# High-Frame-Rate 3-D Vector Flow Imaging in the Frequency Domain
## 4 Discussion and Conclusions
In addition, the CMUT probe results are consistent with those (\(10\%\pm 5\%\)) obtained for the 2-D method with a 6 MHz linear array probe [12]. This is particularly remarkable considering the noticeable differences between the arrays used in the two cases, which involve elements of dimensions passing from \(0.2\times 6\) mm for the linear array to about \(0.3\times 0.3\) mm for the 2-D arrays. The Vermon experimental results confirmed the simulation results, with a slightly better bias but a slightly higher standard deviation, which was probably due to the nonideality of the measurement setup. In particular, the peristaltic pump introduced flow fluctuations that could not be sufficiently attenuated with the flow stabilization system. Even if a bigger pulsation dampener or a higher hydraulic resistance might have been used to limit the flow fluctuations, we believe that the nonideality of the experimental setting had a limited impact on the final results and did not affect their value. The computational cost of the proposed method can be assessed by referring to the analysis carried out for the 2-D method [12]. Indeed, as the 3-D VFI is a two-step extension of the 2-D method, the overall computational cost remains of the order O(N). This is much lower than the 3-D cross-correlation-based algorithms, natively O(N\({}^{3}\)), which may be decreased through appropriate optimizations to reach an order of complexity of at least O(N\({}^{2}\)). Even if the proposed method was tested offline (i.e., by postprocessing the acquired raw data), it is part of the \"high-frame-rate\" family, since one full (Doppler) frame can be obtained after each TX event. Furthermore, the algorithm for 3-D VFI is highly parallelizable and is suited for implementation on GPUs, as already shown for 2-D VFI in [40]. The main limitations of parallel computing are linked to data transfer rates and the number of global memory accesses, which both need the proper and careful design of memory coalescing. Considering also the increasing computational power of GPU-based devices, in the near future, the implementation of 3-D VFI should be feasible in real time, targeting input frame rates of at least 7 kHz (i.e., sufficient to estimate flow speeds up to 100 cm/s). One of the main limitations of this work was using unsteered plane waves, which limited the extension of the investigated region due to the small size of the probe aperture. For example, as seen in the right panel of Figure 6, the paraboloid representative of the velocity module was partially cut off, since only 3.6 mm (about one-half of the pipe diameter) could be investigated along both the x and y axes. A possible alternative to extend the region of interest could be the transmission of diverging waves, although it should be taken into account that they involve a worse signal-to-noise ratio (SNR) [41; 42]. Finally, it has to be highlighted that, in this study, the novel method was implemented by using sparse spiral arrays rather than full 2-D arrays [24], which allowed us to perform the first experimental tests. However, it unavoidably involved a reduced SNR and a lower sensitivity to blood flow. This could be partially compensated by using coded imaging techniques [43] or in-probe preamplifiers. Likewise, doubling the number of active elements by exploiting two synchronized ULA-OP 256 systems [34; 35] could improve the system sensitivity and, at the same time, counteract the unavoidable grating lobes in the transmitted ultrasound beams associated with spiral arrays [42]. By taking these aspects into considerations, the results of this paper appear extremely encouraging and confirm that sparse arrays may be a relatively cheap solution for the implementation of 3-D imaging on systems based on a limited number of channels. Conceptualization, S.R., A.R., and P.T. ; Formal Analysis S.R. and A.R. ; Investigation S.R., A.R., and F.F. ; Methodology, S.R. ; Validation, S.R. ; Visualization, S.R. ; Writing--Original Draft, S.R. ; Writing--Review and Editing, A.R. and P.T. ; Supervision, P.T.