## 2 Materials and Methods
### Proposed Compressed Sensing Process in the Wavenumber Domain
This property ensures that the original space has a one-to-one mapping relationship to the sparse space, which requires that the sampling matrix randomly selected from the observation matrix must be non-singular. The measurement matrix can measure the signal to obtain the measurement vector, and then use the reconstruction algorithm to reconstruct the full signal from the measured value. When designing the measurement matrix, it is required that the measured value will not affect the information of the original signal during the sparse expression of the signal, so as to ensure that the signal can be accurately reconstructed. The theory proves that the Gaussian random sampling matrix and any fixed transformation matrix can make \(\mathrm{P}\) satisfy the RIP condition with a high probability [36], and the Gaussian random matrix can be easily obtained, so this paper adopts the Gaussian random measurement matrix. The process to design a sampling matrix \(\Lambda\) is as Figure 2[33]:Define \(Y(k_{x})=[y_{1}(k_{x}),y_{2}(k_{x}),\cdots,y_{M}(k_{x})]\) as the output of the compressed sampling array elements. The compressed sampling vector is  
\[Z(k_{x})=\Gamma Y(k_{x})=\text{PS}(k_{x}) \tag{16}\]  
After obtaining the compressed sample vector \(Z(k_{x})\), the re-weight minimum-focal underdetermined system solver algorithm (RM-FOCUSS) [34] is used to estimate the projection coefficient vector \(S(k_{x})\). Then the original signal \(G_{\theta}(k_{x})\) can be reconstructed by [12]. For \(L\) plane wave emitted from transducer, the cost function \(J\) is established as follow in order to obtain the sparse signal \(S(t)\)  
\[J^{(p)}(S)=\sum_{i=1}^{2N}\Bigl{(}\sum_{l=1}^{L}\bigl{|}s_{i,l}\bigr{|}^{2} \Bigr{)}^{p/2} \tag{17}\]  
when \(p\) is closer to 0, the \(S(k_{x})\) expressed by \(J^{(p)}(S)\) is more sparse. The Lagrange operator is defined as  
\[L(S,\Omega)=J^{(p)}(S)+\sum_{l=1}^{L}\omega_{l}{}^{T}(\text{PS}[:,l]-Z[:,l]) \tag{18}\]  
where \(\omega_{l}\) is Lagrange operator vector, \(l=1,2,\cdots L\). \(S(t)\) can be obtained by minimum Lagrange operator \(L(S,\Omega)\). The solution of RM-FOCUSS algorithm was proposed in [34] as follow  
\[\left. \begin{array}{l}W_{k+1}=diag\Bigl{[}c_{k}[i]^{1-p/2}\Bigr{]}\\ c_{k+1}[i]=\|s_{k}[i,:]\|=\Bigl{[}\sum_{l=1}^{L}\Bigl{(}s_{k}[i,l]^{2}\Bigr{)} \Bigr{]}^{1/2},p\in[0,2]\\ S_{k+1}=W_{k+1}(PW_{k+1})^{H}\Bigl{(}(PW_{k+1})(PW_{k+1})^{H}+\sigma I\Bigr{)} ^{-1}Z\end{array} \right\} \tag{19}\]  
where \(p\) is related to the sparsity of echo signal. \(\sigma\) is a parameter reflecting the noise. We could eliminate part of the noise by adjusting \(\sigma\). After the RM-FOCUSS algorithm, we obtain the sparse signal \(S(k_{x})\). This sparse signal is the compressed solution after compressed sensing because of lots of zero elements. Finally, apply [12] to obtain the full-array signal. This signal is no longer the original signal but with some amelioration and modification, whether all the information can be recovered and whether the noise can be eliminated depending on the performance of this compressed sensing process. The beamforming based on compressed sensing can become a method of sparsity by reducing the number of transducer elements without reducing the image performance. It uses the sparseness of the ultrasonic signal to randomly extract part of the signals from the transducer, restores the original signal with a restoration algorithm, and then performs beamforming. In beamforming based on the compressed sensing, the transfer matrix, the sampling matrix and the sparsity solution algorithm all influence whether the reconstructed signal can retain all the image information. Figure 2: The process of designing a sampling matrix.