## 4 Experiment
### Results and Discussion
#### 4.2.3 Performance of Elastic-Window-Based Algorithm  
Figure 8 shows the underdetermined pixel proportions of the test between the regular window and elastic window approaches for the land cover map with a size of 4000 pixels by 4000 pixels from tile N11_60. The underdetermined pixel proportions of the regular window approach increase with the mixed scale. On the other hand, the results of the elastic window approach remain at a low level for all ten mixed scales and are almost unaffected by the mixed scale size. For a mixed scale of 5, only 0.003% of mixed pixels cannot be inverted using the elastic window algorithm. This proportion decreases to 0.001% for mixed scale 10 and becomes zero for mixed scales ranging from 15 to 50. This indicates that the elastic-window-based algorithm is effective and robust for all ten mixed scales when inverting the inner part values using the linear spectral mixing model.  
In contrast, when inverting the inner part values based on regular windows with fixed sizes, the underdetermined pixel proportions vary with the window size. The lowest proportion is 0.223% with a window size of 3, while the highest proportion is 14.799% with a window size of 45. On each mixed scale, the number of underdetermined pixels increases with the expansion of the window size.  
Traditionally, it is believed that a larger window size provides more adjacent pixels to construct the unmixing linear system, leading to improved accuracy with the help of the least squares method. However, this result indicates that a larger regular window does not help reduce the number of mixed pixels encountering the underdetermined problem. A larger regular window involves more mixed pixels that consist of land covers different from the unmixing mixed pixel. These different land covers introduce additional unknowns to the unmixing linear system, making the inversion more challenging. With the increase in the mixed scale, more and more mixed pixels are likely to encounter the underdetermined problem. Therefore, compared to a regular window with a fixed size, the elastic-window-based algorithm, which approximates the land surface iteratively, is preferable for preventing the underdetermined problem.  
Figure 7: The average proportions of mixed pixels and underdetermined pixels among sixty land cover maps, ranging from 5 to 50 mixed scales.  
Figure 9 shows a comparison of time consumption between the elastic window and the fixed window (size increasing from 3 to 45) on 10 mixed scales separately. It shows that the unmixing process based on a regular window with a size of 3 is the fastest in terms of time consumption for all mixed scales. With an increase in the mixed scale, the number of mixed pixels in the study area decreases, leading to a reduction in the total unmixing time. The reason why the decrease in time consumption appears with the increase in mixed scales is the reduction in the total number of mixed pixels in a fixed experimental region. Therefore, the time consumption decreases sharply in exponent form for either a fixed window or an elastic window in each mixed scale. It can be seen that the elastic window takes a relatively small amount of time compared with other fixed windows in statistics. Until the mixed scale reaches 45, the time consumption of the elastic window stands at a low level, despite not being the lowest value. At the mixed scale 50, the elastic window spends extra time on iterative approximation of the local region compared with the least time-consuming fixed window. The elastic-window-based algorithm demonstrates high performance, indicating that the iterative approximation of the land surface does not require significant time, making it a practical approach.