# Phase-Sensitive 2D Motion Estimators Using Frequency Spectra of Ultrasonic Echoes
## 4 Discussion
Figure 8(2-a,2-b) show those of the Gaussian window with the lateral and axial sizes of (\(\pm 1.5\) mm (\(M=15\)), \(\pm 0.2464\) mm (\(K=10\))), respectively. The lateral widths at half maximum of the main lobe of the power spectra shown in Figure 8(1-b,2-b) are \(1.24\) mm\({}^{-1}\) and \(0.42\) m\({}^{-1}\), respectively. Therefore, the results obtained in the present study showed that the main lobe width of less than or equal to \(0.42\) mm\({}^{-1}\) was required to obtain accurate lateral velocity estimates by the conventional 2D motion estimator. The main lobe width was compared with the size of the point spread function of the beamformed ultrasonic data. The lateral width at half maximum of \(0.42\) m\({}^{-1}\) corresponds to about three times of the lateral width at half maximum of the point spread function shown in Figure 3 [\(1/(3\times(0.81\) mm\())=0.41\) mm\({}^{-1}\)]. Figure 8: 2D Gaussian spatial windows used for discrete Fourier transform (**a**) and their power spectra (**b**). Lateral and axial window sizes are (\(\pm 0.5\) mm (\(M=5\)), \(\pm 0.2464\) mm (\(K=10\))) (**1**) and (\(\pm 1.5\) mm (\(M=15\)), \(\pm 0.2464\) mm (\(K=10\))) (**2**). On the other hand, the accurate axial velocity estimates could be obtained even with a smaller axial window size of \(\pm\)0.1232 mm, which almost corresponds to the axial size of the point spread function. Figure 9 shows the beamformed RF signal obtained along the scan line, which pass through the peak of the point spread function shown in Figure 3. As can be seen in Figure 9, in the axial direction, there are about three cycles of the ultrasonic signal in the axial width at half maximum of 0.27 mm of the point spread function, which almost corresponds to an axial window size of \(\pm\)0.1232 mm. In the lateral direction, it can be considered that there is only one cycle of signal variation within the point spread function. Therefore, a window size of three times of the point spread function was required in the lateral direction to include three cycles of the ultrasonic signal variation. From these results, it was supposed that the conventional 2D motion estimator gives accurate velocity estimate when the spatial window includes about three cycles of the ultrasonic signal variation in both lateral and axial direction. As described above, the conventional 2D motion estimator requires a relatively large spatial window to obtain accurate lateral and axial motion estimates. On the other hand, the proposed 2D motion estimator realizes accurate estimation of the lateral and axial motion with a smaller spatial window. Such a property of the proposed 2D motion estimator is preferable for more regional measurement of the tissue motion by means of ultrasound. In this study, the basic concept of the proposed 2D motion estimator was presented, and the proposed method was implemented with a linear array probe. Also, the feasibility of the proposed method for 2D motion detection was shown by a primitive phantom experiment. In future work, an experimental validation using a more realistic phantom should be performed and, also, the proposed motion estimator should be implemented with phased and curved arrays.