# Multilevel Regularization Method for Building Outlines
## 2 Materials and Methods  
### Principles and Specific Steps  
The principle of the proposed regularization method is as follows: First, based on the overall morphological characteristics of the building outline, the MBR is combined with the Hausdorff distance to perform overall regularization of the outline. This step ensures that the overall morphology and main orientation of the building outline are preserved as effectively as possible before and after regularization. Then, based on its complexity,the building outline is progressively split and locally regularized in a hierarchical manner to preserve the key details of complex outlines. Finally, considering building outline properties, such as parallelism, perpendicularity, normalization, and orthogonality, the building outline is further refined to make it more consistent with the actual characteristics of the building. The specific process consists of three steps (as shown in Figure 1).  
Step 1: Overall Regularization (Process A).  
First, the MBR of the building outline is calculated. Then, the parts of the original building outline whose Hausdorff distance from the MBR is less than a specified threshold are replaced with corresponding line segments on the MBR. This process yields the overall regularization result, as shown in Figure 1c.  
Step 2: Local Regularization (Process B).  
Based on the convex hull and node distribution characteristics of the building outline, the non-regularized parts of the outline are divided into multiple line chains and classified into three categories: point-like, line-like, and area-like. Then, each category is regularized separately. Complex line chains are iteratively split and classified for further regularization. As shown in Figure 1, after the splitting and regularization process from B\({}_{1}\) to B\({}_{2}\), locally unregularized parts remain in the building outline. Thus, it is necessary to continue classifying and regularizing the unregularized parts, as shown in the process from B\({}_{3}\) to B\({}_{4}\) in Figure 1. After processing all of the nodes, the local regularization result is obtained, as shown in Figure 1d.  
Step 3: Detail Regularization (Process C).  
Based on the characteristics of the building outline, acute convex nodes, duplicate nodes, collinear nodes, and missing nodes are processed. Parallel and perpendicular relationships are also strengthened to obtain the detailed regularization result, as shown in Figure 1e.  
In Figure 1, a comparison between Figure 1b,e reveals that the initial building outline extracted using deep learning has redundant nodes and jagged edges and lacks clear  
Figure 1: Flowchart of the proposed method. The connected arrows represent the regularization processes of the building outline. (**a**) The initially extracted building outline overlaid on a remote sensing image; (**b**) The distribution of nodes in the initial building outline; (**c**) The overall regularization result; (**d**) The local regularization result; (**e**) The morphology of the building outline and node distribution after regularization; (**f**) The outline regularization result overlaid on the remote sensing image.  
corner points. After regularization, the number of nodes is significantly reduced, the edges are more regular, and the corner points are clear, highlighting the parallel and orthogonal elements.