## 3 Proposed Method
### Space-Spectrum Adaptive Multi-Manifold Analysis
Since the pixels in the local spatial pixel block in which \(x_{i}\) is located are located on the same spatial sub-manifold as \(x_{i}\), the neighboring pixels of \(x_{i}\) can be represented as \(\mathrm{M}_{r}^{i1},\mathrm{M}_{r}^{i2},\cdots\mathrm{M}_{r}^{im},\cdots\mathrm{ M}_{r}^{ik}\) where \(k\) is the total number of spatial neighbors. For any spatial sub-manifold \(\mathrm{M}_{r}\), we assumed that the \(i\) pixel on the \(r\) manifold is represented as \(\mathrm{M}_{r}^{i}\). The set of pixels on the \(r\) sub-manifold can be represented as \(\left\{\mathrm{M}_{r}^{1},\mathrm{M}_{r}^{2},\cdots\mathrm{M}_{r}^{n_{r}}\right\}\). \(n_{r}\) is the total number of samples in the \(r\) class. In the manifold intra-graph, the weighted scatter matrix of this spatial block within the manifold can be represented as follows:  
\[\mathrm{S}_{r}^{up}=\sum\limits_{i=1}^{n_{r}}\sum\limits_{m=1}^{k}\frac{w_{m }}{\sum\limits_{j=1}^{k}w_{j}}(\mathrm{M}_{r}^{i}-\mathrm{M}_{r}^{im})( \mathrm{M}_{r}^{i}-\mathrm{M}_{r}^{im}){}^{T} \tag{12}\]  
where \(w_{m}\) is the weight between the nearest neighbor image element \(\mathrm{M}_{r}^{im}\) and the central image element \(\mathrm{M}_{r}^{i}\), which is defined as follows:  
\[w_{m}=\exp\Big{\{}-\gamma\Big{|}\Big{|}\mathrm{M}_{r}^{i}-\mathrm{M}_{r}^{im }\Big{|}\Big{|}_{2}\Big{\}} \tag{13}\]  
In the inter-manifold graph, the scatter matrixes from vertex \(\mathrm{M}_{r}^{i}\) to submanifold \(\mathrm{M}_{s}(s\
eq r)\) and from vertex \(\mathrm{M}_{s}^{j}\) to submanifold \(\mathrm{M}_{r}\) can be defined via Equations (14) and (15). \[\mathrm{H}_{1}(\mathrm{M}_{r}^{i},\mathrm{M}_{s})=\sum\limits_{i=1}^{n_{s}} \sum\limits_{m=1}^{k}\frac{w_{m}}{\sum\limits_{j=1}^{k}w_{j}}(\mathrm{M}_{r}^{ i}-\mathrm{M}_{s}^{im})(\mathrm{M}_{r}^{i}-\mathrm{M}_{s}^{im}){}^{T} \tag{14}\]  
\[\mathrm{H}_{2}(\mathrm{M}_{s}^{j},\mathrm{M}_{r})=\sum\limits_{t=1}^{n_{r}} \sum\limits_{m=1}^{k}\frac{w_{m}}{\sum\limits_{p=1}^{k}w_{p}}(\mathrm{M}_{r}^{tm }-\mathrm{M}_{s}^{j})(\mathrm{M}_{r}^{tm}-\mathrm{M}_{s}^{j}){}^{T} \tag{15}\]Therefore, the inter-class scatter matrix between sub-manifold \(\mathrm{M}_{r}^{i}\) and sub-manifold \(\mathrm{M}_{s}(s\
eq r)\) can be represented via Equation (16). \[\mathrm{H}(\mathrm{M}_{r},\mathrm{M}_{s})=\sum_{i=1}^{n_{r}}\mathrm{H}_{1}( \mathrm{M}_{r}^{i},\mathrm{M}_{s})+\sum_{i=1}^{n_{s}}\mathrm{H}_{2}(\mathrm{M} _{s}^{j},\mathrm{M}_{r}) \tag{16}\]  
The scatter matrix between sub-manifold \(\mathrm{M}_{r}\) and all other sub-manifolds can be calculated via Equation (17). \[\mathrm{H}_{r}^{b}=\sum_{s=1,s\
eq r}^{c}\mathrm{H}(\mathrm{M}_{r},\mathrm{M} _{s}) \tag{17}\]  
The objective function of sub-manifold \(\mathrm{M}_{r}\) in the manifold intra-graph can be represented by Equation (18). \[\mathrm{J}^{w}(\mathrm{V}_{r})=\mathrm{tr}(\mathrm{V}_{r}^{T}\mathrm{S}_{r}^{ w}\mathrm{V}_{r}) \tag{18}\]  
The objective function of sub-manifold \(\mathrm{M}_{r}\) in the inter-manifold graph can be represented by Equation (19):  
\[\max_{\mathrm{V}_{1},\mathrm{V}_{2},\cdots,\mathrm{V}_{c}}J(\mathrm{V}_{1}, \mathrm{V}_{2},\cdots,\mathrm{V}_{c}) \tag{10}\]0  
To increase the aggregation of spatial neighborhood data features in each manifold and make the features of data between manifolds as distinguishable as possible, the condition shown in Equation (20) must be satisfied in each manifold space. \[\max_{\mathrm{V}_{1},\mathrm{V}_{2},\cdots,\mathrm{V}_{c}}J(\mathrm{V}_{1}, \mathrm{V}_{2},\cdots,\mathrm{V}_{c}) \tag{10}\]1  
Using the Lagrange multiplier method, the optimization problem in Equation (20) can be transformed into Equation (21). \[\max_{\mathrm{V}_{1},\mathrm{V}_{2},\cdots,\mathrm{V}_{c}}J(\mathrm{V}_{1}, \mathrm{V}_{2},\cdots,\mathrm{V}_{c}) \tag{10}\]2  
In HSIs, the spectral reflection characteristics between different categories of objects were represented by spectral information, and the feature information on the spatial dimension reflected the spatial distribution of objects.