# Quadratic Clustering-Based Simplex Volume Maximization for Hyperspectral Endmember Extraction
## 3 Results and Discussion
### Parameter Analysis
#### 3.2.1 Spatial Clustering Analysis  
Spatial clustering involves the grid step s and the weighting factor for spatial measurement \(\lambda_{1}\). Two experiments are designed to determine the optimal values of them. To determine s, set \(\lambda_{1}\) to 0.9, and calculate Scatt, AvgPI and SvdPI on the input datasets clustered with s from 5 to 30. The results are shown in Figure 5. To determine \(\lambda_{1}\), set s to 6, and Scatt, AvgPI and SvdPI are calculated on the input datasets clustered with \(\lambda_{1}\) from 0 to 1 at an interval of 0.05. The results are shown in Figure 6.  
Figure 5: The results of (**a**) Scatt; (**b**) AvgPI; (**c**) SvdPI with the change of s.  
Figure 6: The results of (**a**) Scatt; (**b**) AvgPI; (**c**) SvdPI with the change of \(\lambda_{1}\).  
From Figure 5, the indexes fluctuate significantly with the increase in s, and from Figure 6, the indexes keep stable with the increase in \(\lambda_{1}\). It shows that s has a greater impact on the clustering performance than \(\lambda_{1}\).  
Of all the datasets, Cuprite is in sharp contrast with Samson and Jasper Ridge. The indexes on Cuprite turn out to be the most stable. Especially in Figure 5, the indexes have a relatively slight change, with s rising from 5 to 30. This has relations with the spectral contrast of endmembers in datasets. As shown in Figure 4, Cuprite contains endmembers with minor difference, while endmembers in Samson and Jasper Ridge have high spectral contrast.  
With the rise in s, both AvgPI and SvdPI trend down and Scatt trends up. It can be inferred that a large s will lead to high dispersion degree and low spectral correlation as each partition is expanded. However, a small s will increase the computational cost. As is presented in Figure 5, the indexes keep steady as s rises from 5 to 10, so the best range for s is set to 5-10. In our experiments, s is set to 6.  
When \(\lambda_{1}\) is 0.05, the indexes have a sharp change, as the spatial measurement begins to constrain clustering. Then, the indexes tend to be stable because of the limitation of the search scope. Indexes on Samson and Cuprite have little change, but it is different on Jasper Ridge. Specifically, AvgPI and SvdPI fall steadily and Scatt rises slowly. It is away from our requirement of regional homogeneity. Hence, \(\lambda_{1}\) is set to 0.1 to keep the stability as well as good performance of spatial clustering.  
Figure 7 shows the results of spatial clustering with the parameters set above. For better viewing, part of each image is enlarged and shown in the red box.  
Pt represents the percentage of pixels reserved from each partition for representative endmember generation. To analyze the impact of Pt, an experiment is conducted. After completing the spatial clustering with s and \(\lambda_{1}\) being 6 and 0.1, respectively, choose pixels in each partition with Pt from 0.05 to 0.95 at an interval of 0.01, and the indexes are calculated each time. The results are shown in Figure 8.  
Compared to Samson and Cuprite, indexes on Jasper Ridge have the most obvious change. It is caused by its poor spatial homogeneity. The increase in Pt results in the decline in AvgPI and SvdPI and the rise in Scatt as more mixed pixels within each partition are preserved. In terms of dispersion degree and spectral correlation, the smaller the Pt, the better the updated partitions. Considering the prevalence of spectral variability, a small Pt cannot help dealing with it by average the reserved pixels, otherwise it brings errors. In our experiments, Pt is set to 0.4, and the good performance will be demonstrated in subsequent experimental results.  
Figure 7: The results of spatial clustering. (**a**) Samson; (**b**) Jasper Ridge; (**c**) Cuprite.  
Moreover, compared to the indexes in Figure 6, AvgPI and SvdPI increase to above 0.99 and 0.85, respectively, and Scatt decreases to below 0.1. It further shows the important role of spectral purity analysis on regional homogeneity.