# 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 \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) & \(-0.26\pm 0.65\) & \(0.04\pm 0.44\) & \(0.20\pm 0.36\) & \(0.16\pm 0.32\) & \(0.18\pm 0.30\) \\ \hline \(\pm 0.2464\) (10) & \(-0.22\pm 0.43\) & \(-0.13\pm 0.32\) & \(0.01\pm 0.26\) & \(0.00\pm 0.24\) & \(0.01\pm 0.23\) \\ \hline \(\pm 0.3696\) (15) & \(-0.23\pm 0.30\) & \(-0.15\pm 0.29\) & \(-0.04\pm 0.24\) & \(0.01\pm 0.23\) & \(0.02\pm 0.22\) \\ \hline \(\pm 0.4928\) (20) & \(-0.17\pm 0.28\) & \(-0.10\pm 0.27\) & \(0.00\pm 0.25\) & \(0.03\pm 0.24\) & \(0.03\pm 0.23\) \\ \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 \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) & \(0.03\pm 0.24\) & \(0.02\pm 0.23\) & \(0.03\pm 0.23\) & \(0.03\pm 0.23\) & \(0.03\pm 0.23\) \\ \hline \(\pm 0.2464\) (10) & \(0.00\pm 0.24\) & \(0.00\pm 0.24\) & \(0.02\pm 0.23\) & \(0.01\pm 0.23\) & \(0.01\pm 0.23\) \\ \hline \(\pm 0.3696\) (15) & \(0.00\pm 0.24\) & \(0.00\pm 0.23\) & \(0.01\pm 0.23\) & \(0.01\pm 0.23\) & \(0.01\pm 0.23\) \\ \hline \(\pm 0.4928\) (20) & \(0.01\pm 0.23\) & \(0.00\pm 0.23\) & \(0.01\pm 0.23\) & \(0.01\pm 0.23\) & \(0.00\pm 0.23\) \\ \hline \end{tabular}
\end{table} 
Table 1: Bias errors \(e_{\rm bias}\) and standard deviations \(\sigma\) of lateral and axial displacements at nine red points in Figure 4a estimated by the conventional 2D motion estimator without frame averaging. Beamformed echo signals were obtained with rectangular apodization. Actual lateral and axial velocities were 2 mm/s and -1 mm/s, 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) & \(-0.28\pm 0.57\) & \(0.03\pm 0.37\) & \(0.18\pm 0.29\) & \(0.14\pm 0.2  
Table 3 shows the bias errors \(e_{\text{bias}}\) and standard deviations \(\sigma\) of the lateral and axial velocities estimated by the proposed paired 1D motion estimators. Using the proposed paired 1D motion estimators, the bias errors \(e_{\text{bias}}\) of the estimated lateral velocities at small lateral window sizes of less than \(\pm 1.5\) mm (\(M=15\)) were improved significantly compared with those of the conventional 2D motion estimator shown in Table 2, but the standard deviations \(\sigma\) were increased. The bias errors \(e_{\text{bias}}\) of the estimated axial velocities were very small (equal to or less than 2%) at all the examined window sizes. Table 4 shows the bias errors \(e_{\text{bias}}\) and standard deviations \(\sigma\) of the lateral and axial velocities estimated by the proposed 2D motion estimator. At all the examined lateral window sizes, the bias errors \(e_{\text{bias}}\) of the estimated lateral velocities were improved dramatically compared with those of the conventional 2D motion estimator shown in Table 2. Also, the bias errors \(e_{\text{bias}}\) of the estimated axial velocities were very small (equal to or less than 2%) at all the examined window sizes.