## 2 Materials and Methods
### Proposed Compressed Sensing Process in the Wavenumber Domain  
The transducer with \(N\) elements emits a plane wave signal, which equals the number of lateral numbers. The raw data of echo signal is \(\psi_{\theta}(x,z,t)\) in the temporal domain. The echo signal is transferred to \(G_{\theta}(k_{x},z,f)\) in the Fourier domain by 2D-dimensioan FFT. The ultrasonic echo signal in the wavenumber domain can express sparsity by a transfer matrix based on directivity vector. To avoid large transfer matrix, we compress the data load by \(k_{x}\). According to compressed sensing theory, the transfer matrix can be established by the number of transducer elements and size of the transfer matrix. To ensure that the projection vector is sparse, which means there are only few elements non-zero, the number of the rows of the transfer matrix should be less than columns. The projection coefficient vector is obtained by projecting the echo signal on the transfer matrix. Then the echo signal of all transducer elements can be expressed as follows  
\[G_{\theta}(k_{x})=\mathrm{HS}(k_{x}) \tag{12}\]  
where H is the transfer matrix. \(S(k_{x})\) is the projection coefficient vector where echo signal \(G_{\theta}(k_{x})\) project on transfer matrix H. There are only few non-zero elements in the vector \(S(k_{x})=[0,0,\cdots,s_{1}(k_{x}),0,\cdots,0,s_{K}(k_{x}),0,\cdots,0]\), namely \(S(k_{x})\) is sparse. The transfer matrix H can be constructed as follows to make the echo signal in the wavenumber domain \(G_{\theta}(k_{x})\) sparse. Divide the space \([-90^{\circ},90^{\circ}]\) into \(2N\) parts on average to obtain \(\varphi_{k}=k\pi/N\) and establish directivity vector \(\delta(\varphi_{k})\) as follow  
\[\delta(\varphi_{k})=[1,e^{-i2\pi d\sin(\varphi_{k})/\lambda},e^{-i2\pi 2d\sin( \varphi_{k})/\lambda},\cdots,e^{-i2\pi(N-1)d\sin(\varphi_{k})/\lambda}]^{T} \tag{13}\]  
where \(d\) is the kerf between elements of the transducer. \(\lambda\) is wavelength of the plane wave. Directivity vector can be regarded as the sparse base to construct transfer matrix. Then the transfer matrix is  
\[\mathrm{H}=[\delta(\varphi_{1}),\delta(\varphi_{2}),\cdots,\delta(\varphi_{2 N})] \tag{14}\]  
In order to compress signal \(G_{\theta}(k_{x})\), we would like to obtain \(S(k_{x})\) by constructing measurement matrix and re-weight minimum-focal underdetermined system solver algorithm (RM-FOCUSS) [34]. Compressed sampling is not a direct measurement of \(G_{\theta}(k_{x})\), but a design of a \(M\times N(M<N)\) dimensional sampling matrix \(\Lambda=[\kappa_{1},\kappa_{2},\cdots,\kappa_{M}]\), which is uncorrelated to the transfer matrix H. The projection vector \(Z(k_{x})\) of \(G_{\theta}(k_{x})\) on \(\Lambda\) is  
\[Z(k_{x})=\Lambda G_{\theta}(k_{x})=\Lambda\mathrm{HS}(k_{x}) \tag{15}\]  
Sampling matrix \(\Lambda\) represents the compressed sampling method of the echo signal, which is composed of \(M\) sampling bases. Each sampling base is an \(N\)-dimension vector, such as \(\kappa_{i}=[\kappa_{i1},\kappa_{i2},\cdots,\kappa_{iN}]\). The \(i\)-th row means that the output of all the transducer elements is projected onto the sampling base, corresponding to a compressed sampling point. The sampling matrix \(\Lambda\) has \(M\) rows in total, which means that only \(M\) compressed sampling points are needed. \(M\) transducer elements can be selected for sampling from the \(N\) transducer elements of the original array. The matrix \(\mathrm{P}=\Lambda\mathrm{H}\) in [15] is an \(M\times 2N\) matrix, which is called the measurement matrix. The measurement matrix is used to sample the observations so that the original signal can be reconstructed. Theoretical studies [35] have shown that when the measurement matrix \(\mathrm{P}\) satisfies the restricted isometry property (RIP) condition, the projection coefficient vector \(S(k_{x})\) can be solved, and the original signal \(G_{\theta}(k_{x})\) can be reconstructed accurately by the compressed sampling vector \(Z(k_{x})\).