# Bayesian Compressive Sensing Based Optimized Node Selection Scheme in Underwater Sensor Networks
## 2 System Model and Problem Formulation  
### System Model  
The paper considers an underwater sensor network where a sink node is located in the center of a circular region and \(N\) sensor nodes are uniformly distributed in this range. The radius is \(r\). Each sensor collects information such as temperature, salinity, and so on and then transmits them to a sink node. In the traditional information acquisition scheme, the sink node collects the information of all the sensor nodes that consume too much energy. To overcome this difficulty, a compressed sensing method can be applied to reduce the number of measurements. Suppose that the total signal \(\mathbf{x}=[x(1),x(2),\ldots,x(N)]^{T}\), where \(i=1,2,\ldots,N\) can be expressed as  
\[\mathbf{x}=\sum_{i=1}^{N}s(i)\boldsymbol{\
abla}(i)=\boldsymbol{\
abla} \mathbf{s}, \tag{1}\]  
where \(\boldsymbol{\
abla}=[\boldsymbol{\
abla}(1),\boldsymbol{\
abla}(2),\ldots, \boldsymbol{\
abla}(N)]\) is a basis matrix and \(\mathbf{s}\) is a sparse vector that there are \(K\ll N\) nonzero elements and other elements are zero. The \(K\) is called the sparsity of vector \(\mathbf{s}\). The vector \(\mathbf{x}\) is called compressible if the conditions above are satisfied. According to the compressed sensing theory, the sink node can select \(M\ll N\) measurements to reconstruct the original signal only if it is compressible in the basis \(\boldsymbol{\
abla}\). Let \(\boldsymbol{\Phi}\in\mathbb{R}^{M\times N}\) denote the sensor node selection matrix which is independent from \(\boldsymbol{\
abla}\). The structure of node selection matrix \(\boldsymbol{\Phi}\) can be represented as follows:  
\[\left\{ \begin{array}{l}\sum_{i=1}^{M}\boldsymbol{\Phi}_{ij}\leq 1,j=1,2, \ldots,N,\\ \sum_{j=1}^{N}\boldsymbol{\Phi}_{ij}=1,i=1,2,\ldots,M,\\ \boldsymbol{\Phi}_{ij}\in 0,1,i=1,2,\ldots,M,j=1,2,\ldots,N.\end{array} \right. \tag{2}\]If the \(i\)th element in the \(j\)th row of node selection matrix \(\mathbf{\Phi}\) is 1, it means that the \(i\)th node is selected. Then, the measurement vector can be obtained  
\[\mathbf{y}=\mathbf{\Phi}\mathbf{\Psi}\mathbf{s}+\mathbf{n}=\mathbf{A}\mathbf{s }+\mathbf{n}, \tag{3}\]  
where \(\mathbf{A}=\mathbf{\Phi}\mathbf{\Psi}\) is the projection matrix. The vector \(\mathbf{n}\) is independent identically distributed (i.i.d) additive white Gaussian noise with mean zero and variance \(\sigma^{2}\):  
\[p(\mathbf{n})=\prod_{i=1}^{N}\mathcal{N}(n_{i}|0,\sigma^{2}). \tag{4}\]  
The paper aims at reconstructing the sparse vector \(\mathbf{s}\) instead of reconstructing the original signal vector \(\mathbf{x}\) directly. From Equation (3), there are \(N\) variables and \(M\ll N\) equalities. It is impossible to obtain the unique solution. However, notice that the vector \(\mathbf{s}\) is sparse and has \(K\ll N\) nonzero elements, and it makes the reconstruction possible if the projection matrix \(\mathbf{A}\) meets the restricted isometry property (RIP). The reconstructed solution \(\mathbf{\hat{s}}\) can be obtained from the optimization problem as follows:  
\[ \begin{split}&\arg\min\|\mathbf{s}\|_{0},\\ & s.t.\ \|\mathbf{y}-\mathbf{A}\mathbf{s}\|_{2}\leq\varepsilon.\end{split}  \tag{5}\]  
However, the optimization problem above is non-deterministic polynomial (NP)-hard due to the non-convexity of the objective function. To relax the original problem as a convex problem, a convex \(\|\cdot\|_{1}\) is usually used to replace the \(\|\cdot\|_{0}\). Then, the transformed problem can be represented as follows:  
\[ \begin{split}&\arg\min\|\mathbf{s}\|_{1},\\ & s.t.\ \|\mathbf{y}-\mathbf{A}\mathbf{s}\|_{2}\leq\varepsilon.\end{split}  \tag{6}\]  
The problem above can be solved effectively by the CVX tool cabinet. The original signal \(\mathbf{\hat{x}}=\mathbf{\Psi}\mathbf{\hat{s}}\) can be recovered by using the solution \(\mathbf{\hat{s}}\). In the underwater information sampling, the signal is usually sparse due to the spatial correlation. For example, the temperature signal is sparse in the frequency domain, i.e., it can be expressed sparsely by Fourier basis.