联合稳健S变换与K均值聚类的高崩溃污染率拟稳点选取策略

doi:10.11947/j.AGCS.2025.20240323

测绘学报 ›› 2025, Vol. 54 ›› Issue (9): 1608-1619.doi: 10.11947/j.AGCS.2025.20240323

联合稳健S变换与K均值聚类的高崩溃污染率拟稳点选取策略

刘忠贺¹(), 李宗春¹(), 何华¹, 郭迎钢², 赵文斌³

^1.信息工程大学地理空间信息学院，河南　郑州　450001
^2.西北核技术研究所，陕西　西安　710024
^3.深圳综合粒子设施研究院，广东　深圳　518000

收稿日期:2024-08-12 修回日期:2025-08-06 出版日期:2025-10-10 发布日期:2025-10-10
通讯作者: 李宗春 E-mail:2936054806@qq.com;13838092876@139.com
作者简介:刘忠贺（1995—），男，博士，讲师，研究方向为精密工程测量。E-mail：2936054806@qq.com
基金资助:
国家自然科学基金(41974216)

A strategy for selecting quasi-stable points with a high breakdown point by integrating robust S-transform with K-means clustering

Zhonghe LIU¹(), Zongchun LI¹(), Hua HE¹, Yinggang GUO², Wenbin ZHAO³

^1.Institute of Geospatial Information, Information Engineering University, Zhengzhou 450001, China
^2.Northwest Institute of Nuclear Technology, Xi'an 710024, China
^3.Institute of Advanced Science Facilities, Shenzhen 518000, China

Received:2024-08-12 Revised:2025-08-06 Online:2025-10-10 Published:2025-10-10
Contact: Zongchun LI E-mail:2936054806@qq.com;13838092876@139.com
About author:LIU Zhonghe (1995—), male, PhD, lecturer, majors in precision engineering surveying. E-mail: 2936054806@qq.com
Supported by:
The National Natural Science Foundation of China(41974216)

摘要/Abstract

摘要：

合理选取拟稳点是变形监测网稳定性分析的关键一环。变形点数量较多甚至过半情况下，现有稳定性分析方法在选取拟稳点时稳健性不足，导致拟稳点选取结果不尽合理。为提高拟稳点选取的准确率，提出了一种联合稳健S变换模型与K均值聚类算法的高崩溃污染率选取策略。首先，将两期变形监测网的同名点作为采样集合，从集合中随机选取部分点作为子集，利用稳健S变换模型求解子集中控制点的点位转换残差。然后，依据点位转换残差，采用K均值聚类算法将控制点分为拟稳点、小变形点及大变形点，并通过拟稳点集与大变形点集的点位转换残差质心差别判断稳健S变换模型是否可行，若稳健S变换模型奏效，则将转换残差小的点标记为拟稳点，从中选取频次较高的点作为备选拟稳点。最后，使用稳健S变换模型处理备选拟稳点以获得可靠的点位转换残差，以此为据，用K均值聚类算法准确选定拟稳点。通过仿真试验和实例分析，本文方法与传统相似变换模型、迭代加权相似变换模型、结合RANSAC算法的相似变换模型及平方型M_split相似变换模型进行了比较。结果表明，当变形监测网中存在变形点时，本文方法的变形点判别准确率最高，所得控制点位移更加符合实际。在变形点数量过半时，本文方法仍能保持稳健性，具有更高的崩溃污染率。

关键词: 变形监测网, 稳健S变换, K均值聚类, 稳定性分析, 拟稳点

Abstract:

The reasonable selection of quasi-stable points is a key procedure for stability analysis of deformation monitoring networks. In the case of a large number or even more than half of the deformation points, the existing methods have insufficient robustness in selecting quasi-stable points, leading to unreasonable selection results. To improve the correctness of selecting quasi-stable points, a high breakdown point strategy that integrates the robust S-transform model with the K-means clustering algorithm is proposed. First, the residuals of control points are estimated, using the robust S-transform model, from random subsets of homologous points drawn from two-epoch deformation monitoring networks. Then, based on the residuals of points, the K-means clustering method is used to divide the points into quasi-stable, micro-deformation and macro-deformation classes, and the feasibility of the robust S-transform model is assessed via the centroid separation between quasi-stable and macro-deformation clusters. If the robust S-transform model is valid, the points exhibiting minimal residuals are identified as quasi-stable, and the candidate quasi-stable points are selected from the quasi-stable points with high frequency. Finally, the reliable point transformation residuals are acquired by the robust S-transform model from the candidate quasi-stable points, which are used to reasonably determine the quasi-stable points via the K-means clustering algorithm. Through the simulation experiments and a case analysis, the comparisons were made with the traditional similarity transformation model, the iterative weighted similarity transformation model, the similarity transformation model combined with RANSAC algorithm, and the squared M_split similarity transformation model. The results show that when deformation exists in the network, the proposed method has the highest correctness in identifying the deformation points, and the estimated displacement of control points closely matches the actual situation. In the case where the number of the deformation points exceeds the number of the stable points, the proposed method can still maintain its robustness, which means it has a high breakdown point.

Key words: deformation monitoring network, robust S-transform, K-means clustering, stability analysis, quasi-stable point

中图分类号:

P258

刘忠贺, 李宗春, 何华, 郭迎钢, 赵文斌. 联合稳健S变换与K均值聚类的高崩溃污染率拟稳点选取策略[J]. 测绘学报, 2025, 54(9): 1608-1619.

Zhonghe LIU, Zongchun LI, Hua HE, Yinggang GUO, Wenbin ZHAO. A strategy for selecting quasi-stable points with a high breakdown point by integrating robust S-transform with K-means clustering[J]. Acta Geodaetica et Cartographica Sinica, 2025, 54(9): 1608-1619.

图/表 13

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参考文献 35

[1]	王穗辉. 变形数据处理、分析及预测方法若干问题研究[D]. 上海: 同济大学, 2007.
	WANG Suihui. Research on several problems in deformation data processing, analysis and prediction[D]. Shanghai: Tongji University, 2007.
[2]	陶本藻. 自由网平差与变形分析[M]. 武汉: 武汉测绘科技大学出版社, 2001.
	TAO Benzao. Free network adjustment and deformation analysis[M]. Wuhan: Wuhan University of Surveying and Mapping Press, 2001.
[3]	CASPARY W F, HAEN W, BORUTTA H. Deformation analysis by statistical methods[J]. Technometrics, 1990, 32(1): 49-57.
[4]	VELSINK H. On the deformation analysis of point fields[J]. Journal of Geodesy, 2015, 89(11): 1071-1087.
[5]	AYDIN C. Effects of displaced reference points on deformation analysis[J]. Journal of Surveying Engineering, 2017, 143(3): 04017001.
[6]	MANWILLER P E. Three-dimensional network adjustment of laser tracker measurements for large-scale metrology applications[J]. Journal of Surveying Engineering, 2021, 147: 05020009.
[7]	郭迎钢, 李宗春, 刘忠贺, 等. 加速器隧道控制网变形可监测性及稳定性分析[J]. 原子能科学技术, 2019, 53(9): 1634-1642.
	GUO Yinggang, LI Zongchun, LIU Zhonghe, et al. Deformation detectability and stability analysis for tunnel control network of particle accelerator[J]. Atomic Energy Science and Technology, 2019, 53(9): 1634-1642.
[8]	姚宜斌, 黄承猛, 李程春, 等. 一种适用于大角度的三维坐标转换参数求解算法[J]. 武汉大学学报(信息科学版), 2012, 37(3): 253-256.
	YAO Yibin, HUANG Chengmeng, LI Chengchun, et al. A new algorithm for solution of transformation parameters of big rotation angle's 3D coordinate[J]. Geomatics and Information Science of Wuhan University, 2012, 37(3): 253-256.
[9]	周江文, 黄幼才, 杨元喜, 等. 抗差最小二乘法[M]. 武汉: 华中理工大学出版社, 1997.
	ZHOU Jiangwen, HUANG Youcai, YANG Yuanxi, et al. Robust least squares method[M]. Wuhan: Huazhong University of Science and Technology Press, 1997.
[10]	YANG Y. Robust estimation of geodetic datum transformation[J]. Journal of Geodesy, 1999, 73(5): 268-274.
[11]	欧吉坤. 粗差的拟准检定法(QUAD法)[J]. 测绘学报, 1999, 28(1): 15-20.
	OU Jikun. Quasi accurate detection of gross errors (QUAD)[J]. Acta Geodaetica et Cartographica Sinica, 1999, 28(1): 15-20.
[12]	ROUSSEEUW P J, HUBERT M. Robust statistics for outlier detection[J]. Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery, 2011, 1(1): 73-79.
[13]	CHEN Y. Analysis of deformation surveys-a generalized method[D]. Fredericton: University of New Brunswick, 1983.
[14]	CHEN Y Q, CHRZANOWSKI A, SECORD J M. A strategy for the analysis of the stability of reference points in deformation surveys[J]. CISM Journal, 1990, 44(2): 141-149.
[15]	周江文, 陶本藻, 庄昆元, 等. 拟稳平差论文集[M]. 北京: 测绘出版社, 1987.
	ZHOU Jiangwen, TAO Benzao, ZHUANG Kunyuan, et al. Collection of papers on quasi stable adjustment[M]. Beijing: Surveying and Mapping Press, 1987.
[16]	周江文, 欧吉坤. 名次法及拟稳点的选定[J]. 测绘学报, 1987, 16(2): 88-94.
	ZHOU Jiangwen, OU Jikun. The method of order andselection of quasi-stable points[J]. Acta Geodaetica et Cartographic Sinica, 1987, 16(2): 88-94.
[17]	DUCHNOWSKI R. Median-based estimates and their application in controlling reference mark stability[J]. Journal of Surveying Engineering, 2010, 136(2): 47-52.
[18]	NOWEL K, KAMISKI W. Robust estimation of deformation from observation differences for free control networks[J]. Journal of Geodesy, 2014, 88(8): 749-764.
[19]	NOWEL K. Investigating efficacy of robust M-estimation of deformation from observation differences[J]. Survey Review, 2016, 48(346): 21-30.
[20]	AMIRI-SIMKOOEI A R, ALAEI-TABATABAEI S M, ZANGENEH-NEJAD F, et al. Stability analysis of deformation-monitoring network points using simultaneous observation adjustment of two epochs[J]. Journal of Surveying Engineering, 2017, 143: 04016020.
[21]	刘忠贺, 李宗春, 郭迎钢, 等. 结合RANSAC算法与传统相似变换模型的稳定性分析[J]. 测绘工程, 2021, 30(5): 43-48, 57.
	LIU Zhonghe, LI Zongchun, GUO Yinggang, et al. Stability analysis method of combining RANSAC algorithm and traditional similar transformation model[J]. Engineering of Surveying and Mapping, 2021, 30(5): 43-48, 57.
[22]	WIŚNIEWSKI Z. Estimation of parameters in a split functional model of geodetic observations (Msplit estimation)[J]. Journal of Geodesy, 2009, 83(2): 105-120.
[23]	WIŚNIEWSKI Z. Msplit(q) estimation: estimation of parameters in a multi split functional model of geodetic observations[J]. Journal of Geodesy, 2010, 84(6): 355-372.
[24]	DUCHNOWSKI R, WIŚNIEWSKI Z. Estimation of the shift between parameters of functional models of geodetic observations by applying Msplit estimation[J]. Journal of Surveying Engineering, 2012, 138(1): 1-8.
[25]	ZIENKIEWICZ M H. Application of M_split estimation to determine control points displacements in networks with unstable reference system[J]. Survey Review, 2015, 47(342): 174-180.
[26]	ZIENKIEWICZ M H. Deformation analysis of geodetic networks by applying M_split estimation with conditions binding the competitive parameters[J]. Journal of Surveying Engineering, 2019, 145(2): 04019001.
[27]	NOWEL K. Squared M_split (q) S-transformation of control network deformations[J]. Journal of Geodesy, 2019, 93(7): 1025-1044.
[28]	郭迎钢, 李宗春, 何华, 等. 变形监测网稳定点选取的平方型Msplit相似变换法[J]. 测绘学报, 2020, 49(11): 1419-1429. DOI: . doi: 10.11947/j.AGCS.2020.20200023
	GUO Yinggang, LI Zongchun, HE Hua, et al. A squared Msplit similarity transformation method for stable points selection of deformation monitoring network[J]. Acta Geodaetica et Cartographica Sinica, 2020, 49(11): 1419-1429. DOI: . doi: 10.11947/j.AGCS.2020.20200023
[29]	何选森, 何帆, 徐丽, 等. K-Means算法最优聚类数量的确定[J]. 电子科技大学学报, 2022, 51(6): 904-912.
	HE Xuansen, HE Fan, XU Li, et al. Determination of the optimal number of clusters in K-means algorithm[J]. Journal of University of Electronic Science and Technology of China, 2022, 51(6): 904-912.
[30]	张文君, 顾行发, 陈良富, 等. 基于均值-标准差的K均值初始聚类中心选取算法[J]. 遥感学报, 2006, 10(5): 715-721.
	ZHANG Wenjun, GU Xingfa, CHEN Liangfu, et al. An algorithm for initilizing of K-means clustering based on mean-standard deviation[J]. Journal of Remote Sensing, 2006, 10(5): 715-721.
[31]	ROUSSEEUW P J. Least median of squares regression[J]. Journal of the American Statistical Association, 1984, 79(388): 871-880.
[32]	YANG Ling, SHEN Yunzhong, LI Bofeng. M-estimation using unbiased median variance estimate[J]. Journal of Geodesy, 2019, 93(6): 911-925.
[33]	ZHANG Xudong, CHEN Wenjun, ZHANG Xiaodong, et al. The deformation analysis of the 3D alignment control network based on the multiple congruence models[J]. Journal of Geodesy and Geoinformation Science, 2023, 6(2): 21-31.
[34]	ZHU Jianjun, WANG Leyang, HU Jun, et al. Recent advances in the geodesy data processing[J]. Journal of Geodesy and Geoinformation Science, 2023, 6(3): 33-45.
[35]	杨玲, 沈云中, 楼立志. 基于中位参数初值的等价权抗差估计方法[J]. 测绘学报, 2011, 40(1): 28-32.
	YANG Ling, SHEN Yunzhong, LOU Lizhi. Equivalent weight robust estimation method based on median parameter estimates[J]. Acta Geodaetica et Cartographica Sinica, 2011, 40(1): 28-32.

点名/测站名	X	Y	Z
Station1	－4 000.000	1 000.000	0.000
Station2	0.000	1 000.000	0.000
Station3	4 000.000	1 000.000	0.000
Station4	－4 000.000	－1 000.000	0.000
Station5	0.000	－1 000.000	0.000
Station6	4 000.000	－1 000.000	0.000
A₁	－6 000.000	3 000.000	3 000.000
A₂	－2 000.000	3 000.000	3 000.000
A₃	2 000.000	3 000.000	3 000.000
A₄	6 000.000	3 000.000	3 000.000
A₅	－6 000.000	－3 000.000	3 000.000
A₆	－2 000.000	－3 000.000	3 000.000
A₇	2 000.000	－3 000.000	3 000.000
A₈	6 000.000	－3 000.000	3 000.000
A₉	－6 000.000	3 000.000	－3 000.000
A₁₀	－2 000.000	3 000.000	－3 000.000
A₁₁	2 000.000	3 000.000	－3 000.000
A₁₂	6 000.000	3 000.000	－3 000.000
A₁₃	－6 000.000	－3 000.000	－3 000.000
A₁₄	－2 000.000	－3 000.000	－3 000.000
A₁₅	2 000.000	－3 000.000	－3 000.000
A₁₆	6 000.000	－3 000.000	－3 000.000

点名	方案②			方案③			方案④			方案⑤			方案⑥
点名	X	Y	Z	X	Y	Z	X	Y	Z	X	Y	Z	X	Y	Z
A₁	0.000	0.000	0.000	0.000	0.000	0.300	0.100	0.000	0.000	0.000	0.000	0.300	0.000	0.000	0.000
A₂	0.000	0.000	0.000	0.100	0.000	0.000	0.100	0.000	0.000	0.000	0.000	0.300	0.000	0.000	0.000
A₃	0.500	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.300
A₄	0.000	0.000	0.000	0.000	0.000	0.000	0.100	0.000	0.000	0.000	0.000	0.300	0.000	0.000	0.000
A₅	0.000	0.000	0.000	0.000	0.000	0.200	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.300
A₆	0.000	0.000	0.000	0.000	0.000	0.000	0.100	0.000	0.000	0.000	0.000	0.300	0.000	0.000	0.000
A₇	0.000	0.100	0.000	0.000	0.100	0.000	0.000	0.000	0.300	0.000	0.000	0.300	0.000	0.000	0.000
A₈	0.000	0.000	0.000	0.400	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.300
A₉	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.300	0.000	0.000	0.300	0.000	0.000	0.000
A₁₀	0.000	0.000	0.000	0.300	0.000	0.000	0.000	0.000	0.300	0.000	0.000	0.300	0.000	0.000	0.000
A₁₁	0.000	0.000	0.500	0.000	0.000	0.000	0.000	0.000	0.300	0.000	0.000	0.300	0.000	0.000	0.000
A₁₂	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.300
A₁₃	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.300
A₁₄	0.000	0.000	0.000	0.000	0.500	0.000	0.000	0.000	0.300	0.000	0.000	0.300	0.000	0.000	0.000
A₁₅	0.000	0.000	0.000	0.100	0.000	0.000	0.100	0.000	0.000	0.000	0.000	0.300	0.000	0.000	0.000
A₁₆	0.100	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.300

方案	TST模型		IWST模型		RANSAC_ST模型		SMST模型		本文方法
方案	τ₁	τ₂	τ₁	τ₂	τ₁	τ₂	τ₁	τ₂	τ₁	τ₂
方案①	1.0	1.0	1.0	1.0	1.0	1.0	1.0	1.0	1.0	1.0
方案②	0.3	1.0	1.0	1.0	1.0	1.0	1.0	1.0	1.0	1.0
方案③	0.8	1.0	1.0	1.0	1.0	1.0	0.7	1.0	1.0	1.0
方案④	0.6	1.0	0.9	1.0	0.8	1.0	0.7	0.9	1.0	1.0
方案⑤	0.6	1.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
方案⑥	0.6	1.0	1.0	1.0	1.0	1.0	1.0	1.0	1.0	1.0

方案	TST模型	IWST模型	RANSAC_ST模型	SMST模型	本文方法
方案①	0.008	0.008	0.008	0.008	0.008
方案②	0.014	0.008	0.008	0.007	0.008
方案③	0.009	0.007	0.006	0.015	0.004
方案④	0.022	0.008	0.009	0.010	0.006
方案⑤	0.188	0.294	0.294	0.294	0.294
方案⑥	0.118	0.009	0.008	0.008	0.009

年份	测站数	控制点数	中误差/（″）	自由度
2016	43	401	1.2	1341
2019	41	166	1.7	990

联合稳健S变换与K均值聚类的高崩溃污染率拟稳点选取策略

A strategy for selecting quasi-stable points with a high breakdown point by integrating robust S-transform with K-means clustering

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摘要/Abstract

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图/表 13

参考文献 35

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编辑推荐

Metrics

本文评价

方法	拟稳点数	判别的稳定点数	判别的变形点数	最大变形量/mm	变形点位移的均方根值/mm	稳定点位移的均方根值/mm
TST模型	153	63	90	5.849	1.315	0.648
IWST模型	149	51	102	5.760	1.321	0.573
RANSAC_ST模型	109	55	98	5.972	1.292	0.678
SMST模型	103	77	76	5.667	1.841	0.522
本文方法	90	75	78	5.607	1.735	0.464

[1]	王慧贤, 靳惠佳, 王娇龙, 江万寿. k均值聚类引导的遥感影像多尺度分割优化方法[J]. 测绘学报, 2015, 44(5): 526-532.
[2]	夏列钢,王卫红,胡晓东,骆剑承. 图像分类中基于核映射的光谱匹配度量方法[J]. 测绘学报, 2012, 41(4): 0-590.