# A Global Continuous Multi-Scale Terrain Contour Simplification Method Based on the Level Set
## 1 Introduction
[23] proposed an improved morphing method for two linear features at different scales based on their entire structural information. [PERSON] et al. [9] used simulated-annealing-based morphing (SABM) to implement continuous linear features generalization. [PERSON] et al. [24] proposed a progressive simplification method based on multi-bend groups that can delete bend units as little as possible. This method has advantages in area preservation during the multi-scale representation. [PERSON] et al. [1] proposed a Fourier-based method using head/tail breaks that simplifies polylines via a Fourier approximation according to the global shape. These scale-driven simplification methods have advantages over the two aforementioned strategies (i.e., point set reduction and bending simplification). However, they lack domain-specific constraints in the scale-driven simplification process, thereby producing simplified results that do not completely follow the essential global structure of the initial line at continuous scales. In sum, the three aforementioned simplification strategies still show limitations in continuous multi-scale representation. On the one hand, simplifications occur only through a few fixed scales. This problem makes it impossible to show data from one scale to another smoothly and ignores the actual morphological transformations of the contour line of desired scales between fixed scales. On the other hand, the local structures are overemphasized, and the global structures of linear characteristics are ignored during the simplification of continuous multiple scales, which may lead to a sudden change in the shape of the contour line ('jumps') during transformation. Given these limitations, this study proposes a novel contour simplification method to retain the global structure of linear characteristics and to implement a multi-scale deformation of contours continuously. This method is labeled the continuous changing surface model (CCSM) based on the notion of the level set method. The core idea of the level set method [26; 27] is that the motion of lower-dimensional (e.g., 2D) curves can be derived from a higher dimensional (e.g., 3D) surface. Accordingly, our simplification model derives 2D simplified contours from a meticulously constructed 3D surface labeled the continuous changing surface (CCS). According to the contour features, a CCS is established between two contours and is then intersected with a set of horizontal planes with continuous height values. The generated intersection lines are considered simplified contours with continuous scales. We also compare our proposed method with two traditional significant polyline simplification methods on a real contour dataset in China to validate the effectiveness of CCSM. Results show that the shape changes of contours simplified by CCSM are natural and continuous between scales and that the CCSM maintains the global shape features much better than the other two classical significant simplification algorithms. The major original contributions of this study are twofold. First, we propose the CCSM simplification method guided by the level set framework to derive simplified contours with continuous multi-scale transformation, thereby reducing the loss of deformed details during the continuous multi-scale simplification process. Second, we propose a new contour deformation analysis method, namely, a shape similarity-scale trend line based on the improved Frechet distance, which provides a tool for evaluating the performance of multi-scale polyline simplification methods. The rest of this paper is organized as follows. Section 2 elaborates on the proposed method. Section 3 describes the experiment using the contour dataset and reports the results. Section 4 discusses two key factors affecting the quality of CCSM, the arbitrary scale simplification, and the retention of topographical features. Section 5 concludes the paper.