# Adaptive Beamformer Combined with Phase Coherence Weighting Applied to Ultrafast Ultrasound
## 1 Introduction  
Ultrafast ultrasound imaging with parallel beamforming [1] is now frequently used for functional imaging such as the measurement of shear wave propagation [2; 3; 4], evaluation of myocardial function [5; 6; 7], and blood flow imaging [8; 9; 10; 11; 12; 13; 14]. However, the parallel beamforming degrades the lateral spatial resolution and image contrast because unfocused transmit beams are used to illuminate a wide region [2]. The lateral resolution and contrast are important factors determining image quality. Therefore, signal processing methods for improvement of the lateral resolution and contrast are demanded for the ultrafast ultrasound imaging. Spatial compounding [15; 16] and synthetic aperture imaging [17; 18] have been used as such signal processing methods. However, the spatial compounding and the synthetic aperture imaging require multiple transmissions and, thus, the frame rate is degraded. It would be beneficial if the spatial resolution were improved without compromising the temporal resolution. Recently, adaptive beamforming has been studied and used in many applications [19; 20]. In the delay-and-sum (DAS) beamforming, element echo signals are delayed based on the geometrical information of the focal point and each transducer element. The weights applied to element echosignals in the DAS beamforming are predetermined and independent of the received data. On the other hand, in the adaptive beamformer, weights are adaptively optimized using received echo signals [19]. The minimum variance (MV) beamforming [20], a kind of adaptive beamformer, determines the weights to minimize the power of the beamformer output while maintaining unit gain of a signal of interest. Many researchers have previously attempted to introduce the MV beamformer to the field of the medical ultrasound imaging [21; 22; 23], and the MV beamformer provides a significant improvement of the lateral spatial resolution. Also, [PERSON] et al. demonstrated the capability of the MV beamformer to compensate for the degradation of the lateral resolution arising from the parallel beamforming [24]. [PERSON] et al. proposed the amplitude and phase estimation (APES) beamformer [25], which eliminates the desired signal from the spatial covariance matrix by DAS beamforming, where the desired signal means the signal which is coherent with the echo from the focal point. We modified the APES beamforming for more accurate estimation of the desired signal by considering the directivity of the array transducer element to discard sub-array averaging [26; 27; 28; 29]. On the other hand, adaptive weighting methods based on the coherence factor have also been studied for improvement of the lateral spatial resolution. The coherence factor works as a metric to evaluate the focusing error in receive beamforming, and it is evaluated from echo signals received by individual transducer elements. [PERSON] and [PERSON] proposed the generalized coherence factor (GCF) [30], which is the ratio of the energy of the low spatial frequency components of element echo signals to the total energy. The direct current (DC) component and the high-frequency components were regarded as the coherent and incoherent signals, respectively and, hence, the GCF represents the degree of the focusing error. [PERSON] et al. proposed the phase coherence factor (PCF) and the sign coherence factor (SCF) [31], which are evaluated from the phases of the delay compensated echo signals in the DAS beamforming. The PCF is determined by the phase variance among received signals, and the SCF is determined by changes in polarities of the received signals. For further improvement of the lateral resolution and contrast, in the present study, we examined two strategies to combine the adaptive beamforming and the coherence-based imaging. In one of the two examined strategies, the PCF is estimated using element echo signals before application of the weight vector determined by the adaptive beamformer.