## 3 Spatial-Spectral-Emissivity Land-Cover Classification Based on Conditional Random Fields (SSECRF)
### Methodology of SSECRF
#### 3.2.4 Classification Based on SSECRF
The abundant spatial features of the visible imagery play an important role in identifying the types of certain pixels. In our algorithm, the pairwise potential energy simultaneously considers the observed image and the label image to model the spatial-contextual relationship of each pixel and its associated neighborhood [41]. This can ensure that the adjacent pixels in the homogeneous areas obtain the same label, while retaining the boundaries of adjacent areas at the same time. The pairwise potential function take the form of [39; 42] and can be defined as:  
\[\psi_{ij}(x_{i},x_{j},S_{VIS})=\left\{ \begin{array}{ll}0&\text{if }x_{i}=x_{j}\\ g_{ij}(S_{VIS})+\theta&\text{otherwise}\end{array} \right. \tag{13}\]  
\[g_{ij}(S_{VIS})=dist(i,j)^{-1}\exp(-\beta\|S_{VIS,i}-S_{VIS,j}\|^{2}) \tag{14}\]  
where \(g_{ij}\) models the spatial interaction of adjacent pixels \(x_{i}\) and \(x_{j}\), measuring the difference in appearance between them, and where \(dist(i,j)\) is the Euclidean distance between the adjacent pixels, and \(S_{VIS,i}\) and \(S_{VIS,j}\) represent the spectral feature vector in sites \(i\) and \(j\), respectively. The parameter \(\beta\) is set to the mean square deviation of the feature vector variance between the adjacent pixels in the image, which is written as \(\beta=(2\Big{\langle}\|S_{VIS,i}-S_{VIS,j}\|^{2}\Big{\rangle})^{-1}\), where \(\langle.\rangle\) indicates the mean calculation over the image. The pairwise potential energy makes it possible to consider the spatial interaction between local pixels in the classification. When minimizing the energy function \(E(x|y)\), it is still expected to retain the continuity of the labels and make the adjacent pixels in the homogeneous regions have the same label in the classification map. 4. Inference Algorithm  
After the construction of the potential function in SSECRF, it is critical to optimize the energy function \(E(x|y)\) and obtain the final labels. The inference algorithm is designed to predict the optimal class labels, which correspond to calculating the minimum value of the energy function. A number of different approximate inference algorithms have been put forward to obtain the optimal labels, including iterated conditional modes and graph cuts. In this paper, the \(\alpha\)-expansion algorithm based on graph cuts [43] is used for the multi-class labeling problem. A local search strategy is designed to solve the problem of the small moving space easily falling into local optima for the energy function satisfying the multivalued variables. The local search strategy of the algorithm continues to calculate the global minimum values in the binary labeling problem through the graph cuts algorithm in the loop. The graph cuts based on the \(\alpha\)-expansion algorithm can be considered to simplify the multi-class labeling problem to a sequence of optimization subproblems with binary labels, which can be easily optimized by the graph-cut method. Also, the \(\alpha\)-expansion inference algorithm is described in the following. The current label is initialized as \(x_{p}=\left\{x_{i}^{p},i\in V\right\}\), and the \(\alpha\)-expansion algorithm gives each pixel the following choices: keeping the current label unchanged or converting to a specific label \(\alpha\in\mathrm{L}=\left\{1,2,\ldots,K\right\}\). All the pixels of the image make a choice simultaneously; the algorithm has exponential-level moving space with respect to any particular \(\alpha\), which ensures that the algorithm has a strong local minimum property. For particular \(\alpha\), the binary labeling problem in the \(\alpha\)-expansion algorithm can obtain the optimal value \(x^{n}=\left\{x_{i}^{n},i\in V\right\}\) through the graph cuts algorithm to optimize the objective function \(E_{\alpha}(x^{p})\).