## 2 DeepLab v3+ and Proposed Framework
### Proposed Framework  
Figure 3b shows the framework we proposed of hyperspectral image classification. The framework can be roughly divided into three parts: PCA dimensionality reduction, feature extraction and fusion, and the use of classifiers to achieve pixel-level classification. The specific details and mathematical principles of the framework are as follows. The unprocessed hyperspectral image is regarded as a series of samples. Assume that the matrix size represented by the hyperspectral image is \(h\times w\times n\), where \(h\times w\) represents the resolution of the image and \(n\) represents the number of spectral channels. If \(x_{i}\in R^{n}\) is represented by a column vector, then the dimensionality reduction goal is to reduce the dimensionality of \(X\in R^{n\times h\times w}\) composed of \(h\times w\) samples to \(\hat{X}\in R^{k\times h\times w}(k\leq n)\). The process of principal component analysis is as follows. Let \(u_{i}\) be the \(i-th\) principal component. The optimization is:  
\[\underset{u_{i}}{\operatorname{argmax}}(\sum_{i=1}^{n}\bigl{|}proj\bigl{(}x_{ j},u_{i}\bigr{)}\bigr{|}),\ \bigl{(}x_{j},u_{i}\in R^{n}\bigr{)} \tag{3}\]  
and the projection length of \(x_{j}\) on \(u_{i}\):  
\[proj\bigl{(}x_{j},u_{i}\bigr{)}=\frac{x_{j}\cdot u_{i}}{||u_{i}||^{2}}=\frac{x _{j}{}^{T}u_{i}}{||u_{i}||^{2}} \tag{4}\]  
Let \(||u_{i}||=1\), and we already know \(X=[x_{1},x_{2},\cdots,x_{h\times w}]\). Hence, it can be shown:  
\[(\sum_{j=1}^{n}\bigl{|}proj(x_{j},u_{i})\bigr{|})=\underset{u_{i}}{\operatorname {argmax}}({u_{i}}^{T}XX^{T}u_{i}) \tag{5}\]  
The matrix \(XX^{T}\in R^{n\times n}\) is positive definite. Therefore, \({u_{i}}^{T}XX^{T}u_{i}\) is positive definite quadratic form, and it has a maximum value. Under the constraint of \({u_{i}}^{T}u_{i}=1\), we obtain \(XX^{T}u_{i}=\alpha u_{i}\) by Lagrange multiplier method. \(\alpha\) is the eigenvalue of the matrix \(XX^{T}\), and the objective function takes the maximum value when \(u_{i}\) takes the eigenvector corresponding to \(\alpha\). We can get:  
\[\underset{u_{i}}{\operatorname{argmax}}({u_{i}}^{T}XX^{T}u_{i})=\alpha{u_{i}} ^{T}u_{i}=\alpha \tag{6}\]  
Figure 3: Schematic depictions of the architecture of (**a**) DeepLab v3+ and overall architecture of the (**b**) proposed framework. where \(u_{i}\) is the \(i-th\) principal component. Taking the eigenvectors of the first k eigenvalues and the corresponding matrix, \(U=[u_{1},u_{2},\cdots,u_{k}]\in R^{n\times k}\). Finally, a dataset reduced from \(n\) dimensions to \(k\) dimensions can be obtained by  
\[\hat{X}=U^{T}X\in R^{k\times h\times w} \tag{7}\]  
Using an established method [16], the dimensionality of original hyperspectral dataset is reduced by PCA. As a method of dimensionality reduction, PCA only reduces the dimensionality of the data while there is very little information loss. The introduction of PCA is to avoid the use of three-dimensional convolution in the feature extraction process which is computationally intensive. If we do not use PCA, three-dimensional convolution has to be used in all data, including training set and test set generally. Therefore, the introduction of PCA theoretically does not lead to biased results. We take the first three principal components, namely \(k=3\), as the dataset, and take the first principal component, namely \(k=1\), as the label. All data processed by PCA are converted into uint8 format; therefore, in theory, they are all integers in the range of \([0,255]\). The main reason for using the first principal component as the label is that the data after dimensionality reduction remove the redundant information in the spectral domain and only a single channel is retained. The process retains the feature relationship of the object represented by each pixel in space, which is also an essential element for extracting spatial features. To expand the number of samples so as to extract the spatial feature information abundantly, we use an established method [20] to crop the reduced-dimensional images and labels.