# Methods for Grating Lobe Suppression in Ultrasound Plane Wave Imaging
## 1 Introduction  
Plane wave imaging (PWI) has drawn a large amount of attention from researchers in the field of medical ultrasound imaging [1; 2; 3]. First, PWI can provide ultra-fast ultrasound imaging that is essential for a growing number of applications, such as the estimation of shear elasticity [1; 4; 5] and vector Doppler [6; 7] as well as high-frame-rate B-mode imaging [8]. In PWI, plane waves (PWs) with different travelling angles are successively transmitted instead of traditionally focused ultrasound waves; after each firing, the returned ultrasound waves are received at all array elements. Synthetic transmit (Tx) focusing at each imaging point is achieved by compounding PWs with proper delays, while receive (Rx) focusing is performed in the conventional manner. As a result, ultrasound beams are focused at all imaging points for transmission and reception. Theoretically, the Tx beam pattern of PWI maintains the same main lobe width at all depths; the width is determined by the range of compounded PW angles.  
When using a finite number of PWs with uniformly distributed steering angles, the synthetically focused beam has not only the main lobe, but also side lobes and grating lobes (GLs), which create artifacts in ultrasound image and deteriorate the image quality [2]. A large number of compounded PWs allow for the mitigation of the side lobe and GLs in the synthetic beam pattern and provide better image contrast, as illustrated in Figure 1. However, the frame rate of PWI decreases as the number of PWs increases. To reduce the side lobe without compromising the frame rate, various adaptive beamforming methods have been proposed. [PERSON] et al. proposed a minimum variance beamforming method for PWI [9]. In this method, the optimized weighting factors are applied when compounding the low-resolution images of different steered PWs. The joint Tx and Rx adaptive beamformer has also been proposed to apply the data-dependent weighting factors to both the receiving array domain and PW angle domain (i.e., frame domain) [10]. In addition, as the side lobesfrom different angles are uncorrelated, some beamformers have been suggested to reject less coherent signals [11; 12; 13]. A global effective distance-based side lobe suppressing method has also been proposed to achieve high quality images of PWI with a small number of PWs [14].  
Only a few studies, however, have been reported that consider GLs in PWI. Though PW angles with a constant angle interval are employed in most studies, they introduce uniformly spaced grating lobes (GLs), the interval of which is governed by the angle interval, as shown in Figure 1[2]. If the angle interval is sufficiently small to locate GLs far from the main lobe, the GLs might have little effect on the image quality. However, to preserve the ultra-fast frame rate without compromising the resolution (i.e., to use a small number of PWs for the given PW angle range), the angle interval should be large, which introduces GLs close to the main lobe and deteriorates the image quality.  
Here, we investigate the conditions of GL occurrence by analyzing the continuous wave (CW) synthetic transmit beam pattern. Based on the conditions of GL occurrence, the threshold of the angle interval for the elimination of GLs is presented with the use of uniformly distributed PW angles. In addition, we propose a method for GL level reduction using non-uniformly distributed PW angles, which consist of subsets of uniformly distributed angles, each of which has different angle intervals. To verify and evaluate the methods, simulation experiments are conducted using synthetic Tx beam patterns and round-trip point spread functions (PSFs).