## 3 Fuzzy Kernel Nearest Feature Line Embedding (FKNFLE)
### Fuzzification of NFLE  
Consider \(N\) samples in the reduced space \(Y=\left[y_{1},y_{2}...,y_{N}\right]\) and their corresponding fuzzy membership grades, \(\pi(y_{i})\), for each sample, \(y_{i}\). The objective function is re-defined as follows:  
\[ \begin{split}& O=\sum_{i}\pi\big{(}y_{i}\big{)}\Bigg{(}\sum_{i,i=n \
eq n}\left\|y_{i}-L_{m,n}\big{(}y_{i}\big{)}\right\|^{2}I_{m,n}\big{(}y_{i} \big{)}\Bigg{)}\\ &=\sum_{i}\pi\big{(}y_{i}\big{)}\left\|y_{i}-\sum_{j}M_{i,j}y_{j} \right\|^{2}\\ &=tr\Big{(}Y^{T}\big{(}FEI-FEM\big{)}^{T}\big{(}FEI-FEM\big{)}Y \Big{)}\\ &=tr\Big{(}Y^{T}\big{(}FED-FEW\big{)}Y\Big{)}\\ &=tr\Big{(}Y^{T}\big{(}D_{j,i=2,j}-W_{i,i=2,j}\big{)}Y\Big{)}\\ &=tr\Big{(}w^{T}XL_{j,i=2,j}X^{T}w\Big{)}\end{split}  \tag{12}\]  
Here, each sample is assigned a fuzzy grade, \(\pi(y_{i})\). Element \(M_{i,j}\) denotes the connectivity relationship between point \(y_{i}\) and line \(L_{m,n}\) which is the same as that in Equation (1). Two non-zero terms, \(M_{i,n}=t_{m,n}\) and \(M_{i,m}=t_{n,m}\), are set, and \(\sum_{j}M_{i,j}=1\). Using simple algebra operations, the objective function with fuzzification is represented in a Laplacian matrix in which the fuzzy terms, \(\pi\big{(}y_{i}\big{)}\), constitute the column vector, \(F\),with size \(N\times 1\), and \(E\) is a row vector of all those with size \(1\times N\). Similarly, given \(N\) samples \(\phi(X)=\left\{\phi\big{(}x_{1}\big{)},\phi\big{(}x_{2}\big{)},\ldots,\phi \big{(}x_{N}\big{)}\right\}\) in a Hilbert space, the membership grade of a specified sample, \(\phi\big{(}x_{i}\big{)}\), and its \(K_{3}\) neighbors, is designed in the following equation for computing the within-class scatter:  
\[\pi(x_{i})= \begin{cases}0.51+\big{(}0.49*\big{(}q_{i}/K_{3}\big{)}\big{)},& \text{if }q_{i}\geq\theta_{within};\\ \\ 0.49*\big{(}q_{i}/K_{3}\big{)}&\text{otherwise}.\end{cases}  \tag{13}\]  
Here, value \(q_{i}\) is the number of samples whose labels are the same as that of \(\phi\big{(}x_{i}\big{)}\) among \(K_{3}\) nearest neighbors, and \(\theta_{within}\) is a manual threshold. If \(q_{i}=K_{3}\), then \(\pi(x_{i})\) returns to 1, _i.e._, all neighbors are in the same class. Adding the fuzzy term \(\pi(x_{i})\), the within-class scatter matrix becomes:  
\[\mathbf{S}_{w}^{\mathscr{H}}=\sum_{k=1}^{N_{C}}\Bigg{(}\sum_{\phi(x_{i})\in C_ {3}}\pi(x_{i})\times\sum_{L_{m,n}\in F_{k_{1}}(\phi(x_{i}),C_{3})}\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\  
Pines Site (IPS) image, was generated from AVIRIS (Airborne Visible/Infrared Imaging Spectrometer), which was captured by the Jet Propulsion Laboratory and NASA/Ames in 1992. The IPS image was captured from six miles in the western area of Northwest Tippecanoe County (NTC). A false color IR image of dataset IPS is shown in Figure 1a. The IPS dataset contained 16 land-cover classes with 220 bands, e.g., Alfalfa(46), Corn-notill(1428), Corn-mintill(830), Corn(237), Grass-pasture(483), Grass-trees(730), Grass-pasture-mowed(28), Hay-windowed(478), Oats(20), Soybeans-notill(972), Soybeans-mintill(2455), Soybeans-cleantill(593), Wheat(205), Woods(1265), and Bldg-Grass-Tree-Drives(386), and Stone-Steel-Towers(93). The numbers in parentheses were the collected pixel numbers in the dataset. The ground truths in dataset IPS of 10,249 pixels were manually labeled for training and testing. In order to analyze the performance of various algorithms, 10 classes of more than 300 samples were adopted in the experiments, e.g., a subset IPS-10 of 9620 pixels. Nine hundred training samples of 10 classes in subset IPS-10 were randomly chosen from 9,620 pixels, and the remaining samples were used for testing. The other two HSI data sets adopted in the experiments were obtained from the Reflective Optics System Imaging Spectrometer (ROSIS) instrument covering the City of Pavia, Italy.