Simplification and Regularization Algorithm for Right-Angled Polygon Building Outlines with Jagged Edges

Abstract

Building outlines are important for emergency response, urban planning, and change analysis and can be quickly extracted from remote sensing images and raster maps using deep learning technology. However, such building outlines often have irregular boundaries, redundant points, inaccurate positions, and unclear turns arising from variations in the image quality, the complexity of the surrounding environment, and the extraction methods used, impeding their direct utility. Therefore, this study proposes a simplification and regularization algorithm for right-angled polygon building outlines with jagged edges. First, the minimum bounding rectangle of the building outlines is established and populated with a square grid based on the smallest visible length principle. Overlay analysis is then applied to the grid and original buildings to extract the turning points of the outlines. Finally, the building orientation is used as a reference axis to sort the turning points and reconstruct the simplified building outlines. Experimentally, the proposed simplification method enhances the morphological characteristics of building outlines, such as parallelism and orthogonality, while considering simplification principles, such as the preservation of the direction, position, area, and shape of the building. The proposed algorithm provides a new simplification and regularization method for right-angled polygon building outlines with jagged edges.

1. Introduction

Automated generalization of residential areas is a popular and challenging research area in geographic information system (GIS) mapping [1]. In large-scale mapping of urban areas, residential areas are generally represented as polygons, which occupy a significant proportion of the map area [2]. Various techniques are used for generalization, including selection, simplification, aggregation, displacement, exaggeration, and classification. Among these, simplification is one of the most commonly used methods [3]. Building outlines data extracted from remote sensing images and raster maps can be adversely affected by factors such as equipment accuracy, image resolution, and climate. Consequently, directly extracted buildings often have irregular shapes and jagged outlines, and building data are highly redundant. Therefore, appropriate simplification algorithms should be used to denoise and regularize extracted data to comply with mapping standards and adapt them to the representation capability of the feature map.

Various simplification algorithms have been proposed and can be classified into three types: vector-, raster-, and vector–raster-based algorithms. Vector-based algorithms are most commonly used for building outlines simplification and include progressive simplification [4,5,6,7,8], dimensionality reduction [9,10], least-squares adjustment [11,12,13], constrained Delaunay triangulation [14,15,16], feature edge extraction [17,18], template matching [19,20,21], and the utilization of machine learning methods [22]. Progressive simplification methods use the sides adjacent to the concave and convex parts of a building to construct adjacent rectangles and perform simplification based on an area comparison to preserve the area and shape of the building. This method is primarily used for polygons with right-angled turns. Dimensionality reduction methods extract the skeleton of a surface, simplify it using linear processing, and restore it to building outlines surface with enhanced orthogonality. In least-squares adjustment methods, the boundary points are grouped using a set threshold value, and a least-squares adjustment model is used to simplify and regularize the building outlines. Least-squares simplification is an optimization process. Constrained Delaunay triangulation methods partition the polygon into a set of triangles and, after identifying the triangle types, simplify the building outlines by removing or moving triangle vertices while preserving, as much as possible, the surface area of the building. Adjacent four-point methods select a starting point and successively form a processing unit consisting of four adjacent points by moving clockwise or counterclockwise, gradually simplifying the building outlines. This method involves simple operations and can be easily implemented to solve local problems. Template matching methods replace the original building with the closest matching shape from the template database. This method preserves the shapes and distribution patterns of buildings; however, its results are overly reliant on template databases. By increasing the number of samples and the number of candidate algorithms, the machine learning method can achieve a simplification effect with higher accuracy. However, it is time consuming and requires large data and a large number of codes and programs, making it more difficult to perform than the traditional method. The method based on mathematical morphology [23,24] is a representative raster-based simplification method. This simplification is achieved by combining basic morphological operations, including dilation, erosion, opening, and closing. This method is efficient and easy to control; however, it is primarily used for building outlines with evident orthogonal features. Shen et al. proposed a raster-based simplification method, where the building is simplified globally using the superpixel segmentation algorithm, followed by local simplification based on the geometric characteristics of the building [25]. This method is mainly used to simplify high-resolution building outlines. In addition, some studies have focused on vector–raster-based simplification methods, such as those involving the use of backpropagation neural networks [26] and deep convolutional neural networks [27]. A backpropagation neural network model can be used to acquire cartographic knowledge and simplify the localized perception of raster elements. Deep convolutional neural networks, such as U-net, residual U-net, and generative adversarial networks (Gans), can also achieve good results in learning simplified knowledge and simplifying operations. However, vector–raster-based methods generally require the conversion of data between the vector and raster formats, which can result in a certain loss of accuracy.

These algorithms are important tools that enable us to approach the right-angled polygon building outlines with jagged edges simplification process from different perspectives. However, many simplification and regularization algorithms cannot meet mapping standards for the automated extraction of highly irregular building outlines directly from remote sensing images and raster maps. Moreover, most simplification algorithms cannot satisfy the comprehensive generalization rules related to redundant point processing, building orthogonality, and overall outlines regularization or require auxiliary operations, such as coordinate rotation, data conversion, or manual trimming. If multiple algorithms are used successively, the simplification process becomes overly complex, and algorithm cohesion and coupling are adversely affected, leading to gradual data loss and poor results.

In the present study, we propose a simplification and regularization algorithm for extracting building outline vector data from remote sensing and raster images, simulating the concept of raster data simplification. Simplification is achieved by evenly dividing the minimum bounding rectangle (MBR) of the building outlines into blocks and then selecting and combining the parts classified as buildings. The proposed method populates the MBR of building outlines with square grids and successively analyzes the overlay of the grids and the original building outlines to identify its valid feature points, eliminate redundant points, and extract critical turning points. Finally, the proposed algorithm sorts and uses extracted points to reconstruct simplified building outlines. Compared to the current popular regularization methods, such as machine learning and deep learning, this study adopts the traditional geometry method to operate the building data in vector form, which has the advantages of an easy implementation process, low requirements on computer hardware, and strong reliability. Moreover, compared to traditional simplification algorithms, our proposed algorithm utilizes jagged building outline data instead of generally regularized large-scale data. By analyzing the simplification and regularization effect of the proposed method on various resolution data and comparing the proposed method with other methods, we demonstrated that the proposed method fully considers the requirements of direction, position, shape, and area preservation in the simplification and regularization process of residential land elements and strengthens the parallel and right-angled characteristics of building outlines with a good regularization effect.

2. Methodology

2.1. Principle and Characteristics of the Method

When building outlines are extracted from remote sensing images, due to misextraction and missing extraction, there is a deviation between the extracted building outlines and the real outlines, and they often have irregular boundaries: straight line boundaries could be extracted as jagged lines and right-angled corners could be extracted as unclear and inaccurate corners, as shown in Figure 1a. The principle of the proposed method is as follows: first, the smallest external rectangle of the building outline is divided into multiple small rectangles of the same size. Then, these small rectangles are superimposed over the original building outline, after which the small rectangles are divided into two categories, building and non-building, using calculations, as shown in Figure 1b. Finally, the small rectangular blocks belonging to the building class are combined to get the simplified and regularized building outline, as shown in Figure 1c.

The biggest difference between the proposed method and the existing methods is that the proposed method does not depend on the node distribution of the building outlines, it primarily considers the overall and local surface features of the building outline, and it adopts the idea of “overall control and local treatment.” Overall, the changes in shape, area, direction, and position are controlled by the minimum external rectangle and the main direction of the building. Locally, the smallest external rectangle is cut into pieces and then classified and reorganized. To a certain extent, the proposed method can complete the missing extraction part (the green part in Figure 1c) and eliminate the wrong extraction part (the yellow part in Figure 1c).

2.2. The Concrete Realization Process of the Method

Building outline simplification should follow the principles of area preservation, shape similarity, and boundary regularity [12]. Based on a comprehensive understanding of building structure characteristics, synthesis principles, and cartographic representations, the proposed method divides the simplification and regularization process of right-angled polygon building outlines into turning-point extraction and polygon reconstruction. A flowchart of this process is shown in Figure 2.

Part 1: Extraction of turning points. This part is divided into four steps: (1) constructing the MBR; (2) generating a grid mesh inside the MBR using the minimum visibility threshold as a constraint; (3) grid-by-grid intersection of the overlay of the grid with the original building, and preserving the region consisting of grid cells with an area of not less than half a cell; (4) extracting the building outlines points from the selected overall region and removing redundant points. The turning points extracted using these steps comprised the smallest point set of the building outlines. However, because this is an unordered point set, it cannot be directly used to reconstruct the polygons. The building outlines can be reconstructed using an appropriate algorithm to sort the points in a clockwise or counterclockwise order.

Part 2: Building outline reconstruction. This part is divided into three steps: (1) determining the orientation axis of the building, (2) selecting the point with the smallest abscissa in the turning point set as the starting point, and (3) sorting the new nodes based on whether each side is parallel or perpendicular to the building orientation axis and reconstructing the building outlines.

3. Extraction Algorithm of Building Outline Points

3.1. Construction of the Minimum Bounding Rectangle

The MBR of a polygon can be classified into two types: the minimum-area bounding rectangle (MABR) and the minimum-perimeter bounding rectangle (MPBR). As buildings are generally represented on maps as rectangular polygons, the difference between the two types of bounding rectangles is typically insignificant. In this study, we used MABR to better represent the directional characteristics of buildings. The MABR was obtained using the method described in Doytsher [28], which consists of the following steps:

(1) The minimum convex hull of the polygon is calculated using the Graham scan algorithm [29], as illustrated in Figure 3a,b.

(2) One side of the convex hull is selected as the base side, and the convex hull is rotated about the left endpoint of the base side until it becomes parallel to the horizontal axis. The MBR area enclosed by the minimum and maximum coordinates of the rotated convex hull is calculated and stored, and the base side name and rotation angle are stored. Figure 3c,d shows the MBR calculation process with segments 1–2 as the base edges and point 1 as the left endpoint.

(3) The remaining sides are successively selected, and step 2 is repeated. For example, Figure 3e shows the results of the MBR calculation with segments 2–3 as the base edge and point 2 as the left endpoint.

(4) The MBR with the smallest area is selected, and the left endpoint of the corresponding rotated side is used as the pivot point to rotate the MBR in the opposite direction by the corresponding angle, thereby obtaining the MABR.

3.2. Mesh Construction

Typically, opposite sides of a right-angled polygon building outline are parallel, and adjacent sides are perpendicular [12]. Therefore, this study constructed a grid composed of perpendicular and parallel segments as the operating unit for simplification. The proposed grid improves the geometric accuracy of the building outlines and ensures effective simplification.

3.2.1. Principles and Threshold Values of Grid Construction

The aim of building outline simplification is to eliminate detailed features of the building that are smaller than the minimum visible length (which is generally the field length of 0.3 mm [30] on the target scale map), thereby ensuring a consistent visual perception of the object [26]. This study used overlay analysis to construct a grid. The grid spacing should be greater than the minimum visible length to ensure that all the building edges are visible to the human eye after simplification. The threshold “r” is set to be no less than the ground distance corresponding to the smallest visible length at a given map scale. For example, the threshold value for a map scale of 1:5000 should be no less than 1.5 m, and for a scale of 1:10,000, it should be no less than 3 m.

3.2.2. MBR Node Selection and Grid Construction

The grid is constructed by populating each side of the building outline MBR with nodes using the following steps:

(1) The ground distance corresponding to the long and short edges of the MBR is calculated; for example, in Figure 4a, the long edges 1–2 and 3–4 and the short edges 2–3 and 4–1 can be used.

(2) The ground distance corresponding to each side is divided by the visible length threshold, and the result is rounded down to obtain the number of grid rows and columns. If the result is 0, the ground distance of the corresponding MBR side is less than r, and a value of 1 should be selected, which is equivalent to an exaggeration operation.

(3) The ground distance is divided by the corresponding number of rows and columns to obtain the grid and column spacing.

(4) Each side of the MBR is populated with nodes using the calculated row and column spacing, as shown in Figure 4b.

(5) The corresponding nodes of the long and short edges of the MBR are connected to construct the grid, as shown in Figure 4c.

3.3. Selection of Turning Points through Overlay Analysis

Area preservation is another important rule to consider when using a simplified algorithm [2]. This study proposes a grid-by-grid overlay analysis method that analyzes the overlay of each grid with the original building outlines, eliminates elements whose intersection is less than half the area of a single grid cell, and retains and combines the remaining grid cells to generate simplified outlines.

3.3.1. Construction of the Initial Grid

Any point of the MBR is selected as the starting point; for example, in Figure 5, point A is selected as the starting point from which the four vertices of the grid cell are generated clockwise or counterclockwise, constructing the initial grid cell. In Figure 5, the initial grid cell ABCDA is constructed clockwise from starting point A.

3.3.2. Sequential Traversal of All Grid Cells

After selecting one control point, the grid surface is constructed by moving the control point successively until the entire grid has been traversed as follows:

(1) The starting point described above is selected as the first control point, and the same method used to construct the initial surface is used to generate the control surface. In Figure 5, point A is selected as the initial control point, and ABCDA is the initial control surface.

(2) One step along the long or short side (row or column) is traversed to obtain the second control point. As shown in Figure 5, the control point is moved along the long side in one step (column) from point A to point B, and BEFCB is the second control surface.

(3) After traversing all grid cells in the first row, the control point is moved along the short side over one step (row) from point A to point D, and all grid cells in the second row are traversed successively in the same manner as those in the first row.

(4) This process is repeated until the entire grid is traversed.

3.3.3. Overlay Analysis and Turning Point Selection

In traversing the grid, the following algorithm is proposed to extract the effective turning points and remove redundant points.

A coordinate set $V_{turn}$ is created to store the coordinates of the turning points. Starting from the initial control point, the intersection of each grid with the original building outlines overlay is calculated. In this step, three types of conditions may be encountered: (1) the intersection is empty; (2) the intersection is not empty, but the area is less than half of the grid cell area; (3) the intersection is not empty and is greater than or equal to half of the grid cell area. When the first two circumstances are encountered, the control surface points are not stored, and the next control surface is calculated. When the third condition is encountered, each of the four points defining the control surface is first compared with the elements contained in $V_{turn}$; if any of the points have already been stored in $V_{turn}$, it is deleted from $V_{turn}$; if any point has not yet been stored in $V_{turn}$, then it is stored in $V_{turn}$.

For example, in Figure 6, the initial control area A₁A₂B₂B₁A₁ is the third type of intersection. After processing, points A₁, A₂, B₂, and B₁ are stored in $V_{turn}$, thereby obtaining $V_{turn}=\left\{\left.A_1,A_2,B_2,B_1\right\}\right.$. The second control area, A₂A₃B₃B₂A_2, is the third type of intersection. However, because points A₂ and B₂ have already been stored in $V_{turn}$, they are deleted from $V_{turn}$, whereas points A₃ and B₃, which have not yet been stored in $V_{turn}$, are added to $V_{turn}$, thereby obtaining $V_{turn}=\left\{\left.A_1,A_3,B_3,B_1\right\}\right.$. After completing the above iterations for the first row, we obtain $V_{turn}=\left\{\left.A_1,A_6,B_6,B_1\right\}\right.$. Evidently, the proposed algorithm skillfully eliminates redundant points along the building outlines and stores only right-angled turning points.

After the overlay analysis method is successively applied to the remaining rows, turning point set $V_{turn}=\left\{\left.A_1,A_6,B_8,B_{10},C_6,C_8,C_{10},C_{11},E_1,E_{11},E_5,E_9,G_5,G_9\right\}\right.$ is obtained.

It should be noted that the turning point C₆ was first stored in $V_{turn}$ during the analysis of the control surface B₅B₆C₆C₅B₅ and was later removed from $V_{turn}$ during the analysis of C₅C₆D₆D₅C₅; however, it was added again to $V_{turn}$ during the analysis of C₆C₇D₇D₆C₆. Points C₈ and C₁₀ undergo a similar store–delete–store process.

4. Reconstruction Algorithm of the Building Outlines

As the elements in the turning point set $V_{turn}$ obtained in Section 3.3.3 have not been arranged in a clockwise or counterclockwise order, they cannot be used directly to reconstruct the building outlines. In this section, we propose a sorting algorithm for the outline points of a rectangular polygon and analyze the characteristics of turning points with reference to building orientation. By arranging the turning points of the outline in order, the proposed algorithm allows for a regularized reconstruction of the building outlines.

4.1. Basic Algorithm

4.1.1. Calculation of Building Orientation

The orientation direction of a building is an important spatial constraint in map generalization. Some commonly used methods for calculating building orientation include the longest edge, weighted bisector, statistical weighting, MBR, and wall average [31]. In this study, the MBR method was used to construct the grid and obtain the turning point set $V_{turn}$, and hence, we also used it to calculate building orientation.

4.1.2. MBR-Based Calculation of Building Orientation

Using the coordinates of the four vertices of the MBR obtained in Section 3.1, two long edges of the MBR are computed, and the direction of either long edge is chosen as the direction of the building. In Figure 7, PP′ is the orientation of the MBR, which is also the orientation of the building.

4.1.3. Calculation of the Angle between Two Lines

Based on the coordinates of the points P and P′ on the main direction line of the building, the main direction vector of the building a = ($x_a$, $y_a$) is calculated, and based on the coordinates of the two endpoints of the arbitrary line segment, the direction vector b = ($x_b$, $y_b$) of the line segment can be calculated. Then, the angle $\theta$($\theta \in \left[0, 180\right]$) between any line segment and the principal direction can be calculated according to Equation (1).

$$\theta =180\times \frac{\mathit{arccos}\left(\frac{\overrightarrow{\mathit{a}}\times \overrightarrow{\mathit{b}}}{\left|\overrightarrow{\mathit{a}}\right|\times \left|\overrightarrow{\mathit{b}}\right|}\right)}{\pi }=180\times \frac{\mathit{arccos}\left(\frac{x_a{x}_b+y_a{y}_b}{\sqrt{x_a{}^2+y_a{}^2}\times \sqrt{x_b{}^2+y_b{}^2}}\right)}{\pi }$$

(1)

4.2. Sorting Algorithm for Rectangular Building Outline Points

Because all turning points extracted in the preceding section of this study are grid vertices, and grid lines are either perpendicular or parallel to the orientation axis of the building, the segment between any two adjacent turning points will also be either perpendicular or parallel to the orientation axis. Based on this characteristic of the above building outlines, this study proposes a sorting algorithm for the outline points of a rectangular building, which consists of the following steps:

(1) Traverse the turning point coordinate set $V_{turn}$ to find the point with the smallest abscissa value and use it as the control point $P_{control}$. If more than one point is found, select the one with the smallest ordinate value and use it as the control point $P_{control}$ and as the starting point of the polygon generalization result set $V_{result}$. In addition, remove from $V_{turn }$ any elements that have the same value as the starting point.

(2) Traverse all points in $V_{turn}$ and find any points that form a segment with point $P_{control}$, that is parallel/perpendicular (when iterating steps 2–4, exchange the parallel/perpendicular relationship with each iteration) with or to the orientation axis and do not intersect the skeleton of the original building outlines; then copy and place those points in a temporary set $V_{temp}$.

(3) If $V_{temp}$ contains only one element, it is designated as the temporary point $P_{temp}$. If $V_{temp}$ contains multiple elements, calculate the Euclidean distance ($S_{edis}$) from each element to $P_{control}$ and use the element corresponding to $Minimum S_{edis}$ as the temporary point $P_{temp}$. Clear $V_{temp}$.

(4) Input $P_{temp}$ at the end of $V_{result}$ and set $P_{temp}$ as the new control point $P_{control}$; delete from $V_{turn}$ any elements with the same value as $P_{temp}$.

(5) Repeat steps 2 to 4 until $V_{turn}$ becomes empty.

(6) Copy the first element in set $V_{result}$ to the end of $V_{result}$ so that the point set forms a closed figure.

The proposed sorting algorithm uses the orientation axis of the original building as a reference and operates based on the principle that two adjacent points on the building outline form a segment that is parallel or perpendicular to the orientation axis. After selecting the starting point, it systematically searches for the next node that satisfies the above condition and repeats this process until all outline nodes are arranged sequentially. The turning point set extracted using the method illustrated in Figure 6 is sorted as shown in Figure 8. First, point A₁, which has the lowest abscissa value, is set as the starting control point. Next, the point set is searched to find another point that forms a segment with A₁, which is parallel with PP′; only A₆ satisfies this condition. When A₆ is used as the control point, the point set is searched to find a point that forms a segment with A₆ perpendicular to PP′; only C₆ satisfies this condition. When C₆ is used as the control point, the point set is searched to find the point that forms a segment with C₆, which is parallel to the orientation axis PP’. Three points, namely, C₈, C₁₀, and C₁₁, satisfy this condition; however, point C₈ is selected because it is the closest to the control point, and the segment formed by these two points does not intersect the original building outlines.

After all iterations are completed, the sorted point set is $V_{result}$ = $\left\{\left.A_1,A_6,C_6,C_8,B_8,B_{10},C_{10},C_{11},E_{11},E_9,G_{9,}{G}_5,E_5,E_1,A_1\right\}\right.$.

5. Experiment and Analysis

The proposed algorithm was developed in GenerMap V3.0, a secondary independent development system, on the Visual Studio 2019 platform using the C++20 programming language. Building outline data extracted from remote sensing images using machine-learning or deep-learning methods were selected for the experiment. To verify the performance of the proposed algorithm, we used the simplification results of different-precision data under different thresholds and same-precision data processed by other simplification algorithms to perform a comparative analysis.

5.1. Quality Indicators for the Evaluation of Results of Simplification

To evaluate the performance of the proposed simplification and regularization algorithm, we selected five common indicators of simplification algorithms and three common indicators of regularization algorithms to comprehensively assess the effect of our method.

The evaluation indexes of the five simplification algorithms are as follows: orientation retention rate (OT), area change rate (OS), right angle rate (OR), data compression rate (OD), and the position change (OP) of the building outline before and after simplification. They are calculated as follows:

(1) OT measures the change in the building orientation and is given by the following equation:

$$OT=1-\frac{2\left|O_a-O_b\right|}{\pi }$$

(2)

In Equation (2), $O_a$ and $O_b$ represent the angles between the long edges of the building outline MBR and the x-axis before and after simplification, respectively. When $O_a$ = $O_b$, the value of OT is 100%, which is equivalent to the same direction before and after the simplification; when $\left|O_a-O_b\right|$ = $\pi /2$, the value of OT is 0%, which means that the direction shifted by 90°.

(2) OS measures the change in the area of the building outlines before and after simplification. It is given by the following equation:

$$OS=\frac{|S_a-S_b|}{S_a}$$

(3)

In Equation (3), $S_a$ and $S_b$ represent the area of the building outlines before and after simplification, respectively.

(3) OR measures the orthogonality of the building outlines after simplification based on the number of right angles among the interior angles of the polygon. It is used to test whether the algorithm has enhanced performance on the orthogonality of building outlines, which is given by the following equation:

$$OR=\frac{R_a}{R_b}$$

(4)

In Equation (4), $R_a$ and $R_{b }$ represent the number of right angles and the total number of interior angles, respectively, after simplification.

(4) OD represents the effectiveness of data compression by calculating the reduction in the number of nodes after simplification. It is given by the following equation:

$$OD=\frac{D_a-D_b}{D_a}$$

(5)

In Equation (5), $D_a$ and $D_b$ represent the number of nodes that make up the building outlines before and after simplification, respectively.

(5) OP is the change in the position of the building outlines centroid after simplification. It is given by the following equation:

$$OP=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$$

(6)

In Equation (6), P₁($x_1$, $y_1$) and P₂($x_2$_, $y_2$) are the centroid coordinates before and after the simplification, respectively.

The evaluation indexes of three commonly used building outline regularization algorithms are as follows: precision (P), recall (R), and intersection over union (IOU) [32]. They are calculated as follows:

$$P=\frac{P_T}{P_T+P_F}$$

(7)

$$R=\frac{P_T}{P_T+N_F}$$

(8)

$$IOU=\frac{P_T}{P_T+P_F+N_F}$$

(9)

$P_T$ is the area judged to be a building both before and after regularization; $N_F$ is the area that was a building before regularization and was judged to be a non-building after regularization; $P_F$ is the area that was not a building before regularization and was judged to be a building after regularization. Vector data are calculated using the area of the enclosed area and raster data are calculated using the number of pixels contained.

5.2. Algorithm Performance Based on the Simplification of Different-Source Data

Our proposed method focuses on simplifying and regularizing buildings that are approximately rectangular or composed of a combination of rectangles. To better validate the reliability of the method, we used the open-source WHU building dataset to evaluate our proposed method. The dataset covers an area of 450 km² in Christchurch, New Zealand, and includes about 220,000 buildings, most of which are regular rectangles or combinations of rectangles. The dataset also contains a small number of non-rectangular buildings, which can be used to validate the applicability and limitations of our proposed method. The spatial resolution of the image is 0.075 m. It has the characteristics of wide coverage, high image quality, and a large amount of data. We randomly selected a region between latitude 43°32′48″ to 43°33′17″ south and longitude 172°41′14″ to 172°41′58″ east as image #1, which contains 932 buildings, and we also downsampled image 1# to 0.5 and 1 m spatial resolution, respectively, as images #2 and #3. We used the deep learning method proposed by Qiu et al. [33] to extract building outlines from images #1, #2, and #3 and used them as experimental data to evaluate the effect of simplification, as shown in Figure 9.

The simplified results of the three resolutions under the thresholds of 1, 3, 6, and 10 m are shown in Figure 10. The simplification effect of data with different resolutions under different threshold conditions is shown in Figure 11.

In Figure 10, comparing the extraction results of remote sensing images with different resolutions of the same area, it can be seen that the higher the resolution of the image, the smaller the error of the building outline data extracted from it, and the closer it is to the real value of the manually labeled ground. Comparing the simplification results of different resolution data, it can be seen that the main direction of the buildings is maintained better when simplifying the data with 0.075 m resolution; when simplifying the data with 0.5 m resolution, the main direction of the individual buildings is shifted to different degrees, and the larger the extraction error, the larger the main direction of the buildings is shifted; for the remote sensing image with 1 m resolution, the larger the extraction error, the larger the main direction of the buildings is shifted after simplification. Therefore, we concluded that the level of data resolution has a direct effect on the maintenance of the main direction of the building. The higher the data resolution, the closer the main direction of the simplified result to the real main direction of the building.

When high-resolution images of buildings are used, although the amount of extracted original data is large, the shapes of the simplified building outlines are closer to the original outlines and are more orthogonal. The proposed simplification algorithm can reduce the number of nodes and perform well even under low-threshold conditions. When different-resolution data are simplified under different threshold conditions, the area remains stable, indicating that data resolution has little effect on area preservation.

Furthermore, Figure 10 shows that the results of simplification are directly related to the resolution of the remote-sensing image. The horizontal comparison of the simplification results of the three resolutions shows that the higher the data resolution, the higher the edge expression accuracy of the simplification results, and the closer to the manually labeled ground truth from the vertical comparison of the simplification results of a single resolution under different thresholds. It can be seen that the smaller the threshold, the higher the edge expression accuracy of the simplification results, and the closer it is to the manually labeled ground truth. Therefore, an appropriate threshold range should be selected for images with different resolutions. For example, when a 1 m threshold was used for the 0.075 m data, the simplified outlines had a small number of jagged edges, and when the threshold was set to 3 m, the building outlines were more regular. Compared with the 3 m threshold, the building outlines were oversimplified when a 6 m threshold was used; therefore, some outline details were lost. For the 0.5 m and 1 m resolution data, when a threshold of 1 m was used, the simplified outlines had clearly visible jagged edges. Under a threshold of 3 m, the number of jagged edges was small; under the 6 m threshold, the simplified outlines were relatively regular, and their area was well preserved. When a threshold of 10 m was selected, although the simplification algorithm produced regular building outlines for all three resolutions, the building areas changed significantly. Therefore, this threshold was not recommended.

During the simplification process, by establishing the MBR and the internal grid, the proposed algorithm constrains the orientation of the building outlines and ensures that its interior angles are orthogonal. It uses a from-large-to-small processing strategy, suggesting that it approaches the overall simplification process by operating only in areas close to the building outlines, preserving, to a reasonable extent, the spatial and positional similarities between the original and simplified buildings. Therefore, only two indicators, OS and OD, are required to sufficiently reflect the effectiveness of the simplification process under different thresholds. Figure 11 shows the simplified OS and OD evaluation indexes of different resolution data under different threshold conditions. The closer the OS is to 0, the more effective the simplification, and the closer the OD is to 1, the better the simplification effect. Therefore, high-resolution data allow the use of a wider interval of threshold values; as the data resolution decreases, the range of threshold values that produce reliable results gradually decreases. The ranges of reliable threshold values for all three data resolutions are listed in

Table 1. Reliable threshold range for different resolutions of data.

Image resolution	0.075 m	0.05 m	1 m
Reliable threshold range	1–6 m	3–6 m	3–6 m

.

5.3. Comparison with Other Simplification Algorithms Based on Same-Source Data

In this section, we used building data extracted from 0.5 m resolution image data to compare the simplification results of the proposed method with those of the rectangular fitting [34] (Method A), template matching [20] (Method B), iterative [35] (Method C), and superpixel segmentation [25] (Method D) methods. According to the experimental results presented in Section 5.2, the proposed algorithm can select a threshold between 3 m and 6 m. Considering that the smaller the parameter was set, the more details of the building outline were maintained, which could better evaluate the advantages of simplification and regularization capability of algorithms. Therefore, we set the threshold of our proposed method to 3 m. To ensure objectivity in the evaluation results, we also set the threshold value of the three simplification methods Method A, B, and C to 3 m and set the number of iteration times of method D to 1, sNum = 40, lNum = 1, sRatio = 1. The simplified results are shown in Figure 12. The evaluation indicators were obtained by measuring each building and taking the average value. The eight evaluation indicators for each method are shown in

Table 2. Evaluation indicators of different simplification methods.

Method	Evaluation Index
	OT (%)	OS (%)	OR (%)	OD (%)	OP (m)	P (%)	R (%)	IOU (%)
Method A	100	0.22	100	92.63	0	84.54	84.36	73.08
Method B	100	0.67	98.25	91.52	0	85.23	85.79	74.69
Method C	97.32	1.55	100	92.58	0.218	91.74	90.31	83.52
Method D	98.11	1.16	92.56	90.18	0.216	95.82	84.68	81.66
Proposed	100	2.24	100	90.66	0.209	92.34	94.41	87.55

Method A, Rectangular fitting; Method B, Template matching; Method C, Iterative; Method D, Superpixel segmentation. OT, orientation retention rate; OS, area change rate; OR, right angle rate; OD, data compression rate; OP, position change; P, precision; R, recall; IOU, intersection over union.

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The experimental results are as follows:

Although most of the experimental data we selected belong to right-angled buildings, there are still a relatively small number of non-right-angled. Therefore, the OR index is only used to determine whether or not the algorithms indicate the performance of reinforcing right-angled relationships, and cannot be completely used to quantitatively evaluate the performance of the algorithms. The OR indexes of the five methods are in the range of 92.56% to 100% (Table 2), indicating that the five algorithms have considered the orthogonality of building outlines.

The rectangular fitting method replaces the building outlines with a scaled MBR. During the scaling process, constraining rules are imposed on the building area and its centroid position to maintain consistency with the original building. Therefore, this method exhibited the best performance in terms of OT, OS, OD, and OP. However, it exhibited low P, R, and IOU indicators, indicating that the simplified outlines significantly differ from the original building outlines and that the spatial similarity between the simplified and original buildings is low. Therefore, this method is suitable only for simplifying rectangular buildings and should not be used for complex outlines. The template matching method is an improved rectangular fitting method, with a rectangular outline supplemented with a variety of templates of other shapes. However, the disadvantage of this method is that its simplification effectiveness depends on the template database. For example, the simplification method produces poor results when the template does not match the outlines of the original building well. Therefore, although the first five evaluation indicators of rectangular fitting and template matching methods are slightly better than our proposed method, the last three evaluation indicators are considerably lower, indicating that the two methods are far less versatile than ours.

The iterative method adopts a processing strategy that considers the entire shape before local adjustments and can control the change in direction, shape, and position during simplification, yielding good results in indicators such as OT, OS, OD, and P. However, the iterative method tends to oversimplify when simplifying relatively complex building outlines, resulting in slightly lower R and IOU indexes. By increasing the number of iterations of segmentation and adjusting sNum and lNum parameters, the superpixel segmentation method can improve the precision of building outline simplification and regularization and maintain good building characteristics during simplification. However, this method relies heavily on data resolution. When simplifying the building outline data with a resolution of 0.5 m, part of the outlines will be irregular when the number of iterations is 1. Although the regularity degree of the building outlines can be improved by adjusting the sNum and lNum parameters, the simplification accuracy is also reduced, resulting in low index R and IOU. This method can optimize the simplification and regularization effect by repeatedly adjusting the number of iterations and sNum and lNum parameters, but it requires more manual operation.

The simplification process should be based on the premise that all evaluation indicators of the simplification method meet the required mapping standards, producing results that fully preserve the details of the building outlines and its other characteristics, including orthogonality, orientation, and position. Table 2 and Figure 12 show that the proposed algorithm exhibits the best performance in terms of OT, R, and IOU. The OD index of our method is slightly lower because the proposed method can control the change in building outline shape by retaining more nodes under the condition of minimum visible length. Although the number of nodes used to define building outlines was slightly higher, it resulted in more detailed outlines and more effective preservation of the building shape. The OS metrics of our proposed method are slightly worse than those of the other methods, owing to the fact that our proposed method uses the MBR to limit the operating range, and the outline boundaries of the MBR tend to go beyond the boundaries of the building outline polygons, thus resulting in the amplification of the area of the building outline after the simplification. In order to solve this problem, OS* = (S_b − S_a)/S_a (where S_b denotes the area of the building outline after simplification, and S_a denotes the area of the building outline before simplification) can be used to calculate the amount of change in the area before and after simplification, to determine whether it is in line with the area preservation requirements of the mapping rules, and if it is not in line with the requirements, it can be improved by choosing a higher resolution of the data or by adjusting the threshold value of the simplification. Based on the above comprehensive analysis of the results of simplification using various evaluation indicators, the proposed algorithm meets the required mapping standards and produces good simplification results.

5.4. Limitations and Future Work

5.4.1. Suitability Analysis for Complex Types of Buildings

Our proposed method mainly aims to simplify building outlines composed of rectangles or combinations of multiple rectangles. We have also conducted experiments for simplifying non-right-angle buildings. As shown in Figure 13, some line segments deviate from the actual direction when our method simplifies non-rectangular buildings, indicating that the desired effects cannot be achieved yet, which will be the focus of our next study.

5.4.2. Analysis of the Applicability of Data at Different Resolutions

When the data resolution is high, since the corners and edges of the extracted building outlines are closer to the real values of the buildings, using our proposed algorithm of simplification and regularization, the building direction can be well maintained, and the edges and corners of the outlines can be accurately expressed (Figure 14b). However, as the resolution of the data decreases, the error of the extracted building outline increases. Using our proposed method of simplification and regularization, there may be problems such as the main direction of the building deviating from the true direction and the corners and edges of the building outline not being expressed accurately (Figure 14c,d). It shows that the computational accuracy of our proposed method has a certain dependence on the data resolution, which needs to be further improved in future research.

6. Conclusions

In this study, we propose a simplification and regularization algorithm that can be used to simplify vector data of jagged building outlines extracted from remote sensing images using machine learning or deep learning methods. This algorithm can be mainly used to simplify the outlines of right-angle polygon buildings. The proposed method consists of the following steps: (1) The MBR of the building outlines is constructed and used as a “working area” to limit the change in the position of the building outlines during the analysis. (2) Based on the from-large-to-small processing strategy, the MBR is populated with perpendicular intersecting segments that form a square grid to improve the orthogonality of the simplified building outlines while controlling the changes in the area and structural morphology of the building outlines during simplification. (3) Ordering outlines nodes using the orientation axis of the building outlines as a reference to control the orientation of the building before and after simplification. Moreover, our experimental results showed that the proposed method could simultaneously and effectively preserve the structure, area, position, and orientation of a simplified outline and strengthen the parallel and vertical characteristics of building outlines, providing a feasible method for simplifying and regularizing building outlines with jagged edges extracted from remote sensing images and raster maps. Our proposed method is more applicable to higher-resolution data, and in the future, the advantages of this method will become more and more obvious as the resolution of remote sensing images increases and the extraction algorithms become more and more mature. However, the proposed simplification algorithm does not consider buildings with non-right-angle types and buildings containing gaps in the interior, warranting further research.