# A Learning-Based Image Fusion for High-Resolution SAR and Panchromatic Imagery
## 2 Materials and Methods
### Methods
#### 2.2.4 Learning-Based Image Fusion  
As mentioned above, there are significant differences in imaging mechanisms between SAR and panchromatic images. To consider the differences in the imaging mechanisms, compositive characteristics should be utilized, and non-linear relationships are required. Therefore, this study employed RF regression, which is a representative algorithm that considers compositive characteristics and models non-linear relationships. RF regression is based on the classification and regression tree (CART) model, which is an ensemble-based algorithm that combines several decision trees and obtains results [52]. For the classification, the results are obtained by most votes from the tree results, whereas for regression, the tree results are averaged [53]. In particular, each tree is created independently through a process called bootstrap aggregation, or bagging, to avoid correlations with other trees; in this process, training data subsets are drawn by randomly resampling the subsets with replacement from the original training dataset [54; 55]. Thus, this process is robust to the presence of noise or slight variations in the input data, has greater stability, and increases the prediction accuracy [56; 57]. Furthermore, in each tree, approximately 30% of the data is excluded from the training process, which is called out-of-bag (OOB) data. The mean squared error (MSE) between the OOB data and the data used for growing the regression trees is obtained; then, a prediction error called the OOB error is calculated for each variable [53]. This error estimates the importance of every variable, such that the influence of each input variable can be further analyzed. To determine the importance of the input variables, each variable is permuted, and regression trees are grown on the modified dataset [58]. The variable importance is calculated based on the difference in the MSE between the original OOB dataset and the modified dataset. In other words, if the exclusion of a variable leads to a significant reduction in prediction accuracy, the variable is considered important.  
In addition, the RF algorithm requires the specification of two parameters: the number of variables to be used for the best split at each node (\(m_{try}\)) and the number of trees in the forest (\(n_{tree}\)). In regression problems, the standard value of \(m_{try}\) is one-third of the total number of input variables; thus, in this study, \(m_{try}\) was selected as 5 [59]. Regarding \(n_{tree}\), previous studies have shown that using a large value for \(n_{tree}\) provides better stability. However, recent studies have revealed that \(n_{tree}\) has no significant effect on performance; thus, in this study, \(n_{tree}\) was selected as 32 considering both the performance and training time [28; 34; 52].  
Using the two parameters, the RF is modeled and generated independently for each class, which leads to a reduction in the complexity of the algorithm and allows more information to be retrieved. For each class, supervised learning is performed by setting the features extracted from the SAR image as independent variables and the pixel values of the panchromatic image as dependent variables for the positions corresponding to previously obtained invariant pixels. Then, the features of the SAR image corresponding to all the pixels of each class are extracted and utilized as input values of the obtained RF regression model. Finally, the fusion result is generated by integrating the predicted values for each class.