## 1 Introduction  
Cardiovascular diseases account for 31% of global deaths, and blood velocity estimation with ultrasound has become a commonly used tool for diagnosing them [1]. To increase the chances of diagnosing critical diseases at an early stage, a better understanding of the blood flow dynamics is needed. Currently, the velocity estimation in most commercial scanners is limited to 1D or 2D. However, the vascular flow propagates in all three dimensions with large temporal variations, and 3D vector flow imaging (VFI) at a very high frame rate is necessary for visualizing the complex velocity field in time and space. Several methods have been proposed for estimating the 2D vector flow: speckle tracking [2], transverse oscillation (TO) [3], spatial quadrature [4], multibeam Doppler [5], synthetic aperture flow [6], plane waves [7], and Doppler vortography [8]. 2D vector flow gives a realistic estimate of the flow, but does not provide information about the out-of-plane velocity component. The transition from 2D to 3D requires a 2D distribution of transducer elements to steer the ultrasound beam in 3D. The first to make 3D vector flow imaging was [PERSON] in 1978 [9]. A crossed-beam method was used to obtain calibrated 3D Doppler velocimetry information. The system used continuous-wave transmit/receive, sacrificing depth information and had to be manually adjusted for a specific area of interest. [PERSON] [10] implemented a triple-beam lens transducer, where three piezoelectric elements mounted on the surface of a lens produced three identical beams with the same focus depth but at different azimuthal and elevation positions. These three beams created a equilateral triangle, and the velocity component estimation was performed by tracking the receive signal from scattering and correlating it with the signal received from one of the two other beams. [PERSON] et al. [11] presented a 2D matrix array of transducer elements for real-time 3D volumetric imaging. [PERSON] et al., combined a \(32\times 32\) matrix array with the TO method in simulations [12] and measurements [13]. The method requires five beamformed IQ data samples to estimate the three velocity components simultaneously and from the same data set. [PERSON] et al. used the same array to estimate peak velocities and flow rates in-vivo in the carotid artery [14] and validated the measured velocities against magnetic resonance imaging [15]. [PERSON] et al. [16] showed cardiac volumetric 3D VFI using a commercial matrix array requiring electrocardiogram (ECG) gating. [PERSON] et al. [17] presented volumetric tensor velocity imaging (TVI) using a 1024 element fully addressed transducer array and computationally heavy synthetic aperture (SA) flow estimation. In TVI, the full 3D velocity vector is estimated throughout the imaged volume, providing complete flow information. An example of a tensor velocity image is shown in Figure 1.  
One major drawback of the matrix probe is that the total number of interconnections in an \(N\times N\) element transducer scales with \(N^{2}\). This causes issues with both fabrication and with data rates that easily exceed 100 GB/s. [PERSON] and [PERSON] [18; 19] proposed to address the rows and the columns of the matrix array rather than the individual elements to have a linear scaling of the interconnect. [PERSON] et al. [20] and [PERSON] et al. [21] showed how to obtain good images using such row-column (RC) addressed arrays at interconnect complexities and data rates similar to those found in current 2D systems. [PERSON] et al., demonstrated 3D VFI in a plane [22] and in a volume [23] for stationary, laminar flow, but with a very low aliasing limit of a few cm/s in the axial direction. [PERSON] et al. [24] demonstrated volumetric 4D (3D+time) VFI without this aliasing limit by using a computationally very expensive synthetic aperture (SA) imaging sequence, making real-time implementation extremely difficult at best. Also, [PERSON] et al. presented volumetric flow in 4D with RC addressed array by using SA.