On the basis of Alpha Shapes boundary extraction algorithm for discrete point set, a grid partition variable step Alpha Shapes algorithm is proposed to deal with the shortcomings of the original Alpha Shapes algorithm in the processing of nonuniform distributed point set and multiconcave point set. Firstly, the grid partition and row-column index table are established for the point set, and the point set of boundary grid partition is quickly extracted. Then, the average distance of the
From the traditional electronic total station, handheld satellite positioning collector, to mobile (vehicle-mounted/airborne) three-dimensional laser radar, modern spatial information acquisition technology has entered the era of massive data at the GB and TB level. How to store, process, and express massive data efficiently has become a new challenge to the related fields such as computer information technology (IT), computer-aided design/manufacturing (CAD/CAM), geographic information (GIS) and remote sensing (RS), and even building information modeling (BIM) [
To construct the boundary information of point set, it is necessary to study the shape of point set composed of two-dimensional or three-dimensional discrete points. From the literature reading analysis, it is known that, since the 1970s, scholars and experts at home and abroad have successively carried out research work in this field and got good research results. Graham [
In literature [
Simplified AS algorithm steps are as follows: Step 1: Input two-dimensional point set Step 2: Traverse Step 3: Traverse Step 4: After traversing
The simplified flow chart of Alpha Shapes algorithm is shown in Figure
Simplified flow chart of Alpha Shapes algorithm.
Among these steps, the third step, “construct
Draw a circle with radius
Sketch map of
The formula for calculating the coordinates of the center point
In the formula,
The process of constructing point set boundary by AS algorithm can be understood as a circle with radius
Building boundary of point set by Alpha Shapes algorithm.
Compared with the previous similar algorithms, AS algorithm has the advantages of rigorous mathematical definition, the ability to deal with complex two-dimensional point set boundary and three-dimensional point set surface, and a wide range of applications, but there are also shortcomings. The only parameter For point sets with nonuniform density distribution per unit area, such as the discrete points at the farmland boundary collected by handheld or vehicle-mounted satellite navigation receiver, and the three-dimensional laser point cloud of bare tree branches, the processing effect is not very ideal. For the point set composed of many concaves, such as complex buildings, roads, water flow, and other linear features, the processing effect in the concave corner area is not perfect. Therefore, it is necessary to improve and perfect the algorithm.
When applying AS algorithm to extracting building contour, it is found in [
The main improvements of the GPVAS algorithm are as follows: The point set For points in point set
Simplified flow chart of GPVAS algorithm.
This step consists of two steps: “grid partition” and “extracting point set of boundary grid partition.” The steps are as follows.
The envelop rectangle
The envelop rectangle
A row-column index table is established for the point set
Extracting point set of boundary grid partition by GPVAS algorithm.
Traverse the grid, and quickly determine whether the grid contains a point from the row-column index table of point set
Traverse the grid, extract the point set
Once the value of parameter
The GPVAS algorithm proposed in this paper still adopts the calculation steps of VAS algorithm, first calculating the
Sketch map of layer-by-layer search by GPVAS algorithm.
To intuitively verify the time efficiency of the algorithm, two kinds of data are designed for experimental verification based on AS algorithm, VAS algorithm and GPVAS algorithm: one is the data point set of computer numerical simulation (hereinafter referred to as the simulated point set), and the other is the data point set of engineering measurement (hereinafter referred to as the measured point set).
The data of random point set generated randomly by circular analytic formula (
Circular nonuniform simulation point set.
The effective range of parameter
Comparison table of experimental results of different algorithms (simulated point set).
Algorithm | Number of points | Execution time (ms) | |
---|---|---|---|
AS algorithm | 1000 | 0.360 | 245 |
5000 | 0.072 | 5128 | |
10,000 | 0.036 | 26124 | |
20,000 | 0.018 | 128356 | |
VAS algorithm | 1000 | 0.240/0.781 | 326 |
5000 | 0.048/0.156 | 4289 | |
10,000 | 0.024/0.075 | 19356 | |
20,000 | 0.012/0.039 | 96786 | |
GPVAS algorithm | 1000 | 0.240/0.781 | 861 |
5000 | 0.048/0.156 | 3243 | |
10,000 | 0.024/0.075 | 6934 | |
20,000 | 0.012/0.039 | 10189 |
According to the comparison of experimental results in Table
In order to verify the effectiveness of the algorithm applied to the measured point set, this paper selects the highway strip terrain data (Figure
Boundary of strip terrain points on mountain highway.
Comparison table of experimental results of different algorithms (measured point set).
Algorithm | Execution time (ms) | |
---|---|---|
AS algorithm | 10 m | 16287 |
AS algorithm | 30 m | 21458 |
VAS algorithm | 8.6 m/41.2 m | 11356 |
GPVAS algorithm | 8.6 m/41.2 m | 7127 |
Boundaries extracted by different algorithms at highway curve. (a) AS algorithm,
The experimental results show that all the three algorithms can be used to construct the boundary of experimental data, and the construction efficiency is GPVAS > VAS > AS. Regarding the construction results, GPVAS algorithm and VAS algorithm are almost the same with good effect (Figure
Based on the Alpha Shapes algorithm for extracting the boundary of discrete point sets, this paper analyzes and summarizes the previous research work. In view of the shortcomings of Alpha Shapes algorithm in processing nonuniform distributed point sets and multiconcave point sets, this paper proposes the grid partition variable step Alpha Shapes algorithm, which is used to quickly construct the boundary of point sets. This algorithm has two main advantages: Establish grid partition and row-column index table for point set, quickly filter nonboundary point partition, and extract boundary grid partition point set involved in subsequent The average distance of
The algorithm is verified by simulated point set and measured point set, and the execution efficiency of the algorithm is very high. Compared with similar algorithms, the larger the number of point sets is, the more obvious the efficiency improvement is. As an alternative algorithm, this algorithm has been effectively verified in engineering scenarios such as land area statistics and road earthwork calculation. In the field of 3D point cloud surface reconstruction with broader application scenarios, this algorithm has not been verified, which is also the follow-up research direction of this paper.
The data used to support the findings of this research were generated from experiments.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This research was funded by the National Natural Science Foundation of China (41906168), Natural Science Research Project of Anhui Education Department (KJ2018JD04), and AHJZU-Anhui Huali Construction Co., Ltd., Joint Research Project (HYB20190152).