# Surface Reconstruction Planning with High-Quality Satellite Stereo Pairs Searching
## 3 Quality Estimation of Earth Surface Reconstruction  
### Earth Surface 3D Reconstruction  
In this paper, we utilize a classical 3D reconstruction pipeline to generate point clouds from pairs of images along with their associated camera poses. This pipeline consists of two main steps of feature-based image rectification and triangulation-based dense reconstruction [2] as shown in Figure 2. Given that space-borne imaging platforms are typically positioned at high elevations, the objects on the ground can be reasonably approximated as being coplanar. Therefore, homography transformation is employed as the core method to establish geometric relationships between the two image planes in each stereo pair.  
Figure 1: Framework of quality-driven stereo planning.  
During the image rectification, Scale-Invariant Feature Transform (SIFT) is a commonly used method [20]. First, SIFT is employed to detect and describe feature points in a pair of images \(A\) and \(B\). For image \(A\), we consider it as the reference image for rectification and set its homography matrix \(H_{1}\) to the identity matrix. As for image \(B\), the homography matrix \(H_{2}\) relative to image \(A\) is estimated based on feature matching. Sparse feature point sets are selected as initial inputs, combined with the Random Sample Consensus (RANSAC) algorithm for robust estimation [21]. Through iterative solving, the optimal homography matrix \(H_{2}\) is obtained, which is then applied to geometrically transform the feature points from image \(B\) to align with that of image \(A\). This process can be described as \(x_{A}=H_{2}\cdot x_{B}\), where \(x_{A}\) and \(x_{B}\) denote the homogeneous coordinates of feature points from images \(A\) and \(B\), respectively. Due to the inability of homography transformation to handle the rectification of local pixels with significant spatial differences, the SiftFlow algorithm is employed for dense optical flow estimation to refine the initial rectification (top-right dashed box in Figure 2), which can effectively capture subtle motion information between images to improve the coverage of rectification [22]. Afterwards, the dense matching results of two-dimensional coordinates, excluding occlusion effects, are obtained to support the triangulation-based dense reconstruction.  
Absolute scale information is necessary for the task of 3D reconstruction from satellite images. The RPC (Rational Polynomial Coefficients) model is widely used in optical satellite imagery because it not only incorporates scale information but also allows for the description of complex acquisition systems without depending on physical modeling specific to the satellite [23]. Each RPC is defined by a 3D-2D projection function to map Earth coordinates to image pixels and its inverse function, which maps image pixels back to Earth coordinates.  
After the rectification, the dense triangulation is conducted for the generation of a high-quality dense reconstruction, which generates the 3D coordinates of ground points by solving the intersection of back-projected rays from matched pixel pairs obtained in the previous step. First, the pixel coordinates are converted to normalized coordinates according to the RPC model. Using these normalized coordinates, an initial estimation of the 3D point position is derived through linear least squares fitting. This estimation is then iteratively refined by minimizing the reprojection error, which is the difference between the observed pixel coordinates and the predicted coordinates obtained by projecting the  
Figure 2: Three-dimensional surface reconstruction using satellite image pair.  
estimated 3D point back onto the image planes. The process continues until the reprojection error converges to a sufficiently small value or the maximum number of iterations is reached. Once the dense triangulation is completed, the reconstruction results are output, which accurately represent the 3D structure of the scene.