## 5 Gas Data Reconstructed Based on TVS-MH  
Reconstruction of gas data is the core of compressed sensing of gas data flow. It can be theoretically proved that, if the gas data are \(K\)-sparse and the measurement matrix satisfies certain conditions, the signal \(x\) can be accurately reconstructed from the measured value of \(y\) by solving the \(l_{0}\)-norm problem:  
\[\hat{x}=\arg\min\|x\|_{0}\quad s.t.\quad\Phi x=y \tag{9}\]  
However, [PERSON] pointed out that the minimum \(l_{0}\)-norm problem is also an NP-hard problem and all possible arrangements of non-zero values in (\(C_{N}^{K}\)) should be exhausted. Therefore, scholars proposed a series of algorithms for near-optimal solutions. Gas signals are generally slowly varying and only exhibit occasional rapid variations at mutation points, and the mobile measurement points measure the gas data at the same time and in the same space; this means that \(x_{i}\approx x_{i+1}\) in most cases. Therefore, the total variation of \(x\in R^{N}\) can be defined as follows:  
\[||Dx||_{1}=\sum_{i=1}^{N-1}|D_{i}x|=\sum_{i=1}^{N-1}|x_{i+1}-x_{i}| \tag{10}\]  
where \(D\in R^{(N-1)\times N}\) is a dual-diagonal matrix, and \(D_{i}\) denotes the element in the \(i\)-th row of \(D\). \(D\) can be written as follows:  
\[D=\left[\begin{array}{cccccc}-1&1&0&\cdots&0&0&0\\ 0&-1&1&\cdots&0&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ 0&0&0&\cdots&-1&1&0\\ 0&0&0&\cdots&0&-1&1\end{array}\right] \tag{11}\]  
Figure 5: Compression collection model of gas flow data.  
The total variation function assigned larger values to rapidly-varying \(x\). Accordingly, the reconstruction problem can be rewritten as follows:  
\[\underset{x}{\text{argmin}}||Dx||_{1},\quad s.t.\,\Phi x=y \tag{12}\]  
To acquire the minimum, the following augmented Lagrangian function was introduced:  
\[L(x,\lambda,\mu)=||Dx||_{1}-\lambda^{T}(\Phi x-y)+\frac{\mu}{2}(\Phi x-y)^{T}( \Phi x-y) \tag{13}\]  
Equation (13) can be solved using an iterative algorithm, and the unknown parameters \(\lambda\) and \(\mu\) can be selected according to the following Equations (14) and (15):  
\[\lambda^{k+1}=\lambda^{k}-\mu^{k}(\Phi x^{k+1}-y) \tag{14}\]  
\[\mu^{k+1}\geq\mu^{k} \tag{15}\]  
where the superscript \(k\) denotes the \(k\)-th iterated value.  
In contrast to the standard Lagrangian function, the augmented Lagrangian function included a quadratic penalty function.  
Algorithm 1 presents the detailed procedures of data reconstruction using the TVS-MH algorithm.  
```
1.Input:\(x_{j}\), \(\Phi_{ij}\), \(i=1,2\cdots M\), \(j=1,2\cdots N\).
2.Step1: Select \(M\) time-independent multi-hop communication links with different initial nodes in a cluster
3.Step2: Assign the \(i\)-th multi-hop signal received by the wireless access point as \(y_{i}=\sum\Phi_{ij}x_{j}\) (\(i=1,2\cdots M\)).
4.Step3: Calculate all of the \(M\) multi-hop communication links to form a gas data compressed sensing collection model \(y\) based on the multi-hop network
5.Step4: Calculate the total variation of \(x\):
6.\(||Dx||_{1}=\sum\limits_{i=1}^{N-1}|D_{i}x|=\sum\limits_{i=1}^{N-1}|x_{i+1}-x_{ i}|\).
7.Step5: Solve the reconstruction problem that can be rewritten as follows:
8.\(\underset{x}{\text{argmin}}||Dx||_{1},\quad s.t.\,\Phi x=y\).
```  
**Algorithm 1**The TVS-MH Algorithm.