# Phase-Sensitive 2D Motion Estimators Using Frequency Spectra of Ultrasonic Echoes
## 3 Experimental Results
\begin{table}
\begin{tabular}{c c c c c c} \hline \multicolumn{5}{c}{**Bias Errors and Standard Deviation of Estimated Lateral Velocities (mm/s)**} \\ \hline \hline \multirow{2}{*}{axial window size (mm) (\(K\))} & \multicolumn{5}{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) & \(-0.01\pm 0.46\) & \(0.04\pm 0.29\) & \(0.06\pm 0.22\) & \(0.04\pm 0.19\) & \(0.04\pm 0.19\) \\ \hline \(\pm 0.2464\) (10) & \(0.00\pm 0.28\) & \(-0.02\pm 0.21\) & \(0.01\pm 0.18\) & \(0.02\pm 0.17\) & \(0.02\pm 0.17\) \\ \hline \(\pm 0.3696\) (15) & \(-0.01\pm 0.24\) & \(-0.01\pm 0.20\) & \(0.01\pm 0.17\) & \(0.01\pm 0.16\) & \(0.01\pm 0.16\) \\ \hline \(\pm 0.4928\) (20) & \(0.04\pm 0.22\) & \(0.02\pm 0.18\) & \(0.03\pm 0.17\) & \(0.03\pm 0.16\) & \(0.03\pm 0.16\) \\ \hline \multicolumn{5}{c}{**Bias Errors and Standard Deviation of Estimated Axial Velocities (mm/s)**} \\ \hline \hline \multirow{2}{*}{axial window size (mm) (\(K\))} & \multicolumn{5}{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) & \(0.00\pm 0.19\) & \(-0.01\pm 0.18\) & \(-0.01\pm 0.18\) & \(-0.01\pm 0.19\) & \(-0.02\pm 0.19\) \\ \hline \(\pm 0.2464\) (10) & \(-0.02\pm 0.20\) & \(-0.02\pm 0.19\) & \(-0.02\pm 0.19\) & \(-0.02\pm 0.19\) & \(-0.02\pm 0.19\) \\ \hline \(\pm 0.3696\) (15) & \(-0.02\pm 0.19\) & \(-0.02\pm 0.19\) & \(-0.02\pm 0.19\) & \(-0.02\pm 0.19\) & \(-0.02\pm 0.19\) \\ \hline \(\pm 0.4928\) (20) & \(-0.01\pm 0.19\) & \(-0.02\pm 0.19\) & \(-0.02\pm 0.19\) & \(-0.02\pm 0.19\) & \(-0.02\pm 0.19\) \\ \hline \end{tabular}
\end{table} 
Table 4: Bias errors \(\epsilon_{\text{bias}}\) and standard deviations \(\sigma\) of lateral and axial displacements at nine red points in Figure 4a estimated by the proposed 2D motion estimator 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. Figure 6: Lateral (**a**) and axial (**b**) velocity estimates obtained by respective methods. Table 5 shows the errors \(e_{\text{bias}}\) and standard deviations \(\sigma\) of the lateral and axial velocities estimated by analyzing the RF signals beamformed with Hanning apodization using the proposed 2D motion estimator. Although the bias errors \(e_{\text{bias}}\) and standard deviations \(\sigma\) of the estimated axial velocities are very similar to those obtained with rectangular apodization shown in Table 4, the bias errors \(e_{\text{bias}}\) and standard deviations \(\sigma\) of the lateral velocity estimates are degraded. The lateral velocity estimates obtained with lateral modulation (Table 6) are much better than those obtained with Hanning apodization (Table 5), but the lateral velocity estimates obtained with rectangular apodization (Table 4) are significantly better than those obtained with lateral modulation when the lateral window size is smaller than \(\pm\)1.5 mm (\(M=15\)). In ultrasonic imaging, the point spread function varies with depth, i.e., the spatial resolution in a deep region is worse than that in a shallow region. To examine the effect of the change in the point spread function on the accuracies of the estimators, errors in the displacements estimated at nine red points shown in Figure 4b were also evaluated. Tables 7 and 8 show the bias errors \(e_{\text{bias}}\) and standard deviations \(\sigma\) of the lateral and axial velocity estimates obtained by the conventional and proposed 2D motion estimators, respectively, with rectangular apodization. The lateral velocity estimates obtained by the conventional 2D motion estimator were much more affected by degradation of the spatial resolution than those obtained by the proposed 2D motion estimator. For both methods, the axial velocity estimates were less dependent on depth.