## 3 Spatial-Spectral-Emissivity Land-Cover Classification Based on Conditional Random Fields (SSECRF)
### Background
#### 3.1.2 Conditional Random Fields (CRF)  
CRF is a probabilistic discriminative model, and is the statistical model first used to mark and segment serialized data in 2001 by [PERSON] et al. [26]. It directly models the posterior probability of the given observation data so that the contextual information can be considered in both the observed data and the labeled data. As a context classification model, [PERSON] and [PERSON] [27] successfully extended this model to two-dimensional image classification and processing in 2003, and the CRF method has also been successfully applied in hyperspectral imagery classification [28; 29], man-made scene interpretation [30], building detection from high-resolution interferometric synthetic aperture radar (InSAR) data [31], change detection [32], and high-resolution image classification, integrating the spectral, spatial, and location information by adding the additional higher-order potential in the pairwise CRF model [33].  
CRF constructs the conditional probability distribution model of the whole label variables under the condition that the observation variable is given, i.e., to model the maximum a posteriori probability (MAP) distribution directly. In order to build distributions over the combined set of _input variables_y that are always called the observed field, and the _output variables_x that are the label field that we expect to predict, the probabilistic discriminative model can model the posterior probability distribution under the condition of the given observation data as a Gibbs distribution [34] with the following form:  
\[P(\mathrm{x}|\mathrm{y})=\frac{1}{Z(\mathrm{y})}\exp\left\{-\sum_{c\in C} \psi_{c}(\mathrm{x}_{c},\mathrm{y})\right\} \tag{1}\]  
where y represents the observed data from the input image, and x is the corresponding class labels for the entire image. \(Z(\mathrm{y})=\sum_{x}\exp\{-\sum_{c\in C}\psi_{c}(\mathrm{x}_{c},\mathrm{y})\}\) is the partition function, and \(\psi_{c}(\mathrm{x}_{c},\mathrm{y})\) is the potential function. Theoretically, the potential function based on the size of the variables in the cliques can be divided into unary potentials, pairwise potentials, and even high-order potentials, and the corresponding Gibbs energy function can be defined as follows:  
\[E(\mathrm{x}|\mathrm{y})=-\log P(\mathrm{x}|\mathrm{y})-\log Z(\mathrm{y})= \sum_{c\in C}\psi_{c}(\mathrm{x}_{c},\mathrm{y}) \tag{2}\]In the classification of remote sensing imagery, the commonest model is called pairwise CRF, which includes unary and pairwise potentials [35], as shown in the Figure 2 and following formula:  
\[E(\mathrm{x}|\mathrm{y})=\sum_{i\in V}\psi_{i}(x_{i},\mathrm{y})+\lambda\sum_{i \in V,j\in\mathrm{N}_{i}}\psi_{ij}(x_{i},x_{j},\mathrm{y}) \tag{3}\]  
The unary potential \(\psi_{i}\) is a single point of potential energy which constructs the relationships between observation variables and the corresponding label field. At the same time, pairwise potential \(\psi_{ij}\), a two-position potential energy, considers the spatial-contextual interaction between the observed variables and the label variables, so it can be regarded as a model that can establish the spatial interaction of the paired random variables in the local neighborhood.  
CRF is used to model the posterior probability \(P(\mathrm{x}|\mathrm{y})\), as mentioned above. Based on the Bayesian MAP rule, the image classification is designed to find the label image \(\mathrm{x}\) by maximizing the posterior probability distribution function \(P(\mathrm{x}|\mathrm{y})\), and it can be defined as [35]: \(\mathrm{x}_{MAP}=\mathrm{argmax}_{\mathrm{x}}P(\mathrm{x}|\mathrm{y})\). Thus, the largest posterior category marker of the CRF can be expressed as:  
\[\mathrm{x}_{MAP}=\mathrm{argmax}_{\mathrm{x}}P(\mathrm{x}|\mathrm{y})= \mathrm{argmin}_{\mathrm{x}}E(\mathrm{x}|\mathrm{y}) \tag{4}\]  
Thus, maximizing the posterior probability distribution \(P(\mathrm{x}|\mathrm{y})\) is equivalent to minimizing the energy function \(E(\mathrm{x}|\mathrm{y})\).