## 3 Proposed Method
### Space-Spectrum Adaptive Multi-Manifold Analysis  
A new image was obtained after a regionally adaptive weight representation of each pixel in the image was performed. The spectral curves of pixels of the same category in the new image were more similar. Then, \(c\) manifold graphs were constructed for each class of ground objects in the new image to find the low-dimensional feature representation of each class of ground object, where \(c\) denotes the number of the classes of ground object in the image. Supervised multi-manifold learning (SMML) can reduce the dimensionality of data by constructing multi-manifold graphs. However, SMML only considers inter-class relationships between HSI data and does not take intra-class relationships into account. In view of this, we proposed a spatial-spectral adaptive multi-manifold analysis method. The steps of the algorithm were as follows. Firstly, the new image obtained from the adaptive weight representation of the region was divided into different subsets according to the class labels of the samples. Each subset was a separate sub-manifold. The intra- and the inter-manifold maps were constructed in the spatial and spectral dimensions of each sub-manifold. By adaptively fusing the intra- and inter-manifold maps constructed for each sub-manifold in the spatial and spectral dimensions, the spatial-spectral intra-manifold maps and spatial-spectral inter-manifold maps on each sub-manifold map were obtained, meaning that the projection matrix corresponding to each sub-manifold was found. Finally, the pixels in each sub-manifold were downscaled. The features of all samples in the low-dimensional subspace were combined to obtain the feature representation of the entire hyperspectral data in the low-dimensional subspace. Based on this approach, the spatial-spectral adaptive graph embedding algorithm effectively fused the low-dimensional embedding of each sample point in each sub-manifold and used it to perform classification, thus achieving a better classification result. In the HSI spectral dimension, the set of ground objects in each category was considered to be a sub-manifold. We assumed that the \(i\) pixel on the \(r\) manifold was represented as \(\mathrm{M}_{r}^{i}\). The set of pixels on the \(r\) sub-manifold can be denoted as \(\left\{\mathrm{M}_{r}^{1},\mathrm{M}_{r}^{2},\cdots,\mathrm{M}_{r}^{n_{r}}\right\}\), where \(n_{r}\) is the total number of samples in category \(r\). Firstly, based on the spectral similarity relationship between the samples, the set of intra-manifold near neighbors \(\mathrm{N}_{in}\) and the set of inter-manifold nearest neighbors \(\mathrm{N}_{out}\) for each pixel of each sub-manifold in the spectral dimension could be obtained. We assumed that \(\mathrm{V}_{1},\mathrm{V}_{2},\cdots,\mathrm{V}_{c}\) denotes the projection matrix of \(c\) in different sub-manifolds of the high-dimensional data, where \(c\) is the number of classes of the high-dimensional data. In the spectral dimension, in order to minimize the aggregation of intra-manifold data and maximize the separation of inter-manifold data in the low-dimensional space, the optimization problem can be expressed via Equation (4) [35],  
\[ \begin{array}{l}\max\limits_{\mathrm{V}_{1},\mathrm{V}_{2},\cdots,\mathrm{V }_{c}}J(\mathrm{V}_{1},\mathrm{V}_{2},\cdots,\mathrm{V}_{c})\\ =J_{1}(\mathrm{V}_{1},\mathrm{V}_{2},\cdots,\mathrm{V}_{c})-J_{2}(\mathrm{V}_{ 1},\mathrm{V}_{2},\cdots,\mathrm{V}_{c})\\ =\sum\limits_{r=1}^{c}(\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_{r=1}^{c}(\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})\end{array}