# High-Frame-Rate 3-D Vector Flow Imaging in the Frequency Domain
## 4 Discussion and Conclusions  
A novel 3-D high-frame-rate vector flow imaging method has been presented. The method is an extension of the 2-D approach proposed in [12], which estimates the velocity through the 2-D \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Vermon Probe**} \\ \hline \(\theta_{\mathbf{D}}\) & \(\theta_{\mathbf{A}}\) & \(\overline{B}_{\mathbf{v}_{2}}\pm\overline{\sigma}_{\mathbf{v}_{2}}\) [\%] & \(\overline{B}_{\mathbf{v}_{2}}\pm\overline{\sigma}_{\mathbf{v}_{2}}\) [\%] & \(\overline{B}_{\mathbf{v}_{2}}\pm\overline{\sigma}_{\mathbf{v}_{2}}\) [\%] & \(\overline{B}_{\mathbf{v}_{2}}\)\(\pm\overline{\sigma}_{\mathbf{v}_{2}}\) [\%] & \(\overline{B}_{\mathbf{v}_{2}}\)\(\pm\overline{\sigma}_{\mathbf{v}_{2}}\) [\%] \\ \hline \(91^{\circ}\) & \(0^{\circ}\) & \(0.6\pm 1.3\) & \(11\pm 11.8\) & \(-2\pm 12.6\) & \(10.2\pm 9.5\) \\ \(85^{\circ}\) & \(0^{\circ}\) & \(1.1\pm 1.4\) & \(10.5\pm 13.3\) & \(-2\pm 12.5\) & \(9.8\pm 10.6\) \\ \(80^{\circ}\) & \(0^{\circ}\) & \(1.4\pm 1.5\) & \(9.8\pm 14.2\) & \(-2.7\pm 14.1\) & \(8.8\pm 11.4\) \\ \(91^{\circ}\) & \(44^{\circ}\) & \(0.6\pm 1.4\) & \(9.3\pm 13.7\) & \(10.1\pm 13.7\) & \(13.3\pm 11.6\) \\ \(85^{\circ}\) & \(45^{\circ}\) & \(1.3\pm 1.2\) & \(8.7\pm 13.2\) & \(9.6\pm 12.9\) & \(12.6\pm 10.8\) \\ \(81^{\circ}\) & \(43^{\circ}\) & \(1.9\pm 1.4\) & \(11.4\pm 14.1\) & \(10.5\pm 13.6\) & \(14.6\pm 11.4\) \\  & & Mean of \(\overline{B}_{\mathbf{v}_{2}}\pm\overline{\sigma}_{\mathbf{v}_{2}}\) [\%] & & \(11.5\pm 10.9\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Relative bias and mean standard deviation for experiments. Figure 7: Velocity profiles reconstructed by Vermon acquisitions, along the line crossing the probe axis with different probe-to-flow orientations. The black solid line is the reference velocity profile. frequency domain phase shifts corresponding to the displacement of 2-D kernels extracted from consecutive RF frames. The proposed approach involves the transmission of unsteered plane waves from a bidimensional probe and parallel beamforming in reception. Then, the volume of beamformed data is divided into 3-D kernels and, for each of them, the 3-D vector flow is estimated. Simulations were performed for two 256-element spiral probes having comparable dimensions (\(\sim\)1 cm): the first one was based on a 1024-element piezoelectric probe manufactured by Vermon; the second one was based on a CMUT prototype that is currently under development [39]. These probes were chosen since they are illustrative of the performance achievable by the proposed method at different frequencies (3.7 and 7.0 MHz). Furthermore, it was possible to connect the Vermon probe to the ULA-OP 256 research scanner to perform experimental tests. Simulations and experiments were conducted at depths around 16 mm, which are compatible with both probes and with possible peripheral vascular applications. The performance was assessed, for different beam-to-flow angles, in terms of bias and standard deviation. Even if both experiments and simulations were conducted for a maximum flow speed of 7.5 cm/s--suggested by the limited entrance length of the experimental flow phantom--the PRF was accordingly set to a low value of 500 Hz so that, in the worst case (\(\theta_{\text{D}}=80^{\circ}\) for the high-frequency CMUT probe), the produced Doppler shift along the z direction was about 50% the aliasing limit. Hence, the obtained results indicate that the possible proximity to the aliasing limit is not a crucial issue in the proposed method. As reported in Table 2 simulations have shown that the CMUT probe, in the presented cases, performed better than the Vermon one (mean bias \(\pm\) standard deviation: \(6.5\%\pm 8.0\%\) vs \(14.4\%\pm 8.7\%\)). This difference can be attributed to the characteristics of the two probes, namely, the similar equivalent apertures (\(\sim\)6 mm) and different central frequencies, which are known to give better spatial resolution for higher-frequency (CMUT) probes.