\pm\)0.2464 (10) & –0.77 \(\pm\) 0.42 & –0.36 \(\pm\) 0.48 & –0.25 \(\pm\) 0.47 & –0.20 \(\pm\) 0.38 & –0.16 \(\pm\) 0.32 \\ \hline \(\pm\)0.3696 (15) & –0.65 \(\pm\) 0.43 & –0.20 \(\pm\) 0.38 & –0.15 \(\pm\) 0.39 & –0.11 \(\pm\) 0.38 & –0.08 \(\pm\) 0.31 \\ \hline \(\pm\)0.4928 (20) & –0.67 \(\pm\) 0.39 & –0.19 \(\pm\) 0.32 & –0.06 \(\pm\) 0.32 & –0.05 \(\pm\) 0.30 & –0.06 \(\pm\) 0.27 \\ \hline \multicolumn{6}{c}{**Bias Errors and Standard Deviation of Estimated Axial Velocities (mm/s)**} \\ \hline \hline \multirow{2}{*}{axial window size (mm) (\(K\))} & \multicolumn{6}{c}{lateral window size (mm) (\(M\))} \\ \cline{2-7}  & \(\pm\)0.5 (5) & \(\pm\)1.0 (10) & \(\pm\)1.5 (15) & \(\pm\)2.0 (20) & \(\pm\)2.5 (25) \\ \hline \(\pm\)0.1232 (5) & 0.02 \(\pm\) 0.18 & 0.03 \(\pm\)