## 3 Spatial-Spectral-Emissivity Land-Cover Classification Based on Conditional Random Fields (SSECRF)
### Methodology of SSECRF
#### 3.2.4 Classification Based on SSECRF  
As noted in Section 2, in the remote sensing imagery classification task, the most widely used model is the pairwise CRF model, which can be written as a sum of the unary potential term and pairwise potential term. It is equivalent to the Gibbs energy in Equation (3), where \(\psi_{i}(x_{i},\mathbf{y})\) is the unary potential term and \(\psi_{ij}(x_{i},x_{j},\mathbf{y})\) is the pairwise potential term. The latter item is defined on the local neighborhood \(N_{i}\) of site \(i\). The non-negative constant \(\lambda\) is the adjusting parameter of the pairwise potential term, which weighs the influence of these two potential terms. After modeling the basic framework, how to model the potential function needs to be considered, as well as optimizing the energy function to find the final class labels. 1. SSECRF Energy Function  
Here, we redefine Equation (3) in the SSECRF algorithm, so the unary potential can be written as \(\psi_{i}(x_{i},\text{SSE})\), integrating the SSE feature obtained in the multi-feature extraction from the visible and thermal infrared hyperspectral imagery data, where SSE is an observation field representing the SSE feature set, and the pairwise potential is \(\psi_{ij}(x_{i},x_{j},\mathbf{S}_{VIS})\), representing the spatial interaction in the visible imagery, where \(\mathbf{S}_{VIS}\) is also an observation field representing the spectral and spatial contextual information of the visible imagery. Therefore, the well-studied pairwise CRF can be reformulated with the sum of the unary and pairwise potentials, and the energy function can be shown as follows:  
\[E(\mathbf{x}|\mathbf{y})=\sum_{i\in V}\psi_{i}(x_{i},\text{SSE})+\lambda\sum _{i\in V,j\in N_{i}}\psi_{ij}(x_{i},x_{j},\mathbf{S}_{VIS}) \tag{11}\]  
2. Unary Potential  
The unary potential \(\psi_{i}(x_{i},\text{SSE})\) is modeled to integrate the emissivity information with the spatial-spectral feature set from the visible imagery. The complementary information in the two types of imagery is then used to conduct the identification of land-cover types. The unary potential models the relationship between the label image and the observed image. In other words, it considers the calculation cost of each pixel belonging to each specific class. Therefore, under the condition that the given image feature vector is known, the classification algorithm can be used to calculate the probability estimation of each pixel independently. Here, the \"given image feature vector\" is defined as the \"fusion image\", which makes full use of the emissivity and spatial-spectral feature set of the visible imagery to help discriminate the various land-cover types. Therefore, the different featuresthat are sensitive to the different ground classes are considered in the classification task. The unary potential can be defined as [39]:  
\[\psi_{i}(x_{i},\text{SSE})=-\ln(P(x_{i}=l_{k}|f_{i}(\text{SSE}))) \tag{12}\]  
where \(f_{i}(\text{SSE})\) represents the feature vector consisting of the emissivity and other feature information at site \(i\), and \(P(x_{i}=l_{k}|f_{i}(\text{SSE}))\) is the probability of pixel \(x_{i}\) belonging to class label \(l_{k}\), based on the feature vector. Minimizing the unary potential function can be considered as image classification. Due to the SVM algorithm being known to perform well in remote sensing imagery classification in the case of limited samples [40], the probability estimation is obtained by SVM. 3. Pairwise Potential  
Due to the spectral variability and image noise in the visible imagery, the spectral vectors of adjacent pixels in the homogeneous regions of the image are not equal. Based on the spatial correlation of the ground features, adjacent pixels can usually be assumed to be the same object. The pairwise potential \(\psi_{ij}(x_{i},x_{j},S_{VIS})\) is modeled to consider the a-priori smoothness and the spatial patterns of the land-cover types, such as the shape and size information, etc.