# Multilevel Regularization Method for Building Outlines
## 1 Introduction  
Buildings constitute important artificial ground objects that are closely related to human activities. Building outline data have high application value in various fields, such as map creation and updating, disaster analysis, emergency rescue, urban planning, change monitoring, 3D modeling, and population assessment [1]. In recent years, the rapid development of \"smart cities,\" spatial big data, and artificial intelligence has accelerated the demand for diverse, accurate, and timely building outline data [2]. High-resolution remote sensing images contain rich ground object features, including color, size, shape, texture, and spatial relationships between objects [3]. By employing deep learning and machine learning techniques, building outlines can be extracted rapidly from high-resolution remote sensing images [4, 5] and building outline data can be updated in a timely manner. However, high-resolution remote sensing images contain many types of ground objects and face problems such as \"same ground objects, different spectra\" and \"same spectra, different ground objects.\" Consequently, accurately identifying the original building outline shapes is difficult, resulting in issues such as omission and misinterpretation [6]. Currently, there are further studies on methods for extracting building outlines from remote sensing images using deep learning and machine learning techniques. However, the extracted outlines are generally in the form of patches and often have problems such as irregular boundaries, unclear inflection points, etc., and high data redundancy. Directly using the extracted initial outline would significantly affect the effectiveness of the use. By contrast, manual repair of the initial outline would significantly reduce work efficiency. To obtain more realistic,accurate, and complete building outlines, studying automatic regularization methods for the outlines is important [7]. Regularization can establish a bridge between \"extraction\" and \"application\" and improve the usability of remote sensing images for extracting building outline data. Currently, building outlines are mainly regularized by image morphology, geometric correction, and model matching. Image morphology methods primarily utilize the texture and shape characteristics of building outlines and regularize building outlines through morphological operations such as erosion, dilation, opening, and closing. [PERSON] et al. [8] first filled the rasterized information of the building outline and then employed morphological algorithms, such as erosion and dilation, to optimize the jagged parts of the outline. Finally, a binary image boundary extraction algorithm was employed to extract the optimized results. [PERSON] et al. [9] initially over-segmented the images through simple linear iterative clustering to obtain super-pixels, and then used seed lines obtained through manual interaction to determine candidate foreground and background images for target buildings. Afterward, a star-shaped shape-constrained image segmentation algorithm was adopted to obtain building patches, and the Harris corner detection algorithm was applied to extract corner points from these patches. The extracted points were grouped, linearly fitted, and rotated to obtain regularized building outlines. [PERSON] et al. [10] first acquired eigenvectors for segmentation based on the angle and texture characteristics of buildings. Then, the graph-cut method was used to determine candidate building points and, after mathematical morphological processing, to obtain building patches. Finally, the corner points of the buildings were obtained by referring to each building's main direction and least squares matching template and were then connected to yield the building outline information. Although image morphology methods improve the accuracy and speed of image processing during extraction, they may introduce problems such as node distortion, data redundancy, and endpoint contraction.