## 3 Ssrhe
### Spatial-Spectral Hypergraph Embedding  
Due to the spatial consistency of HSI, the pixels in HSI are usually spatially related, which means the pixels within a small neighborhood usually possess the spatial distribution consistency of ground objects. Therefore, neighborhood pixels can be utilized to learn spatial-spectral combined features. Suppose that pixel \(\mathbf{x}_{i}\) is the center pixel, and \(T\) is the spatial neighborhood window. \((u_{i},v_{i})\) is the spatial coordinate of pixel \(\mathbf{x}_{i}\) in the image, and the spatial neighborhood set \(\Omega(\mathbf{x}_{i})\) with the window size \(T\) (positive odd) can be recorded as  
\[\Omega(\mathbf{x}_{i})=\left\{\mathbf{x}_{im}:(u_{m},v_{m})\left| \begin{array} []{l}u_{m}\in(u-z,v+z)\\ v_{m}\in(u-z,v+z)\end{array} \right.\right\} \tag{26}\]  
where \(z=(T-1)\big{/}2\), \(\mathbf{x}_{im}:(u_{m},v_{m})\) corresponds to the \(m\)-th pixel in the spatial neighborhood. \(\Omega(\mathbf{x}_{i})\) has a total of \(T\times T\) pixels. Thus, the distance measure in the spatial neighborhood block can be defined as  
\[a_{i}=\sum\
olimits_{m=1}^{T^{2}}\left\|\mathbf{x}_{i}-\mathbf{x}_{im} \right\|^{2}v_{m} \tag{27}\]  
where \(v_{m}=\exp\left\{-\left\|\mathbf{x}_{i}-\mathbf{x}_{im}\right\|^{2}\middle/2 (\frac{1}{T\times T}\sum\limits_{m=1}^{T^{2}}\mathbf{x}_{im})^{2}\right\}\), which measures similarity between central pixel \(\mathbf{x}_{i}\) and its spectral-spatial neighbors. For all training samples in HSI data, the local spatial neighborhood preserving scatter matrix is calculated as follows:  
\[\mathbf{S}^{w}=\sum\
olimits_{i}a_{i}=\sum\
olimits_{i}\sum\
olimits_{m=1}^{ T^{2}}\left\|\mathbf{x}_{i}-\mathbf{x}_{im}\right\|^{2}v_{m}=\sum\
olimits_{i=1}^ {N}\sum\
olimits_{m=1}^{T^{2}}(\mathbf{x}_{i}-\mathbf{x}_{im})(\mathbf{x}_{i} -\mathbf{x}_{im})^{T}v_{m} \tag{28}\]  
To preserve the global structure of samples, a total scatter matrix is defined as  
\[\mathbf{S}^{b}=\sum\
olimits_{i=1}^{N}(\mathbf{x}_{i}-\mathbf{\overline{x}}) (\mathbf{x}_{i}-\mathbf{\overline{x}})^{T} \tag{29}\]  
where \(\mathbf{\overline{x}}\) is the mean of all samples. To extract low-dimensional spatial-spectral joint features, a objective function should be designed to preserve the local neighborhood as well as compact the samples with interclass hypergraph and separate the samples with interclass hypergraph simultaneously. Therefore, Equations (25), (28), and (29) are transformed into the following optimization function:  
\[\left\{ \begin{array}{l}\arg\max_{\mathbf{P}}tr\left\{\alpha\mathbf{P}^{T} \left[(1-\beta)\mathbf{M}^{b}+\beta\mathbf{X}\mathbf{X}^{T}\right]\mathbf{P}+ (1-\alpha)\mathbf{P}^{T}\mathbf{S}^{b}P\right\}\\ \arg\min_{\mathbf{P}}tr\left\{\alpha\mathbf{P}^{T}\left[(1-\beta)\mathbf{M}^{ w}+\beta diag(diag(\mathbf{M}^{w}))\right]\mathbf{P}+(1-\alpha)\mathbf{P}^{T} \mathbf{S}^{w}\mathbf{P}\right\}\end{array} \right. \tag{30}\]The above optimal function can be further simplified as  
\[\arg\max_{\mathbf{P}}\frac{tr\left\{\alpha\mathbf{P}^{T}\left[(1-\beta)\mathbf{M}^ {b}+\beta\mathbf{X}\mathbf{X}^{T}\right]\mathbf{P}+(1-\alpha)\mathbf{P}^{T} \mathbf{S}^{b}\mathbf{P}\right\}}{tr\left\{\alpha\mathbf{P}^{T}\left[(1-\beta) \mathbf{M}^{w}+\beta diag(diag(\mathbf{M}^{w}))\right]\mathbf{P}+(1-\alpha) \mathbf{P}^{T}\mathbf{S}^{w}\mathbf{P}\right\}} \tag{31}\]  
According to Lagrange multiplier method, Equation (31) can be solved by  
\[\left\{\alpha\left[(1-\beta)\mathbf{M}^{b}+\beta\mathbf{X}\mathbf{X}^{T}\right] +(1-\alpha)\mathbf{S}^{b}\right\}\mathbf{P}=\lambda\left\{\alpha\left[(1-\beta )\mathbf{M}^{w}+\beta diag(diag(\mathbf{M}^{w}))\right]+(1-\alpha)\mathbf{S}^ {w}\right\}\mathbf{P} \tag{32}\]  
where \(\lambda\) is an eigenvalue set. With the \(d\) largest eigenvalues corresponding eigenvector, the optimal projection matrix can be denoted as \(\mathbf{P}=\left[\mathbf{p}_{1},\mathbf{p}_{2},\cdots,\mathbf{p}_{d}\right]\).