Evaluation of Endmembers Selection in Linear Spectral Unmixing5. Linear Spectral Unmixing to Achieve Better Classification Accuracy Attaining more accurate classification results is performable just by considering the effect of subpixel spectral gain of sensor. Although registration of sensor images is pixel-based but since we do not know the combination of objects underneath that pixel, it is possible to decompose pixels information using spectral capability of images. It is distinctive that the spectral resolution one sensor has, the better subpixel analysis would be possible, and hence better classification results would be achieved. The attitude of pixels decomposition which is used in this paper is based on linear mixture modeling which briefly described in section 2. Mathematical demonstration of Linear Spectral Unmixing is [4]:   If the number of endmembers selected is less than the expected number, an infinite number of solutions is possible and the problem becomes trivial. The problem can be circumvented by allowing for an error part in the equations that can also account for the errors in the measurements. In this case, the error is represented by the e component in the equation and the total error must be minimized by least squares adjustment method to have an equation of best fit. The solution is [4]:  The theoretical processes discussed are implemented by specialized modules situate in ENVI 4.2 and sample results are presented in form of figures and tables. The subset of the scene was inverted with the methodology discussed above for both ways of selecting the endmembers. Both inversions produced spatially similar results. The fraction maps of Linear Spectral Unmixing are presented in Figure 2.  Figure 2. Four fraction maps resulted from the Linear Spectral Unmixing The high correlation of the two vegetation estimates is shown in the scatterplot of the two vegetation fraction images. The correlation coefficient between the two vegetation endmembers was calculated to be 0.99.  Figure 2. The plot in the left compares the vegetation fractions resulting from the two ways of selecting endmembers. Minimally inclusive is on the x-axis Correlation coefficient is 0.99. The error of the estimate for both inversions is consistently below 0.02 (95%). As the results show, the mean error is very similar for the two inversions. The minimally selected inversion seems to have a somewhat higher mean and higher standard deviation from the mean (Table 1).
The errors indicate that the unmixing using the four selected components is feasible and promising. Without, however, ground-truth assessing of the results the true quality of the inversion could prove trivial. The validity of the inversion, however, is further supported by reasonable correlation of the unmixing results with a Tasseled Cap Greenness and the Normalized Difference Vegetation Index [9]. In addition, the results also indicate that the error consistently reduces with increasing vegetation fraction. A careful examination of the results however, reveals that there is a difference between the two inversions for the vegetation fractions. The two inversions seem to have a deviation of agreement of 0.01 on their means and 0.03 on their standard deviations, with the minimally inclusive inversion to have a lower standard deviation and mean. It also has a higher minimum and a lower maximum. The latter shows that the Minimally inclusive selection produces results that better constrain the vegetation fractions to a unity range (Table 2 / Figure 3).
 Figure 3. Comparison of histograms of the two inversions. The maximally inclusive endmember selection is shown in green color. The Maximally inclusive vegetation fractions shows the highest number of negative value cells, with approximately 2.39% of the corresponding unmixed image to have negative values compared to 0.93% in the minimally inclusive unmixed image. A reconsideration of the image fraction values from the maximally inclusive unmixed image of cells that were included in the minimally inclusive selection shows a consistent overestimation of those cells. The mean fraction value of those pixels seems to be higher than one. A reconstruction of the reflectance spectrum for the components for the unmixing results using the vegetation endmember of the minimally inclusive vegetation fraction showed the true reflectance and the reconstructed to be almost identical. The minimally inclusive selection model seems to honor the endmember regions almost perfectly. An additional assessment of the minimally inclusive endmember selection model was attempted. In this case, pixels with a high error component were selected and averaged from the unmixed images. Then their true reflectance values were compared to the reconstructed from the fraction image spectra. This was attempted to gain an insight of what is causing the error. The results are shown in Figure 4:  Figure 4. Comparison of observed and reconstructed Spectra in Bands 1, 2, 3, 4, 5, and 7 for high error areas. The actual observed spectrum is shown in black. Most of the error is concentrated in Bands 2, 5, and 6. Figure 4 indicates that most of the error misfit is in Bands 2, 5 and 6. Band 2 contributes most to the error misfit. It seems encouraging that the reconstructed spectra replicate the observed values of Band 4 and Band 3 well because most of the vegetation information is conveyed in those two bands. 7. Conclusion A linear spectral unmixing approach appears to be a tractable problem in areas with highlights and shadows. The underlying geology and vegetation can be adequately discriminated and accounted in terms of their percentages. Vegetation shows high sensitivity in the way its pure spectra are selected with a minimally inclusive selection to show the best results. The results need to be validated with ground truth sites. Modeling the shadowed areas as a mixing endmember seems to provide good results although there is no guaranty that shadowed areas are not vegetated. As a result, some of the vegetation content may be lost by this approach. 8. References
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