# Multilevel Regularization Method for Building Outlines
## 5 Conclusions  
Using deep learning and machine learning methods, building patches can be quickly extracted from high-resolution remote sensing images. However, because of the interference of external factors and the errors of the algorithms themselves, the extracted patches have problems such as extraction omissions and extraction errors. After vectorization, these building outlines have low regularization, unclear inflection points, and high data redundancy, which are not suitable for direct use and need regularization. However, there are relatively few studies on regularization methods for building outlines extracted from remote sensing images, and most of the results are unsatisfactory. To improve the application performance of the data and expand the application scenarios, this paper proposes a multi-stage regularization method for the initial building outlines extracted from remote sensing images. Specifically, the proposed method is divided into three steps: overall, local, and detailed regularization. Firstly, an improved Hausdorff distance method was combined with an improved MBR to perform overall building outline regularization. Then, the unregularized parts of the building outline were split into three categories: point-like, line-like, and area-like, based on the convex hull and node distribution of the building outline, and each category was regularized separately. Finally, detailed regularization adjustments and corner orthogonalization were performed based on the characteristics of the building outline to make the regularization results closer to the true outline. Through a large number of experimental verifications, our proposed method can make the data volume of the initial building outline extracted from high-resolution remote sensing images greatly compressed, the degree of regularization greatly improved, and the expression accuracy greatly increased. This can effectively improve the performance and application range of using remote sensing images to extract the building outline. Our proposed  
Figure 20: Regularization effects of building outlines with various shapes. (**a**) Completion of missing parts; (**b**) Removal of erroneously extracted parts; (**c**) Broken line simulation of curved segments.  
method is not only applicable to the building outline with obvious rectangular features but also achieves a better effect on the regularization of the building outline with non-rectangular features. However, for building outlines that contain curved segments, further improvements are needed.  
Methodology, L.K.; software, L.K. and Z.H.; validation, C.B.; formal analysis, H.Q.; investigation, L.K.; resources, Y.Q.; data curation, Y.Q.; writing--original draft preparation, L.K.; writing--review and editing, L.X.; visualization, L.K. and Z.H.; supervision, H.Q.; project administration, C.B.; funding acquisition, H.Q. All authors have read and agreed to the published version of the manuscript.  
This work was supported by The National Natural Science Foundation of China under grant Nos. 42271463, 42101453, and 42371461 and Henan Provincial Science Foundation for Outstanding Young Scholars under grant No. 212300410014.  
Not applicable.  
The datasets used and/or analyzed during the current study are available from the corresponding author upon reasonable request. The data are not publicly available due to privacy.  
The authors declare no conflict of interest.