# High-Resolution Through-the-Wall Radar Imaging with Exploitation of Target Structure
## 2 Signal Model
### Posterior Inference  
We use a Markov Chain Monte Carlo (MCMC) sampler to conduct the Bayesian inference of the algorithm. For simplicity, the set of the hyperparameters is defined as \(\mathbf{\Xi}\triangleq\{a,b,c,d,\mathbf{e},\mathbf{f}\}\), where \(\mathbf{e}\triangleq\{e_{0},e_{1},e_{2}\}\) and \(\mathbf{f}\triangleq\{f_{0},f_{1},f_{2}\}\). The set of random variables is defined as \(\mathbf{\Theta}\triangleq\{\mathbf{\theta},\mathbf{z},\mathbf{\pi},\mathbf{\alpha},\mathbf{\beta}\}\). Thus, the specific form of the joint pdf can be given by  
\[p(\bar{\mathbf{y}},\mathbf{A},\mathbf{\Theta}|\mathbf{\Xi}) =\mathcal{CN}(\bar{\mathbf{y}}|\mathbf{A}(\mathbf{\theta}\circ\mathbf{ z}),\alpha^{-1}\mathbf{I}_{l})\text{Gamma}(\alpha|c,d)\] \[\times\prod_{l=1}^{N_{x}N_{y}}\prod_{k=1}^{K}[\mathcal{CN}(\theta _{kl}|0,\beta_{l}^{-1})]^{z_{l}}\pi_{l}^{z_{l}}(1-\pi_{l})^{1-z_{l}}\] \[\times\prod_{l=1}^{N_{x}N_{y}}\text{Beta}(\pi_{l}|\mathbf{e}, \mathbf{f})\text{Gamma}(\beta_{l}|a,b). \tag{14}\]  
According to the conjugate relationship, an analytical method will be used to determine the posterior distribution of each random variable. Since the difference between the proposed algorithm and the clustered MT-BCS lies only in the measurements, the updated variables independent of the measurements will share the same updated equations. Here we further present the inferences of those variables which are dependent on the measurements and have different updated formulations in [17]. From the signal model, it is observed that only the variables \(\{\mathbf{\theta},\mathbf{z},\alpha\}\) are related to the observation data \(\bar{\mathbf{y}}\). We first update  
sity pattern \(q\), a posterior distribution is obtained analytically owing to the Beta distribution on \(\pi_{l}\) and given by  
\[p(\pi_{l}^{(q)}|\mathbf{e},\mathbf{f},\mathbf{z})=\text{Beta}(e_{q}+\kappa_{l}+z _{l},f_{q}+\
u_{l}+1-\kappa_{l}-z_{l}), \tag{20}\]  
where \(l\in\{1,\cdots,N_{x}N_{y}\}\), and \(\
u_{l}\) is the length of \(\mathbf{z}_{\eta_{l}}\). In the two-dimensional signal instance shown in Figure 2, \(\
u_{l}=8\) indicates that each pixel is taken into account along with its 8 closest neighbors. With the help of the conjugate relationship between the Gaussian and Gamma distributions, the analytical posterior distribution of the precision variable \(\beta_{l}\) is obtained as  
\[p(\beta_{l}|a,b,\mathbf{\theta})=\text{Gamma}(\beta_{l}|\tilde{a},\tilde{b}), \tag{21}\]  
where \(\tilde{a}=a+(K/2)\), \(\tilde{b}=b+\sum_{l}(\theta_{lk}^{2}/2)\), and \(l\in[1,\cdots,N_{x}N_{y}]\). For the noise precision \(\alpha\), we acquire the posterior distribution as  
\[p(\alpha|c,d,\mathbf{y},\mathbf{A},\mathbf{\theta},\mathbf{z})=\text{Gamma} \bigg{(}c+\frac{l}{2},d+\frac{\|\hat{\mathbf{y}}-\mathbf{A}(\mathbf{\theta}\circ \mathbf{z})\|^{2}}{2}\bigg{)}. \tag{22}\]  
Algorithm 1 provides the entire procedure of the proposed clustered BCS algorithm. We employ the Gibbs sampler with consecutive sampling to infer the model parameters. It is observed that the computational complexity is \(\mathcal{O}(J\times N_{x}N_{y}\times K^{3}\times N_{\text{maxiter}})\), where \(N_{\text{maxiter}}\) is the maximum number of iterations. When the maximum number of iterations is reached, the algorithm stops and outputs the result. ```
Input:  Initialize \(\Xi\) and \(\mathbf{\Theta}\);  Assign \(N_{\text{maxiter}}\); for\(n=1,\cdots,N_{\text{maxiter}}\), do for\(l=1,\cdots,N_{x}N_{y}\), do  update \(z_{l}\sim p(z_{l}=1|\theta_{\backslash l},\mathbf{z}_{\backslash l},\hat{ \mathbf{y}})\) [Equation (16)];  update \(\theta_{l}\sim p(\theta_{l}|z_{l}=1,\mathbf{\theta}_{\backslash l},\mathbf{z}_{ \backslash l},\hat{\mathbf{y}})\) [Equation (19)];  update \(\pi_{l}\sim p(\pi_{l}^{(q)}|\mathbf{e},\mathbf{f},\mathbf{z})\) [Equation (20)];  update \(\beta_{l}\sim p(\beta_{l}|a,b,\mathbf{\theta})\) [Equation (21)];  endfor  update \(\alpha\sim p(\alpha|c,d,\mathbf{y},\mathbf{A},\mathbf{\theta},\mathbf{z})\) [Equation (22)]; endfor Output: \(\mathbf{\theta}\) and \(\mathbf{z}\).