# A Probabilistic Hyperspectral Imagery Restoration Method
## 4 Probabilistic LRMA Model for HSI Restoration
### Framework  
Since modeling noise deterministically as Equation (1) or Equation (2) can only represent some kinds of noise, we propose to model noise probabilistically. The basic assumption is that though \(\mathbf{S}\) is difficult to be modeled deterministically for multiple kinds of noise, it will be easier and more flexible to be modeled from the probabilistic perspective. In addition, we can further initialize the rank number by taking advantage of the linearly mixed model, as we analyzed in Section 3. With these considerations, we propose a probabilistic LRMA based HSI restoration method, whose framework is shown in Figure 2.  
It is commonly realized that mixture of Gaussian distribution can depict any continuous distribution. Thus, we probabilistically model noise \(\mathbf{S}\) in HSI with mixture of Gaussian as  
\[s_{ij}\sim\sum\
olimits_{k=1}^{K}\omega_{k}N(s_{ij}|\mu_{k},\tau_{k}^{-1}). \tag{7}\]  
\(s_{ij}\) is the value in the \(j\)-th column and \(i\)-th row of \(\mathbf{S}\). \(K\) represents the number of Gaussian components. Given the mean value \(\mu\) and precision \(\tau\) as \(\mu=\mu_{k}\) and \(\tau=\tau_{k}\), respectively, \(N(s_{ij}|\mu_{k},\tau_{k}^{-1})\) depicts the adopted \(k\)-th Gaussian distribution. \(\omega_{k}\) is the mixing proportion weight which satisfies \(\omega_{k}\geq 0\) and \(\sum_{k=1}^{K}\omega_{k}=1\).  
With such representation for \(\mathbf{S}\), likelihood between \(\mathbf{X}\) and \(\mathbf{L}\) obeys  
\[p(\mathbf{X}|\mathbf{L},\mathbf{\omega},\mathbf{\mu},\mathbf{\tau})=\prod_{(i,j)\in \Omega}\Bigl{(}\sum\
olimits_{k=1}^{K}\omega_{k}N(x_{ij}|l_{ij}+\mu_{k},\tau _{k}^{-1})\Bigr{)}, \tag{8}\]  
Figure 2: Framework of the proposed method.  
where \(l_{ij}\) and \(x_{ij}\) represent the element in the \(i\)-th row and the \(j\)-th column of \(\mathbf{L}\) and \(\mathbf{X}\), respectively. Since \(K\) Gaussian distributions are adopted, \(\mathbf{\omega}=(\omega_{1},\omega_{2},\cdots,\omega_{K})\) has \(K\) elements. Similarly, the mean value \(\mathbf{\mu}=(\mu_{1},\mu_{2},\cdots,\mu_{K})\) and the precision value \(\mathbf{\tau}=(\tau_{1},\tau_{2},\cdots,\tau_{K})\) also have \(K\) elements. \((i,j)\in\Omega\) is a subset composed by all possible values of \((i,j)\).  
Denote \(\theta\) as the parameter which controls the distribution of \(\mathbf{L}\in\mathbf{R}^{n_{b}\times\alpha_{p}}\). Choosing proper distribution and parameter \(\theta\) can guarantee that \(p(\mathbf{L}|\theta)\) is a low-rank matrix [42; 43]. For a probabilistic LRMA model, if \(\mathbf{\omega}\), \(\mathbf{\mu}\), \(\mathbf{\tau}\), and \(\theta\) are known in advance, restoration task turns to recover \(\mathbf{L}\) from \(\mathbf{X}\) as  
\[\mathbf{L}_{\text{opt}}=\underset{\mathbf{L}}{\text{argmax}}p(\mathbf{L}| \mathbf{X})\propto p(\mathbf{X}|\mathbf{L},\mathbf{\omega},\mathbf{\mu},\mathbf{\tau})p( \mathbf{L}|\theta). \tag{9}\]  
However, \(\mathbf{\omega}\), \(\mathbf{\mu}\), \(\mathbf{\tau}\) and \(\theta\) are all unknown in reality. To make those parameters data-dependent, they need to be estimated jointly with \(\mathbf{L}\) as \(p(\mathbf{L},\mathbf{\omega},\mathbf{\mu},\mathbf{\tau},\theta|\mathbf{X})\), which can be seen from Section 4.2.