## 2 Materials and Methods
### Compressive Sensing Framework  
Compressive sampling is a new theory of information acquisition proposed by [PERSON] [23], [PERSON] [24] and [PERSON] et al. [25; 26]. The fundamental mathematical theory of compressive sensing (CS) states that if a real-valued, finite-length, one-dimensional, discrete-time signal \(\mathbf{x}^{\prime}\in\mathbb{R}^{N}\) is S-sparse, i.e., only \(S\) (\(S\ll N\)) components of \(\mathbf{x}^{\prime}\) are non-zero, this signal can be exactly recovered from far fewer randomly chosen samples or linear measurements with an overwhelming probability. In addition, a compressible signal can also be treated as sparse if most of the transform coefficients are zeros or near zero in transform domains (e.g., Fourier, wavelet bases, spatial domain, etc.). CS is a theoretical framework that stands upon two pillars: sparsity and incoherence. Sparsity indicates that a signal's information content can be represented in terms of a proper basis. The proper basis which transfers a compressible signal into sparse is called a \"sparsifying basis\", \(\boldsymbol{\Psi}\in\mathbb{R}^{N\times N}\), which can be expressed as:  
\[\mathbf{x}=\boldsymbol{\Psi}\boldsymbol{\alpha} \tag{1}\]  
where \(\mathbf{x}\) is the compressible signal, \(\boldsymbol{\alpha}\) is the transform coefficients vector with \(S\) nonzero elements (\(S\)-sparse) and \(\boldsymbol{\Psi}\) is the sparsifying basis in a specific domain. The general under-sampling measurements can be expressed as:  
\[\mathbf{y}=\boldsymbol{\Phi}\boldsymbol{\Psi}\boldsymbol{\alpha}+\boldsymbol{n }=\boldsymbol{\Theta}\boldsymbol{\alpha}+\boldsymbol{n} \tag{2}\]  
where \(\mathbf{y}\in\mathbb{R}^{m}\) is the measurement vector, \(\boldsymbol{\Phi}\in\mathbb{R}^{m\times N}\) is the under-sampling linear measurement matrix (\(m<N\)), \(\boldsymbol{n}\) represents the noise during the measuring process, and \(\boldsymbol{\Theta}=\boldsymbol{\Phi}\boldsymbol{\Psi}\) is defined as the final transfer matrix. The under-sampling measurement is the number of linear measurements \(\mathbf{y}\) smaller than the number of unknown variables in compressible signal \(\mathbf{x}\). It is an ill-posed problem to estimate the unknown variables in \(\mathbf{x}\) from few measurements \(\mathbf{y}\)[27; 28]. As we mentioned above, because the unknown signal \(\mathbf{x}\) is compressible and sparse, the sparse vector \(\boldsymbol{\alpha}\) which contains information of signal \(\mathbf{x}\) could be reconstructed by solving an \(\ell_{1}\)-norm optimization problem. In order to reconstruct the unknown compressible signal \(\mathbf{x}\), the second principle: incoherence is introduced due to its robustness in the presence of measurement noise [29]. Thus, the transfer matrix \(\boldsymbol{\Theta}\) is required to satisfy the following restricted isometry property (RIP) with an isometric constant \(\delta\) smaller than unity [24]:  
\[1-\delta\leq\frac{\left\|\boldsymbol{\Theta}\boldsymbol{\psi}\right\|_{2}}{ \left\|\mathbf{v}\right\|_{2}}\leq 1+\delta,\;\delta>0 \tag{3}\]  
where \(\delta\) is the RIP constant which is defined as the smallest value that meets the requirement of Equation (3), \(\mathbf{v}\in\mathbb{R}^{N}\) is all \(S\)-sparse vector. RIP constant \(\delta\) is a parameter representing the character of the nearly orthogonal matrices operating on sparse vectors. \(\delta\approx 0\) indicates that the transfer matrix is a nearly orthonormal matrix, while \(\delta\approx 1\) indicates that the vectors in the transfer matrix \(\boldsymbol{\Theta}\) are redundant. Random selected matrices have been proven to meet the requirement of RIP [24], which guarantees that the CS problem can be reconstructed with an overwhelming probability. In addition, the sampling number should be larger than the required number \(m>\mu\cdot S\cdot\log(N/S)\) where \(\mu\) is a constant (generally, \(\mu=4\)) dependent on the basis.