## 3 Methodology
### Probability Generative Model of SAR Observations  
Inspired by the principle of PPCA, the generative model of SAR observations is carefully constructed in this section. It must be noted that the form of this model is much more complicated than that of the traditional PPCA, as the SAR observations are complex random variables. In this paper, this model is referred to as the complex probabilistic principal component analysis (CPPCA) model:  
\[\mathbf{y}=\mathbf{W}\mathbf{z}+\mathbf{\varepsilon} \tag{14}\]  
where \(\mathbf{W}\) denotes the loading matrix, \(\mathbf{z}\) is the vector of latent variables and \(\mathbf{\varepsilon}\) is the noise component. \(\mathbf{z}\) and \(\mathbf{\varepsilon}\) obey complex circular Gaussian distribution (i.e., \(\mathbf{z}\sim\mathcal{CN}\big{(}0,\mathrm{I}_{q}\big{)}\) and \(\mathbf{\varepsilon}\sim\mathcal{CN}\big{(}0,\sigma^{2}\mathrm{I}_{N}\big{)}\)), respectively. According to the linear property of the complex circular Gaussian distribution, the probabilistic density function of \(\mathbf{y}\) can be deduced and written as:  
\[f(\mathbf{y})=\pi^{-N}|\mathbf{C}|^{-1}\exp\Big{\{}-\mathbf{y}^{H}\mathbf{C}^ {-1}\mathbf{y}\Big{\}} \tag{15}\]  
where \(\mathbf{C}=\mathbf{W}\mathbf{W}^{H}+\sigma^{2}\mathrm{I}_{N}\), and the unknown parameter set is \(\Theta=\big{\{}\mathbf{W},\sigma^{2}\big{\}}\). With the use of Equation (15), the log-likelihood of a given complex-valued observation dataset \(\{\mathbf{y}_{m}\}_{m=1}^{M}\) can be expressed as:  
\[\mathcal{L}(\Theta)=-M\Big{\{}N\ln\pi+\ln|\mathbf{C}|+\text{tr}\Big{(}\mathbf{ C}^{-1}\mathbf{S}\Big{)}\Big{\}} \tag{16}\]  
As previously mentioned, \(\Theta\) can be solved through the EM algorithm, which regards the vector of latent variables \(\mathbf{z}_{m}\) and \(\{\mathbf{z}_{m},\mathbf{y}_{m}\}\) as the \"missing\" data and the \"complete\" data, respectively. The log-likelihood of the \"complete\" data can be expressed as:  
\[\mathcal{L}_{c}=\sum_{m=1}^{M}\ln\{p(\mathbf{y}_{m},\mathbf{z}_{m})\} \tag{17}\]  
Hence, the joint probabilistic density function can be deduced:  
\[f(\mathbf{y},\mathbf{z})=\Big{(}\pi\sigma^{2}\Big{)}^{-N}\exp\biggl{(}-\frac{ 1}{\sigma^{2}}\|\mathbf{y}-\mathbf{W}\mathbf{z}\|^{2}\biggr{)}(\pi)^{-q}\exp \Bigl{(}\|\mathbf{z}\|^{2}\Bigr{)} \tag{18}\]  
where \(\|\cdot\|\) denotes \(L1\)-norm operator. Based on the Bayes' rule, the conditional distribution of \(\mathbf{z}\) can be obtained:  
\[\mathbf{z}|\mathbf{y}\sim\mathcal{CN}\Big{(}\mathbf{M}^{-1}\mathbf{W}^{H} \mathbf{y},\sigma^{2}\mathbf{M}^{-1}\Big{)} \tag{19}\]  
where \(\mathbf{M}=\mathbf{W}^{H}\mathbf{W}+\sigma^{2}\mathrm{I}_{q}\). In general, the EM algorithm contains two processing steps: E-step and M-step. In E-step, the expectation of \(L_{c}\) with respect to the distribution of \(\mathbf{z}|\mathbf{y}\) is calculated based on the following formulation:  
\[Q(\Theta) =\mathrm{E}\{L_{c}|\mathbf{y}\} \tag{20}\] \[=-\sum_{m=1}^{M}\left\{ \begin{array}{l}N\ln\bigl{(}\sigma^{2} \bigr{)}+\text{trace}\bigl{[}\mathrm{E}\bigl{(}\mathbf{z}_{m}\mathbf{z}_{m}^{ H}|\mathbf{y}_{m}\bigr{)}\bigr{]}+\frac{1}{\sigma^{2}}\text{Tr}\sigma^{H} \mathbf{y}_{m}\\ -\frac{2}{\sigma^{2}}\mathrm{E}\bigl{(}\mathbf{z}_{m}^{H}|\mathbf{y}_{m} \bigr{)}\bigl{(}\mathbf{W}^{H}\mathbf{y}_{m}\bigr{)}+\frac{1}{\sigma^{2}}\text{ trace}\bigl{[}\bigl{(}\mathbf{W}^{H}\mathbf{W}\bigr{)}\mathrm{E}\bigl{(}\mathbf{z}_{m} \mathbf{z}_{m}^{H}|\mathbf{y}_{m}\bigr{)}\bigr{]}\end{array} \right\}\]  
where \(\mathrm{E}\{\cdot\}\) represents expectation operator. Based on the [PERSON] rule, the conditional distribution of the latent variable vector can be obtained and expressed as:  
\[\mathbf{z}|\mathbf{y}\sim\mathcal{CN}\Big{(}\mathbf{M}^{-1}\mathbf{W}^{H} \mathbf{y},\sigma^{2}\mathbf{M}^{-1}\Big{)} \tag{21}\]where \(\mathbf{M}=\mathbf{W}^{H}\mathbf{W}+\sigma^{2}\mathbf{I}_{q}\). The expectations involved in Equation (20) hence can be deduced:  
\[\left\{