基于区域正交化分割的平面点集凸包算法

李可; 高清维; 卢一相; 孙冬; 竺德

doi:10.16383/j.aas.c190590

基于区域正交化分割的平面点集凸包算法

doi: 10.16383/j.aas.c190590

李可^{1, 2,},
高清维^{1, 2,},
卢一相^{1, 2,},
孙冬^{1, 2,},
竺德^{1, 2,}

1.
安徽大学电气工程与自动化学院合肥 230601
2.
安徽大学计算智能与信号处理教育部重点实验室合肥 230601

基金项目: 国家自然科学基金(61402004, 61370110), 安徽省自然科学基金(2008085MF183, 2008085MF192)资助

详细信息

作者简介:
李可：安徽大学电气工程与自动化学院硕士研究生. 主要研究方向为图像处理. E-mail: likea310@163.com

高清维：安徽大学电气工程与自动化学院教授. 主要研究方向为数字信号处理, 小波变换与应用, 多尺度几何分析, 混沌与分形理论与应用. 本文通信作者.E-mail: qingweigao@ahu.edu.cn

卢一相：安徽大学电气工程与自动化学院教授. 主要研究方向为图像处理, 信号的多尺度几何分解, 统计信号处理和稀疏表示.E-mail: lyxahu@ahu.edu.cn

孙冬：安徽大学电气工程与自动化学院教授. 主要研究方向为图像处理, 模式识别, 非线性动力学信号处理, 计算机图形学和深度学习.E-mail: sundong@ahu.edu.cn

竺德：安徽大学电气工程与自动化学院博士研究生. 主要研究方向为图像处理和模式识别.E-mail: sundong@ahu.edu.cn

计量
- 文章访问数: 399
- HTML全文浏览量: 238
- PDF下载量: 107
- 被引次数: 0
出版历程
- 收稿日期: 2019-08-17
- 录用日期: 2020-03-11
- 网络出版日期: 2022-11-30
- 刊出日期: 2022-12-23

A Convex Hull Algorithm for Plane Point Sets Based on Region Normalization Segmentation

LI Ke^{1, 2
,},
GAO Qing-Wei^{1, 2
,},
LU Yi-Xiang^{1, 2
,},
SUN Dong^{1, 2
,},
ZHU De^{1, 2
,}

1.
School of Electrical Engineering and Automation, Anhui University, Hefei 230601
2.
Key Laboratory of Computational Intelligence and Signal Processing of Ministry of Education, Anhui University, Hefei 230601

Funds: Supported by National Natural Science Foundation of China (61402004, 61370110) and Natural Science Foundation of Anhui Province (2008085MF183, 2008085MF192)

More Information

Author Bio:
LI Ke　Master student at the Sch-ool of Electrical Engineering and Automation, Anhui University. His main research interest is image processing

GAO Qing-Wei　Professor at the School of Electrical Engineering and Automation, Anhui University. His research interest covers digital signal processing, wavelet transform and applications, multi-scale geometric analysis, and chaos and fractal theory and applications. Corresponding author of this paper

LU Yi-Xiang　Professor at the School of Electrical Engineering and Automation, Anhui University. His research interest covers image processing, multi-scale geometric decomposition of signals, statistical signal processing and sparse representation

SUN Dong　Professor at the Sch-ool of Electrical Engineering and Automation, Anhui University. His research interest covers image processing, pattern recognition, nonlinear dynamic signal processing, computer graphics and deep learning

ZHU De　Ph.D. candidate at the School of Electrical Engineering and Automation, Anhui University. His research interest covers image processing and pattern recognition

摘要

摘要: 为解决实际工程应用中具有超大规模的平面点集的凸包计算问题, 提出了一种基于点集所在区域正交化分割的新算法. 利用点集几何结构的部分极点对平面点集进行正交化分割, 以获取不相干的点集子集簇, 再对所有点集子集分别计算其凸包极点, 最后合并极点得到凸包点集. 在不同层级的正交化分割过程中, 根据已知极点的信息, 逐层舍去对于凸包极点生成没有贡献的无效点, 进而提高算法运行效率. 在与目前常用凸包算法的对比实验中, 该算法处理超大规模的平面点集时稳定性高且速度更快.
- 平面点集 /
- 凸包 /
- 正交化分割 /
- 并行算法
Abstract: In order to solve the convex hull calculation problem of ultra-large scale planar point set in practical engineering applications, a new algorithm based on orthogonal segmentation of the region where the point set is located is designed. The partial point of the point set geometry is used to orthogonalize the plane point set to obtain the incoherent point set subset cluster, and then the convex hull poles are calculated for all the point set subsets, and finally the convex hull point set is obtained by combining the poles. In the process of orthogonalization and segmentation in different levels, according to the information of the existing convex hull poles, many of invalid points are discarded layer by layer, which improves the efficiency of the algorithm. In the comparison experiment with the commonly used convex hull algorithm, the proposed algorithm has high stability and speed when dealing with super large-scale planar point sets.
- Planar point set /
- convex hull /
- orthogonal segmentation /
- parallel algorithm

HTML全文

图 1 极值点结构图

Fig. 1 Structure diagram of extreme points

下载: 全尺寸图片幻灯片

图 2 点集子集第二层级正交化分割图

Fig. 2 Diagram of the second level orthogonalization of the subset of point sets

下载: 全尺寸图片幻灯片

图 3 算法并行化流程图

Fig. 3 Parallel flow chart of the algorithm

下载: 全尺寸图片幻灯片

图 4 点集遍历示意图

Fig. 4 Diagram of point sets traversal

下载: 全尺寸图片幻灯片

图 5 理想凸包极点分布图(h = 2, 3, 5, 9)和h = 5的遍历过程

Fig. 5 Ideal convex hull pole map (h = 2, 3, 5, 9) and a traversal process with h = 5

下载: 全尺寸图片幻灯片

图 6 Iris数据集的部分凸包图形

Fig. 6 Graphs of partial convex hull of Iris data set

下载: 全尺寸图片幻灯片

图 7 肺图像的修复

Fig. 7 Restoration of lung image

下载: 全尺寸图片幻灯片

表 1 点集不同层级分割的子集点数

Table 1 The number of points in the subset corresponding to different levels of segmentation

层级	5000	1 万	5 万	10 万	20 万	50 万	100 万	200 万	500 万	800 万	1000 万
1	438	873	4543	8893	17782	44541	88532	178749	449484	715181	885358
2	65	123	644	1213	2404	6120	11861	24559	61273	97727	120068
3	0	16	88	155	302	730	1398	3054	7967	12473	15307
4	0	0	12	22	44	85	175	403	1032	1630	2053
5	0	0	1	4	5	11	25	55	146	229	282

下载: 导出CSV

表 2 各算法运行时间 (s)

Table 2 Runtime for different algorithms (s)

点集点数	算法运行时间
点集点数	Graham	Jarvis	文献 [19]	算法 1	本文算法
5000	0.266	0.022	0.013	0.021	0.008
10000	1.067	0.045	0.027	0.047	0.011
50000	26.999	0.236	0.054	0.076	0.019
100000	108.645	0.539	0.102	0.448	0.028
150000	244.662	0.891	0.329	0.624	0.039
200000	435.077	1.296	0.585	0.752	0.047

下载: 导出CSV

表 3 不同数量级点集的运行时间 (s)

Table 3 Runtime comparison for different orders of magnitude point sets (s)

类别	20 万	50 万	100 万	200 万	500 万
0	0.586　　0.048	1.574　　0.102	3.315　　0.197	6.964　　0.384	18.509　　0.989
1	0.580　　0.046	1.575　　0.104	3.317　　0.194	6.962　　0.383	18.510　　0.965
2	0.589　　0.047	1.574　　0.103	3.312　　0.201	6.962　　0.386	18.506　　1.021
3	0.579　　0.047	1.568　　0.107	3.321　　0.200	6.960　　0.396	18.518　　0.949
4	0.581　　0.045	1.574　　0.106	3.312　　0.202	6.963　　0.392	18.509　　0.959
5	0.589　　0.048	1.579　　0.108	3.313　　0.190	6.958　　0.386	18.511　　0.962
6	0.585　　0.048	1.573　　0.105	3.317　　0.197	6.988　　0.390	18.513　　0.970
7	0.585　　0.046	1.570　　0.103	3.313　　0.199	6.966　　0.391	18.509　　0.957
8	0.591　　0.050	1.574　　0.105	3.315　　0.200	6.963　　0.405	18.518　　0.959
9	0.586　　0.047	1.582　　0.105	3.315　　0.198	6.968　　0.398	18.499　　0.961
10	0.584　　0.046	1.575　　0.105	3.313　　0.196	6.962　　0.385	18.510　　1.019

下载: 导出CSV

参考文献(21)

[1]	Barber C B, Dobkin D P, Huhdanpaa H. The quickhull algorithm for convex hulls. ACM Transactions on Mathematical Software, 1996, 22(4): 469−483 doi: 10.1145/235815.235821
[2]	Kim Y, Kim J. Convex hull ensemble machine for regression and classification. Knowledge and Information Systems, 2004, 6(6): 645−663 doi: 10.1007/s10115-003-0116-7
[3]	Ding S G, Nie X L, Qiao H, Zhang B. A fast algorithm of convex hull vertices selection for online classification. IEEE Transactions on Neural Networks and Learning Systems, 2018, 29(4): 792−806 doi: 10.1109/TNNLS.2017.2648038
[4]	Long P M, Servedio R A. Random classification noise defeats all convex potential boosters. Machine Learning, 2010, 78(3): 287−304 doi: 10.1007/s10994-009-5165-z
[5]	Chau A L, Li X O, Yu W. Convex and concave hulls for classification with support vector machine. Neurocomputing, 2013, 122: 198−209 doi: 10.1016/j.neucom.2013.05.040
[6]	Sadahiro Y. Number of polygons generated by map overlay: the case of convex polygons. Transactions in Gis, 2001, 5(4): 345−353 doi: 10.1111/1467-9671.00087
[7]	Chazelle B. An optimal convex hull algorithm in any fixed dimension. Discrete & Computational Geometry, 1993, 10(4): 377−409
[8]	Yao A C. A lower bound to finding convex hulls. Journal of the ACM, 1981, 28(4): 780−787 doi: 10.1145/322276.322289
[9]	Akl S G, Toussaint G T. A fast convex hull algorithm. Information Processing Letters, 1978, 7(5): 219−222 doi: 10.1016/0020-0190(78)90003-0
[10]	Kim Y J, Lee J, Kim M S, Elber G. Efficient convex hull computation for planar freeform curves. Computers & Graphics, 2011, 35(3): 698−705
[11]	Park M, Park C W, Park M, Lee C. Algorithm for detecting human faces based on convex-hull. Optics Express, 2002, 10(6): 274−279 doi: 10.1364/OE.10.000274
[12]	Zhou X F, Jiang W H, Tian Y J, Shi Y. Kernel subclass convex hull sample selection method for SVM on face recognition. Neurocomputing, 2010, 73(10−12): 2234−2246 doi: 10.1016/j.neucom.2010.01.008
[13]	Graham R L. An efficient algorith for determining the convex hull of a finite planar set. Information Processing Letters, 1972, 1(4): 132−133 doi: 10.1016/0020-0190(72)90045-2
[14]	Jarvis R A. On the identification of the convex hull of a finite set of points in the plane. Information Processing Letters, 1973, 2(1): 18−21 doi: 10.1016/0020-0190(73)90020-3
[15]	Andrew A M. Another efficient algorithm for convex hulls in two dimensions. Information Processing Letters, 1979, 9(5): 216−219 doi: 10.1016/0020-0190(79)90072-3
[16]	Stein A, Geva E, El-Sana J. CudaHull: fast parallel 3D convex hull on the GPU. Computers & Graphics, 2012, 36(4): 265−271
[17]	Tang M, Zhao J Y, Tong R F, Manocha D. GPU accelerated convex hull computation. Computers & Graphics, 2012, 36(5): 498−506
[18]	Brönnimann H, Iacono J, Katajainen J, Morin P, Morrison J, Toussaint G. Space-efficient planar convex hull algorithms. Theoretical Computer Science, 2004, 321(1): 25−40 doi: 10.1016/j.tcs.2003.05.004
[19]	刘斌, 王涛. 一种高效的平面点集凸包递归算法. 自动化学报, 2012, 38(8): 1375−1379 doi: 10.3724/SP.J.1004.2012.01375 Liu Bin, Wang Tao. An efficient convex hull algorithm for planar point set based on recursive method. Acta Automatica Sinica, 2012, 38(8): 1375−1379 doi: 10.3724/SP.J.1004.2012.01375
[20]	Beltran A, Mendoza S. SymmetricHull: a convex hull algorithm based on 2D geometry and symmetry. IEEE Latin America Transactions, 2018, 16(8): 2289−2295 doi: 10.1109/TLA.2018.8528248
[21]	冯昌利, 张建勋, 梁睿, 代煜, 崔亮. 基于分形几何和最小凸包法的肺区域分割算法. 天津大学学报(自然科学与工程技术版), 2015, 48(10): 937−946 Feng Chang-Li, Zhang Jian-Xun, Liang Rui, Dai Yu, Cui Liang. A lung region segmentation method based on the fractal theory and the minimal convex hull method. Journal of Tianjin University(Science and Technology), 2015, 48(10): 937−946