# High-Frame-Rate 3-D Vector Flow Imaging in the Frequency Domain
## 1 Introduction  
Estimation of blood flow velocity through Doppler ultrasound (US) [1] has been for a long time limited to the axial component of the velocity vector. The lack of knowledge of the so-called Doppler angle between the US propagation direction and blood flow direction has in fact prevented the accurate estimation of the velocity magnitude. Even if several methods were proposed to overcome this limitation by estimating the two components of the velocity vector lying within the scan (x-z) plane (2-D Doppler), such methods only estimate the velocity in a single sample volume by exploiting triangulation [2; 3; 4], Doppler bandwidth [5], and transverse oscillations approaches [6; 7]. Nonetheless, the introduction of high-frame-rate (HFR) imaging methods, based on plane wave transmission [8; 9; 10], has paved the way for the extension of vector flow imaging (VFI) to multiple points (e.g., covering a two-dimensional region of interest (ROI)). Several approaches have been so far proposed based on either 2-D speckle tracking [11; 12], multiangle Doppler analysis [13; 14; 15], transverse oscillations [16; 17], or color Doppler and continuity equations [18].  
The next development step was the extension of Doppler estimation to all three components of the velocity vector. In this case as well, the first 3-D vector studies were limited to the investigation of a single sample volume and exploited multiteam or multitransducer approaches [19; 20; 21]. Extension of VFI to a full volume has been recently made possible by the introduction of bidimensional transducer arrays [22; 23; 24; 25; 26; 27]. The first attempts were based on either row-column-addressed arrays [28; 29] or full 2-D arrays [30] together with transverse oscillations, but they required long acquisition schemes, thus limiting the temporal resolution. More recently, methods exploiting HFR sequences have been developed; they have been based either on a combination of autocorrelation and a least-squares fitting approach [31], on a 3-D block matching procedure with a 3-D similarity function [32], or on particle image velocimetry [33]. However, the latter two methods required multiple heart cycles with electrocardiogram stitching to reconstruct the whole volume of interest. On the other hand, the autocorrelation-based method was characterized by a high computational load requiring more than 1 h to process one heartbeat, despite it being implemented on a powerful graphics processing unit (GPU).  
Of course, when two or three components of the velocity vector have to be estimated, the computational efficiency of the used algorithm plays a fundamental role, as its complexity can become O(N\({}^{2}\)) or O(N\({}^{3}\)), respectively, with N the number of estimation points. In [12], a 2-D HFR VFI method was proposed, which, compared with a cross-correlation-based method [11], achieved a significant computational load reduction with a negligible impact on the estimation accuracy. It was based on the transmission of plane waves and a frequency domain displacement estimation method, which computed spectral phase differences in a limited number of frequency points. Therefore, in this paper, we propose an efficient 3-D HFR VFI method, inheriting the benefits of the previous 2-D VFI method by extending it to the third dimension. Specifically, the displacement estimation along the three dimensions is here obtained through two 2-D HFR VFI estimations on perpendicular planes.  
The paper is organized as follows: Section 2 introduces the new 3-D VFI method, details the simulation and experimental setups, and defines the performance metrics. Section 3 shows and objectively describes simulation and experimental results that are thoroughly discussed, introducing possible perspectives, in Section 4. Appendix A recalls the basics of the 2-D HFR VFI method presented in [12].