## 3 Fuzzy Kernel Nearest Feature Line Embedding (FKNFLE)
### Classification Results  
The proposed methods, NFLE [20, 21], KNFLE, FNFLE [26], and FKNFLE, were compared with two state-of-the-art algorithms, _i.e._, nearest regularized subspace (NRS) [25] and NRS-LFDA [25]. The parameter configurations for both algorithms NRS [29] and NRS-LFDA were as seen in [25]. The gallery samples were randomly chosen for training the transformation matrix, and the query samples were matched with the gallery samples using the nearest neighbor (NN) matching rule. Each algorithm was run 30 times to obtain the average rates. To obtain the appropriate reduced dimensions of FKNFLE, the available training samples were used to evaluate the overall accuracy (OA) _versus_ the reduced dimensions in the benchmark datasets. As shown in Figure 4, the best dimensions of algorithm FKNFLE for datasets IPS-10, Pavia University, and Pavia City Center were 25, 50, and 50, respectively. The proposed FKNFLE and KNFLE algorithms are both extended from algorithm NFLE. From the  
Figure 3: The second toy sample distributions projected on the first three axes using algorithms (**a**) PCA; (**b**) LDA; (**c**) supervised LPP; (**d**) LFDA; (**e**) NFLE; (**f**) FNFLE; (**g**) KNFLE; and (**h**) FKNFLE. classification results as shown in Figure 4, though FKNFLE achieves the best results at the specific reduced dimensions on three datasets, the high variant OA rates are obtained. Moreover, two additional parameters \(K_{3}\) and \(K_{4}\) were needed for training during the fuzzification. On the other hand, the performance of KNFLE is more robust than that of FKNFLE. KNFLE usually achieves a higher performance even at low reduced dimensions, e.g., five or 10. It also outperforms the other algorithms at all reduced dimensions on datasets IPS-10 and Pavia City Center. Compared with NRS-LDA, slightly reduced OA rates were obtained on dataset Pavia University. From this analysis, algorithm KNFLE is more competitive than FKNFLE in HSI classification. Figure 4: _Cont._  
The average classification rates _versus_ the number of training samples on dataset IPS-10 are shown in Figure 5a; algorithms FKNFLE and KNFLE outperformed the other methods. The accuracy rate of FKNFLE was 4% higher than that of FNFLE. The kernelization strategy effectively enhanced the discriminative power. The performance of FKNFLE was better than that of KNFLE to a value of 0.8%, and the rate of FNFLE was higher than that of NFLE to a value of 0.7%. This shows that the fuzzification strategy slightly enhanced the performance. Figure 5b,c also demonstrates the overall accuracy _versus_ the number of training samples in the benchmark datasets of Pavia University and Pavia City Center, respectively. According to the classification rates in these two datasets, algorithm FKNFLE outperformed the other methods. In addition, the classification results were insensitive to the number of training samples. Next, the maps of the classification results for the dataset IPS-10 are given in Figure 6. The classification results of algorithms FKNFLE, KNFLE, FNFLE, NFLE, NRS, and NRS-LFDA are given based on the maps of 145 \(\times\)145 pixels depicting the ground truth. The speckle-like errors of FKNFLE were fewer than those of the other algorithms. Figures 6-8 give the maps of the classification results for datasets Pavia University and Pavia City Center, respectively. Once again, the speckle-like errors of FKNFLE were fewer than in the case of the other algorithms. In addition, the thematic maps of Pavia University and Pavia City Center are shown in Figure 9a,b, respectively, using the proposed FKNFLE method. Observing the results in Figure 9a, the roads, buildings, and the areas in University were clearly classified even though there was some speckle-like noise in the images. The roads, rivers, buildings, small islands, and the areas in the city were classified in the same way. See Figure 9b. Algorithm FKNFLE effectively classified the land cover even in the limited training samples.