# Surface Reconstruction Planning with High-Quality Satellite Stereo Pairs Searching
## 3 Quality Estimation of Earth Surface Reconstruction
### Regression Model for Quality Estimation
#### 3.2.1 Selection for High-Dimensional Features  
The characterization features for quality estimation consist of satellite imaging meta-parameters such as solar elevation angle, solar azimuth angle, time interval, intersection angle, etc., leading to a multi-dimensional regression problem. Because these raw parameters have different influences on the final accuracy of 3D reconstruction, a new form of features is constructed by combining the raw features of an image pair. For example, the difference of two solar azimuth angles is an intersection of two raw features, leading to a new representation. Besides the difference, the operations of the minimum, maximum, root of sum of squares, etc., are also applied to enhance the features. Because not all constructed features are equally important, the Pearson Correlation Coefficient map is calculated for filtering and enhancing the features as shown in Figure 3, in which various features are depicted and compared.  
From Figure 3 we can find that the features derived from solar elevation and azimuth angles are more strongly correlated with DSM accuracy, including \(s\_a\_diff\) (difference of sun azimuth angle), \(s\_e\_diff\) (difference of sun elevation angle), etc., which will be further utilized for feature enhancement. Logarithmic transformation is applied to these chosen parameters to capture the nonlinear relationships [26; 27; 28]. The transformed features are then appended to the previous features. These efforts construct a richer and more diverse feature space, thus providing a more solid dataset for model training, to better capture complex patterns in the data and improve the quality estimation accuracy, together with the LASSO-based feature selection.  
LASSO can automate feature selection by introducing an absolute value-based penalty term (L1 regularization) on top of the traditional least squares method, which not only effectively shrinks the model coefficients, but also reduces unimportant feature coefficients to zeros [25]. This characteristic makes LASSO well suited to deal with high-dimensional data, reducing overfitting while improving the interpretive and predictive performance of the model. Inspired by reference [29], the optimization for quality linear estimation can be formalized as follows:  
\[\min_{C\in\mathbb{R}^{d}}\frac{1}{2}\
orm{Y-X^{T}C}_{2}^{2}+\lambda\
orm{C}_{1} \tag{1}\]  
where \(C=[c_{1},\dots,c_{d}]^{T}\) is the regression coefficient vector, and \(c_{k}\) is the regression coefficient for the \(k\)-th feature; \(1\leq k\leq d\). \(\lambda\
orm{C}_{1}\) is a penalty specific to regression, using L1 norm \(\
orm{\cdot}_{1}\). This encourages some coefficients to become 0 to achieve the effect of feature selection. \(\lambda\) is the regularization parameter and controls the penalty intensity. In Equation (1), each sample \((\mathbf{x}_{i},y_{i})\) (\(1\leq i\leq n\)) contains the enhanced features \(\mathbf{x}_{i}\) of a stereo pair and the corresponding quality value \(y_{i}\).  
The above LASSO-based optimization achieves sparse features used for quality estimation, as shown in Figure 4. The positive feature value indicates that the feature is positively correlated with the reconstruction quality, and vice versa. As shown in Figure 4, there are two features whose coefficients become 0, indicating their irrelevance or redundancy. It can be seen that according to the current feature selection, most of the feature extensions are selected according to the correlation coefficient. Compared to the original features, the feature with enhanced data has a higher selection coefficient in the model, which indicates that data enhancement has a significant positive effect on improving the quality of modeling data.  
Figure 3: Feature correlation heatmap.