# A Spatial Compounding Method for Non-Delayed Sequential Beamforming
## 1 Introduction  
Synthetic aperture focusing technique (SAFT) was first introduced for the single-element transducer. A 'virtual source' concept is used by treating the focus point as a point source emitting spherical waves towards and away from the transducer [1, 2]. Improved resolution is obtained for areas away from the focus. For linear array, similar algorithms have been introduced [3, 4, 5, 6, 7, 8].  
Although the image quality is significantly improved, the computational cost of synthetic aperture beamformer is heavy. The work of [PERSON] [6] showed that the size of the beamformed image is 30 mm \(\times\) 30 mm. The computation time for a traditional synthetic aperture beamformer is 59 min, and the advanced synthetic aperture beamformer takes 145 min, whereas dynamic focusing takes 42 s with an O(\(n_{t}n_{t}n_{s}\)) computation complexity (\(n_{s}\) denotes the number of transducer element and \(n_{t}\) denotes the number of time samples). The requirement of real-time imaging demands the beamformer to be computationally efficient while retaining the image quality.  
A dual-stage beamformer combining an advanced beamformer and traditional delay-and-sum (DAS) beamformer has been proposed to enhance the resolution and contrast [9, 10]. A dual-stage beamformer (or termed'sequential beamforming') has also been addressed in the context of synthetic aperture imaging [11, 12, 13]. A sequential beamforming algorithm termed non-delayed sequential beamforming (NDSB) is developed to simplify the front-end electronics design [13]. In the first stage, the received signals are directly summed without delay. In the second stage, the high-resolution image is reconstructed based on the two-way travel path. The authors proved this new approach can achieve a comparable imaging quality with a simpler receive electronic front-end than dynamic receive focusing (DRF). Due to the lack of focusing in the first beamformer, NDSB requires a large transmit opening angle to combine more data to generate a high-resolution image in the secondstage. This means a large number of active elements. Even so, the contrast of NDSB is still inferior to dynamic receive focusing.  
Spatial compounding can enhance image contrast by combining limited-correlating images of the same region-of-interest (ROI). To generate the limited-correlating images, we can mechanically translate the transducer, or electrically activate different transmit elements, or apply different transmit apodization [14; 15]. To date, spatial compounding methods have not been applied to sequential beamforming in synthetic aperture imaging.  
Here, a new spatial compounding method is proposed for NDSB. Spatiotemporal analysis of the pressure field is performed. By modeling the transmit sub-aperture as a continuous curved aperture, the pressure field can be characterized by three excitation signals with different transmit origins and different time delays. Sub-images can be reconstructed based on the propagation path of the corresponding excitation signals. Because these excitation signals are spatially separated, the reconstructed sub-images carry different spatial frequencies and thus decorrelate from each other. Spatial compounding can be done via summing these sub-images. The proposed compounding method improves contrast ratio and contrast-to-noise ratio in-vitro, and in-vivo.