## 3 Solution
### The Elastic-Window-Based Algorithm  
As an implementation of the iterative approximating strategy, an elastic-window-based algorithm was developed to approach the adjacent pixels around the coarse pixel and address the underdetermined problem of linear spectral unmixing. The entire workflow of the algorithm is shown in Figure 3.  
In the algorithm, the inner part types and numbers within the mixed pixel were extracted from the fine spatial resolution land cover map. The initial window size was estimated using Formula (4). Each mixed pixel created its own initial window centered around it to select the adjacent pixels. Then, pixels containing different land cover types compared to the central mixed pixel were discarded. The abundances of all inner parts among these selected pixels were extracted, and a corresponding linear system was constructed. A simulated examination was also conducted to assess the presence of the underdetermined problem.  
In the simulated examination, all inner part values were assumed to be known by assigning constant values. The values of the pixels could be calculated using the linear spectral mixing model with abundances and zero residual error. The linear system, treating the inner part values as unknown parameters, was then inverted using the ordinary least squares method. If the inverted values matched the constant values, the selected pixels could be used to estimate the inner part values of the mixed pixel. If not, the window size would be expanded by two-pixel units, and the additional pixels within the expanded  
Figure 3: The workflow of the elastic-window-based algorithm.  
Figure 2: Adjacent region iterative approximating strategy.  
window would be included in the new linear system. The simulated examination would be repeated until the inverted values matched the constant values. This simulated examination serves as a practical alternative to Formula (5).  
The matrix form of the algorithm's linear system is given in Formula (6), where \(\mathbf{F}\) represents the coefficient matrix, \(r\) represents the matrix of inner part values, \(\varepsilon\) represents the residual error matrix of the linear system, and \(R_{E}\) represents the matrix of coarse pixel values. The limiting conditions for \(R_{E}\) are shown in Formula (7), where \(R_{\mathrm{i}}\) represents the value of a coarse pixel located within the spatial range of the elastic window \(S_{W}\), and \(k\) represents a coarse pixel that contains different inner part types compared to the central pixel.  
\[\mathbf{F}\cdot\mathbf{r}+\varepsilon=\mathbf{R}_{E}, \tag{6}\]  
\[\mathbf{R}_{E}=\{R_{\mathrm{i}}|R_{\mathrm{i}}\in S_{W},i\
eq k\}, \tag{7}\]  
The least-squares solution of Formula (6) is given in Formula (8). The superscript \(\mathbf{F}^{T}\) is the transformation of the matrix, and \(\left(\mathbf{F}^{T}\cdot\mathbf{F}\right)^{-1}\) is the inverse of the matrix.  
\[\mathbf{r}=\left(\mathbf{F}^{T}\cdot\mathbf{F}\right)^{-1}\cdot\mathbf{F}^{T }\mathbf{R}_{E} \tag{8}\]  
It should be noted that achieving an exact equivalency between the inverted values and constant values during the simulated examination may be unlikely due to the inherent computational errors of computers. Therefore, considering a minor difference between the values is a more practical approach. To prevent the situation where more than two inner parts of the mixed pixel never appear in the entire image, leading to the process getting stuck in a loop, it is necessary to set a maximum number of iterations for the loop. This ensures that the algorithm does not continue indefinitely and allows for a predetermined number of iterations before reaching a stopping condition.