# Multilevel Regularization Method for Building Outlines
## 1 Introduction
Geometric correction methods primarily optimize building outlines using geometric features, such as the direction, angle, and orthogonality of the straight lines constituting the building outlines. [PERSON] et al. [11] firstly used the Mask R-CNN algorithm to solve the initial outlines of buildings, then employed the Douglas-Peucker algorithm [12] to simplify the outlines. Finally, they iteratively obtained the adjacent three nodes on the building outline in order, combining them with the minimum description length model for stepwise regularization. [PERSON] et al. [13] separately extracted a building outline from high-resolution remote sensing images from two different periods and obtained its minimum bounding rectangle (MBR). To this end, the Hausdorff distance algorithm was adopted for corner point matching and optimal edge selection. Finally, the obtained edges were reorganized to obtain the building outline. [PERSON] et al. [14] preliminarily simplified the outline using the convex hull method, the perpendicular distance method, and an improved Douglas-Peucker algorithm. Then, the least squares method was used for line fitting and finding the intersections of the lines. This step was followed by edge regularization and corner orthogonalization. Finally, the maximum area overlap degree was utilized to optimize the building outline further. Geometric correction methods fully utilize the point and line features of building outlines for outline reconstruction, alleviating problems such as jagged outlines, ambiguous corner positions, and high data redundancy caused by spectral characteristics. However, these approaches have limitations when regularizing the outlines of buildings with curved or overly complex shapes. Model matching methods fit and replace the overall or partial outline of a building using regular geometric models, thereby regularizing the outline. [PERSON] et al. [15] first utilized the shifted shadow segmentation and classification method to perform polygon fitting on the building outline. Then, the building outline and MBR were divided respectively into equal segments. Each segment was optimized using the Hausdorff distance method. [PERSON] et al. [16] regularized the building outline using MBR fitting and the external circle method for adjacent points on the outline. Model matching methods increase the regularization accuracy by adding more models but reduce the regularization efficiency. The regularization effect significantly depends on the accuracy of the initial extraction results, limiting the practicality of these approaches. In addition, researchers have used auxiliary information, such as point clouds, digital elevation map data, and surrounding environments, or incorporated deep learning methods to optimize building outlines further. For example, [PERSON] et al. [17] utilized digital surface model data to obtain elevation information and combined data-driven and model-driven approaches to regularize the building outline, effectively solving the problem of omitting some parts of the building. [PERSON] et al. [18] combined building outlines extracted from point cloud data and high-resolution remote sensing images with edge segments. They selected edge lines, corrected rough outlines, and repaired outline corners to regularize the outlines. [PERSON] et al. [19] first used a fully convolutional network model [20] to segment building clusters into individual buildings. Next, individual building outlines were vectorized into polygons, and geometric methods were employed for coarse and fine regularization. [PERSON] et al. [21] introduced pyramid pooling, residual structure, and convolutional block attention modules into the U-Net model and established the PRC-UNet model to obtain the building outline. Subsequently, the MBR and Hausdorff distance algorithm were utilized to optimize the outline further. Owing to the efforts of many researchers, the process of regularizing building outlines from high-resolution remote sensing images has been significantly improved.