Quantitative Relations between Morphostructural Similarity Degree and Map Scale Change in Contour Clusters in Multi-Scale Map Space

Abstract

This paper aims to propose a new approach to calculate the quantitative relations between morphostructural similarity degree and map scale change in multi-scale contour clusters for automatic contour generalization. Terrain lines were extracted by pre-processing of unclosed contour lines, and an indirect quantitative expression method of morphostructural similarity relation was proposed based on terrain line hierarchical trees. Thirteen groups of multi-scale contour clusters with different drainage areas of loess geomorphy were employed to explore the changing regularity of morphostructural similarity indices with map scale. Finally, the quantitative relations between morphostructural similarity degree and map scale change were calculated using 52 groups of points. The results show that power function is the best fit among the candidate functions, and the quantitative relations between the morphostructural similarity degree and map scale change can be expressed using the same power function, which facilitates the automation of contour generalization.

1. Introduction

The quantitative expression of spatial relations has been studied by researchers in the cartography [1,2] and geographic information science [3,4] communities for years. Spatial similarity relation [5], including graphic similarity [6], topological similarity [7], morphostructural similarity, and semantic similarity [8], is an essential component of spatial relations. It has been extensively used in human spatial cognition [9,10], pattern recognition [11,12], and spatial data matching [6,12,13,14]. Spatial similarity relation is particularly important in automating map generalization [5,15], which is the process of reducing the level of detail from larger scale maps (e.g., 1:5000) to smaller scale maps (e.g., 1:25,000) [16,17]. Contour is an effective means for portraying the three-dimensional topography of the real world on two-dimensional surfaces [18]. Its automatic generalization is crucial for downsizing small-scale topographic maps, topographic analysis based on terrain, and construction of multi-scale vector topographic map database. Contour generalization is essentially a kind of spatial similarity transformation between multi-scale contour clusters [5]. Cartographers [15,19] contend that the descriptions of spatial relation and geomorphic morphological structure are the key and core of contour generalization. Spatial relation between contours primarily pertains to topological relation [15,20]. Existing research [5,7] has manifested that change in topological relations and map scale has a quantitative relation in map generalization. However, the quantitative relation between morphostructural similarity degree and map scale change in multi-scale contours has not been explored, hindering the automation of contour generalization, including automation of generalization algorithms [21,22], automatic control of generalization process [23,24], and automatic evaluation of generalization results. Therefore, quantitative expression of morphostructural similarity relation is a crucial area of study concerning multi-scale contour spatial similarity relations.

Previous studies on spatial similarity relations of contour have primarily concentrated on the multi-source spatial similarity relations between raster dataset, such as ASTER GDEM, SRTM, and spatial similarity relations between multi-source contours [14,25]. However, contour expresses the geomorphological characteristics of the Earth’s surface through a closed curve cluster, in contrast to the simplification of individual polyline. Therefore, contour cluster must be represented, understood, and processed as a group [20,21]. Existing studies only consider traditional geometric indices of contour lines, such as length, sinuosity, and longest common subsequence [14,25], etc., while ignoring the “clustering nature” [15] and “bending nesting relations” between them [20], as shown in

Table 1. Literature overall review of contour cluster.

Research Fields	Data Structure (Specific Research Orientation)	Key Issues
Contour generalization	DEM or contours (mainly focuses on raster datasets, including automation of generalization algorithm [19,20,21,22,23,24,26], semi-automatic control of generalization algorithm and process [23,27,28], curve bends recognition [22,29], and line simplification [18,29].	Research in this field has reached a mature stage. (1) However, most existing algorithms rely on the Töpfer rule, which is based on cartographic experience to determine the selection quantity. (2) The reliability of the generalization results is heavily dependent on the individual experience of the cartographer. (3) Previous research has mainly focused on DEM and lacks exploration based on vector contours. The semi-automatic generalization algorithm falls short in terms of automation, intelligence, and generalization efficiency of the software.
Spatial similarity relation and its application of other individual objects or object groups	Raster or vector (basic theories [1,2,5], mixed similarity [5,30], semantic similarity [8,31], topological similarity [7], geometric similarity [6,11], spatial similarity assessment [9], human spatial cognition [10], pattern recognition [11,12], and spatial data matching [6,12,13,14,32].	Some achievements have been reached. (1) However, researchers have mainly focused on the quantitative measurement of spatial similarity relations using geometric indices of objects, such as length, area, sinuosity, curvature, etc.; (2) In the process of map generalization, the different geographical features of different objects have been ignored.
Spatial similarity relation between contours and its application	Raster or contours (mixed similarity [5,25], topological similarity [7], geometric feature similarity of multi-source contours, and contour matching [14].	Relevant researches are rare. (1) Additionally, previous research has primarily concentrated on the multi-source spatial similarity relations of contours generated from raster datasets, such as ASTER GDEM and SRTM. (2) Only traditional geometric indices of contour line, such as length, sinuosity, and longest common subsequence, etc., were considered, whereas the “clustering nature” and “bending nesting relations” between contour lines were ignored.

. This inevitably leads to wrong expression of geomorphic features in the measurement of multi-scale spatial similarity relations. In conclusion, research has yet to produce any findings on the quantitative relations between changes in map scale and the morphostructural similarity degree among vector contour clusters on maps at multiple scales. This not only hinders the use in human spatial cognition, pattern recognition, and spatial data matching, but also leads to contour generalization failure of full automation, and further reduces the implementation efficiency of existing cartographic generalization software.

Therefore, the rest of this article is organized as follows: Section 2 outlines the datasets utilized in this study and their pre-processing. Section 3 introduces a quantitative expression method for the similarity relations of multi-scale contour clusters based on terrain line hierarchical trees. Section 4 demonstrates the validity of the proposed methods using experiments. Section 5 discusses the proposed methods and the experiments presented in this study. Finally, Section 6 concludes the study and provides directions for future research.

2. Datasets and Their Pre-Processing

2.1. Experimental Datasets

Loess geomorphy is the predominant geomorphological type in the Loess Plateau, largely formed through aeolian accumulation and water erosion processes, resulting in unique landforms [33], such as loess tablelands, hills, and ridges, as well as well-developed gullies. It is approximately 4~6 km/km², as estimated from 1:50,000 topographic maps [33,34]. Therefore, we selected 13 groups of multi-scale contour clusters of loess geomorphy for this study. Each contour cluster covers different drainage areas on the Earth’s surface (

Table 2. Thirteen groups of multi-scale contour clusters used in the study. (A refers to drainage area, N_u refers to the number of unclosed contour lines, and N_c refers to the number of closed contour lines, respectively).

Sample	Map Scale	A (km2)	NU	NC	Sample	A (km2)	NU	NC	Sample	A (km2)	NU	NC
1	1:10,000	1.2105	162	42	6	1.8084	107	51	11	2.3252	258	46
	1:50,000		34	10			26	13			21	11
	1:100,000		16	5			10	6			8	5
	1:250,000		4	2			4	3			7	3
2	1:10,000	3.4408	103	51	7	3.9368	263	61	12	2.8006	199	54
	1:50,000		23	12			35	16			22	13
	1:100,000		7	6			13	8			10	6
	1:250,000		3	3			5	3			3	3
3	1:10,000	5.3838	124	54	8	2.5028	103	45	13	2.3345	185	53
	1:50,000		27	13			30	11			22	13
	1:100,000		13	6			10	5			8	6
	1:250,000		3	2			3	3			5	3
4	1:10,000	2.4095	99	42	9	2.0133	259	49
	1:50,000		28	13			31	13
	1:100,000		9	6			12	6
	1:250,000		5	3			6	3
5	1:10,000	3.0428	99	52	10	1.7728	94	51
	1:50,000		28	13			18	12
	1:100,000		9	6			8	6
	1:250,000		5	3			5	3

). The source map scale of each contour cluster is 1:10,000, and the corresponding generalized contours have target scales of 1:50,000, 1:100,000, and 1:250,000, with contour intervals at 5 m, 20 m, 40 m, and 100 m, respectively.

To test and validate the accuracy of terrain line extraction results, three groups of multi-scale contour clusters were randomly selected from the 13 groups. The datasets for these groups were manually generalized by cartographers at multiple scales (1:10,000, 1:50,000, 1:100,000, and 1:250,000) and are available in vector format from the National Geomatics Center of Gansu (NGCG), China.

2.2. Pre-Processing of the Dataset

Contours in reality are continuous and closed. However, a topographic map provides only a partial representation of the real world [26], and contour lines are truncated into unclosed contours at the map’s border due to being stored according to sheet [27,28]. Additionally, unclosed contour lines can cause uncertainty in recognizing curved clusters [27], which may complicate terrain line extraction. Therefore, it is necessary to pre-process these unclosed contours to close them [19,27].

Considering positive topography as an example, Figure 1 illustrates the unclosed contour line closure process, which involves three steps: (i) Constructing an adjacency matrix (R_{N × N =} {R_(i,j)}) based on Voronoi polygons of contours. The adjacency relation of contours can be defined by the corresponding Voronoi polygon of contours [35]. In addition, there are three types of adjacency relation for any contour line L_i: Neighbors contained by L_i, neighbors containing Li, and neighbors parallel with L_i. In order to avoid repetition, only the latter two are considered; (ii) matching unclosed contours with the elevation field (ID_1) and the adjacency matrix (R_(i,j)), including matching trunk contours and subtree contours; (iii) dissolving unclosed contours according to the matching field (ID_1) into closed contours, and reconstructing the binary adjacency triangular matrix (R_{M × M}(M = max{ID_1} + 1)) of closed contours.

3. Methodology

Terrain lines not only can effectively express the “clustering nature” and “bending nesting relations” of the contour cluster, but also form a tree structure in itself [27,36] as the “representative” or “substitute” of geomorphologic morphology, since they are easily evaluated [1,20,23]. This paper aims to propose an indirect quantitative method of expressing morphostructural similarity between multi-scale contour clusters, based on terrain line extraction results and its corresponding hierarchical tree. Therefore, the extraction of terrain lines and the construction of its hierarchical tree are essential prerequisites for this paper. Furthermore, to explore the change in trends of morphostructural similarity degree with map scale, we present an indirect quantitative expression method of morphostructural similarity relations, which is based on the constructed terrain line hierarchical tree. Figure 2 illustrates the research ideas and technical route.

3.1. Extraction of Terrain Line

Compared with the respective extraction results of constrained Delaunay triangulation (D-TIN) [37] or curvature recognition method [29], a combination of both can avoid their individual deficiency and improve the recognition results of terrain feature points through secondary detection.

3.1.1. Recognition of Contour Curve Cluster

Contour represents surface terrain through closed curves [3,16]. Therefore, contour curve cluster recognition is a prerequisite to detect terrain feature points, which involves judging the convexity and concavity of node groups on closed curves. Compared with the D-TIN global recognition method [38] and local feature point segmentation method [39], the vector cross product can merge pseudo bent nodes caused by abnormal jitter through determining the convexity and concavity of adjacent nodes. This allows for the effective recognition of concave and convex segments of contours. Considering positive topography as an example, the nodes on the contour line is assumed to move counterclockwise, concave and convex points correspond to valley line and ridge line points, respectively, while concave and convex segment clusters correspond to catchment area and watershed area, respectively. Its recognition principle is taking three sequential adjacent points {$P_{i-1}\left(x_{i-1},y_{i-1}\right),P_i\left(x_i, y_i\right),P_{i+1}\left(x_{i+1},y_{i+1}\right)$} of the contour line and calculating the cross produce of $\overrightarrow{P_{i-1}{P}_i}$ and $\overrightarrow{P_i{P}_{i+1}}$. If three adjacent points are collinear, then the curvature of the intermediate node P_i is 0, and if $\overrightarrow{P_{i-1}{P}_i}\times \overrightarrow{P_i{P}_{i+1}}>0$, then the corresponding middle point P_i is the convex point; otherwise, P_i is the concave point [20]. Therefore, continuous concave and convex nodes form concave and convex segment clusters of the contours, which correspond to catchment and watershed area, respectively.

3.1.2. Detection of Terrain Feature Points

Terrain lines express the “clustering” relation of contour cluster [17], as they are the structuralization of contours. When the nodes of the valley segment or ridge segment are dense, and each bend segment is represented by multiple ridge points or valley points, the terrain feature points corresponding to each concave and convex segment curve can be preliminarily screened by combining extreme points of curvature. Due to the fact that concave and convex points are local feature points calculated using the coordinates of sequential adjacent three points on the contour line, abnormal jitter bending points of contours can be removed by determining the concavity and convexity of adjacent three points. Assuming that the coordinates of three non-collinear adjacent points $\{P_{i-1},P_i,P_{i+1}\}$ on the convex segment contour line are known, and according to the derivation curvature calculation principle, the curvature of the intermediate node P_i can be approximated to the reciprocal of the circumscribed circle radius of adjacent three points [20]. If the extreme points of contours in a concave or convex segment are not unique, and there are two adjacent concave or convex points with equal extreme values, such as points A and B (red dotted lines) in Figure 2d, then the midpoint of the two points is used as the feature point.

3.1.3. Tracking and Connection of Terrain Line

The tracking and connection of terrain lines involve determining the multi-resolution wedge tracking region, performing a secondary inspection of terrain feature points, and finally, tracking and connection of terrain lines.

Determination of multi-resolution wedge tracking region. Curved clusters corresponding to the secondary valley lines (ridge lines) are constantly simplified during the contour generalization process, and the adjacent ridge (valley) is constantly merged. This results in significant differences in the level of detail on the terrain skeleton lines at different map scales. For contour curve cluster, if their concavity has been marked, the water flows downstream along the descending direction of declivity, and thus forms a wedge-shaped tracking area of the valley line. Otherwise, a wedge-shaped tracking region of the ridge line is formed [21].

Secondary inspection of terrain feature points and tracking and connection of terrain lines. Connection models of terrain line are different corresponding to the different types of constrained D-TIN of contour cluster. Considering Figure 3a as an example, a class I triangle must correspond to one or multiple bends (1 ≤ N ≤ 3) of contour line and at least one class III triangle, and thus the terrain feature point must be a vertex of III class triangle (Figure 3b). Therefore, tracking and connection of models of terrain lines are as follows [21,26]:

Rule 1: If a class I triangle corresponds to two or three bends, the terrain lines should be orthogonal to the ridge or valley line [40] (Figure 3a). The tracking and connection models of terrain lines are shown in Figure 3a,b.

Rule 2: According to the principle that the motion or slope direction of a particle located on the terrain surface is the fastest in the direction of elevation decline [20], when a contour line passes through a ridge or a valley, the connection line of the midpoint of a class II triangle is the dividing line of the catchment or watershed area [40] (Figure 3c).

Rule 3: If there are only class II triangles between two adjacent contour lines, then the contour lines are dense. The tracking and connection models of terrain lines are shown in Figure 3d. If the extreme values of adjacent concave or convex points are equal, for example, A and B in Figure 3d, then the feature point is the midpoint of A and B.

3.2. Construction of Hierarchical Tree of Terrain Lines

Terrain lines serve as a “representative” or “substitute” of geomorphic morphology, establishing a “clustering” relationship between contours [27]. For example, a valley line represents a group of contour curve clusters that correspond to the valley. Therefore, the evaluation and treatment of geomorphic structures can be transformed into the processing of their “substitute”. Additionally, terrain lines are tree structures. As a result, to evaluate and process terrain lines as a whole, it is necessary to organize them organically. The primary way to achieve this is through the construction of a corresponding hierarchical tree.

Considering positive topography as an example, the principle of contour generalization involves eliminating secondary valleys and expanding corresponding ridges [27]. The elimination of valleys indicates the merger of adjacent ridges (black bold dotted line in Figure 4a). Therefore, the processing of valley lines entails the modification of corresponding ridge lines. Therefore, geomorphological generalization can be viewed as a generalization of valley information [27]. According to the principles of hydrology, the hierarchical tree of terrain lines reflects the inheritance relations between trunk and branch valley lines (Figure 4b). The more branches a valley line has, the higher the hierarchical encoding of its trunk valley line [41]. Contour curve cluster corresponding to the more highly encoded valley lines is more likely to be retained during the process of contour generalization; otherwise, they are simplified to emphasize the main geomorphological morphology [28]. However, the hierarchical encoding of terrain lines reflects their relative levels (Figure 4b).

Compared with Horton, Strahler, and Shreve [42,43,44], the Branch rule [45] is the most common encoding method of tree structure. This is due to the fact that it reflects the number of incoming tributaries along the entire river from its source to its mouth, and the difference in the number of tributaries per unit distance [30,45]. Therefore, this study uses the Branch rule to build the hierarchical tree of terrain line. Considering loess geomorphy as an example, Figure 5 shows the Branch codes of valley line hierarchical trees corresponding to multi-scale contour clusters.

3.3. Quantitative Expression Method of Multi-Scale Contour Cluster Morphostructural Similarity

Hydrological geomorphic characteristic parameters include topological and geometrical characteristic parameters [46,47]. The former focuses on stream bifurcation and stream order, while the latter includes stream length, stream length law, chain length, inlet angle, river density, and stream gradient, etc. Stream length law and chain length essentially describe stream length, and stream gradient describes three-dimensional characteristics, which can be excluded. Stream inlet angles are generally unchanged after generalization [30]. Therefore, based on the parameters mentioned above [46,47] and previous research on generalizing drainage networks [30,45], stream bifurcation, river density, stream length, and stream order have been selected. Therefore, the degree of morphostructural similarity between multi-scale contour clusters can be indirectly measured from the following four aspects.

3.3.1. Topological Similarity

The topological relation of a tree-like network mainly refers to the descendent relation [30], with stream bifurcation describing the descendant relations between the main stream and its tributaries in hydrology. Therefore, the valley line bifurcation ratio can be used to indirectly express the topological similarity between multi-scale contour clusters quantitatively. Assuming that the contour cluster M₁ at scale S₁ is generalized to contour cluster M₂ at scale S₂, then the formula for calculating the topological similarity degree ($S_{\mathit{Topo}}{}_{S_1}^{S_2}$) is as follows:

$$S_{\mathit{Topo}}{}_{S_1}^{S_2}=R_{\mathrm{b}}^{S_2}/R_{\mathrm{b}}^{S_1}={\displaystyle \sum _{g_2=1}^n{N}_{g_2}^{g_2+1}}/{\displaystyle \sum _{g_1=1}^m{M}_{g_1}^{g_1+1}}$$

(1)

where $g_1{, g}_2$ refers to the level of valley line before and after generalization, respectively. $N_{g_2}^{g_2+1},M_{g_1}^{g_1+1}$ represent the level of valley line and bifurcation number between the higher level ($g_i+1$)$(g_1=1,2,\dots ,m;g_2=1,2,\dots ,n)$ valley line and $g_i$ level valley line of contour cluster at scale S₁, S₂, respectively. Meanwhile, m, n are the total numbers of valley line levels and $R_{\mathrm{b}}^{S_1},R_{\mathrm{b}}^{S_2}$ represent the total numbers of valley line bifurcation before and after generalization, respectively. According to the Branch encoding principle [47], $R_{\mathrm{b}}^{S_i}=N-1$, N refers to the branch code of the first-level valley line at map scale S_i, $S_{\mathit{Topo}}{}_{S_1}^{S_2}\in (0,1]$.

Considering Figure 5 as an example, the Branch codes of first-level valley lines corresponding to contour clusters at map scales of 1:10,000, 1:50,000, 1:100,000, and 1:250,000 are 173, 45, 22, and 9, respectively. This indicates that the total numbers of valley line bifurcations are 172, 44, 21, and 8, respectively. Therefore, the topological similarity ($S_{\mathit{Topo}}{}_{S_1}^{S_2}$) of valley line hierarchical trees corresponding to multi-scale contour clusters is 0.2558, 0.1221, and 0.0465, when the contour clusters are generalized from 1:10,000 to 1:50,000, 1:100,000, and 1:250,000, respectively.

3.3.2. Directional Similarity

In contrast to the discrete settlement groups [5], the direction of which is changed after generalization, the contours on topographic maps are spatial targets that have high positional accuracy and are often used for spatial positioning. In map generalization processing, it is generally not allowed to move contour lines on maps [2]. Therefore, the directional relationships between contour clusters are not changed after map generalization, and the corresponding directional similarity degree ($S_{\mathit{Dir}}{}_{S_1}^{S_2}$) should be viewed as 1.

3.3.3. Distance Similarity

Gully density refers to the lengths of gullies per unit area, and the greater its value, the more the watershed cuts into the geomorphology. Therefore, gully density is one of the most important indicators to represent regional geomorphic features [33], and can be used to measure the distance similarity between valley line hierarchical trees. Due to the fact that the drainage area remains unchanged after contour generalization; therefore, the distance similarity degree ($S_{\mathit{Dis}}{}_{S_1}^{S_2}$) can be calculated as follows:

$$S_{\mathit{Dis}}{}_{S_1}^{S_2}={\displaystyle \sum _{g_2=1}^n{\displaystyle \sum _{q=1}^{t_2}{L}_{g_{2}q}^{S_2}}}/{\displaystyle \sum _{g_1=1}^m{\displaystyle \sum _{p=1}^{t_1}{L}_{g_{1}p}^{S_1}}}$$

(2)

where $S_{\mathit{Dis}}{}_{S_1}^{S_2}\in (0,1]$, $g_1, g_2$ represent the level of valley line; $t_1,t_2$ are the total numbers of valley lines; and $p, q (p=1,2,\dots ,t_1; q=1,2,\dots ,t_2)$ refers to the p-th or q-th valley line of $g_1$ or $g_2$ level before and after generalization, respectively.

3.3.4. Attribute Similarity

During the process of contour structural generalization, contour curve clusters corresponding to the main valley lines are retained, but those corresponding to the secondary valley lines, such as ditches, gullies, etc., are gradually “eroded” [23]. Therefore, attribute similarity ($S_{Att}{}_{S_1}^{S_2}$) can be expressed by Formula (3):

$$S_{\mathit{Att}}{}_{S_1}^{S_2}={\displaystyle \sum _{i=1}^{t_2}{A}_i^{S_2}}/{\displaystyle \sum _{i=1}^{t_1}{A}_i^{S_1}}$$

(3)

where $i$ refers to the i-th valley line; $t_1,t_2$ refer to the total numbers of valley line before and after generalization, respectively; $A_i^{S_2}$ refers to the Branch code of the i-th valley line in the generalized contour cluster; $A_i^{S_1}$ refers to the Branch code of the i-th valley line in the original contour cluster, and exists in the generalized contour cluster, $S_{Att}{}_{S_1}^{S_2}\in (0,1]$.

For example, when the contour cluster is generalized from 1:10,000 to 1:100,000, the sum of valley line Branch codes in the generalized contour cluster is 51 (Figure 5c), and the sum of valley line Branch codes in the original contour cluster as well as that existing in the generalized contour cluster (red valley lines in Figure 5a) is 310. Therefore, the multi-scale attribute similarity degree between multi-scale contour clusters corresponding to the valley line hierarchical trees is 0.1645.

3.3.5. Multi-Scale Morphostructural Similarity Degree

The analysis in Section 3.2 shows that the morphostructural similarity relation between multi-scale contour clusters can be indirectly and quantitatively expressed by the corresponding multi-scale valley line hierarchical trees. Therefore, the multi-scale contour cluster morphostructural similarity degree ($S_{S_1}^{S_2}$) can be calculated by Formula (4):

$$\begin{gathered} S_{S_1}^{S_2}=w_1{\ast S}_{\mathit{Topo}}{}_{S_1}^{S_2}+w_2{\ast S}_{\mathit{Direction}}{}_{S_1}^{S_2}+w_3{\ast S}_{\mathit{Distance}}{}_{S_1}^{S_2}+w_4\ast S_{\mathit{Attribute}}{}_{S_1}^{S_2} \\ {S}_{S_1}^{S_2}=w_1{\ast S}_{\mathit{Topo}}{}_{S_1}^{S_2}+w_2{\ast S}_{\mathit{Dir}}{}_{S_1}^{S_2}+w_3{\ast S}_{Dis}{}_{S_1}^{S_2}+w_4\ast S_{Att}{}_{S_1}^{S_2} \end{gathered}$$

(4)

where $w_1,w_2,w_3,w_4$ are 0.22, 0.25, 0.31, and 0.22, respectively, and these weight parameters are obtained through a large number of spatial psychological tests conducted by our team [5].

4. Experiments and Results

4.1. Results Analysis of Pre-Processing

The pre-processing of unclosed contour lines, which includes topology checking, matching, and closure of unclosed contour lines, is the prerequisite for extraction terrain lines from contour cluster.

Table 3. Elevation and code matching of unclosed trunk contour lines. (“Elev” and “ID_1” refer to elevation and matching field, respectively).

FID	ID_1	Elev	FID	ID_1	Elev	FID	ID_1	Elev
110	0	4460	127	10	4280	154	14	4200
108	1	4440	137	11	4260	159	14	4200
7	2	4420	134	11	4260	171	15	4180
20	3	4400	135	11	4260	178	15	4180
8	4	4380	145	12	4240	175	15	4180
11	5	4360	148	12	4240	173	15	4180
107	6	4340	147	12	4240	172	15	4180
23	7	4320	146	12	4240	170	15	4180
113	8	4320	185	13	4220	167	16	4160
116	9	4300	188	13	4220	168	16	4160
115	9	4300	187	13	4220	166	16	4160
124	9	4300	189	13	4220	112	17	4140
125	10	4280	155	14	4200	99	18	4120
128	10	4280	158	14	4200	105	19	4100

and

Table 4. Elevation and code matching of unclosed trunk contour lines (FID=113). (“Elev” and “ID_1” refer to elevation and matching field, respectively).

FID	ID_1	Elev
14	21	4360
15	20	4340
113	8	4320

show the elevation and code matching results of unclosed contour lines in Figure 6.

Table 3 and Table 4 reveal that the 44 unclosed contour lines in Figure 6 are matched and formed 22 closed contour lines after pre-processing, which include 19 trunk contour lines (Table 3) and three branch contour lines (Table 4). The closure results of 13 groups of unclosed multi-scale contours are shown in Table 2.

4.2. Extraction Results Accuracy Evaluation of Terrain Lines

Typically, 10–20% of experimental datasets are used to evaluate the accuracy of extraction results. Therefore, to evaluate the geometry and position accuracy of three groups of valley line extraction results, three sets of ranked samples of multi-scale contour clusters at map scale 1:10,000, 1:50,000 are selected [48].

Table 5. Comparison of geometry and position accuracy errors. (“LR”, “SD”, “PE”, and “SMD” refer to length ratio, sinuosity ratio, position error, and standard deviation of displacement, respectively).

Samples	1:10,000				1:50,000
	LR	SD	PE\km	SMD	LR	SD	PE\km	SMD
Sample 1	1.0150	0.9852	0.0024	0.6532	0.9727	1.0279	0.0102	0.4028
Sample 5	0.9686	0.9987	0.0026	0.8945	0.9195	1.1130	0.0098	0.7207
Sample 9	0.9691	1.1363	0.0100	0.4785	0.9327	1.2007	0.0027	0.4564

shows the geometry and position accuracy errors of the valley line extraction results of the three groups of samples, compared with the standard river dataset of the corresponding map scale.

Table 5 indicates that compared with the standard river dataset of 1:10,000 and 1:50,000, the maximum errors ratio is only {0.0314,0.1363}, {0.0805,0.2007}, respectively. Overall, the local position accuracy (PE) of valley line extraction results corresponding to the 1:10,000 contour clusters is higher than that of 1:50,000, but the overall position accuracy (SMD) of the latter is higher than the former, with a maximum overall position accuracy error of 0.8945.

Considering sample 1 as an example, Figure 7 shows the overlap of the valley line extraction results with the standard river data of the corresponding map scale.

Figure 7 demonstrates that the position accuracy SMD, PE [48,49], and geometric accuracy LR, SD [48] of the valley line extraction results are {0.6532, 0.0024, 1.0150, 0.9852}, {0.4028, 0.0102, 0.9727, 1.0279}, respectively (Table 5), compared with standard river data of 1:10,000, 1:50,000. The geometric accuracy errors ΔLR, ΔSD that tend toward zero are only {0.0150, 0.0148}, {0.0276, 0.0279}, respectively, and the geometric accuracy is higher. The local position accuracy of the valley line extraction results corresponding to 1:10,000 contour cluster is higher than that of 1:50,000 (Figure 7a), but the overall position accuracy of the latter is higher than the former (Figure 7b), with a smaller position error margin. In conclusion, the accuracy of the valley line extraction results can be used to express the morphostructural similarity between multi-scale contour clusters quantitatively.

4.3. Quantitative Trends of Multi-Scale Morphostructural Indices

4.3.1. Neighbor Scale Morphostructural Indices

Based on the analysis in Section 3.3, the morphostructural similarity relation between multi-scale contour clusters can be indirectly and quantitatively evaluated through their corresponding valley line hierarchical trees. The measurement indices that relate to the quantitative expression of morphostructural similarity relations of multi-scale contour clusters mainly include the length of valley line, bifurcation number, and Branch codes of valley lines, etc. Figure 8 displays the variation trends of the neighbor scale morphostructural indices of 13 groups of contour clusters of loess geomorphy.

Some insight can be gained from Figure 8.

(1)

In contrast to the bifurcation ratio and length ratio of neighbor scale valley lines, the average of closed contour lines number ratio is equal to the contour interval ratio (CI) of neighbor scale contour clusters. However, the number ratio of neighbor scale closed contour lines is not a constant; therefore, it fluctuates around the average of neighbor scale contour interval ratio due to differences in surface relief, steep slope, and surface fragmentation [7]. For instance, as the contour cluster is generalized from 1:100,000 to 1:250,000, the neighbor scale closed contour lines number ratio fluctuates around 2.5 (Figure 8a).

(2)

Although the length ratio and bifurcation ratio of neighbor scale valley lines both fluctuate around their average, the fluctuation degree of the latter among different samples is greater than the former (Figure 8c). The maximum fluctuation range of the latter is [1.1, 8.0588] with a difference of 6.9588 (Figure 8b). There is no specific proportional relationship between the neighbor scale morphostructural indices and the corresponding contour interval ratio or contour interval distance ratio of neighbor scale contour clusters. Consequently, the following sections further investigate the changing trends of multi-scale morphostructural similarity indices with map scale.

4.3.2. Multi-Scale Morphostructural Indices

By analyzing the distribution characteristics of morphostructural indices, including average value ($\stackrel{-}{\mathrm{X}}$), standard deviation ($\mathsf{\delta }$), and variation coefficient (${\mathrm{C}}_{\mathrm{V}}$), etc., the variation trends of these indices can be discovered. Standard deviation reflects the dispersion of a set of statistics relative to its average, and the variation coefficient represents the relative fluctuation of statistical data [50].

Table 6. Statistical indices of multi-scale morphostructural similarity indices of contour cluster.

Indices	C	${\mathit{S}}_{\mathit{Topo}}{}_{{\mathbf{S}}_1}^{{\mathit{S}}_2}$	${\mathit{S}}_{\mathit{Dis}}{}_{{\mathbf{S}}_1}^{{\mathit{S}}_2}$	${\mathit{S}}_{\mathit{Att}}{}_{{\mathbf{S}}_1}^{{\mathit{S}}_2}$	${{\mathit{S}}_{\mathit{im}}}_{{\mathit{S}}_1}^{{\mathit{S}}_2}$	Indices	C	${\mathit{S}}_{\mathit{Topo}}{}_{{\mathit{S}}_1}^{{\mathit{S}}_2}$	${\mathit{S}}_{\mathit{Dis}}{}_{{\mathit{S}}_1}^{{\mathit{S}}_2}$	${\mathit{S}}_{\mathit{Att}}{}_{{\mathit{S}}_1}^{{\mathit{S}}_2}$	${{\mathit{S}}_{\mathit{im}}}_{{\mathit{S}}_1}^{{\mathit{S}}_2}$
$\stackrel{-}{\mathrm{X}}$	5	0.2835	0.3041	0.4769	0.5271	${\mathrm{C}}_{\mathrm{V}}$	5	0.7421	0.3334	0.2922	0.1265
	10	0.1106	0.1219	0.2956	0.3927		10	0.9973	0.3202	0.2481	0.0884
	25	0.0675	0.0754	0.2177	0.3489		25	1.1865	0.3529	0.3002	0.0923
$\mathsf{\delta }$	5	2.2103	0.1014	0.1393	0.0666
	10	0.1103	0.0391	0.0733	0.0347
	25	0.0801	0.0266	0.0653	0.0321

shows the statistical indices of 13 groups of multi-scale morphostructural similarity indices with the same map scale change in loess geomorphy.

Figure 9 shows the variation trends of morphostructural similarity indices of multi-scale contour clusters at the same or different map scale change(s) in loess geomorphy.

The following three conclusions can be drawn from Table 6 and Figure 9.

(1)

Multi-scale morphostructural similarity indices of contour cluster gradually decrease with the increasing map scale change, regardless of multi-scale morphostructural similarity degree or other measures, such as distance similarity degree. For example, the average of multi-scale morphostructural similarity degree decreases from 0.5271 to 0.3849 as the map scale change increases from 5 to 25 (Figure 9a).

(2)

Compared with the same index of different map scale changes, the standard deviation of multi-scale morphostructural similarity indices decreases with the increasing map scale change. For example, the multi-scale morphostructural similarity degree decreases from 0.0666 to 0.0321 as the map scale change increases from 5 to 25. However, the variation coefficients of other indices decrease at first and then increase, except for topological similarity degree (Table 6). This indicates that the dispersion degree between multi-scale morphostructural similarity degree and its average gradually decreases with the increase in map scale change (black line > red > gray) (Figure 9). However, the degree of relative fluctuation among different samples decreases from 0.1265 to 0.0884 at first, and then increases from 0.0884 to 0.0923 (Table 6). Therefore, when the map scale change (C) is 10, the fluctuation degree of morphostructural similarity degree is the gentlest, which is confirmed by Figure 9.

(3)

Compared with different indices of the same map scale change, the standard deviation and variation coefficient of multi-scale morphostructural similarity degree are both minimum (Table 6). This indicates that the agglomeration degree of multi-scale morphostructural similarity degree is the largest, and all fluctuate around its average. For example, when the map scale change is 10, all of the multi-scale morphostructural similarity degree fluctuates around 0.3927 (Figure 9a). This conclusion further indicates that the relations between morphostructural similarity degree and map scale change between multi-scale contour clusters can be expressed quantitatively by the same function. Therefore, the following contents further formulate the quantitative relations between morphostructural similarity degree and map scale change in multi-scale contour clusters.

4.4. Quantitative Relations between Multi-Scale Morphostructural Similarity Degree and Map Scale Change

4.4.1. Quantitative Relations

Based on the above analysis, the functions that conform to the trend can be listed as follows:

$$\{\begin{array}{l}\begin{array}{c}S{ =\mathrm{aC}}^{-\mathrm{b}}\left(\mathrm{a}>0,\mathrm{b}>0\right)\\ S{ =\mathrm{ae}}^{-\mathrm{bC}}\left(\mathrm{a}>0,\mathrm{b}>0\right)\end{array}\hfill \\ S=\mathrm{aIn}\left(\mathrm{C}\right){+\mathrm{b}}_{ }\left(\mathrm{a}>0,\mathrm{b}>0\right)\hfill \\ S=\mathrm{aC}+\mathrm{b}\left(\mathrm{a}<0,\mathrm{b}>0\right)\hfill \\ S{ =\mathrm{aC}}^2+\mathrm{bC}+\mathrm{d}(\mathrm{a}>0)\hfill \end{array}$$

(5)

Figure 10 shows the fitting results between the multi-scale morphostructural similarity degree and map scale change, considering 52 pairs of <C, S> fitting points of loess geomorphy as an example.

It is observed in Figure 10 that although the quadratic function has the highest fitting accuracy among the candidate functions, it has two inflection points and is not always monotonic in the change range of map scale, and thus it can be excluded. Consequently, the power function (Formula (6)) provides the best fit for the quantitative relations between morphostructural similarity degree and map scale change in multi-scale contour clusters of loess geomorphy, R² = 0.9235.

$$\{\begin{array}{l}S=1\left(\mathrm{C}=1\right)\hfill \\ S={\mathrm{aC}}^{-b}\left(\mathrm{a}>0,\mathrm{b}>0\right)\hfill \end{array}$$

(6)

Figure 10 also indicates that the morphostructural similarity degree of multi-scale contour clusters decreases sharply when the map scale change is less than 5, and then levels off gradually with an increase in map scale change. To investigate whether the quantitative relations between morphostructural similarity degree and map scale change in 13 groups of multi-scale contour clusters can be expressed quantitatively by the same power function, the following contents further explore the influence of sample size on the types and precisions of the fitting function.

4.4.2. Influence of Sample Size on the Types and Precisions of Fitting Function

It is still unknown whether the types of optimal fitting function change constantly with the increase in sample sizes. Therefore, this study considers multi-scale contour clusters with different drainage areas of loess geomorphy as research objects. Figure 11 shows the influence of sample size on the types of quantitative function relations between multi-scale morphostructural similarity and map scale change.

Figure 11 also demonstrates that the fitting accuracy of the power function is always the highest among the candidate functions with the increase in sample sizes. Therefore, the power function is always the best fit for the quantitative relation between multi-scale morphostructural similarity degree and map scale change. Their fitting accuracies (R²) are not less than 0.8986, and the maximum fitting accuracy is up to 0.9401. This experimental result further confirms that the power function (Formula (6)) is the best to express the quantitative relation between morphostructural similarity degree and map scale change in multi-scale contour clusters.

Theoretically, a complete fit is R² = 1, but with the increase in sample size, R² decreases and eventually converges to a certain value. Therefore, considering the contour cluster with different contour interval changes as an example, Figure 12 shows the influence of sample sizes on the accuracy and coefficients of fitting results of power function.

As shown in Figure 12, with the increase in sample sizes, the fitting precision (R²) of the power function increased at first and then decreased, and eventually stabilized. The fitting accuracy (R²) of the power function converges to 0.93 (Figure 12a). The fitting coefficient “a” basically remains unchanged with the increase in sample size, while the fitting coefficient “b” decreases initially and then tends to stabilize. The fitting coefficients “a” and “b” converge to 0.95 and −0.35, respectively (Figure 12b,c).

$$S=0.95{\mathrm{C}}^{-0.34}\left({\mathrm{R}}^2=0.93\right)$$

(7)

The above analysis further indicates that the quantitative relation between morphostructural similarity degree and map scale change in 13 groups of samples of multi-scale contour clusters with different drainage areas of loess geomorphy can be expressed quantitatively by the same power function (Formula (7)). This conclusion demonstrates that it is reasonable and feasible for contour cluster to realize the automation of map generalization based on the multi-scale spatial similarity relations.

5. Discussion

Insights can be obtained from the experimental results presented above.

First, the ratio of neighbor scale contour line number to the corresponding neighbor scale contour interval distance ratio is constant. The reason is that the process of contour generalization includes two parts: Thinning and simplification [27]. Contour lines are thinned according to a certain contour interval in the generalization process, which is in essence a generalization of contours in quantity [28]. However, the number ratio of neighbor scale closed contour lines fluctuates around the average of the neighbor scale contour interval ratio, due to variations in geomorphological characteristics, such as surface relief, steep slopes, and surface fragmentation degree. For instance, large mountainous and broken hilly topographies have distinct morphological characteristics. Similarly, this is true for contour clusters with identical map scale and geomorphological type [7].

Second, the fluctuation degree of the bifurcation ratio of valley lines diminishes with the decrease in map scale change. Contour structure generalization involves the constant deletion of secondary valley lines in the large map scale, the continuous merging of adjacent ridge, and the preservation of main valley lines with the decrease in map scale. When the map scale reduces to a certain extent, the valley lines, which highlight the primary geomorphological features in the contour generalization process, remain unchanged. Consequently, the corresponding valley line bifurcation ratio tends to be more stable with the increase in map scale change in Figure 8b. These findings align with the principle of contour structural generalization and can provide guidance for improving spatial cognition, spatial reasoning, and map design.

Third, the results of valley line extraction, which only considered curvature, reveal that numerous feature points are extracted in the U-shaped region of gentle bending, increasing the complexity of valley line tracking and being significantly affected by the threshold value [27]. The geometric accuracy and position accuracy of valley line extracted through the combination of constrained Delaunay triangulation and curvature are generally higher. The primary source of error is that the theoretically extracted valley lines are the connecting lines of the midpoint of constrained Delaunay triangulation, while rivers in reality are naturally formed catchment lines. Consequently, the error of local position accuracy between valley line extraction results and rivers is more significant in areas with larger river bends, as demonstrated in Figure 7.

Fourth, the method proposed in this paper can also pertain to the contour cluster extracted from raster data. However, there are significant differences between the simulated contour cluster extracted from raster data using existing software modules, such as “Contour” module tool in ArcMap, and the vector contour cluster generated by a cartographer. Contour cluster extracted from corresponding raster data with different resolutions by existing software modules only refer to rarefaction of contour lines, and ignore the process of simplification, the merging of secondary positional hilltops, and displacement, etc. These deficiencies can lead to mistakes in the extraction of terrain lines, which will future lead to unreliable calculation results of multi-scale morphostructural similarity. Currently, our research team mainly focuses on the research of spatial similarity relations in multi-scale map spaces and its application in automatic map generalization. Our future research direction is to study the spatial similarity between raster dataset, such as DEM with different resolutions.

Finally, the power function is the best to express the quantitative relations between morphostructural similarity degree and map scale change in multi-scale contour clusters. When the sample size increases to a certain amount, the fitting accuracy and coefficients of power function tend to be stable. Therefore, the quantitative relations between morphostructural similarity degree and map scale change in multi-scale contour clusters with different drainage areas of loess geomorphy can be expressed by the same power function.

This conclusion has significance in the automation of contour generalization. First, it demonstrates the feasibility of realizing the automatic generalization of contour cluster based on multi-scale spatial similarity relations. Second, Formula (7) can be used in automatic contour generalization, including automation of contour generalization algorithm, automatic control of generalization process, and automatic evaluation of contour generalization results. For example, in automatic control of contour structural generalization process, if a large scale contour topographic map is generalized to any small scale contour topographic map, and the map scale change (C) in contour clusters is given, the theoretical morphostructural similarity degree S_T between the original contour cluster and generalized one can be calculated by Formula (7). The practical morphostructural similarity degree S_i between the original contour cluster and the intermediate result generalized by the contour generalization algorithm can also be calculated using the proposed model in Section 3.3. When $\left|S_i-S_T\right|\to min$, the intermediate result of contour generalization can be regarded as the target scale topographic map, and the generalization process of contour generalization algorithm should be stopped. This may facilitate the automation of map generalization software.

6. Conclusions and Future Works

6.1. Conclusions

Obtaining morphostructural similarity degrees among the same contour clusters on multi-scale maps is important in automatic map generalization. However, the quantitative expression of the morphostructural similarity relationship between contour clusters in multi-scale map spaces is a precondition for exploring the trends between morphostructural similarity degree and map scale change in multi-scale contour cluster. This paper proposes an indirect and quantitative expression model of morphostructural similarity relations of multi-scale contour clusters by constructing terrain line hierarchical trees. On this basis, the different quantitative trends of morphostructural similarity degrees of multi-scale contour clusters with the change in map scale were explored. It can be concluded that power function ($S=\mathrm{a}{\mathrm{C}}^{-\mathrm{b}}(\mathrm{a}>0,\mathrm{b}>0,)$) is the best fit to express the quantitative relations between morphostructural similarity degree and map scale change in multi-scale contour clusters. The quantitative relations of 13 groups of multi-scale contour clusters with different drainage areas of loess geomorphy can be expressed using the same power function. This conclusion indicates that it is reasonable and feasible to realize the automation of contour generalization based on multi-scale spatial similarity relations, which may facilitate the automation of map generalization software.

6.2. Future Works

Currently, only four kinds of map scale and multi-scale vector contour clusters were considered in this study. However, experiments indicate that the morphostructural similarity of multi-scale contour clusters changes drop drastically at small map scale change (C ≤ 5). Additionally, morphostructural similarity relations of multi-scale contour cluster have not been well applied to the processing of contour generalization. Therefore, our future study will increase the sample sizes of large scale datasets of different geomorphologic morphology types, such as mountainous topography, Aeolian landform, and fluvial landform, etc. We will also apply multi-scale contour spatial similarity relations to the processing of contour generalization, which will provide a foundation for human spatial cognition, matching of spatial data, and pattern recognition, especially the automation of map generalization for contour clusters. In addition, the spatial similarity relations between multi-resolution raster DEMs are a direction worth studying.

Future research on this issue may focus on the following areas:

(1)

Conducting more experiments to improve the accuracy and adaptability of the proposed models and formulas. Scale changes are an inductive behavior of spatial thinking, and it is necessary to obtain universal, unified, and high-level spatial knowledge from a large number of specific, individual, and underlying data through granularity filtering, hierarchical merging, and linguistic generalization [51]. Therefore, it would be more useful to perform the analysis on more map samples. Additionally, it is essential to explore the extent to which the preservation of geomorphological characteristics at smaller scales is possible, as the nature of maps is changing from topographic to general geographic.

(2)

Is it possible to design one model that can calculate the spatial similarity degree of individual point objects, and/or object groups, such as point clouds, parallel line clusters, intersected line networks, tree-like networks discrete polygon groups, by taking map scale change as an independent variable? Previous studies on models for spatial similarity relations in multi-scale map spaces indicated that there is a quantitative relationship between map scale change and spatial similarity degree. However, this quantitative relationship has not been determined, and some studies consider it to be a power function, while others consider it to be an exponential function.

(3)

It is worth exploring approaches for automatically obtaining the parameters used in the algorithms and operators that are not parameter free with the help of the models and formulae of multi-scale spatial similarity relations. Progress in this area will lay a good foundation for full automation of map generalization.