## 3 Proposed Method
### Space-Spectrum Adaptive Multi-Manifold Analysis
 \tag{4}\]  
where \(\mathrm{V}_{1},\mathrm{V}_{2},\cdots,\mathrm{V}_{c}\) denotes the objective function used to minimize the aggregation of intra-manifold data and maximize the separation of inter-manifold data. \(J_{1}(\mathrm{V}_{1},\mathrm{V}_{2},\cdots,\mathrm{V}_{c})\) denotes the objective function used to maximize the separation of inter-manifold data, and \(J_{2}(\mathrm{V}_{1},\mathrm{V}_{2},\cdots,\mathrm{V}_{c})\) denotes the objective function used to minimize the aggregation of intra-manifold data. \(\mathrm{M}_{r}^{p}\) denotes \(p\) as a inter-manifold near neighbor of \(\mathrm{M}_{r}^{i}\). \(\mathrm{M}_{r}^{i}\) denotes \(q\) as a intra-manifold near neighbor of \(\mathrm{M}_{r}^{i}\). \(k_{1}\) and \(k_{2}\) denote the number of inter- and intra-manifold near neighbors. \(A_{r}^{ip}\) denotes the weight matrix for the similarity between \(\mathrm{M}_{r}^{i}\) and its inter-manifold near neighbor. \(B_{r}^{iq}\) denotes the weight matrix for the similarity between \(\mathrm{M}_{r}^{i}\) and its intra-manifold near neighbor. They are defined via Equations (5) and (6). \[A_{r}^{ip}=exp(-\left|\left|\mathrm{M}_{r}^{i}-\mathrm{M}_{r}^{p}\right| \right|/\sigma^{2}) \tag{5}\]  
\[B_{r}^{ip}=exp(-\left|\left|\mathrm{M}_{r}^{i}-\mathrm{M}_{r}^{q}\right| \right|/\sigma^{2}) \tag{6}\]  
where \(\sigma\) is the heat kernel. This algorithm iterates to find the projection matrix on the \(c\) sub-manifold. For the \(i\) manifold, the objective function is expressed as shown in Equation (7). \[ \begin{array}{l}\max\limits_{\mathrm{V}_{r}}J(\mathrm{V}_{r})\\ =\left(J_{1}(\mathrm{V}_{r})+F_{1}\right)-\left(J_{2}(\mathrm{V}_{r})+F_{2} \right)\\ =\left(\left.\sum\limits_{i=1}^{n_{r}}\sum\limits_{p=1}^{k_{1}}\left|\left| \mathrm{V}_{r}{}^{T}\mathrm{M}_{r}^{i}-\mathrm{V}_{r}{}^{T}\mathrm{M}_{r}^{p} \right|\right|^{2}A_{r}^{ip}+F_{1}\right)-\left(\left.\sum\limits_{i=1}^{n_{ r}}\sum\limits_{q=1}^{k_{2}}\left|\left|\mathrm{V}_{r}{}^{T}\mathrm{M}_{r}^{i}- \mathrm{V}_{r}{}^{T}\mathrm{M}_{r}^{q}\right|\right|^{2}B_{r}^{iq}+F_{2}\right) \end{array}  \tag{7}\]  
where \(F_{1}\) and \(F_{2}\) are two constant matrices that can be neglected in the process of computation of \(\mathrm{V}_{r}\), meaning that Equation (8) can be obtained. \[ \begin{array}{l}J_{1}(\mathrm{V}_{r})\\ =\sum\limits_{i=1}^{n_{r}}\sum\limits_{p=1}^{k_{1}}\left|\left|\mathrm{V}_{r}{}^ {T}\mathrm{M}_{r}^{i}-\mathrm{V}_{r}{}^{T}\mathrm{M}_{r}^{p}\right|\right|_{2}A _{r}^{ip}\\ =\sum\limits_{i=1}^{n_{r}}\sum\limits_{p=1}^{k_{1}}\mathrm{tr}((\mathrm{V}_{r} ^{T}\mathrm{M}_{r}^{i}-\mathrm{V}_{r}^{T}\mathrm{M}_{r}^{p})(\mathrm{V}_{r}{}^ {T}\mathrm{M}_{r}^{i}-\mathrm{V}_{r}{}^{T}\mathrm{M}_{r}^{p}){}^{T}A_{r}^{ip}) \\ =\sum\limits_{i=1}^{n_{r}}\sum\limits_{p=1}^{k_{1}}\mathrm{tr}(\mathrm{V}_{r} {}^{T}[(\mathrm{M}_{r}^{i}-\mathrm{M}_{r}^{p})(\mathrm{M}_{r}^{i}-\mathrm{M}_{ r}^{p}){}^{T}A_{r}^{ip}]\mathrm{V}_{r})\\ =\mathrm{tr}(\mathrm{V}_{r}^{T}\sum\limits_{i=1}^{n_{r}}\sum\limits_{p=1}^{k_ {1}}[(\mathrm{M}_{r}^{i}-\mathrm{M}_{r}^{p})(\mathrm{M}_{r}^{i}-\mathrm{M}_{ r}^{p}){}^{T}A_{r}^{ip}]\mathrm{V}_{r})\\ =\mathrm{tr}(\mathrm{V}_{r}^{T}Q_{1r}\mathrm{V}_{r})\end{array}  \tag{8}\]  
where \(Q_{1r}{}^{\triangle}\Big{(}M_{r}^{i}-M_{r}^{p}\Big{)}\Big{(}M_{r}^{i}-M_{r}^{ p}\Big{)}{}^{T}A_{r}^{ip}\). \[J_{2}(\mathrm{V}_{r})=\sum\limits_{i=1}^{n_{r}}\sum\limits_{q=1}^{k_{1}}\left\| \mathrm{V}_{r}{}^{T}\mathrm{M}_{r}^{i}-\mathrm{V}_{r}{}^{T}\mathrm{M}_{r}^{q} \right\|^{2}B_{r}^{iq}\!\!=\mathrm{tr}(\mathrm{V}_{r}{}^{T}\mathrm{Q}_{2r} \mathrm{V}_{r}) \tag{9}\]  
Thus, in the spectral dimension, Equation (10) can be obtained. \[\max_{\mathrm{V}_{1},\mathrm{V}_{2},\cdots,\mathrm{V}_{c}}J(\mathrm{V}_{1}, \mathrm{V}_{2},\cdots,\mathrm{V}_{c}) \tag{10}\]  
The above equation can be translated into Equation (11). \[(\mathrm{Q}_{1r}-Q_{2r})\mathrm{V}_{r}=\lambda_{r}\mathrm{V}_{r} \tag{11}\]  
In the HSI space dimension, if the pixel \(x_{i}\) is located on the \(r\) spectral sub-manifold, it can be represented as \(\mathrm{M}_{r}^{i}\).