## 1 Introduction  
Medical ultrasound is a widely used clinical diagnostic method due to its convenience, safety, and real-time nature. During the past few decades, the line-per-line acquisition has become the mainstream method for clinical ultrasound imaging [1]; however, the frame rate is generally limited. There are some applications which require kHz frame rates to record the tissue motion [2; 3] exceeding the ability of the conventional line-per-line method. In order to solve the low frame rate problem, ultrasound plane-wave imaging (PWI) has been applied to some advanced fields that need high frame rates [4; 5] and PWI usually has low imaging quality due to insufficient transmit focusing. [PERSON] et al. proposed to use several tilted-plane-wave transmissions to overcome this restriction, which is called coherent compounding [2]. Since then, coherent plane-wave compounding (CPWC) has become a promising technique in ultrafast ultrasound imaging [2; 6]. However, there is a tradeoff between the frame rate and image quality in CPWC: the higher the number of angles to obtain an image, the better the image quality but the lower the frame rate [6]. In theory, the frame per second (fps) of one-angle CPWC could reach 15,000 Hz at a depth of 5 cm, while it drops down to 1500 Hz using 11-angle CPWC. Thus, it is important to improve the image quality without increasing the number of angles. [PERSON] et al. used the delay-and-sum (DAS) algorithm for image reconstruction in each plane wave, which is now widely accepted as a standard reconstruction method for CPWC [2; 6]. DAS uses the simple addition of the channel signal data in the delay line. It does not take into account signal coherence and can only obtain limited imaging quality. In order to improve the spatial and contrast resolution, various adaptive beamformers have been proposed besides the conventional DAS method. The minimum variance beamforming and coherence-based beamforming are two important parts of the beamformers [7]. [PERSON] [8] proposed the minimum variance (MV) and in 1969 and [PERSON] [9] proposed eigenspace-based minimum variance (ESBMV) beamforming on the basis of MV. The minimum variance beamforming could achieve better resolution than DAS, but its computational amount is also large. Meanwhile, the coherence-based beamforming is an important part of the adaptive beamformers as well [7], which could increase detail resolution and contrast with less computational amount. Recently, many beamformers based on the machine learning [10] and compressed sensing [11] also made much progress. The coherence-based beamforming is still of great significance today because of its ease of calculation and improved image quality. Previous theory on the coherence based beamformer of medical ultrasound signals was adapted from the van Cittert Zernike (VCZ) theorem to describe the spatial coherence of ultrasound radio-frequency (RF) signal [12]. It works as a spatial filter to enhance or attenuate the signal from the location information. [PERSON] and [PERSON] first introduced the coherence factor beamformer in 1994 [13]. [PERSON] used this theory to develop a phase aberration correction method [14; 15]. Subsequently, a variety of techniques utilizing spatial coherence have been developed [16; 17; 18; 19]. Among these spatial coherence beamformers, the Phase Coherence Factor (PCF) beamformer were first introduced by [PERSON] based on the analysis of the phase diversity of the aperture data [18]. For phase quantization, SCF can be treated as the'simplified' version of PCF. In addition to the spatial method, the angular theory could also improve image quality. The angular coherence theory has been proposed as an extended form of the van Cittert-Zernike theorem [20]. It was first introduced by [PERSON] et al. in the context of optics [21]. In recent years, [PERSON] introduced the angular coherence theory in medical ultrasound imaging and analyzed the autocorrelation function of the CPWC signal [20].