## 1 Introduction  
In conventional medical ultrasound imaging, numerous focused ultrasound beams are utilized to scan a region of interest (ROI) [1]. Since only a single or few image scanlines are formed in each beam transmission, approximately one-hundred insoinifications are generally required to reconstruct an entire image frame. Considering the speed of sound in body tissue is approximately 1540 m/s [2], the maximum frame rate is fundamentally limited (e.g., 40 frames per second when scanning a depth of 15 cm with 128 beam transmissions). In other words, the total number of ultrasound beams per frame must be reduced in order to increase the frame rate. For high-frame-rate imaging, various beam profiles and transmission schemes have been presented, such as broad-beam [3], multiline [4], and plane [5; 6; 7] or diverging beam [8] transmissions. The plane-wave imaging technique shown in Figure 1, which utilizes an unfocused wave to interrogate the entire ROI in a single transmission, has especially revealed the potential for very high-frame-rate imaging. In addition, by coherently compounding multiple frames of images acquired using several plane-waves with various steering angles, the spatial resolution, contrast, and signal-to-noise ratio (SNR) of plane-wave imaging can be enhanced while attaining a high enough frame rate for the applications [9]. As a result, a number of new applications have been enabled. For example, tissue elasticity mapping by monitoring a shear wave propagating through the medium using a very high frame rate [10; 11; 12], very sensitive color or power Doppler imaging using a very large number of data samples obtained within a short acquisition time [13; 14; 15; 16], stable or very fast monitoring of contrast agents [17; 18], and super-resolution imaging based on the localization of contrast bubbles [19] were all made possible by the plane-wave compounding technique. Although plane-wave compound imaging has mainly been utilized in linear array imaging, it can be applied to curved array imaging (e.g., curvilinear, endocavitary, and microconvex) for high-resolution abdominal ultrasonography [20]. However, curved array imaging generally requires a larger number of plane-waves to obtain the wider ROI compared to linear array imaging since the region inside of the compounded plane waves (i.e., the area that is fully synthesized) is narrow [6]. Moreover, in terms of transmit delay calculation for curved array imaging, a nonlinear delay for each element of the curved array (e.g., 128-elements) should be individually calculated due to the structurally curved shape (the elements are aligned along the radius of the transducer), whereas a linear delay can be simply calculated by adding up the increment in a linear array transducer. Therefore, as the number of transmissions per frame increases, more computations are required to calculate the transmit delays of the plane beams and reconstruct the images in curved array imaging. These increments in terms of computational complexity can be considered a challenge for real-time imaging. In addition, various plane-wave imaging applications, such as shear wave elastography and microvascular imaging, commonly utilize an ROI map to visualize biological function in the finite range of interest regions [5]. By changing the position and size of the ROI map along the axial and lateral directions, the transmit delays corresponding to the number of plane-wave transmissions and the degree of steering angles must be adjusted to provide uniform image quality and field of view [9]. These transmit delays can be pre-calculated and stored in the look-up table (LUT). However, since the number of transmission cases varies depending on the target application, it is advantageous to recalculate the transmit delays in real time. Moreover, for the fast image reconstruction in the real-time system, a parallel beamformer architecture based on graphics processing units (GPUs) can be adopted [21] and the operations required for calculating transmit delays should be considered as well.