## 2 Materials and Methods
### Monomer and Contour Extraction
#### 2.5.3 Geometric Similarity Indicators of Contours  
The following three indicators were proposed to measure the spatial geometric similarity between two different contours, acting as quantified indicators for the subsequent monomer determination and contour selection:  
1. Positional deviations of contour planes. This refers to the spatial distance between contour polygons in the XOY plane, as measured by the distance between the geometric centers of two contour polygons. Contour lines of the same building should exhibit relatively small plane space deviations at separate elevations;
2. Area deviations of contour polygons. This refers to the difference in surface areas between two contour polygons. The more the areas of two different contour lines resemble each another, the higher their geometric similarity;
3. Geometric deviations of contours. This reflects the global similarity between two contours, mainly characterized by contour shape and size. To increase the computational efficiency and facilitate the description of the above indicators, an outer bounding box of a building contour was first established when determining the geometric shape of the plane space position and contour. The plane space position of the contour is replaced by the space position of the outer bounding box, and the geometric shape of the contour is determined by the length and width of the outer bounding box. As shown in Figure 8, a series of contour sets were obtained for each building via horizontal \"slicing\" at different elevation planes. Subsequent monomer and contour selections were calculated and solved based on the geometric information of this contour series. Table 1 lists the formulas used to calculate each of the three indicators. First, the outer bounding boxes were solved for each contour. \(P_{1}\) and \(P_{2}\) are the geometric centers of the outer bounding boxes of source contour C\({}_{1}\) and target contour C\({}_{2}\), respectively. S\({}_{1}\) and S\({}_{2}\) represent the areas of these two contours. L\({}_{1}\) and L\({}_{2}\) are the heights, and W\({}_{1}\) and W\({}_{2}\) are the widths of the outer bounding boxes of these contours, respectively. Using the above three indicators, conditional strategies can be applied to determine whether two contours are similar. The indicators had to meet predetermined thresholds simultaneously before two contours could be labeled \"similar\". These conditions were as follows:  
1. The deviation of the plane position is not more than 0.1 m, the threshold value is empirical, and the position similarity (SimP) is calculated as follows: \[\text{SimP}=\left\{ \begin{array}{l}\text{{True},\;\Delta\;\leq\;0.1\text{m} }\\ \text{{False},\;\Delta\;\text{D}>0.1\text{m}}\end{array} ;\right.\] (9)
2. The difference between the length and width of the outer bounding box adopts a relative threshold that does not exceed 1/20 of the sum of the compared quantities when geometric similarity (including length and width similarity) is calculated as \[\text{SimL}=\left\{ \begin{array}{l}\text{{True},\;\Delta\;\leq\;(L_{1}+L_{2} )/20}\\ \text{{False},\;\Delta\;\text{D}>(L_{1}+L_{2})/20}\end{array} ,\right.\] (10) \[\text{SimW}=\left\{ \begin{array}{l}\text{{True},\;\Delta\;\text{W}\;\leq\;( W_{1}+W_{2})/20}\\ \text{{False},\;\Delta\;\text{W}>(W_{1}+W_{2})/20}\end{array} ;\right.\] (11)
3. The contour area deviation adopts the relative threshold of no more than 1/20 of the sum of the comparison contests. The similarity in the contour areas (SimS) is determined as follows: \[\text{SimS}=\left\{ \begin{array}{l}\text{{True},\;\Delta\;\text{S}\;\leq\;( S_{1}+S_{2})/20}\\ \text{{False},\;\Delta\;\text{S}>(S_{1}+S_{2})/20}\end{array} ;\right.\] (12)
4. When the values obtained for SimP, SimL, SimW, and SimS are all \"True\", the two contour profiles were considered similar.