## EMD_CS_ST Denoising Method
### Compressed Sensing  
CS theory has been widely used in information theory, image optimization, and other fields. It has accurate reconstruction ability for compressible target signals and has low requirements on sampling frequency. The basic idea of compressed sensing is the following: If the sparsity of an unknown signal is K or is transformed into K sparse by a sparse basis, then based on K sparse, combined with non-linear transformation, the original signal can be accurately reconstructed [21].  
Let the expression of the noisy MS signal be:  
\[x=X+e \tag{3}\]  
where \(x\) is a noisy MS signal, \(X\) is a valid signal, and \(e\) is random noise.  
The basic ideas of CS are as follows:  
In the first step, a set of orthogonal basis matrices \(\Psi(\text{N}\times\text{N})\) is found to thin the MS signal \(x\) and obtain the sparse vector \(X_{0}\):  
\[X_{0}=\Psi x \tag{4}\]  
In the second step, select a group of measurement matrix bases \(\Phi\) (M \(\times\) N, K \(<\) M \(<\) N), which are not related to the orthogonal basis, to project MS signal in low dimension, from N-dimension signal to M-dimension signal. Measured value \(y\):  
\[y=\Phi x \tag{5}\]  
Combining Equations (4) and (5) gives:  
\[y=\Phi\Psi^{-1}X_{0}=\Phi\theta \tag{6}\]  
where \(\theta\) is the coefficient to be reconstructed.  
In the third step, in the case where the measured value \(y\) and the matrix \(\Phi\) are known, the MS signal \(x\) is recovered from the measured value \(y\) with a high probability by an optimization method.  
In the sparse transform domain, the effective signal \(X\) has sufficient sparsity, and the noise \(e\) does not have sparsity, so the noise cannot be restored in the process of signal reconstruction. This is the denoising principle of compressed sensing [22].  
If the matrix \(\Phi\) meets the Restricted Isometry Property (RIP) criterion, K sparse can obtain an optimal solution from the M measured values \(y\). Since the dimension N of the coefficient to be reconstructed \(\theta\) is much larger than the dimension M of the measured value \(y\), the Equation (6) is an underdetermined system of formulas with an infinite set of solutions. Taking advantage of the sparsity of theta and using L1 norm regularization formula to find the optimal solution, it is a typical optimization algorithm to solve the underdetermined formula [29]:  
\[\hat{\theta}=\text{argmin}\{\|y-\Phi\theta\|_{2}^{2}+\alpha\|\theta\|_{1}\} \tag{7}\]  
Equation (7) is the theoretical basis of the compressed sensing algorithm, where \(\|\ \|_{\text{P}}\) denotes the L\({}_{\text{P}}\) norm; the first term in the braces is the constraint measurement term; the second term is the sparse constraint term, and the parameter \(\alpha\) is the balance parameter. This paper chooses a sparse Bayesian learning algorithm that is more suitable for solving Equation (7) than the traditional L\({}_{1}\) norm regularization algorithm [30].  
At present, commonly used measurement matrices include Gaussian random measurement matrices, random Bernoulli matrices, partially orthogonal matrices, Topliz and cyclic matrices and sparse random matrices, and other matrices. The Gaussian random measurement matrix is used as a universal compressed sensing matrix [31]. For this reason, it is used as the measurement matrix in this paper. The core idea is to construct the M \(\times\) N measurement matrix \(\Phi\) and make each element in the measurement matrix \(\Phi\) obey the N(0,1/M) Gaussian distribution and conform to the RIP criterion. There is no correlation with most orthogonal bases [32]. The classical sparse transform methods include discrete cosine transform [33], discrete Fourier transform [34], discrete wavelet transform [35], and other transforms. In this paper, to organically combine with the following soft-thresholding denoising methods, a discrete wavelet transform is selected.