An Improved ANN-Based Label Placement Method Considering Surrounding Features for Schematic Metro Maps

Abstract

On schematic metro maps, high-quality label placement is helpful to passengers performing route planning and orientation tasks. It has been reported that the artificial neural network (ANN) has the potential to place labels with learned labeling knowledge. However, the previous ANN-based method only considered the effects of station points and their connected edges. Indeed, unconnected but surrounding features (points, edges, and labels) also significantly affect the quality of label placement. To address this, we have proposed an improved method. The relations between label positions and both connected and surrounding features are first modeled based on labeling natural intelligence (i.e., the experience, knowledge, and rules of labeling established by cartographers). Then, ANN is employed to learn such relations. Quantitative evaluations show that our method reaches lower percentages of label–point overlap (0.00%), label–edge overlap (4.12%), and label–label overlap (20.58%) compared to the benchmark (4.17%, 14.29%, and 35.11%, respectively). On the other hand, our method effectively avoids ambiguous labels and ensures labels from the same line are placed on the same side. Qualitative evaluations show that approximately 75% of users prefer our results. This novel method has the potential to advance the automated generation of schematic metro maps.

1. Introduction

Schematic metro maps are abstract representations of metro networks where the complexity is reduced by removing geographical features, simplifying lines, reorienting lines, etc. [1,2]. With such maps, passengers can quickly and accurately perform metro route planning and orientation tasks. One prominent example is the London Underground map (see Figure 1), where simplified metro lines are in vertical, horizontal, and diagonal directions. On the other hand, the congested areas have been expanded, and most details of geographic features have been removed. At present, these design principles usually serve as a basis for guiding automatic schematic metro map design [3].

Generally, the automatic schematic metro map design is divided into two subtasks: metro map layout and metro map labeling [4,5,6,7,8]. The research on metro map layout has received much attention and achieved remarkable results. In the early stage, a series of criteria for map layouts have been defined, and some methods for automatically generating schematic maps have been proposed with these criteria [4,9,10]. Later, researchers aimed to improve metro map layouts from different perspectives. It has been pointed out that the edge lengths should be maintained equal to ensure regular spacing between stations [1,11,12]. On the other hand, the directions of the edge are commonly along vertical, horizontal, and 45-degree diagonals, named octolinearity [13]. Recently, it has been found that other direction design styles (e.g., multilinear and curvilinear) may be preferred by passengers in some scenes [2,13]. Moreover, the topological relation between features and the main structure of the network should be preserved for high-quality schematic metro maps [14,15,16,17,18].

Figure 1. Schematic metro map of London was designed by Harry Beck in the 1930s [19].

Figure 1. Schematic metro map of London was designed by Harry Beck in the 1930s [19].

The research on metro map labeling has received relatively limited attention [1,6,7,11]. In theory, the automatic labeling method should be developed based on the labeling natural intelligence of schematic metro maps (i.e., the experience, knowledge, and rules in labeling established by cartographers). In practice, such labeling natural intelligence is complex, and it is challenging to model them with mathematical expressions. The point labeling natural intelligence on topographic maps (which have formed a series of mathematical principles [20,21,22]) is commonly used as an alternative [11]. However, the applicability of these principles to schematic metro maps may be inappropriate in some situations [6]. For example, one crucial aspect of metro map labeling states: “labels of those points touching the same passing line are placed together on the same side” (see Figure 2a). By contrast, on topographic maps, points are usually positioned adjacent to rather than touching the line, and one labeling principle states: “a label and the corresponding point should be placed at the same side” (see Figure 2b).

The modeling of labeling natural intelligence for schematic metro maps has been reconsidered in recent years [6,7,23]. Labeling natural intelligence has been modeled from official schematic metro maps in two ways: to count labeling positions for point–edge situations in a statistical manner [6] and to learn the characteristics of point–edge situations by an artificial neural network (ANN) [7]. The first way depends on the pre-defined point–edge situations. In other words, if a specific relation is not pre-defined, it may be difficult to identify appropriate labeling positions for this relation. Theoretically, this problem can be solved by pre-defining all relations, but it requires a lot of time and resources. The second way employs an artificial neural network (with good generalization ability) to learn the characteristics of point–edge situations and infer labeling positions with examples. Although it is a good idea, the basic unit used for learning, i.e., point–edge situations, only considers the point referred to by the label and the edges connected to the point. Indeed, the unconnected but surrounding points and edges also significantly affect the quality of label placement. Figure 3a shows the fixed label position of a point–edge situation, and no overlaps are observed. However, overlaps occur in Figure 3b when other features (points and edges in this case) surround this fixed label. To address this, we aim to develop an improved labeling method by integrating relations between label positions and both connected and surrounding features into the ANN model.

The remainder of this article is organized as follows: In Section 2, we analyze and model the labeling natural intelligence of schematic metro maps. In Section 3, we propose an ANN-based label placement method with labeling natural intelligence. In Section 4, the experimental evaluation is presented. Some discussion and conclusions are made in Section 5 and Section 6, respectively.

2. Analysis and Modeling of Labeling Natural Intelligence

To learn the new relations, we first conduct a thorough analysis of labeling natural intelligence. Then, we model them with a set of variables.

2.1. Analysis of Labeling Natural Intelligence

Through a thorough survey of the literature on metro map labeling (see

Table 1. Common labeling natural intelligence in the existing methods.

Labeling Natural Intelligence	Source
Place labels without overlaps	References [1,4,5,6,7,9,11,12,24,25,26,27,28,29,30]
Place labels without ambiguity	References [1,6,11,12,24,29,30]
Place labels from the same passing line on the same side	References [1,5,6,7,11,12,24,26,27,28,30]
Place labels in the most preferred position	References [1,4,6,7,9,11,12,26,27,28,30]

), we have identified four common labeling natural intelligences: (1) place labels without overlaps; (2) place labels without ambiguity; (3) place labels from the same passing line on the same side; and (4) place labels in the most preferred positions.

2.1.1. Place Labels without Overlaps

The avoidance of overlapping labels is the most basic and important aspect [1,10], which aims to ensure the clarity and legibility of the map. Overlapping labels can be categorized into three kinds of cases: label–point overlap, label–edge overlap, and label–label overlap, as shown in Figure 4.

2.1.2. Place Labels without Ambiguity

Labels are often the only way to identify stations on a metro map. Therefore, ambiguous associations must be forbidden to ensure that passengers can accurately identify the location of each station. Two examples of ambiguous labels are shown in Figure 5. In these examples, it is unclear which point the labels associate.

2.1.3. Place Labels from the Same Passing Line on the Same Side

To maintain consistency and continuity on the metro maps, it is necessary to place labels for points touching the same passing line on the same side as much as possible, as shown in Figure 6. Additionally, if the labels are arbitrarily placed on the two sides of the passing line, this can lead to visual confusion and make the map difficult to read.

2.1.4. Place Labels in the Most Preferred Position

Labels should be placed in the most preferred position among all acceptable positions. Such a preferred position means the visual and cognitive preferences when viewing schematic metro maps. Some examples of preferred label placements are shown in Figure 7 (see reference [6] for the preferred label positions of all point–edge situations).

Conflicts may be caused among the four labeling natural intelligences mentioned above. In Figure 8a, all labels are placed on the right of the passing line, and this satisfies the third labeling natural intelligence but violates the first one. Indeed, such overlapped results are most unacceptable, and some possible improvements (e.g., Figure 8b,c) will be made by designers. This indicates that there exists a labeling natural intelligence that determines the prioritization among labeling principles.

2.2. Modeling of Labeling Natural Intelligence

To ensure the labeling natural intelligence to be learned by the ANN model, it is necessary to model them mathematically. As the basic unit to be learned is the labels, we first define the candidate labels for station points. Generally, the direction of labels can be horizontal and vertical. For each direction, eight fixed positions are commonly used. As a result, 16 candidate labels are defined for each station point, as shown in Figure 9.

2.2.1. Modeling of Overlaps

For label–point and label–edge overlaps, we respectively define 16 binary variables (($x_1, x_2, x_3, \cdots , x_{16}$) and ($x_{17}, x_{18}, x_{19}, \cdots , x_{32}$)) to denote whether candidate labels 1 to 16 are overlapped or not. If a candidate label is overlapped, it is recorded as 1, and vice versa as 0. For label and label overlap, we define 16 variables ($x_{33}, x_{34}, x_{35}, \cdots , x_{48}$) to denote the percentage of overlapped area, which ranges from 0 to 1. Some examples are shown in Figure 10. In Figure 10a, the second candidate label of $p_1$ is overlapped by $p_2$, hence, the variable $x_2$ of $p_1$ is recorded 1.

2.2.2. Modeling of Ambiguity

The ambiguity of a label can be measured by the distances of the label from the points. The distance between a label and its point is usually fixed in the given metro schematic map. The label is ambiguous when the distances between the label and other points approximate this fixed distance. Therefore, we define 16 variables ($x_{49}, x_{50}, x_{51}, \cdots , x_{64}$) to denote the minimum distances between the candidate labels and other points. It is noted that the distance measure used in this paper is the minimum Euclidean distance between the point and the label contour, as shown in Figure 11.

2.2.3. Modeling of Passing Line Direction

To model the principle “labels for points touching the same passing line are placed on the same side”, the identification of the type of passing line is required. In this paper, we adopt the previous method [7], which employs direction to classify passing lines into five types: horizontal, vertical, left diagonal, right diagonal, and hybrid types. Then, a five-dimensional binary variable ($x_{65}, x_{66}, \cdots , x_{69}$) is used to represent these five passing line directions, as shown in Figure 12.

As the official schematic metro maps are well-designed productions by cartographic experts, they are already considered the most preferred position and the priority among labeling principles. Therefore, this labeling natural intelligence will be observed by learning the relations between label positions and map features. In summary, the mathematical model of labeling natural intelligence for each station point’s 16 candidate labels can be defined as the following matrix ($N\times 69$, where $N$ is the number of points, see

Table 2. Characteristic matrix modeling labeling natural intelligence.

	Variables Modeling Label–Point Overlap	Variables Modeling Label–Edge Overlap	Variables Modeling Label–Label Overlap	Variables Modeling Ambiguity of Labels	Variables Modeling Direction of Passing Lines
	${\mathit{x}}_1, {\mathit{x}}_2, {\mathit{x}}_3, \cdots , {\mathit{x}}_{16}$	${\mathit{x}}_{17}, {\mathit{x}}_{18}, {\mathit{x}}_{19}, \cdots , {\mathit{x}}_{32}$	${\mathit{x}}_{33}, {\mathit{x}}_{34}, {\mathit{x}}_{35}, \cdots , {\mathit{x}}_{48}$	${\mathit{x}}_{49}, {\mathit{x}}_{50}, {\mathit{x}}_{51}, \cdots , {\mathit{x}}_{64}$	${\mathit{x}}_{65}, {\mathit{x}}_{66}, \cdots , {\mathit{x}}_{69}$
Point 1	$x_1^1,x_2^1,x_3^1,\cdots ,x_{16}^1$	$x_{17}^1,x_{18}^1,x_{19}^1,\cdots ,x_{32}^1$	$x_{33}^1,x_{34}^1,x_{35}^1,\cdots ,x_{48}^1$	$x_{49}^1,x_{50}^1,x_{51}^1,\cdots ,x_{64}^1$	$x_{65}^1,x_{66}^1,\cdots ,x_{69}^1$
Point 2	$x_1^2,x_2^2,x_3^2,\cdots ,x_{16}^2$	$x_{17}^2,x_{18}^2,x_{19}^2,\cdots ,x_{32}^2$	$x_{33}^2,x_{34}^2,x_{35}^2,\cdots ,x_{48}^2$	$x_{49}^2,x_{50}^2,x_{51}^2,\cdots ,x_{64}^2$	$x_{65}^2,x_{66}^2,\cdots ,x_{69}^2$
Point 3	$x_1^3,x_2^3,x_3^3,\cdots ,x_{16}^3$	$x_{17}^3,x_{18}^3,x_{19}^3,\cdots ,x_{32}^3$	$x_{33}^3,x_{34}^3,x_{35}^3,\cdots ,x_{48}^3$	$x_{49}^3,x_{50}^3,x_{51}^3,\cdots ,x_{64}^3$	$x_{65}^3,x_{66}^3,\cdots ,x_{69}^3$
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
Point N	$x_1^n,x_2^n,x_3^n,\cdots ,x_{16}^n$	$x_{17}^n,x_{18}^n,x_{19}^n,\cdots ,x_{32}^n$	$x_{33}^n,x_{34}^n,x_{35}^n,\cdots ,x_{48}^n$	$x_{49}^n,x_{50}^n,x_{51}^n,\cdots ,x_{64}^n$	$x_{65}^n,x_{66}^n,\cdots ,x_{69}^n$

), which is used as the input to the ANN model.

3. ANN-Based Label Placement Method with Natural Intelligence

In this section, we propose an ANN-based label placement method. To implement this method, the construction of the ANN model is required. Then, the training and testing datasets are developed.

3.1. Construction of the ANN Labeling Model

As mentioned in the previous section, ANN has a good generalization capability to perform well on unseen or new data after being trained on a dataset [31,32,33]. Several common architectures or models of ANN (e.g., multilayer feedforward, MLF, and convolutional neural network, CNN) have been developed over the years to address various tasks and challenges, such as image and speech recognition [34,35], natural language processing [36], and recommendation systems [37]. In cartography, ANN can be applied to enhance various aspects, such as map generalization [38,39,40,41] and map production [42,43].

In this study, we treat the label placement as a supervised learning task. Then, the MLF model is selected for its suitability for supervised tasks [44]. A complete MLF model typically contains the following components: input, hidden, and output layers. The number of hidden layers and the number of neurons in the hidden layers are determined by a “trial-by-error” method [7]. This method involves initially training neural networks with varying numbers of hidden neurons, followed by an assessment based on generalization error to determine the optimal number. For activation functions, preference is given to nonlinear options. Among these, the Rectified Linear Unit (ReLU) is employed in this study for its computational efficiency and nonlinear properties. Finally, we construct the ANN model using PyTorch, which is a popular open-source machine learning library developed by Facebook’s AI Research lab (FAIR). Figure 13 shows the constructed ANN model.

The model’s initial values of parameters and hyperparameters are required before the training and testing phase. Usually, the parameters are randomly initialized and dynamically optimized in the training process. The hyperparameters (e.g., learning rate) are set manually, and their initial values are listed in

Table 3. Initial values of parameters.

Hyperparameters	Initial Values	Explanation
Learning rate	0.005	Controls the balance between convergence speed and stability
Batch size	5	The number of training examples utilized in one iteration of the training process
Momentum	0.9	Accelerates optimization, smooths gradient updates, and helps overcome noisy gradients and oscillations
Step size Gamma	7 0.1	Affects the change in the learning rate
Number of epochs	50	Determines how many times the network learns from the entire training dataset

.

3.2. Training and Testing the ANN Labeling Model

The original training and testing datasets are collected as pictures from the official schematic metro maps, which are well-designed productions by cartographic experts. To fill the characteristic matrix ($N\times 69$), we vectorize pictures of metro maps by ArcGIS 10.6.1 software. It is noted that the sizes of vector pictures are various, leading to significant differences in the size of the labels. To handle this, we keep the label size similar, i.e., the same size of language characters in name labels. Specifically, the lengths of characters on schematic maps are utilized as benchmarks to guide the “zoom in” or “zoom out” function of these vectorized pictures. That is, the side lengths of the minimum bounding rectangle (MBR) of the characters on different schematic metro maps are equal. This standardization is achieved by dividing all coordinates of the points on a schematic metro map by the length of the MBR of a character. Then, we generate the candidate labels for each station point, as shown in Figure 14. Finally, we fill the characteristic matrix with ArcGIS functions, such as “Select Layer by Location” and “Near”.

It is noted that the common metric “accuracy” is limited when evaluating the testing labeling outcomes because the same input may produce multiple correct outputs. This may be attributed to the individual preferences of schematic network map designers [7].

4. Experimental Evaluation

Quantitative and qualitative evaluations have been conducted in this section. Five evaluation metrics of the labeling results have been compared between our method and the benchmark [7]. On the other hand, a survey has been conducted by asking users to select the more pleasing labeling results.

4.1. Experimental Data

The official schematic metro maps were collected from 15 cities (Chengdu, Guangzhou, Hong Kong, Nanjing, Shenzhen, Suzhou, Tianjin, Wuhan, Beijing, Xiamen, Chongqing, Singapore, Madrid, New York, and London) in different countries, as shown in Figure 15. A total of 3048 station points are sampled from these maps. The training set is from 12 cities (Chengdu, Guangzhou, Hong Kong, Shenzhen, Tianjin, Wuhan, Beijing, Xiamen, Chongqing, Madrid, New York, and London) with 2635 station points. The test set is from three cities (Nanjing, Suzhou, and Singapore) with 413 stations.

4.2. Experimental Evaluation Method

Four metrics corresponding to the most common labeling natural intelligence (see Section 2) are used for quantitative evaluation, i.e., the number of overlapping labels ($Nu{m}_{\mathrm{overlapping}}$), the sum of the closest distance from each label to other points ($Di{s}_{\mathrm{minimum}}$), the number of labels placed at the most preferred position ($Nu{m}_{\mathrm{preference}}$), and the number of labels placed on the same side ($Nu{m}_{\mathrm{consistency}}$). On the other hand, the map-load level ($MLL$) is a crucial aspect of map design, and it was used to evaluate schematic metro maps.

$Nu{m}_{\mathrm{overlapping}}$ includes overlap between label and point ($Overla{p}_{\mathrm{label}-\mathrm{point}}$), overlap between label and edge ($Overla{p}_{\mathrm{label}-\mathrm{edge}}$), as well as overlap between label and label ($Overla{p}_{\mathrm{label}-\mathrm{label}}$). $MLL$ can be calculated by the Graphic Map Load Measuring Tool (GMLMT) [45]. In order to calculate the $Nu{m}_{\mathrm{consistency}}$, we define the side where the labels should be placed, which is determined by the majority.

Qualitative evaluation asks users to select the more pleasing labeling results from our method and benchmark. In this survey, we obtained the selection of 42 participants through the “Wenjuanxing” online platform.

4.3. Experimental Results

Figure 16, Figure 17 and Figure 18 show the labeling results of our method and the benchmark.

Table 4. The number of overlapping labels produced by our method and the benchmark.

City	Method	$\mathit{O}\mathit{v}\mathit{e}\mathit{r}\mathit{l}\mathit{a}{\mathit{p}}_{\mathbf{l}\mathbf{a}\mathbf{b}\mathbf{e}\mathbf{l}-\mathbf{p}\mathbf{o}\mathbf{i}\mathbf{n}\mathbf{t}}$	$\mathit{O}\mathit{v}\mathit{e}\mathit{r}\mathit{l}\mathit{a}{\mathit{p}}_{\mathbf{l}\mathbf{a}\mathbf{b}\mathbf{e}\mathbf{l}-\mathbf{e}\mathbf{d}\mathbf{g}\mathbf{e}}$	$\mathit{O}\mathit{v}\mathit{e}\mathit{r}\mathit{l}\mathit{a}{\mathit{p}}_{\mathbf{l}\mathbf{a}\mathbf{b}\mathbf{e}\mathbf{l}-\mathbf{l}\mathbf{a}\mathbf{b}\mathbf{e}\mathbf{l}}$	$\mathit{N}\mathit{u}{\mathit{m}}_{\mathbf{o}\mathbf{v}\mathbf{e}\mathbf{r}\mathbf{l}\mathbf{a}\mathbf{p}\mathbf{p}\mathbf{i}\mathbf{n}\mathbf{g}}$
Nanjing	Our method	0	6	33	39
	Benchmark	9	24	58	91
Suzhou	Our method	0	2	39	41
	Benchmark	1	7	54	62
Singapore	Our method	0	9	12	21
	Benchmark	7	28	33	68

lists the $Nu{m}_{\mathrm{overlapping}}$, $Overla{p}_{\mathrm{label}-\mathrm{point}}$, $Overla{p}_{\mathrm{label}-\mathrm{edge}}$, and $Overla{p}_{\mathrm{label}-\mathrm{label}}$. The results show that the $Nu{m}_{\mathrm{overlapping}}$ of our method is significantly lower than that of the benchmark. More precisely, the $Overla{p}_{\mathrm{label}-\mathrm{point}}$, $Overla{p}_{\mathrm{label}-\mathrm{edge}}$, and $Overla{p}_{\mathrm{label}-\mathrm{label}}$ of our method are lower than those of the benchmark, accounting for 0.00%, 4.12%, and 20.58% of the total number of labels (413), compared with 4.17%, 14.29%, and 35.11% for the benchmark, respectively. These results indicate that our method is effective in avoiding overlapping labels.

Table 5. The $Di{s}_{\mathrm{minimum}}$ of our method and the benchmark.

City	Method	$\mathit{D}\mathit{i}{\mathit{s}}_{\mathbf{m}\mathbf{i}\mathbf{n}\mathbf{i}\mathbf{m}\mathbf{u}\mathbf{m}}$
Nanjing	Our method	2234.7
	Benchmark	1807.3
Suzhou	Our method	1939.7
	Benchmark	1792.6
Singapore	Our method	3542.1
	Benchmark	3111.6

shows the $Di{s}_{\mathrm{minimum}}$ of our method and the benchmark. The results show that the $Di{s}_{\mathrm{minimum}}$ of our method is larger than that of the benchmark. This indicates that our method is good at avoiding ambiguous labels, as shown in Figure 19.

Table 6. The $Nu{m}_{\mathrm{preference}}$ and $Nu{m}_{\mathrm{consistency}}$ of our method and the benchmark.

City	Method	$\mathit{N}\mathit{u}{\mathit{m}}_{\mathbf{preference}}$	$\mathit{N}\mathit{u}{\mathit{m}}_{\mathbf{consistency}}$
Nanjing	Our method	104	135
	Benchmark	138	136
Suzhou	Our method	89	111
	Benchmark	120	116
Singapore	Our method	121	129
	Benchmark	135	125

lists the $Nu{m}_{\mathrm{preference}}$ and $Nu{m}_{\mathrm{consistency}}$ of our method and the benchmark. It was found that the $Nu{m}_{\mathrm{preference}}$ of our method is lower than that of the benchmark. This is because our method prioritizes the avoidance of overlapping labels at the expense of preference. On the other hand, $Nu{m}_{\mathrm{consistency}}$ is comparable between our method and the benchmark.

Figure 20, Figure 21 and Figure 22 show the map-load level (ranging from 0 to 100) distribution on the resulting maps of our method and the benchmark. Specifically, the maps are divided into 100 grids, and the map-load level for each grid is calculated. We focus on the grids that contain overlapping labels (marked by green boxes) or ambiguous labels (marked by blue boxes) in the benchmark but do not in our method.

It was found that the map-load levels of the blue boxes and most of the green boxes decreased in our resulting maps. This finding indicates that our method can place labels with a lower map-load level. Further, we found that the map-load levels decrease after eliminating label–label overlaps (green solid boxes) but increase after eliminating label–point or label–edge overlaps (green dotted boxes, see Figure 22). This is because the visual complexity of the label–point or label–edge overlap seems lower than that of the label–label overlap. However, it is clear that our label placements in green dotted boxes, despite having larger map-load levels, are more readable. This implies that the map-load level calculated by GMLMT is only an objective estimation of visual complexity. Therefore, a subjective evaluation of readability for label placement is required.

Table 7. Preferences of schematic metro maps between our method and the benchmark.

City	Method	Number	Proportion
Nanjing	Our method	35	83.3%
	Benchmark	7	16.7%
Suzhou	Our method	30	71.4%
	Benchmark	12	28.6%
Singapore	Our method	31	73.8%
	Benchmark	11	26.2%

presents the preferences of 42 participants regarding label placement generated by our method compared to the benchmark. Results show that 35 participants (83.3%), 30 participants (71.4%), and 31 participants (73.8%) preferred our labeling results for Nanjing, Suzhou, and Singapore, respectively. These results reveal that the label placements of our method are more pleasing than the benchmark.

5. Discussion

5.1. Impact Analysis of Different Characteristics and Settings

An ablation study has been conducted to analyze the impact of different components or settings of our method on the resulting schematic metro maps. Specifically, hidden layers-based and characteristic variables-based ablation experiments have been performed, respectively (see

Table 8. Hidden layers-based and characteristic variables-based ablation experiments.

	Experiment	Explanation
Hidden layers-based	$Remove\_H1$	Remove hidden layer 1 ($69\times 512$)
	$Remove\_H2$	Remove hidden layer 2 ($512\times 1024$)
	$Remove\_H3$	Remove hidden layer 3 ($1024\times 512$)
Characteristic variables-based	$Remove\_C1$	Remove variables of label–point overlap
	$Remove\_C2$	Remove variables of label–edge overlap
	$Remove\_C3$	Remove variables of label–label overlap
	$Remove\_C4$	Remove variables of ambiguity of labels
	$Remove\_C5$	Remove variables of direction of passing lines

for details on these experiments).

Table 9. Results of hidden layers-based ablation experiments.

City	Experiment	$\mathit{O}\mathit{v}\mathit{e}\mathit{r}\mathit{l}\mathit{a}{\mathit{p}}_{\mathbf{l}\mathbf{a}\mathbf{b}\mathbf{e}\mathbf{l}-\mathbf{p}\mathbf{o}\mathbf{i}\mathbf{n}\mathbf{t}}$	$\mathit{O}\mathit{v}\mathit{e}\mathit{r}\mathit{l}\mathit{a}{\mathit{p}}_{\mathbf{l}\mathbf{a}\mathbf{b}\mathbf{e}\mathbf{l}-\mathbf{e}\mathbf{d}\mathbf{g}\mathbf{e}}$	$\mathit{O}\mathit{v}\mathit{e}\mathit{r}\mathit{l}\mathit{a}{\mathit{p}}_{\mathbf{l}\mathbf{a}\mathbf{b}\mathbf{e}\mathbf{l}-\mathbf{l}\mathbf{a}\mathbf{b}\mathbf{e}\mathbf{l}}$	$\mathit{N}\mathit{u}{\mathit{m}}_{\mathbf{overlapping}}$	$\mathit{D}\mathit{i}{\mathit{s}}_{\mathbf{minimum}}$
Nanjing	$Remove\_H1$	0	6	32	38	2178.9
	$Remove\_H2$	1	6	37	44	2207.1
	$Remove\_H3$	0	7	35	42	2252.9
	Reference	0	6	33	39	2234.7
Suzhou	$Remove\_H1$	0	2	37	39	1961.3
	$Remove\_H2$	0	2	40	42	1958.5
	$Remove\_H3$	0	2	41	43	2001.7
	Reference	0	2	39	41	1939.7
Singapore	$Remove\_H1$	0	9	14	23	3531.4
	$Remove\_H2$	0	9	14	23	3474.7
	$Remove\_H3$	0	8	8	16	3550.7
	Reference	0	9	12	21	3542.1

shows the results of hidden layers-based ablation experiments. No significant impact was found after removing any hidden layers.

Table 10. Results of characteristic variables-based ablation experiments.

City	Experiment	$\mathit{O}\mathit{v}\mathit{e}\mathit{r}\mathit{l}\mathit{a}{\mathit{p}}_{\mathbf{l}\mathbf{a}\mathbf{b}\mathbf{e}\mathbf{l}-\mathbf{p}\mathbf{o}\mathbf{i}\mathbf{n}\mathbf{t}}$	$\mathit{O}\mathit{v}\mathit{e}\mathit{r}\mathit{l}\mathit{a}{\mathit{p}}_{\mathbf{l}\mathbf{a}\mathbf{b}\mathbf{e}\mathbf{l}-\mathbf{e}\mathbf{d}\mathbf{g}\mathbf{e}}$	$\mathit{O}\mathit{v}\mathit{e}\mathit{r}\mathit{l}\mathit{a}{\mathit{p}}_{\mathbf{l}\mathbf{a}\mathbf{b}\mathbf{e}\mathbf{l}-\mathbf{l}\mathbf{a}\mathbf{b}\mathbf{e}\mathbf{l}}$	$\mathit{N}\mathit{u}{\mathit{m}}_{\mathbf{overlapping}}$	$\mathit{D}\mathit{i}{\mathit{s}}_{\mathbf{minimum}}$
Nanjing	$Remove\_C1$	5	53	39	97	1973.2
	$Remove\_C2$	4	15	40	59	2131.2
	$Remove\_C3$	0	6	37	43	2151.2
	$Remove\_C4$	0	5	41	46	2158.2
	$Remove\_C5$	0	7	33	40	2207.0
	Reference	0	6	33	39	2234.7
Suzhou	$Remove\_C1$	3	16	44	63	1894.3
	$Remove\_C2$	0	4	42	46	1948.0
	$Remove\_C3$	0	2	41	43	1939.8
	$Remove\_C4$	0	2	39	41	1927.8
	$Remove\_C5$	0	2	40	42	1957.3
	Reference	0	2	39	41	1939.7
Singapore	$Remove\_C1$	3	76	22	101	2960.0
	$Remove\_C2$	5	23	27	55	3354.9
	$Remove\_C3$	0	9	16	25	3483.5
	$Remove\_C4$	0	8	10	18	3516.9
	$Remove\_C5$	0	8	12	20	3497.8
	Reference	0	9	12	21	3542.1

shows the results of characteristic variables-based ablation experiments. It was found that $Overla{p}_{\mathrm{label}-\mathrm{point}}$, $Overla{p}_{\mathrm{label}-\mathrm{edge}}$, and $Overla{p}_{\mathrm{label}-\mathrm{label}}$ significantly increase while $Di{s}_{\mathrm{minimum}}$ decreases after removing characteristic variable 1 (label–point overlaps) and characteristic variable 2 (label–edge overlaps). These results indicate that these two characteristics play an important role in avoiding overlapping and ambiguous labels. On the other hand, this implies that the remaining characteristics may be redundant in the original experiment.

5.2. Reduce Overlapping Labels Using an Optimization Algorithm

The overlapping labels still exist in our labeling results, and most of them are caused by adjacent station points touching a horizontal passing line, as shown in Figure 23a. On official schematic metro maps, labels of such points are usually progressively placed crosswise up and down based on heuristic information, as shown in Figure 23b. However, such heuristic information heavily depends on human intelligence, and it is difficult to integrate them into the characteristic matrix.

The traditional method was employed to reduce these remaining overlapping labels automatically. Specifically, the label placement is treated as a combinatorial optimization problem in a limited cartographic space. To solve the optimization problem means to maximize or minimize one or more objective functions.

In this process, an initial label placement is required to input, and a random placement is commonly used due to simplicity. However, such placement is inappropriate for schematic metro maps because these maps require special constraints, e.g., labels from the same passing line should be placed on the same side as much as possible.

The labeling result of our method can be used as an appropriate initial label placement. In order to verify this idea, we employed a genetic algorithm (one commonly used optimization algorithm) to optimize the following objective function:

$$Objective function=Min\left(Overla{p}_{\mathrm{label}-\mathrm{point}}+Overla{p}_{\mathrm{label}-\mathrm{edge}}+Overla{p}_{\mathrm{label}-\mathrm{label}}\right)$$

(1)

Figure 24 shows the results optimized from our label placement and the random label placement. No overlapping labels appear in both optimized results. It is clear that the results from our label placement exhibit better preference and consistency.

On the other hand,

Table 11. The number of overlapping labels and the number of iterations performed for two initial label placements.

Initial Placement	Overlapping Labels	Number of Iterations
Our method	41	18
Random method	105	58

shows the overlapping labels of two initial label placements and the iterations required for optimization. The findings suggest that the optimization from our label placement performs fewer iterations, implying its effectiveness for practical application.

6. Conclusions

On schematic metro maps, high-quality name labels are helpful to passengers performing metro route planning and orientation tasks. It has been reported that an artificial neural network (ANN) is an effective and efficient way to place high-quality labels. In this study, we developed an improved labeling method by integrating relations between label positions and both connected and surrounding features into the ANN model.

Quantitative and qualitative evaluations with a benchmark have been conducted using three official schematic metro maps. Quantitative results show our method can produce schematic metro maps with fewer overlapping labels than the benchmark. Specifically, the number of three types of overlaps (label–point, label–edge, and label–label) by our method is lower than that of the benchmark, accounting for 0.00%, 4.12%, and 20.58% of the total number of labels, compared with 4.17%, 14.29%, and 35.11% for the benchmark, respectively. On the other hand, our method is good at avoiding ambiguous labels and also trying to ensure labels from the same passing line on the same side. It should be noted that the reduction in overlapping and ambiguous labels is achieved at the cost of the label position preference. Qualitative results show that most users (around 75%) are more satisfied with our schematic maps than those of the benchmark. Thus, based on the experimental results, it may be concluded that the improved ANN-based label placement method is effective.

On the other hand, our method can be extended to place labels for topographic maps, which usually include multiple features (i.e., points, lines, and areas). The key to developing an ANN-based method for labeling topographic maps is to define effective relationships between labels and features based on labeling natural intelligence. It is believed that this method can significantly improve the quality and efficiency of label placement on topographic maps.

There are limitations to this new method. First, a particular type of overlapping label (i.e., occurring at adjacent station points touching a horizontal passing line) still exists in our labeling results. Although an optimization algorithm-based method has been discussed to address this limitation, future work will focus on how to model it to achieve completely automatic label placement based on artificial neural networks. Second, the developed characteristic matrix may include redundant variables, and an improved matrix will be explored.