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地球信息科学学报 >

2018 , Vol. 20 >Issue 4: 414 - 421

DOI: https://doi.org/10.12082/dqxxkx.2018.170489

全国激光雷达大会特约稿件

机载LiDAR点云数据的二面角滤波算法

  • 刘凯斯 ,
  • 王彦兵 , * ,
  • 宫辉力 ,
  • 李小娟 ,
  • 余洁
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  • 1. 首都师范大学 三维信息获取与应用教育部重点实验室,北京 100048
  • 2. 首都师范大学城市环境过程与数字模拟国家重点实验室培育基地,北京 100048
  • 3. 首都师范大学资源环境与旅游学院,北京 100048
*通讯作者：王彦兵（1972-）,男,博士,副教授,主要从事GIS理论与方法研究。E-mail:

作者简介：刘凯斯（1989-）,女,博士生,研究方向为基于LiDAR点云数据的三维空间建模与分析。E-mail:

收稿日期: 2017-10-17

  要求修回日期: 2018-02-28

  网络出版日期: 2018-04-20

基金资助

国家自然科学基金面上项目（41671417）

北京市教委科技计划一般项目（KM201610028013）

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Dihedral Angle Filtering Algorithm for Airborne LiDAR Point Cloud Data

  • LIU Kaisi ,
  • WANG Yanbing , * ,
  • GONG Huili ,
  • LI Xiaojuan ,
  • YU Jie
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  • 1. Key Lab of 3D Information Acquisition and Application, Ministry of Education, Beijing 100048, China
  • 2. The State Key Laboratory Breeding Base of Processing of Urban Environment and Digital Simulation, Beijing 100048, China
  • 3. College of Resource Environment and Tourism, Capital Normal University, Beijing 100048, China
*Corresponding author: WANG Yanbing, E-mail:

Received date: 2017-10-17

  Request revised date: 2018-02-28

  Online published: 2018-04-20

Supported by

National Natural Science Foundation of China, No.41671417

General Project of Science and Technology Plan of Beijing Municipal Education Committee, No.KM201610028013.

Copyright

《地球信息科学学报》编辑部 所有
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摘要

机载LiDAR是获取地表DEM的重要技术之一。本文针对机载LiDAR点云数据在复杂城区环境下的大型建筑及低矮地物滤波问题,提出一种新的二面角滤波法。利用空间二面角的平面角可以表达空间两相交平面相对位置的原理,实现机载LiDAR点云数据滤波。首先,算法提取点云数据中的高程突变点,以非突变点的二面角余弦均值稳定性作为判定迭代结束的条件;其次,分别统计高程突变和非突变点集的二面角余弦值频率分布,以交点处对应余弦值和最后一次迭代的坡度值作为LiDAR点云滤波的判定条件;最后,利用数学形态学“开”算子,去除残留低矮植被,得到可靠的滤波结果。对同一区域机载LiDAR点云数据,通过“二面角法”与“渐进三角网法”进行滤波处理。实验结果表明,二面角滤波法能有效地降低地物点错分为地面点的百分率,且在去除地物信息的同时能良好地保留地形特征。

关键词： 机载LiDAR; 二面角滤波法; 坡度; 开算子; DEM

本文引用格式

刘凯斯 , 王彦兵 , 宫辉力 , 李小娟 , 余洁 . 机载LiDAR点云数据的二面角滤波算法[J]. 地球信息科学学报, 2018 , 20(4) : 414 -421 . DOI: 10.12082/dqxxkx.2018.170489

Abstract

Airborne LiDAR is one kind of the technologies for obtaining ground surface DEM. On the analysis of the airborne LiDAR point cloud filtering algorithms, this paper proposes a new filtering algorithm-dihedral filtering. The algorithm is based on the theory that can express the relative position of two intersect planes in space, to achieve the airborne LiDAR point cloud data filtering process. Firstly, the elevation-mutate points are extracted from point cloud data. The iteration ends when the stability of the cosine of non-mutated points′ dihedral angle reaches required level. Then, the frequency distributions of the cosine of both mutated and non-mutated points′ dihedral angle are counted, and draws a line chart. Ground points and non-ground points are classified based on the intersection′s cosine of line chart and slope value of the last iteration. Finally, the open operator of the mathematical morphology is used to remove low vegetation, and the reliable results are obtained. Comparing with ′Progressive TIN Method′, the misjudged percentage of the non-ground points were effectively reduced. Dihedral method can retain topographical information while filtering terrestrial object information.

Key words： airborne LiDAR; dihedral angle filtering; slope; open operator; DEM

1 引言

机载激光测距技术可高效、快速获取高空间分辨率的地表三维信息,广泛应用于地形测绘、城市建模等多个领域[1,2,3]。原始机载LiDAR点云的数据结构是离散、不规则的,包括地面点和非地面点[4]。高精度分离地面点与非地面点的过程即滤波处理,是制作DTMs的关键技术[5]。现有的机载LiDAR点云数据滤波方法,主要分为：坡度滤波法[6,7]、移动窗口法[8]、数学形态学法[9,10]、渐进三角网（Triangulated Irregular Network, TIN）滤波法[11,12]及其他滤波方法[13,14,15,16]。
Foerstner等[17]提出基于灰度值的数学形态学滤波算法,使用“开”算子去除局部窗口内的地物点,实现滤波;该算法程序简单、易于实现,但需获取最大建筑物尺寸、建筑物最低高程等先验知识,自动化程度较低。Vosselman[18]提出坡度滤波算法,引入坡度因子判定非地面点;Sithole等[19]、Yi Chenshao等[20]改进了该算法,以自适应地形坡度变化;坡度算法需针对不同地形设置有效阈值,在处理地形不连续、低矮植被等情况时误差较大。门林杰等[6]采用规则格网方式组织数据,通过比较格网最低点与邻域最低点的坡度得到预设地形,再对格网任一点与邻域最低点的坡度进行判断,获得可靠的地形信息;该方法提高了滤波效率且可良好处理地形不连续情况,但低矮植被的识别误差仍较大。张宁宁等[21]结合区域生长原理改进坡度滤波法,用于地形较为平坦的城市区域,但对低矮植被、汽车等小型地物的滤波精度不高。综上,经典滤波算法易于滤除大型建筑,但难于识别低矮地物。
本文针对复杂城市区域低矮地物滤波精度不高的现状,提出一种机载LiDAR点云数据滤波的方法—二面角滤波法。利用空间二面角的平面角可以表达空间两相交平面相对位置的原理,实现点云数据的滤波。采用二面角余弦均值的稳定性作为高程突变与非突变点提取的终止条件;基于坡度滤波法在低矮地物识别方面的劣势,提出以首次高程非突变点二面角余弦值频率大于突变点频率时对应的二面角余弦,结合高程突变点迭代结束时对应的坡度,共同作为点云滤波的判定条件;最后,利用数学形态学“开”算子,去除残留低矮植被,得到高精度、可靠的LiDAR点云滤波结果。

2 二面角滤波算法原理和流程

2.1 二面角算法原理

从空间任一直线（l）出发的2个半平面（α、β）所组成的图形叫作二面角。以l上任一点P为端点,在两半平面α和β内分别作垂直于l的2条射线PA、PB,∠APB（θ）称之为二面角的平面角。本文限定二面角的大小介于[0°,180°][22],并以二面角的平面角余弦值判定角度大小（图1）。
Fig. 1 Dihedral angle

图1 二面角

Sithole和Vosselman[19]将地形定义为分段连续的曲面片拼接成的曲面。在实际地形中,相邻两平面的二面角通常位于区间[90°,180°],对于密集植被和建筑物,其内部的高程变化相对平缓,而其边界处,存在高程突变。
已知A（x1, y1, z1）、B（x2, y2, z2）、p（x3, y3, z3）3点坐标, AB → 长度为：
AB → = c = x 1 - x 2 2 + y 1 - y 2 2 + z 1 - z 2 2 （1）
二面角θ余弦值的计算公式为：
cos ( θ ) = AP → 2 + BP → 2 - AB → 2 2 ∙ AP → ∙ BP → （2）
根据空间中任意相交两平面的二面角平面角大小可确定两平面的相对位置,如图2所示。对于规则格网的机载LiDAR数据,每两相交线所成的夹角θ,即空间二面角的平面角;其越接近180°,地表高程变化越平缓,越接近90°,高程变化越急剧。其余弦值cosθ则随着地面变化由平缓到起伏在[0, 1]间单调递减。
Fig. 2 Schematic view of surface elevation change

图2 地表高程变化示意图

2.2 二面角滤波算法

根据上述原理,设计了二面角滤波算法流程（图3）。
Fig. 3 Flowchart of airborne LiDAR point cloud filtering based on dihedral angel

图3 二面角法点云滤波流程图

（1）原始LiDAR数据采用规则格网方式存储,遍历数据集,记录每一格网单元内最低高程值。对于有多次回波的点,仅保留末次回波高程;高程过高或过低的粗差点,不被记录。
（2）设置初始坡度阈值为72°,随着多次迭代高程突变点被逐步提取,坡度阈值逐渐减小,当相邻两2次迭代二面角余弦均值增长小于0.5%时,迭代结束。
（3）计算高程突变点、非突变点的二面角余弦,获得相应频率分布。以首次高程非突变点二面角余弦频率大于突变点频率时对应的二面角余弦值（COSmin）,及步骤（2）中获取的坡度值（Smax）,共同作为滤波的判定条件。
（4）采用数学形态学“开”算子,滤除残余低矮地物,得到可靠的滤波结果。
2.2.1 数据预处理
二面角滤波法首先将点云数据规则格网化,使用一个格网间距等于点云平均间隔的二维格网覆盖在点云数据上,判断每个机载LiDAR点所在格网单元的编号,公式如下：
X i = INT ( x - X min ) d Y i = INT ( y - Y min ) d （3）
式中：（Xi, Yi）代表格网编码;（x, y）是点云平面坐标;（Xmin, Ymin）代表实验区最小坐标;d为格网单元尺度。
2.2.2 高程突变点提取
基于Vosselmanly等[19]提出的两点间坡度越大且高程值越大的点是地面点几率越小的原理,利用坡度阈值进行高程突变点提取。首次坡度阈值为72°（斜率约为3）[21],在后续迭代过程中,随着高程变化剧烈的地物点被逐步识别,坡度阈值逐渐减小。
以非高程突变点的二面角余弦均值稳定性,作为迭代结束的判定条件。当相邻2次迭代的二面角余弦均值增长小于0.5%时,迭代结束（表1）。
Tab. 1 The cosine of dihedral angle

表1 二面角余弦值

迭代次数/次
1 2 3 4 5
斜率阈值 3.3 2.5 2 1.6 1.3
二面角余弦均值 0.909 0.917 0.925 0.929 0.932
单次高程突变点提取算法步骤如下：
（1）设置坡度阈值S,判定标识Label,0为非高程突变点,1为高程突变点（初始值Label=0）。
（2）顺序遍历格网,计算dk与左邻点和下邻点的坡度值,当坡度最小值大于S时,Label=1。
2.2.3 二面角滤波阈值确定
经高程突变点提取的机载LiDAR点云数据被分成2个点集。非高程突变点间的高程变化较为平缓,如图4实验数据1统计结果所示（0.2处直方图表示二面角余弦值出现在(0.1,0.2]间数据点的个数）,地面、建筑区域内部LiDAR数据点间相对位置变化小,二面角余弦值接近1,且二面角频数分布相似。因此,本文算法将建筑内部的相对位置变化与地面引起的位置改变等同处理。而高程突变点邻域的二面角余弦值小,这是地物与地面之间过渡导致的高程突变。
Fig. 4 Distribution range of dihedral angle cosine for buildings and grounds

图4 建筑物和地面的二面角余弦分布范围

分别统计高程突变和非突变点集的二面角余弦频率分布,如图5所示（实验数据1统计结果）：高程突变点集的二面角余弦集中分布在[0, 0.7]之间,仅不到3%的数据点二面角余弦在(0.7, 1]之间;而高程非突变点集的二面角余弦虽集中分布于 [0.7, 1]之间,呈单调递增趋势,但仍有大于6%的数据点余弦值小于0.7。
Fig. 5 Frequency statistics of dihedral angle cosine for elevation mutation and non-mutation points

图5 高程突变和非突变点二面角余弦值频率统计

由此可知：经坡度阈值确定的高程突变、非突变点具有一定程度的错误分类,尤其非突变点的误判概率较大,且单一坡度阈值滤波法对低矮地物的识别能力较差,因此本文提出以首次高程非突变点二面角余弦值频率大于突变点频率时对应的二面角余弦（COSmin）,结合2.2.2节高程突变点迭代结束时对应的坡度（Smax）,共同作为LiDAR点云滤波的判定条件。
因坡度（Smax）和二面角余弦（COSmin）迭代存在误差,设坡度值标准差为αs,二面角余弦标准差为θs,增加一个95%的置信区间（式（4））,滤波函数为：
ΔS = S max + 1.65 2 ∙ α s ΔCOS = CO S min - 1.65 2 ∙ θ s （4）
具体滤波步骤如下：
（1）设置未遍历的格网数据集P=T={t1, t2, …, tn}。统计每个格网单元四角点高程之和Ci,设置P中最小Ci对应的格网单元四角点为种子点[10]。如图6（a）所示,4个红色角点为种子点。
Fig. 6 Dihedral angle filtering

图6 二面角滤波

（2）采用自内向外扩散的遍历方式。如图6（b）所示,计算tk的水平（竖直同理）二面角余弦、坡度值。若cosi大于ΔCOS,且Si小于ΔS,则待定点为地面点;若cosi小于ΔCOS,或Si大于ΔS,待定点为非地面点,将该点高程用8邻域内已判定为地面点的高程均值替代。
（3）将新判定的点作为种子点,并将这些点从数据集P中去除（图6（c）外围的点）。重复步骤（2）,直至全部格网点判定完毕,P为空集。
2.2.4 小地物的去除
经二面角法滤波后的LiDAR点云数据,大地物已基本移除,但仍存在少量低矮地物（如小灌木等）。对于这些小地物,本文使用数学形态学“开”算子（式（5））进行滤除,设置4 × 4的滤波窗口,高差阈值设为研究区小灌木高度（0.5 m）[20]。
f ∘ g i , j = f ⊗ g ⊕ g i , j （5）
式中： f ∘ g 为“开”算子运算; f ⊗ g 为腐蚀算子运算; f ⊕ g 膨胀算子运算。

3 滤波实验与分析

3.1 实验数据与程序实现

本文选择了2块机载LiDAR点云数据作为试验区,数据1为海地太子港市区的局部点云,数据2为美国印第安纳州诺布尔斯维尔市区的局部点云数据（图7）。数据包含空间坐标、强度、GPS时间、回波次数、第几次回波等12个字段,平均点距约为1.5 m。其中,实验区1的范围为278 m × 239 m,共含有283 713个数据点,实验区2的范围为621 m × 628 m,共含有637 456个数据点。2个实验区内均包含不同高度的植被、建筑物等地物。
Fig. 7 Filtering results

图7 LiDAR点云滤波效果图

本文采用C++语言在VS 2013平台上实现了基于二面角的机载LiDAR点云数据滤波算法（电脑配置为Windows 7, 2.4 GHz主频,8 GB内存）,实验区1滤波时间为3.1 s,实验区2滤波时间为15.7 s。

3.2 结果分析

分析一种滤波方法性能的好坏,重要指标是统计出现的I类、II类误差概率,I类误差（Type I）是地面点被错误分为地物点的百分比率,II类误差（Type II）是地物点被错误分为地面点的百分比率,总误差是I类、II类误差之和与总机载LiDAR点数的百分比率[4]（式（6）-（8））。
第一类误差： a b + a （6）
第二类误差： c c + d （7）
总误差： a + c a + b + c + d （8）
式中：a是错误分类为地物点的地面点数目;b是正确分类的地面点数;c是将地物点分类为地面点的点数;d是正确判断的地物点数目。
对滤波后的实验区点云数据进行误差统计 （式（6）-（8））,并与“渐进三角网法[12]”进行对比,得到的精度评价结果如表2所示。
Tab. 2 Accuracy assessment for filtering algorithm

表2 滤波精度评价表

滤波结果误差统计 实验区1 实验区2
二面角滤波法 渐进三角网法 二面角滤波法 渐进三角网法
a 678 3518 1583 7961
b 28 171 144 296 63 298 324 108
c 760 4824 1686 10 840
d 25 761 131 075 58916 294 547
Type I Error/% 2.35 2.38 2.43 2.4
Type II Error/% 2.87 3.55 2.78 3.55
Total Error/% 2.60 2.94 2.61 2.95
如表2、图8（误差分布点云图）所示,由误差分布图可知,I类误差主要分布于区域内高程较高处,II类误差的产生主要受低矮植被影响。本文算法的I类误差和渐进三角网法相当,但II类误差有所改善,这表明二面角算法能在正确滤除地物点,保留地面信息的同时,兼顾小面积、低矮地物的去除。
Fig. 8 Distribution of error points

图8 误差点分布图

3.3 对比分析

为进一步验证本文算法的实用效果,以数据2为例进行对比试验。实验区中的地物主要包含建筑物、汽车、高矮不一的树、灌木。如图9所示,A区域地物为汽车,B、C、D区域地物为低矮植被。图10为实验数据2的局部俯视图和分别采用渐进三角 网[12]、二面角法滤波后的高程值剖面图。
Fig. 9 Comparison of two kinds of filtering methods: Dihedral and TIN filters

图9 二面角滤波和渐进三角网滤波方法对比分析

Fig. 10 Top view and sectional view of the area for experiment 2

图10 数据2的局部俯视图、剖面图

比较2种方法的滤波结果及剖面图可知,2种算法均能有效滤除建筑等大型地物,但相较于三角网滤波法,二面角法滤波后的地面点高程变化更为缓慢、精细,对植被的滤除效果更好。
图9中A、B、C区域中的汽车、低矮灌木,以及图10剖面图中右侧的低矮植被,渐进三角网算法均未能识别、去除;而本文二面角滤波法,对于地形起伏程度的敏感性好,能较好地保留地形信息,滤除低矮地物。当然,本文算法同样存在着不足之处,如图8中D区域内条状植被（高差42 cm）,本文算法与渐进三角网算法均未能识别此地物,从而将地物点错误地判断为地面点,这说明本文方法在二面角滤波后,进一步“精”滤波时的判定规则仍需加强。

4 结论

本文针对机载LiDAR点云数据在复杂城区环境下的大型建筑及低矮地物滤波问题,提出一种新的二面角滤波法。利用空间二面角的平面角可以表达空间两相交平面相对位置的原理,实现机载LiDAR点云数据滤波。得到以下结论：
（1）采用二面角余弦均值的稳定性作为高程突变与非突变点提取的终止条件;基于坡度滤波法在低矮地物识别方面的劣势,提出以首次高程非突变点二面角余弦值频率大于突变点频率时对应的二面角余弦,结合高程突变点迭代结束时对应的坡度,共同作为点云滤波的判定条件;最后,利用数学形态学“开”算子,去除残留低矮植被。得到高精度、可靠的LiDAR点云滤波结果。
（2）对海地太子港市区点云数据进行“二面角法”、“渐进三角网法”滤波实验,结果表明,二面角滤波法能有效地降低地物点错分为地面点的百分率,且在去除地物信息的同时能良好地保留地形 特征。

The authors have declared that no competing interests exist.

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