## 2 Materials and Methods
### Wavenumber Domain Compressive Beamforming  
To improve the compressed rate without sacrificing image quality, we propose the framework to combine compressed sensing with beamforming in the frequency-wavenumber domain. According to the characteristic of ultrasonic signal, the energy of the echo signal concentrates on low frequency. Assume that \(\psi_{\theta}(x,z=j,t)\) represents the \(j\)-th line of the RF data of echo signal. To transform the echo signal to the frequency-wavenumber domain, the spectrum can be expressed as follows  
\[G_{\theta}(x,z=j,f)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\psi_{ \theta}(x,z=j,t)e^{-i2\pi ft}e^{-i2\pi k_{x}x}dtdx \tag{20}\]  
After transforming the echo signal into the frequency-wavenumber domain, the spectrum of signal parallel to the transducer direction become symmetric because of the characteristic of FFT. We choose the 32 nd sampling line and the 10 th line of the echo signal parallel to the transducer as the example, and the spectrum is shown in Figure 3. We can see that the information of the signal is repeated twice, when the length of the signal in the wavenumber domain is the same as that in the temporal domain. Thus, we can select the first half of the data in the sampling direction for beamforming, which make the amount of data be reduced to 50%. Moreover, the signal in the frequency-wavenumber domain becomes more centralized with a rapid decay. With these two features of the signal in the wavenumber domain, compressing echo signal in the wavenumber domain will lose less information than that in the temporal domain. Thus, for the same compressing rate, beamforming with compressed sensing in the wavenumber domain will obtain better image quality.  
In order to improve the image quality further, the multi-angle compound imaging method is also suitable for this framework. In this way, we introduce the general case of the algorithm, in which the plane waves are emitted at different angles. The process of this frequency-wavenumber compressive beamforming method can be summarized as follows:  
1. Apply 2-dimensional FFT to the RF raw data \(\psi_{\theta}(x,z,t)\) to transform the echo signal into the frequency-wavenumber domain as \(G_{\theta}(k_{x},z,f)\).
2. Select the first half of the data in the sampling direction for beamforming.
3. For plane waves emitted at different angles, the emission delay caused by angles should be removed by [1] at first.
4. Compress the echo signal in the Fourier domain by the sampling matrix \(\Lambda\) and obtain the sparse solution \(\mathrm{S}(k_{x})\) by RM-FOCUSS algorithm [19]. The ameliorative signal can be obtained by [12].
5. The [PERSON]'s migration described by the procedure from [8; 9; 10; 11] is applied with the ERM velocity \(\widetilde{c}\) as defined in [5].  
Figure 3: (**a**) Spectrum of the 32 nd sampling line after FFT, (**b**) spectrum of the 10 th line of echo signal parallel to transducer in the wavenumber domain after FFT.  
Restore the data in sampling direction before inverse FFT is applied to the data after migration to transform the echo signal back to the temporal domain for imaging.  
This new framework is proposed to reduce data load and memory by applying the compressive sensing on the lateral wavenumber, as well as to guarantee the quality and stability of the imaging signal after compressed sensing. The pseudo-code for tilted plane wave is shown in Figure 4. The compression rate in this frame work is discussed in Section 3 by applying this algorithm to both simulated data and in vivo experiments. Part of the information of the original echo signal will usually be lost after compressing, so the sampling matrix and the reconstruction algorithm have higher requirements to reconstruct high quality image. Therefore, the algorithm and the reconstruction algorithm can be further explored in these two aspects to find a better compressing method.