## 3 Fuzzy Kernel Nearest Feature Line Embedding (FKNFLE)
### A Toy Example  
Two toy examples are given to illustrate the discriminative power of FKNFLE in the following. Firstly, 561 samples with 220 dimensions of the three classes (Grass/pasture, Woods, and Grass/trees) were collected from a hyperspectral image. The samples were projected onto the first three axes using eight algorithms: PCA, LDA, supervised LPP, LFDA [28], NFLE, FNFLE, KNFLE, and FKNFLE, as shown in Figure 2. These class samples are represented by green triangles (class G), blue stars (class B), and red circles (class R). A simple analysis was done by observing the sample distributions in the reduced spaces. Since the global Euclidean structure criterion was considered during the PCA and LDA training phases, the samples from three classes in the reduced spaces were mixed after the PCA and LDA projections as shown in Figure 2a,b. Since the samples were distributed in a manifold structure in the original space, the manifold learning algorithms, e.g., supervised LPP, LFDA, and NFLE, were executed to preserve the local structure of the samples. The sample distributions projected by supervised LPP, LFDA, and NFLE are displayed in Figure 2c-e, respectively. Three classes were efficiently separated and contrasted with those in Figure 2a,b. The class boundaries, however, were unclear due to the non-linear and non-Euclidean sample distributions in the original space. Kernelization and fuzzification were pre-performed to extend the original non-Euclidean and non-linear space to a higher linear space. Consider the sample distributions in Figure 2e,h, the boundaries of classes G and R in Figure 2e being still unclear using the NFLE transformation. The sample distributions of FNFLE and KNFLE as shown in Figure 2f,g were the results when the kernelization and fuzzification strategies were used, respectively. Obviously, classes G and R were more effectively separated than those in Figure 2e. The local structures of the samples from the observed sample distribution were preserved, and the class separability improved. Several points located at the boundaries were misclassified in these cases. When both strategies were further adopted in FKNFLE, only one red point was mis-located at class G, and classes G and R were clearly separated. From the analysis, both fuzzification and kernelization strategies enhanced the discriminative power of manifold learning methods. Figure 2: _Cont._  
Secondly, 561 samples with 220 dimensions of the three classes (Corn-no till, Soybeans-min till, Soybeans-no till) were collected from a hyperspectral image. The samples were projected onto the first three axes by eight algorithms: PCA, LDA, supervised LPP, LFDA, NFLE, FNFLE, KNFLE, and FKNFLE, as shown in Figure 3. These class samples are also represented by green triangles (class G), blue stars (class B), and red circles (class R). A simple analysis was also done by observing the sample distributions in the reduced spaces. Since the global Euclidean structure criterion was considered during the PCA and LDA training phases, the samples of three classes in the reduced spaces were mixed after the PCA and LDA projections as shown in Figure 3a,b. Since the samples were distributed in a manifold structure in the original space, the manifold learning algorithms, e.g., supervised LPP, LFDA, and NFLE, were executed to preserve the local structure of the samples. The sample distributions projected by supervised LPP, LFDA, and NFLE are displayed in Figure 3c-e, respectively. Due to the strong overlapping in classes G, R, and B, they were mixed, and the separation was relatively low compared with those in Figure 2c-e. However, when the kernelization and fuzzification strategies were used, class B was more effectively separated than those shown in Figure 3c-e. According to the analysis, in the case of strong overlapping, both fuzzification and kernelization strategies enhanced the discriminative power of manifold learning methods.