Progressive Collapse of Dual-Line Rivers Based on River Segmentation Considering Cartographic Generalization Rules

Abstract

Collapse is a common cartographic generalization operation in multi-scale representation and cascade updating of vector spatial data. During transformation from large- to small-scale, the dual-line river shows progressive collapse from narrow river segment to line. The demand for vector spatial data with various scales is increasing; however, research on the progressive collapse of dual-line rivers is lacking. Therefore, we proposed a progressive collapse method based on vector spatial data. First, based on the skeleton graph of the dual-line river, the narrow and normal river segments are preliminarily segmented by calculating the width of the river. Second, combined with the rules of cartographic generalization, the collapse and exaggeration priority strategies are formulated to determine the handling mode of the river segment. Finally, based on the two strategies, progressive collapse of dual-line rivers is realized by collapse and exaggeration of the river segment. Experimental results demonstrated that the progressive collapse results of the proposed method were scale-driven, and the collapse part had no burr and topology problems, whereas the remaining part was clearly visible. The proposed method can be better applied to progressive collapse of the dual-line river through qualitative and quantitative evaluation with another progressive collapse method.

1. Introduction

Multi-scale representation and cascade updating of vector spatial data are important aspects of research on geographic information systems. At present, surveying and mapping departments in numerous countries, such as the National Map Agency of China (1:50,000, 1:250,000, 1:50,000, and 1:1,000,000, etc.) and the Federal Cartographic and Geodetic Bureau of Germany (1:50,000, 1:100,000, 1:200,000, 1:250,000, 1:500,000, and 1:1,000,000, etc.), use fixed scales to describe the real world. With the development of information communication technology, map representation and creation have become more extensive and popular [1,2]. The existing vector spatial data production mode with fixed scales can no longer meet the user’s needs for vector spatial data with various scales. Meeting these needs depends on the further development of cartographic generalization technology. Cartographic generalization is a scale- and knowledge-driven spatial data processing mode [3,4]. Collapse is the basic operation of cartographic generalization [5,6,7].

As an important part of a map, rivers exhibit numerous morphological characteristics: they have self-similarity and non-smoothness [8]; father-son relationships between the mainstream and influent [7,9]; and they could be multi branch, multi-level, and multi-curve rivers [6]. River cartographic generalization includes river selection, river simplification, and dual-line river collapse. Influenced by the regional geology, topography, climate, river inflow, and other factors, rivers have different width and shape characteristics in different river segments. With the reduction of the map scale, dual-line rivers are preferentially collapsed from the narrow part to the line representation, reflecting the progressive collapse process to ensure visual progressive and graphical clarity. Simultaneously, the progressive collapse of dual-line rivers conforms to the basic connotation of the multi-scale representation of vector spatial data from detailed to rough [3,5]. Therefore, the collapse of dual-line rivers is not only a problem of skeleton line extraction, but also involves numerous constraints, such as geometric characteristics, geographical characteristics, and spatial relationships. However, most of the existing skeleton line extraction methods are based on geometric characteristics and rarely consider geographical characteristics and spatial relationships; they also lack consideration for the progressive collapse process [6,10]. Since DeLucia and Black [11] first proposed the Delaunay triangulation method for polygon skeleton line extraction, Delaunay triangulation has been widely used owing to its spatial proximity, computability, and other advantages [7,12,13,14,15,16]. Considering the characteristics of the geometric, geographical, and spatial relationship of rivers and the cartographic generalization rules of collapse processing, a method based on Delaunay triangulation is proposed to realize the progressive collapse of dual-line rivers. The main objectives of this study are as follows: (1) from the perspective of vector spatial data, we aim to realize the progressive collapse process of the dual-line river and ensure that the collapse results are free of burrs and topological problems and that the other parts are visible at the target scale; (2) considering the cartographic generalization rules of the dual-line river, reasonable collapse strategies are designed to ensure that the progressive collapse results meet the practical application requirements.

The rest of the paper is structured as follows. In Section 2, the related work is described. In Section 3, a new method for the progressive collapse of dual-line rivers is proposed, which mainly includes structural expression of dual-line rivers, progressive pre-segmentation of rivers, and determination of the river segment handling mode and progressive collapse. In Section 4, the use case is realized through a real dual-line river example, and the experimental results of the proposed method and the comparison method are compared and evaluated. Finally, Section 5 describes the conclusions and discusses potential improvements.

2. Related Work

This study involves two aspects: skeleton line extraction and progressive collapse, which are introduced as follows.

2.1. Skeleton Line Extraction Method

As a one-dimensional characteristic expression of two-dimensional entities, skeleton line extraction methods have been widely used by scholars in computer graphics, computer vision, and other interdisciplinary fields, and numerous results have been obtained, including:

• Delaunay triangulation and Voronoi diagram [7,11,12,13,14,15,16,17,18,19,20];
• Straight skeletons [21,22,23,24];
• Thinning [25,26,27,28,29];
• Distance transformation [30,31,32];
• Superpixel segmentation [6,33];
• Skeleton line extraction based on deep learning [34,35,36,37].

Thus, with the development of deep learning technology, researchers have also focused on skeleton line extraction based on deep learning. In the study of cartographic generalization, the extracted skeleton lines can conform to the results of manual extraction [7]. However, the actual operation involves the partial collapse of the dual-line river, and the existing skeleton line extraction methods are not competent for progressive collapse.

2.2. Progressive Collapse Method of Dual-line Rivers

The progressive collapse of dual-line rivers is more in line with the requirements of multi-scale expression of spatial data, resulting in relevant research by several scholars. For instance, Liu et al. [10] segmented the river into different segments by fixing the step length, detected the spatial graphic conflicts by establishing a buffer zone of the central axis of the dual-line river, resolved the spatial graphic conflicts according to the mapping rules, and modified the comprehensive results to realize the partial collapse operation of the dual-line river. Shen et al. [6] segmented the dual-line river by multi-level superpixel segmentation, extracted the center and midpoint of the boundary of the superpixel using image processing technology, and realized progressive collapse of the dual-line river by formulating a skeleton line connection strategy and polynomial approximate interpolation.

At present, research on the progressive collapse of dual-line rivers and related research findings are lacking. Simultaneously, the application of existing progressive collapse research results to multi-scale expression of vector spatial data warrants further demonstration.

3. Methodology

3.1. Framework

In the proposed method, geometric characteristics are used as the basis for the segmentation of dual-line rivers. After segmentation, two strategies for collapse of dual-line rivers are formulated by considering the rules of cartographic generalization based on which the progressive collapse of dual-line rivers is realized. Figure 1 shows the basic process of the proposed method. The specific implementation will be explained in detail in the following sections.

3.2. Structural Expression of Dual-Line Rivers

Constrained Delaunay triangulation (CDT) was used to detect the narrow segment of dual-line rivers, calculate its length, and extract its skeleton line. The structural model refers to the skeleton graph [7]. First, a CDT is constructed for the shoreline of a dual-line river, the edge of the triangle that coincides with the dual-line river shoreline is defined as a constrained edge, and the edge shared by two adjacent triangles is defined as an unconstrained edge. Triangles are divided into three types according to the number of unconstrained edges (as shown in Figure 2a): type I, a triangle with only one unconstrained edge; type II, a triangle with two unconstrained edges; and type III, a triangle with three unconstrained edges.

Second, triangles are regarded as the node, and the adjacent spatial relationships between triangles are considered as links; the CDT of a dual-line river is abstracted as a graph, $SG=(V,E)$, where $V$ is a set of all the nodes and $E$ is a set of all the links. The weight of the links between two adjacent nodes is calculated as the distance between their corresponding triangle centroids. According to the degree of nodes, nodes are divided into three types (as shown in Figure 2b): nodes with degree 1 are defined as end nodes, which correspond to type I triangles; nodes with degree 2 are defined as connection nodes, which correspond to type II triangles; and nodes with degree 3 are defined as joint nodes, which correspond to type III triangles.

After $SG$ is constructed, some nodes in the graph are labelled according to the spatial relationship. The triangle where the linear river contacts the dual-line river is extracted, and the corresponding node of the triangle is labelled (as shown in Figure 2c).

The establishment of $SG$ lays a solid foundation for the application of graph theory, including the traversal of nodes in the graph, establishment of hierarchy between paths, and pruning of short paths. Moreover, graph node labelling can avoid the topological problem of extracted skeleton lines attributed to the deletion of short paths with linear river connection.

3.3. Progressive Pre-Segmentation of the River Segment

Rivers in the same river system in nature are presented as a whole. For identification and navigation, human beings name and distinguish rivers and maintain this distinction in the spatial data of river system, such as the Yangtze River and its tributary, Hanjiang River. The segmentation of the river segment needs to be based on different application needs. For example, the Yellow River is divided into the upper, middle, and lower segments according to the natural environment and hydrological conditions of the river flow area. The Yangtze River is segmented into the Chongqing segment and Hubei segment based on the current administrative region of the river. Therefore, river segmentation is a process based on human cognition, which needs to be combined with the actual application needs.

Ai and Guo [38] proposed the use of the average height of each Delaunay triangle as the average width or to calculate the weighted average value of the central axis as the average width of dual-line rivers. In this study, based on the advantage of the Delaunay triangulation, the width of each triangle was calculated to obtain the local width of the dual-line river, and the adjacent triangles were connected in series to generate the river segment according to the width threshold $W_{thd}$. The river segment is divided into two types: normal river segment and narrow river segment. The specific division process is as follows:

• The $SG$ is traversed, and the $Len(tr{i}_i)$, which is the width of triangle $tr{i}_i$ corresponding to node $v_i$ in the graph, is calculated in turn. According to the different types of triangles, the widths of type I, II, and III triangles are calculated as the length of the unconstrained edge, shortest distance from the constrained edge to the relative vertex, and the average of the lengths of three edges, respectively.
• According to the width calculation result, the attribute issmall of the node $v_i$ corresponding to the triangle meeting $Len(tr{i}_i)<W_{thd}$ is labelled as True; otherwise, the attribute issmall of the node $v_i$ is labelled as False (as shown in Figure 3b). The attribute issmall of the node $v_i$ is used to distinguish whether it meets $Len(tr{i}_i)<W_{thd}$.
• All links in the $SG$ are traversed in turn, and the $SG$ can be divided into different subgraphs by judging whether the nodes at both ends of the link are associated with different issmall attributes (as shown in Figure 3c). These subgraphs can be subdivided into two types: the issmall attribute of the included nodes is True, which is defined as narrow subgraphs; and the issmall attribute of the included node is False, which is defined as a normal subgraph.
• The subgraphs are abstracted as new nodes, and the connection relationship between subgraphs is abstracted as links to obtain a more abstract skeleton graph $S{G}^{\prime }$ (as shown in Figure 3d). By traversing different subgraphs, the triangle sets corresponding to the nodes of subgraphs are obtained, and the pre-segmented river segments are obtained by merging them respectively. Among them, the merging result of triangles in a narrow subgraph corresponds to a narrow river segment, whereas the merging result of a normal subgraph corresponds to a normal river segment.

Owing to the zigzag change of the river shoreline, a situation may arise where the three vertices of the triangle are located on the same side of the shoreline. Most of these triangles are small in width, and their collections can be easily identified as narrow river segments. However, these river segments are small in area and width, and should be classified as adjacent bigger river segments in normal cognition. As shown in Figure 4, the existence of such river segments will result in a short linear river during collapse, affecting the accuracy of the collapse results. Therefore, it is necessary to use a small length threshold $L_{prune}$ to pre-prune the skeleton graph, label such short river segments, and refrain from calculating their geometric information. Before dividing the subgraph, the issmall attributes of the pre-pruned nodes are synchronized with the issmall attributes of the adjacent nodes.

According to different scales, different width thresholds $W_{thd}$ (100 m, 150 m, 200 m, and 300 m) are set respectively. The $SG$ is traversed, the width of the triangles are calculated and labelled, and progressive pre-segmentation of the river segment through subgraph division is achieved. The pre-segmentation results of the river segment are shown in Figure 5.

3.4. Determination of River Segment Handling Mode

In cartographic generalization, the collapse of a narrow river segment into a line is a basic operation. However, the complexity of the river shape results in uneven changes in the width of the river, and the narrow river segment and normal river segment have different lengths and interwoven distributions (as shown in Figure 5b,c). Unreasonable results are possible by merely judging the width of the river segment for collapse handling. In cartographic generalization, it is necessary to exaggerate the narrow river segment in some cases; simultaneously, cases of collapse of the normal river segment to the skeleton line also exist. Therefore, for a certain river segment, its potential handling modes include collapse, exaggeration, and keeping it unchanged. The determination of the handling mode is a complex judgment process, which should consider not only the geometric information of the current river segment, but also the geometric information of the adjacent river segments.

In Section 3.3, progressive pre-segmentation of a dual-line river is realized, which is an accurate segmentation. This section quantifies the geometric information of the segmented dual-line river, and the quantitative index is the length of the dual-line river segment to determine the handling mode. In this study, the river segment length is represented as the skeleton line length of the current dual-line river segment but does not include the skeleton line of the pre-pruning part. As shown in Figure 6, for the dual-line river segment $RiverSe{g}_i$ corresponding to subgraph $SubS{G}_i$, the skeleton line is extracted by CDT, and the length is calculated as the extracted skeleton line length, which is represented by $Len(RiverSe{g}_i)$. When $Len(RiverSe{g}_i)<L_{thd}$ is met, $RiverSe{g}_i$ is a short river segment and $SubS{G}_i$ is a short subgraph; otherwise, $RiverSe{g}_i$ is a long river segment and $SubS{G}_i$ is a long subgraph. $L_{thd}$ is the length threshold to measure whether the dual-line river segment needs to be collapsed or exaggerated.

After the division of long and short river segments, for a narrow and short river segment, its potential handling methods include exaggeration and collapse, while for a normal and short river segment, its potential handling methods include collapse and remaining unchanged. Based on the analysis, for an isolated short river segment, that is, $SubS{G}_i$ which is a short subgraph in $S{G}^{\prime }$, the adjacent subgraphs are long. Three cases of distinguishing narrow and normal attributes of dual-line river segments exist, which should be handled separately. To ensure clarity and continuity of graphic expression following the collapse of the dual-line river, the handling mode of the isolated short river segment is designed. As shown in Figure 7a, when both ends of the narrow short river segment are adjacent to the normal long river segment, it is necessary to exaggerate it; when one end of a narrow short river segment is adjacent to a normal long river segment and the other end is adjacent to a linear river, it needs to be collapsed; and when only one end of a narrow short river segment is adjacent to a normal long river segment, it also needs to be collapsed. As shown in Figure 7b, the normal short river segment needs to be collapsed in all three cases.

For the non-isolated short river segment, that is, when multiple short subgraphs are connected in series in $S{G}^{\prime }$, as shown in Figure 8, eight cases are obtained according to the actual situation. The determination of handling modes for each short segment is complex, and the modes interact with each other. According to the requirements of topographic map compilation specification, the dual-line river segments with frequent changes of line and dual-line are represented by line or dual-line as a whole. Therefore, two strategies are formulated to deal with this situation, namely exaggeration priority strategy and collapse priority strategy. The exaggeration priority strategy emphasizes the exaggeration of the short river segment, whereas the collapse priority strategy emphasizes the collapse of the short river segment. The two strategies are opposite to each other, and each has its own emphasis. The specific implementation process of the two strategies is as follows:

First, the total length $L={\displaystyle \sum _{i=1}^{n}Len(RiverSe{g}_i)}$ of the adjacent short river segments is calculated, where $n$ is the number of short river segments and $n\ge 2$; then, according to whether the total length $L$ is greater than $L_{thd}$, the handling mode of the short river segments is determined according to eight cases (as shown in

Table 1. Cases of exaggeration priority strategy.

Dividing Cases	Handling Modes
	$\mathit{L}<{\mathit{L}}_{\mathit{t}\mathit{h}\mathit{d}}$	$\mathit{L}\ge {\mathit{L}}_{\mathit{t}\mathit{h}\mathit{d}}$
Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
Case 8

and

Table 2. Cases of collapse priority strategy.

Dividing Cases	Handling Modes
	$\mathit{L}<{\mathit{L}}_{\mathit{t}\mathit{h}\mathit{d}}$	$\mathit{L}\ge {\mathit{L}}_{\mathit{t}\mathit{h}\mathit{d}}$
Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
Case 8

). When the edge of the adjacent short river segment is adjacent to the linear river or suspended (as shown in cases 4, 5, 6, 7, and 8 in Figure 8), special handling is required. When the short river segment at the edge is narrow, the length of the short river segment is not considered in the calculation of $L$, that is, the suspended narrow river segment needs to be collapsed, and the handling is not exaggerated under any strategy to reflect progressive collapse.

3.5. Progressive Collapse of the Dual-Line River

After the handling mode of the river segment is determined, the skeleton line is required for the handling of the dual-line river segment for collapse or exaggeration, except that the dual-line river segment remains unchanged. First, the hierarchical relationship of $SG$ is established to guide the hierarchical connection of skeleton lines in type III triangles. Second, the breadth first algorithm in graph theory is used to obtain the graph diameter based on $SG$, and the main path is generated. The secondary path is traced from the main path joint node, and so on, to obtain the hierarchical relationship of each path. Finally, the skeleton line of the dual-line river segment to be exaggerated or collapsed is extracted through the hierarchical connection and topological connection. The specific implementation of the extraction process can be seen in the literature [7]. The skeleton lines extracted in the above steps have certain advantages in natural continuity, hierarchy, and topological consistency, and are closer to the results of manual extraction.

For collapse of the dual-line river segment, its skeleton line is directly used to replace it. For exaggeration of the dual-line river segment, the skeleton line is used as a buffer zone with radius $W_{thd}/2$ (as shown in Figure 9a). Then, the buffer zone is merged with the river segment as a result of exaggeration (as shown in Figure 9b).

The collapse priority strategy is considered as an example; progressive collapse is carried out for the dual-line river segment with different thresholds $W_{thd}$ (100 m, 150 m, and 200 m). According to the topographic map compilation specification prepared by the China Institute of Surveying and Mapping Standardization, rivers with a width of more than 0.4 mm on the map are represented by dual-line rivers in scale, and those with a width of less than 0.4 mm are represented by linear rivers. However, the compilation specification does not give the length value required for the collapse or the relationship between $L_{thd}$ and $W_{thd}$. In the actual collapse operation, the collapse operation depends more on the experience of cartographers, and the judgment standards of different cartographers are also different; thus, quantifying the implementation of this process is challenging. Therefore, we set $L_{thd}=5\cdot W_{thd}$ by comprehensively considering the compilation specification and the experience of the cartographer. However, it is not fixed and needs to be set according to the different situation. The progressive collapse results are shown in Figure 10a–c. Simultaneously, we set $W_{thd}=150\mathrm{m}$ to implement the exaggeration priority strategy during progressive collapse, and the implementation results are shown in Figure 10d. By analyzing and comparing 1, 2, 3, and 4 in Figure 10b,d, the progressive collapse results of the two strategies have obvious differences. For the collapse priority strategy, the collapsed results are closer to those of complete collapse; for the exaggeration priority strategy, the astringency of the collapsed results is slow, and after exaggerated handling, the narrow part of the dual-line river is clearly visible at the target scale.

4. Experimental Results and Evaluations

4.1. Experimental Design and Setting

To test the behavior of the proposed method, we used Python programming language to realize the progressive collapse on the shapefile structure of the river. The test was carried out on a computer equipped with Intel Core i7-8565U 1.80 GHz CPU, 16 GB memory, and running a Microsoft Windows 10 operating system. The original rivers of the experiment are from the basic geographic information data of Jiangxi Province, China, with a scale of 1:250,000, including linear and dual-line rivers. The data range is 114.47° E to 115.50° E and 25.69° N to 26.0° N (as shown in Figure 11). To reflect the progressive collapse of dual-line rivers, $W_{thd}$ is set as 100 m, 150 m, 200 m, 300 m, 400 m, and 600 m in combination with the width (0.4 mm) of the compilation specification, and we obtain progressive collapse results at six scales. The collapse priority strategy is determined for the handling mode of the river segment, and the length threshold is set as $L_{thd}=5\cdot W_{thd}$. The skeleton graph pre-pruning threshold is set as $L_{prune}=W_{thd}$.

For better evaluation of the proposed method for dual-line rivers, the superpixel river collapse (SURC) method [6] is selected for comparison. The SURC method is also a progressive collapse method for dual-line rivers, which has a strong contrast with the proposed method. ArcMap software (ESRI, Redlands, CA, USA, Version 10.2) was used to convert the vector river data of the experimental area to raster data, and the progressive collapse results of the SURC method were vectorized. The converted image size was 10,496 × 3054 pixels, and the width of one pixel was equivalent to 9.9 m of the field distance. Combined with the SURC method parameter settings and experimental data characteristics, four levels are segmented according to the superpixel size. The superpixel sizes corresponding to the four levels are 5000, 10,000, 50,000, and 100,000.

4.2. Results and Evaluations

The global progressive collapse results of the proposed method are shown in Figure 12a–c and Figure 13a–c. With an increase in the width threshold $W_{thd}$, the segment with a width of less than $W_{thd}$ in the dual-line river was detected and pre-segmented. The progressive collapse results were obtained through the handling mode of the dual-line river segment determined by the collapse priority strategy. The proposed method can better meet the requirements of the progressive collapse of the dual-line river, and the skeleton line shape of the collapsed part naturally continues and conforms to objective geographical cognition. The non-collapsed part meets the visual requirements under the target scale. The progressive segmentation results of the collapsed and non-collapsed parts of dual-line rivers are objective and regular, meeting the application requirements of cartographic generalization. As shown in the zoom-in areas in Figure 12 and Figure 13, the progressive collapse results of the proposed method can maintain topological consistency with adjacent linear rivers, without redundant burrs. The skeleton lines at the intersection are hierarchical, and some narrow river segments can be seen clearly after exaggerated handling.

4.2.1. Qualitative Evaluation

Figure 14 shows the results of the progressive collapse in the experimental area using the SURC method. Figure 14a shows the collapse result when the fourth level superpixel was collapsed to the skeleton line; Figure 14b shows the collapse result when the third level superpixel was collapsed to the skeleton line; Figure 14c shows the collapse result when the second level superpixel was collapsed to the skeleton line; and Figure 14d shows the collapse result when the first level superpixel was collapsed to the skeleton line. Considering image processing, the SURC method can effectively avoid the burr generated by progressive collapse results and has a good recovery ability for river intersections. Using the smoothing method to adjust the local consistency of vertex distribution, a smooth skeleton line extraction result is obtained.

We observed the following after analyzing and comparing the differences between the two methods. First, when the SURC method is used for progressive collapse, only the geometric and geographical characteristics of the dual-line river itself are considered, and the spatial relationship between the dual-line river and the linear river is not considered. Thus, the linear river connected with the collapsed part of the dual-line river has a problem of suspension (as shown in the zoom-in area in Figure 14), which requires topological handling. Second, the SURC method, after multi-level segmentation of superpixels, is a progressive collapse method based on levels, which still has a certain gap considering the requirements of cartographic generalization. As shown in Figure 14a–c, in the process of dual-line river multi-level segmentation using the SURC method, there are many short and small collapsed river segments and unchanged river segments. The appearance of this type of river segment leads to the fragmentation of progressive collapse results, which does not meet the requirements of the compilation specification. Lastly, the progressive collapse using multi-level superpixels from the raster space shows good hierarchy; however, the target scale of the collapse results has some fuzziness. In comparison, the proposed method does not face the above challenges, and the topological consistency between rivers can be effectively maintained by applying the skeleton line extraction method that takes the spatial relationship constraints into account. The pre-segmentation of river segments and the formulation of handling strategies for dual-line rivers, especially for the handling of non-isolated short river segments, can effectively eliminate the fragmentized progressive collapse results and ensure that they meet the requirements of the compilation specification. Through parameter setting, accurate progressive collapse results in the target scale can be obtained.

In combination with the five aspects of qualitative evaluation indexes (redundant intersections, fractured intersections, burrs, noise, and progressive) proposed by Shen et al. [6], we added three new indexes of topological consistency, fragmentation, and scale accuracy to qualitatively evaluate the results of the two methods. Topological consistency and scale accuracy are often used as qualitative evaluation indexes in cartographic generalization. Fragmentation is an index for evaluation from the perspective of visual perception. The introduction of these three indexes can help evaluate whether the implementation results of the two methods are more consistent with the application requirements of cartographic generalization. As shown in

Table 3. Qualitative evaluation of the two methods.

Index	The SURC Method	The Proposed Method
Redundant intersections	No	No
Fractured intersections	No	No
Burrs	No	No
Noise	No	No
Progressive	Yes	Yes
Topological consistency	No	Yes
Fragmentation	Yes	No
Scale accuracy	No	Yes

, the proposed method can achieve the same effect as the SURC method in the first five indexes. Further comparison of the last three indexes shows that the proposed method has better performance.

4.2.2. Quantitative Evaluation

Because the two methods are progressive collapse from the perspective of vector structure and raster structure, the skeleton line extraction algorithm used in this study has obvious advantages over other methods based on Delaunay triangulation [7]; however, the SURC method compared with three typical image thinning methods, the Pavlidis [25], Rosenfeld [26], and ZS [27] methods, also has obvious advantages [6]. Therefore, the quantitative comparison takes the existing skeleton lines in 1:250,000 basic geographic information data as the standard centerline and uses the method developed by Goodchild and Hunter [39] to evaluate the geometric accuracy of linear features to quantitatively evaluate the skeleton lines extracted by the proposed and SURC methods.

First, the progressive collapse results by the SURC method are vectorized. Second, the lengths of the skeleton line of the standard centerline, the SURC method, and the proposed method are compared. As shown in

Table 4. Length of the skeleton lines extracted by the two methods.

	Standard Centerline (m)	The SURC Method (m)	The Proposed Method (m)
length	235,747.2	224,008.5	235,518.6

, the total length of the skeleton line of the proposed method is very close to the total length of the standard centerline, whereas the total length of the skeleton line of the SURC method is 11,738.7 m shorter than the standard centerline. As shown in Figure 15, the standard centerline and the proposed method both consider the topology consistency and add numerous short skeleton line segments connecting the linear river and the main skeleton line; thus, the total length increases. In contrast, the SURC method does not take this aspect into account, and its total length is shorter.

The standard centerline is used as a reference to build a buffer zone with different radii (including 10 m, 20 m, 30 m, 40 m, and 50 m), and the ratio of the length of the skeleton line is calculated in the buffer zone and the length of the standard centerline (as shown in

Table 5. Quantitative evaluation of geometric accuracy.

Buffer Width (m)	The SURC Method (%)	The Proposed Method (%)
10	33.0	47.44
20	59.73	71.67
30	74.54	83.0
40	81.33	88.68
50	85.19	91.93

). The analysis shows that when the buffer radius (10 m, 20 m) is small, the length ratio of the two methods is small; the length ratio of the two methods increases with an increase in the buffer radius. When the buffer radius was 50 m, the length ratio of the proposed method reached 91.93%. The reasons for the lower length ratio of the SURC method compared to the proposed method are analyzed as follows:

• As shown in 1, 2, 3, and 4 in Figure 15, the short skeleton line branches generated by the proposed method through topological maintenance coincide more with the standard centerline, increasing the coincidence length, whereas the SURC method does not take topological maintenance into account.
• The process of vector-to-raster and raster-to-vector conversion inevitably results in loss of accuracy and reduces the length of coincidence.

Therefore, the proposed method performs better than the SURC method in geometric accuracy.

5. Conclusions

In this study, a progressive collapse method for dual-line rivers from the vector structure was implemented, and the main objectives of the study were completed. Taking advantage of the spatial proximity and computability of Delaunay triangulation, based on the skeleton graph of dual-line rivers, the dual-line river segment can be obtained by calculating the triangle width. Second, based on the width and length of the dual-line river segment and the adjacency between different dual-line river segments, two strategies for determining the dual-line river segment handling mode are formulated, including the collapse priority strategy and the exaggeration priority strategy. Finally, based on the two strategies, the progressive collapse of the dual-line river is realized through dual-line river segments collapse and exaggeration. Actual dual-line river data were used in the experiment, and the conclusions are as follows:

• The proposed method can realize scale-driven progressive collapse handling of dual-line rivers. The progressive collapse results are regular and not fragmented, which is more consistent with the application requirements of cartographic generalization.
• The proposed method has no burr in the collapsed part of the dual-line rivers, maintaining the topological consistency with the linear river and avoiding topology maintenance operations.
• By comparing the standard centerline with the SURC method, the geometric accuracy of the proposed method is higher.

However, the collapse priority strategy and exaggeration priority strategy formulated in this study may not be completely suitable for more actual application needs, and further strategy design is still needed. In addition, the segmentation of dual-line rivers with more complex shapes needs further exploration in future research.