**5. Discussion**

As mentioned in Sections 2 and 3, some parameters should be fixed in the SVM-SSHK method. All of our experiments on HSIs, including those that are not mentioned here, confirmed that the number of the morphological operators *n* and the selected PCs play an important role for the construction of the EMP, and the critical threshold *υ* greatly influences the hierarchical information extraction. Furthermore, the weights in the spectral-spatial-hierarchical kernel make a significant impact on the classification performance of the proposed method. In this section, the impact of all the previously mentioned parameters is further analyzed to better understand the application of SVM-SSHK method for HSI classification.

#### *5.1. Impact of n*

To exploit the spatial kernel in the proposed framework, the number of the opening/closing operators (*n*) should be appropriately selected. In this subsection, the impact of *n* on the performance of the SVM-SSHK method is firstly analyzed. Experiments were performed on the IP and UP images in the case of *M* = 40 and the parameter settings were fixed the same to the previous experiments in Sections 4.3.1 and 4.3.2. Table 6 lists the classification accuracies by the proposed framework under different values of *n*. In this table, the highest classification accuracies for the IP image can be obtained when *n* = 8, and the OA, AA and *κ* for the UP image were very stable when *n* ≥ 8 around 98.1%, 98.6%, and 97.5%, respectively, and the highest OA, AA, and *κ* were achieved when *n* = 16, 14, and 12, respectively. A large value of *n* means that more number of MPs should be computed for spatial information extraction. To ensure computational efficiency of the SVM-SSHK method, we fixed this parameter as *n* = 8 for both the IP and UP images.

**Table 6.** Classification accuracy (%) by the SVM-SSHK method under different values of *n* for the IP and UP images. The best accuracies are indicated in bold in each column.


#### *5.2. Impact of Different Number of PCs*

To present further inspections with respect to the most appropriate number of PCs, three combinations were analyzed for spatial information extraction. Experiments were performed on the two HSIs in the case of *M* = 40 and the parameter settings were fixed the same to the previous experiments in Sections 4.3.1 and 4.3.2. Table 7 lists the classification accuracies by the proposed framework under different number of PCs. In this table, as the number of PCs is increased, which means that more spatial information can be exploited for constructing the EMP of the HSI, the improved classification accuracies can be obtained. For instance, the SVM-SSHK method using the first three PCs can increase the OA for the IP image by 0.84% and 0.88%, and for the UP image by 0.43% and 4.74%, than using PC 1 + PC 2 and PC 1, respectively. For conciseness and efficiency, the first three PCs were exploited for spatial information extraction.


**Table 7.** Classification accuracy (%) by the SVM-SSHK method under different number of PCs for the IP and UP images.

#### *5.3. Impact of υ*

To figure out the impact of *υ*, experiments were performed on the IP and UP images in the case of *M* = 40 and the parameter settings were fixed the same to the previous experiments in Sections 4.3.1 and 4.3.2. Table 8 provides the classification accuracies by the proposed framework under different values of *υ*. As *υ* is increased from 0.05 to 0.1 for the two HSIs, the variation of the classification accuracies is very similar. Specifically, the highest OA, AA, and *κ* of 95.86%, 97.12%, and 95.25% for the IP image and 98.1%, 98.73%, and 97.49% for the UP image were achieved when *υ* = 0.3 and *υ* = 0.2, respectively. To ensure that the SVM-SSHK is capable of achieving the optimal results, the parameter settings were used in the previous experiments for comparison.

**Table 8.** Classification accuracy (%) by the SVM-SSHK method under different values of *υ* for the IP and UP images. The best accuracies are indicated in bold in each column.


#### *5.4. Impact of Weights*

In the SVM-SSK method, the weights in *KSPE*−*SPA*−*HIE* critically determine the classification performance, since their values indicate the contribution of spectral, spatial, and hierarchical structure information for classification. An appropriate combination of their values may obtain better results. To obtain the interaction effect of *<sup>μ</sup>SPE*, *<sup>μ</sup>SPA*, and *<sup>μ</sup>HIE*, we can perform a four-dimensional (4-D) analysis to evaluate the influence of these three weights on our method's performance. Based on the constraint of *μSPE* + *μSPA* + *μHIE* = 1, we converted this 4-D analysis to a problem of analyzing different combinations of *μSPE* and *μSPA* in terms of classification accuracies. Figure 6 illustrates the three-dimensional (3-D) plot of the classification accuracies with the change of *μSPE* and *μSPA* from 0 to 1 with a step size of 0.1. Several conclusions can be observed from this figure.

**Figure 6.** *Cont.*

**Figure 6.** Impact of *μSPE* and *μSPA* using the two images on SVM-SSHK's performance. (**a**) overall accuracy (OA); (**b**) average accuracy (AA); and, (**c**) kappa coefficient (*κ*) for the IP image; (**d**) OA; (**e**) AA and (**f**) *κ* for the UP image.

First, for the IP image, if *μSPE* = 0, it means that the proposed framework includes only two kernels of *KSPA* and *KHIE*. In such case, we can obtain the OA, AA and *κ* with 91.01%~95.59%, 93.53%~96.91%, and 89.69%~94.95%, respectively; if *μSPA* = 0, it indicates that the proposed framework includes only two kernels of *KSPE* and *KHIE*. In such a case, we can obtain the OA, AA and *κ* with 82.06%~95.1%, 88.12%~96.65%, and 79.51%~94.38%, respectively. Specifically, the OA, AA and *κ* of 91.52%, 94.7%, and 90.31% were achieved when *μSPE* = *μSPA* = 0. For the UP image, the OA, AA, and *κ* can be obtained with 61.28%~98.05%, 70.91%~98.68%, and 52.53%~97.42% when *μSPE* = 0, respectively, and 61.28%~92.86%, 70.91%~95.27%, and 52.53%~90.65% when *μSPA* = 0, respectively. Specifically, the very poor OA, AA, and *κ* of 61.28%, 70.91%, and 52.53% were achieved when *μSPE* = *μSPA* = 0, respectively.

Second, the appropriate selection of *<sup>μ</sup>SPE*, *<sup>μ</sup>SPA*, and *μHIE* can result in the best classification accuracies. For instance, the highest OA, AA, and *κ* for the IP and UP images can reach to 95.86%, 97.12%, and 95.25% under the condition of *μSPE* = 0.3, *μSPA* = 0.1, and *μHIE* = 0.6, and to 98.14%, 98.75%, and 97.53% under the condition of *μSPE* = 0.1, *μSPA* = 0.7, and *μHIE* = 0.2, respectively. Compared to Tables 2 and 4, it can be confirmed again that the combination of the spectral, spatial and hierarchical kernels is really essential to produce better classification accuracies than using a single or double kernels in the SVM classifier.

Finally, the SVM-SSHK method can demonstrate very stable classification performance in most cases of different parameter settings on *μSPE* and *<sup>μ</sup>SPA*. According to Figure 6, there are 66 combinations of the two weights in total. For the IP image, the SVM-SSHK method can obtain the OA, AA and *κ* higher than 92%, 94%, and 90% for 53 of 66 (80.30%) different parameters settings, respectively. For the UP image, the proposed method is capable of achieving the OA, AA, and *κ* higher than 95%, 95%, and 90% for 40 of 66 (60.60%) different parameters settings, respectively.
