## 2 Ultrafast Ultrasound Imaging  
The primary factor limiting the temporal resolution in ultrasound imaging is the speed of sound in the body. Ultrasound pulses can be transmitted at pulse repetition frequencies of about 10,000 Hz for superficial organs and about 5000 Hz for deep organs. An ultrasound image is typically composed of 100-250 scan lines and one transmit-receive event is required to create one scan line because a focused transmit beam is used in conventional ultrasound imaging (ultrasonic echoes are coming from only a limited region). As a result, the imaging frame rate is limited to less than 100 fps unless the number or density of scan lines is not reduced.  
The concept of ultrafast ultrasound imaging is not new. It was first developed and examined in the 1970s [1; 2; 3]. In ultrafast ultrasound imaging, unfocused transmit beams, such as plane and diverging beams, are used. As a result, ultrasonic echoes are coming from a wide region illuminated by an unfocused transmit beam. By creating focused beams in reception, a number of scan lines can be created simultaneously. Therefore, the number of emissions required to create one image frame can be reduced significantly. In an extreme case, an ultrasound image can be created by only one transmit-receive event if we can illuminate a region of interest by a single emission of an unfocused transmit beam.  
On the other hand, image quality in ultrafast ultrasound imaging, e.g., lateral spatial resolution and contrast, is inherently worse than that in conventional imaging using focused transmit beams because the directivity is created only in reception and ultrasonic echoes from a wide region produce undesirable echoes. Various attempts have been made for improvement in image quality in ultrafast ultrasound imaging. Spatial coherent compounding is a frequently used method to improve theimage quality in ultrafast ultrasound imaging [4; 5; 6]. By compounding point spread functions (PSFs) created from multiple steered beams, the compounded PSF is sharpened because only the central parts of the PSFs are coherently summed, and other parts of the PSFs are incoherently summed (canceled). The improvement of the image quality is increased by increasing the number of coherently compounded angles and consequently the imaging frame rate is degraded.  
Another approach is to improve the performance of an ultrasound beamformer. One strategy is to use the coherence among ultrasonic echo signals received by individual transducer elements [7; 8]. Such methods utilize the characteristics of received signals, e.g., echoes from a focal point are temporally aligned after delay compensation done by a delay-and-sum (DAS) beamformer, while out-of-focus echoes are not aligned. Coherence evaluation metrics, such as coherence factor (CF) and phase coherence factor (PCF), were developed and demonstrated to improve ultrasound image quality. Adaptive beamforming is an alternative strategy for improvement in the performance of an ultrasonic beamformer. The minimum variance beamformer [9] was introduced in medical ultrasound imaging in the late 2000s [10; 11]. It minimizes the power of received ultrasonic signals (undesired echoes and noise are suppressed) while keeping the all-pass characteristic with respect to the desired direction (focal point). Significant improvements in image quality can be realized by minimum variance beamforming. On the other hand, the computational complexity of the minimum variance beamformer is very high, and developments of efficient implementations of the minimum variance beamformer are still ongoing [12; 13]. In addition, various studies on improvement in the performance of the minimum variance beamformer have been conducted [14; 15].