II. OVERVIEW OF PHOTON-COUNTING LASER ALTIMETRY
【综述】光子计数激光测高仪
Photon-counting laser altimetry records the time position of each individual received photon. Then, the surface elevation of illuminated area can be derived by calculating photon travel time and knowing the altitude of the sensor. Two sources of simulated ICESat-2 data are used in this letter.
Photon-counting[ˈfəʊtɒn]：光子计数
laser altimetry[ˈleɪzə(r) ˈæltɪmiːtə(r)]：激光高度计
elevation[ˌelɪˈveɪʃn]：海拔
illuminated[ɪˈluːmɪneɪtɪd]：
derive[dɪˈraɪv]：获得
altitude[ˈæltɪtjuːd]：海拔高度
simulated[ˈsɪmjuleɪtɪd]：模拟的
光子计数激光测高仪，会记录每个接收到光子的时间位置。接着，通过计算光子的运动时间，了解传感器的海拔高度，可以得到被照射区域的表面高度。本文使用了两种模拟ICESAT-2数据。
A. First Principle Simulation of Photon-Counting Laser Altimetry
光子计数激光测高的第一原理模拟
The first data set is based on the first principle simulation of photon-counting laser altimetry. As mentioned in our previous letter [15], the transmitter and receiver are modeled using a model of the ICESat-2 ATLAS instrument. The laser beam is characterized as circular Gaussian with a diameter of 10 m on the ground.
transmitter[trænzˈmɪtə(r)]：发射机
ATLAS： Advanced Topographic Lidar Altimetry System 先进地形激光测高系统 【onboard ICESat-2】
laser beam[ˈleɪzə(r) biːm]：激光束
be characterized[ˈkærəktəraɪz] as：被认为，被描述
diameter[daɪˈæmɪtə(r)]：直径
第一组数据集基于光子计数激光测高的第一原理模拟。根据之前的论文[15]，发射器与接收器均按照ICESat-2上的ATLAS结构建模。激光束的特征为，地面直径1/e2 10米。
Meanwhile, the temporal shape of laser photons is modeled with a Gaussian distribution with a 1-ns full-width at half-maximum pulsewidth. Laser along-track sampling is 0.7 m based on the latest ICESat-2 design. The number of mean received photoelectrons per shot for typical ice sheets is set as 2.04 for a weak spot and 8.17 for a strong spot [16]. A 3-D synthetic surface is also generated using fractal techniques. Here, the created terrain has a size of 1024 × 1024 m, with a resolution of 1 m.
pulse width：脉冲宽度
Gaussian distribution：高斯分布
temporal[ˈtempərəl]：时间的，短暂的
sampling[ˈsɑːmplɪŋ]：n. 抽样
along-track：沿轨道
photoelectrons ：光电子
synthetic[sɪnˈθetɪk]：合成的
fractal[ˈfræktl] technique：分形技术
terrain[təˈreɪn]：地形
resolution[ˌrezəˈluːʃn]：正式决定，分辨率
同时，用一个半最大脉冲宽度为1-ns全宽的高斯分布，来模拟激光光子的时间形状。根据最新ICESat-2的设计，激光沿轨采样为0.7m。针对典型的冰原，每一次拍摄的平均光电子数为2.04（针对弱拍摄）与8.17（针对强拍摄）。3D合成表面由分形技术生成。创建的地形尺寸为1024×1024 m，分辨率为1 m。
For the surface reflectance model, an analytical snow bidirectional reflectance distribution function presented by Kokhanovsky and Breon [17] in a slightly modified notation is used here. Laser wavelength λ is set as 532 nm with χ = 2.54 × 10−9. Meanwhile, parameter M is set to be 5.5 × 10−8, with a = 1.247, b = 1.186, and c = 5.157, based on Kokhanovsky’s letter [17]. The snow grain size is 200 μm.
reflectance[rɪˈflektəns]：反射
bidirectional[ˌbaɪdəˈrekʃənl] ：双向的
notation[nəʊˈteɪʃn]：符号
wavelength：波长
grain[ɡreɪn]：颗粒
针对地面反射模型，由Kokhanovsky and Breon [17]提出的雪双向反射分布函数在这里应用。激光波长λ设为532 nm，χ=2.54×10−9。同时，根据Kokhanovsky的文章[17]，参数m设置为5.5×10−8，a=1.247，b=1.186，c=5.157。雪粒大小为200μm。
In addition, noise is added to the point cloud with uniform random distribution. With a noise rate of 2 MHz, an ICESat-2 point cloud of 0.1-s flight (700-m distance on the ground) over test 3-D synthetic scene is plotted in Fig. 1.
uniform[ˈjuːnɪfɔːm]：n.制服；adj.统一的。
此外，通过随即均匀分配，在点云中增加噪点。图1显示，在3D合成场景中，一个噪声率2兆赫的ICESat-2点云飞行0.1秒（距地面700米）。
图1
B. High-Altitude MABEL Data Set
The second source of simulated ICESat-2 data is from MABEL [18]. The test data (L2A) were collected in WI, USA, on September 26, 2012, where lots of canopy-covered ground are present, as shown in Fig. 2.
canopy-covered ground[ˈkænəpi]：冠层覆盖地面
第二种模拟ICESat-2 data数据来自于MABEL[18]。测试数据(L2A)于2012年9月26日在美国WI收集。在WI收集到大量的树冠覆盖地面，如图2所示。
图2
III. METHODOLOGY
B. Modified DBSCAN
The key idea of DBSCAN is that, for each point of a cluster, the neighborhood of a given radius has to contain at least a minimum number of points, i.e., the density in the neighborhood has to exceed some threshold. The shape of a neighborhood is determined by the choice of a distance function for two points p and q, denoted by dist(p, q). Two parameters mentioned here are an Eps-neighborhood of a point, defined by dist(p, q) ≤ Eps, and the minimum number of points (MinPts) in that Eps-neighborhood [14].
density[ˈdensəti]：密度
exceed[ɪkˈsiːd]：v.超过
denote [dɪˈnəʊt]：v.表示
DBSCAN的主要思想是，簇中的每一个点，在给定半径内至少有minimum个点。也就是说，该点邻近范围内的密度必须超过某个阈值。邻近范围的形状取决于距离函数的选择，关键点在于p与q，用dist(p, q)表示。提到的这两个参数，决定了该点的Eps领域范围，需要满足dist(p, q) ≤ Eps。同时，在该Eps领域范围内，有至少MinPts个点。
For a data set in two dimensions, the distance between two points p(tp, hp) and q(tq, hq) is defined as
公式1
where t represents delta_time in Fig. 1, which can be considered as an along-track distance, and h represents elevation. tscale and hscale are used for normalization so that the points in the test data set have comparable order over t- and h-axes. Hence, dist(p, q) is now unitless.
unitless：无单位的
对于二维数据集，点p(tp, hp)与q(tq, hq)之间的距离可以表示为公式1。其中，t表示图1中的delta_time，可以理解成沿轨距离。h表示高度。tscale和hscale用于标准化，以便测试数据集中的点在T轴和H轴上具有可比较的顺序。因此，dist（p，q）现在是无单位的。
In our algorithm, since most of the clusters (surface returns) have higher density in the horizontal than the vertical direction, it is reasonable to modify the shape of search area accordingly. Therefore, the distance between points p(tp, hp) and q(tq, hq) is now modified as
公式2
horizontal[ˌhɒrɪˈzɒntl]：水平的
vertical[ˈvɜːtɪkl]：垂直的
在我们的算法中，由于绝大多数簇（地表返回的）在水平方向有比垂直方向更高的密度，因此相应地改变搜索区域的形状是合理的。距离表示公式修改如公式2所示。
As can be seen in Fig. 3, the search area is modified as an ellipse with centroid p, major axis with length 2a, and minor axis with length 2b, while a > b. Due to the change in search area, points in the horizontal direction have more weight with respect to the search area center than points in the vertical direction. Therefore, continuous points in a roughly horizontal direction are more likely to be classified as belonging to the cluster. That is also the same as in the detection of ground for MABEL lidar point clouds.
图3
with respect to：关于、至于
roughly[ˈrʌfli]：大约
由图3看到，搜索区域变成了一个椭圆，中心点为p，长轴长2a，短轴长2b（a>b）。由于搜索区域的改变，水平方向上的点相较于垂直方向上的，有更高的权重。因此，在大致水平方向上的连续点，更容易被划分成属于这一簇。这同样适用于MABEL雷达点云对地面的勘测结果。
C. Estimation of Clustering Parameters
As the ellipse shape is determined by a and b in (2), two parameters are needed for modified DBSCAN implementation: MinPts and Eps. Here, we develop a simple but effective heuristic way to determine the two parameters. For simplicity, Eps = 2 is used all the time so that only MinPts will be modified. It can be done by estimating the average point density within the search ellipse.
heuristic[hjuˈrɪstɪk]：启发式的
simplicity[sɪmˈplɪsəti]：简单
在公式2中， 椭圆的形状由参数a与b决定。两个参数MinPts与Eps，在改进后的DBSCAN中很重要。我们使用了一个简单但是有效的方法去决定这两个参数。简单起见，Eps=2被广泛应用，因而只有MinPts需要被改变。这可以通过估算一个椭圆区域内的平均点云密度实现。
1）A partition of points from the test data set is first extracted. This example covers a flight time of δt and an elevation range of δh. The area S of this sample data set is
公式3
partition[pɑːˈtɪʃn]：分割
extracted[ɪkˈstræktɪd]：提取
首先，从测试数据集中提取一个点分区。它覆盖的飞行时间为δt，高度范围为δh。则该样本数据集的区域S可表示为公式3。
2) For an ellipse with dist(p, q) = Eps, its area s1 is
公式4
where a = 0.5 and b = 0.2. Hence, the number of ellipses within the example data set is roughly estimated as S/s1.
当该椭圆dist(p, q) = Eps时，其区域面积s1可表示为公式4。其中，a=0.5，b=0.2。因此，数据集中椭圆的数量可估计为S/s1。
3）The number of points in the example data set is found to be N . Therefore, the average point density (ρ) within the search ellipse can be calculated
公式5
数据集中的点数量为N。因此，一个搜索椭圆区域内的平均点密度(ρ)可按照公式5计算。
4) To better estimate ρ, more than one of the example data sets are extracted from the test data set, processed through steps 1) to 3), and then averaged. In the proposed clustering method, the point density for clusters should be higher than the average density of the whole data set. MinPts can be empirically estimated as
公式6
proposed：提议，打算
empirically[ɪmˈpɪrɪkl] ：经验性的
为了更好的估算ρ，选取多个样本数据集重复进行步骤一至步骤三，然后取平均值。在提出的聚类方法中，聚类的点密度要比整个数据集的平均密度ρ高。按经验，MinPts估计为公式6。
Practically, we can always start with the minimum integer larger than 4ρ and increase by one gradually. For the simulated photon-counting lidar data sets as in Fig. 1, ρ ≈ 0.3, and MinPts = 4 is finally applied. For the MABEL data sets as in Fig. 2, ρ ≈ 3.7, and MinPts = 16 is used. This proposed clustering algorithm can be quickly implemented and adaptive to photon-counting lidar data sets with different point densities.
Practically[ˈpræktɪkli]：实际上
实际上，我们一般从大于4ρ的最小整数开始，然后逐一增加。图一模拟的光子计数雷达数据，应用ρ ≈ 0.3，MinPts = 4。图二的MABEL数据集，应用ρ ≈ 3.7，MinPts = 16。提出的聚类算法可以改进应用于各种不同点密度的光子计数雷达数据。
IV. PERFORMANCE AND EVALUATION
Our algorithm is tested using the aforementioned two sets of photon-counting laser altimeter data. In the first principle simulation, parameter p in the filter for generating the 3D synthetic surface is 2.0 [15]. The noise rate is set as 2 MHz. As can be seen in Fig. 4, surface returns can be reliably classified as ground returns using the proposed algorithm. A quantitative evaluation on the performance of the proposed method is presented later.
aforementioned[əˈfɔːmenʃənd]：前面提到的
synthetic[sɪnˈθetɪk]：合成的
quantitative [ˈkwɒntɪtətɪv]：定量性的
利用之前提到的两组光子计数激光测高数据，对该算法进行测试。在第一原则模拟中，用于生成三维合成曲面的1/f^p滤波器的参数p为2.0。噪声率设为2兆赫。由图4看出，使用该算法，surface returns可以可靠地被归类为ground returns。随后对该方法的性能进行了定量评价。
图4
This density-based clustering approach is also tested for the MABEL data set. For the experimental data set, the classification result is shown in Fig. 5, which demonstrates that the proposed algorithm is capable of detecting both canopy and ground surface. The adaptive nature of our proposed algorithm allows it to work on a variety of surfaces and with data from a variety of photon-counting lidars. More classification results using the proposed algorithm for surface detection are presented in another letter [20], where point clouds of photon-counting lidar collected from different scenes and atmospheric conditions are studied.
experimental[ɪkˌsperɪˈmentl]：实验性的
demonstrates[ˈdemənstreɪts]：v.证明
atmospheric conditions[ˌætməsˈferɪk]：大气条件
这种基于密度的聚类方法同样被应用于MABEL数据集。实验数据集的分类结果如图5所示，这也证明了该算法同时适用于检测冠层与地表。该算法的自适应性质允许它在各种表面上工作，能够使用来自各种光子计数激光雷达的数据。使用该算法检测表面的更多分类结果在文章[20]中有说明，其中研究了从不同场景和大气条件下收集的光子计数激光雷达的点云。
图5
To quantitatively evaluate the performance of the proposed algorithm, ground truth information is required. From the synthetic terrain, a 2-D profile of illuminated terrain can be directly extracted, which contains ground elevation versus flight distance or time. Note that the laser footprint has a radius of 5 m. Hence, due to the variance of ground within the circular laser footprint, it is hard to designate the returning photon to a specific location within the illuminated area.
terrain [təˈreɪn]：地形，地势
profile[ˈprəʊfaɪl]：概述，简介，侧面轮廓
illuminated[ɪˈluːmɪneɪtɪd] ：被照明的
versus[ˈvɜːsəs]：与...相对
designate[ˈdezɪɡneɪt] ：指定
为了定量评测该算法的性能，需要地面真实信息。从合成地形中，我们可以直接提取到一个被照射地形的二维剖面，其中包含地面高度与飞行距离或者时间。请注意，激光足迹的半径为5m。因此，由于在一个圆形激光足迹中地表的变化，我们很难把返回的光子指定到被照射区域中的一个确定位置。
A statistical method is then necessary to define a region for accuracy evaluation. Here, an upper/lower boundary along the 2-D ground truth is created with a specific height above/below the terrain profile. The two boundaries enable a window which can be regarded as the criterion of true surface returns. Therefore, each photon is assigned to an elevation with respect to flight distance and can be categorized as surface returns if it is within the contour “window.” A height of 10 cm, which is close to the expected elevation bias standard deviation for ICESat-2, is chosen for performance assessment [15], [16].
statistical[stəˈtɪstɪkl]：统计学的
region[ˈriːdʒən]：区域
with respect to ：关于
criterion[kraɪˈtɪəriən]：n.标准
categorize[ˈkætəɡəraɪz] ：分类
bias[ˈbaɪəs]：偏向
deviation[ˌdiːviˈeɪʃn]：偏离
需要一个统计学的方法来定义准确评估的区域。于是，沿二维地面实况创建上／下边界，在地形剖面上／下有一个确定高度。两个边界组成的窗口，可以被认为是正确地面返回的标准。因此，每个光子都被分配到一个与飞行距离有关的高度，并且如果它在轮廓“窗口”内，则可以被归类为表面返回。选择高度10cm来进行性能评估，这个高度接近ICESat-2预期高度偏向标准偏离的值。
In addition, the statistical indicators known as recall and precision are computed. Recall R is the fraction of true signal points that are successfully enclosed within the contour window. Precision P is the fraction of true signal points from all points enclosed within the detected contours. They are defined as follows [21]:
indicators[ˈɪndɪkeɪtəz] ：指标
fraction[ˈfrækʃn]：少量，分数
contour [ˈkɒntʊə(r)] ：轮廓
另外，计算了作为统计指标的召回率和准确度。R=被预测为地面返回且成功落在窗口内的点的数量／所有落在窗口内的信号点的总数，P=被预测为地面返回且成功落在窗口内的点的数量／被预测为是地面返回点的总数。这些在文章[21]中定义。
公式7
where TP , FP , and FN represent the numbers of true positives (hit), false positives (false alarm)m and false negatives (miss), respectively. To be more specific, true positives represent points that are enclosed in the contour window being detected as surface returns, and false positives represent points that are not enclosed in the contour window being detected as surface returns. For better estimation, the proposed algorithm is evaluated for five sets of point clouds, each of which was collected by different test tracks (as shown in Fig. 6).
respectively[rɪˈspektɪvli]：分别地
其中TP表示将正类预测为正类数，FP将负类预测为正类数（误报），FN表示将正类预测为负类数（漏报）。更详细的说，TP表示落在框内且被预测为是地面返回的点数，FP表示没落在框内却被预测为是地面返回的点数。为了更好地估计，对五组点云进行了评估，每个点云由不同的测试轨道收集。（如图六所示）
图6
For each track, a statistical indicator is calculated to find TP, FP, and FN, respectively. As can be seen in Fig. 7, the contour window is labeled as a black dashed line. Returns classified as ground and enclosed inside the window are TP (Hit), and those not enclosed inside the window are FP (False Alarm). Mean- while, classified noise enclosed inside the window is FN (Miss).
dashed line[dæʃt]：虚线
针对每一个轨道，计算一个统一指标去找到TP, FP和FN。如图7所示，轮廓窗口标记为黑色虚线。被预测为地面返回且落在窗口内的是TP (Hit)。被预测为地面返回但未落在窗口内的是FP (False Alarm)。同时，被预测为噪点但落在窗口内的是FN (Miss)。
图7
In order to use a single performance measure that will allow for comparison of results, the harmonic mean of recall and precision will be used
公式8
For all five tracks, the F-measure value is calculated, respectively, and then averaged. Thus, uncertainty caused by ground surface variation will be mitigated. The result of F-measure versus surface roughness parameter p is shown in Fig. 8. Note that, as p increases, the synthetic terrain becomes less rough [15] and the F-measure increases significantly from 0.58 to 0.86. Therefore, the proposed algorithm has better performance on a smoother surface.
harmonic mean[hɑːˈmɒnɪk miːn]：调和平均数
roughness [rʌfnəs] ：粗糙
为了使用一个允许结果比较的单一性能度量，将使用召回和精度的调和平均值。分别计算这五组轨迹的F-measure值，并选取平均数。F-测量值与表面粗糙度参数P的对比结果如图8所示。注意，随着P值的增加，合成地形变得不那么粗糙[15]，F值从0.58显著增加到0.86。因此，该算法在光滑表面上具有较好的性能。
图8
In addition, the impact of noise rate is studied. Noise rate varies based on atmospheric and solar conditions: 0.5 MHz simulates nighttime acquisitions, while 2 and 5 MHz represent daytime acquisitions with clear sky and hazy atmosphere, respectively [19]. As we increase the noise rate from 0.5 to 5 MHz, the F-measure maintains an average of 0.8 (blue curve in Fig. 9), and the elliptical DBSCAN algorithm is seen to be robust. However, it is shown that lower noise rate will lead to slightly better detection performance.
nighttime[ˈnaɪttaɪm]：夜间
acquisition[ˌækwɪˈzɪʃn]：获得，采集
robust[rəʊˈbʌst]：强健的
此外，还研究了噪声率的影响。噪声率根据大气和太阳条件而变化：0.5兆赫模拟夜间采集，而2兆赫和5兆赫分别代表晴空和朦胧大气的白天采集[19]。当我们将噪声率从0.5兆赫增加到5兆赫时，f-测量保持0.8的平均值（图9中的蓝色曲线），并且椭圆DBSCAN算法被认为是稳健的。然而，研究表明，较低的噪声率会使检测性能稍有改善。
图9
Meanwhile, the improvement of ground detection accuracy is studied using the proposed elliptical DBSCAN over the conventional circle DBSCAN method. For comparison, all parameters used in the proposed algorithm remain the same for the circle DBSCAN method, except that in (2), in which a = b = 0.5 is used to change the search area to a circle. The result of ground detection accuracy using circle DBSCAN is plotted in red color in Fig. 9. With a low noise rate (around 1 MHz), both reach the F-measure of around 0.8. As the noise rate increases, the ground detection accuracy is significantly improved while using elliptical DBSCAN method. This quantitative plot also shows that the proposed method using elliptical DBSCAN has better performance despite the solar noise rate. Note that this conclusion works for photon-counting laser altimeter data whose point density of surface returns is higher than the background noise. If the surface return rate is too low to visually distinguish surface returns from noise, it is difficult to achieve good performance of the proposed algorithm.
conventional[kənˈvenʃənl]：常规的
同时，在常规圆DBSCAN方法的基础上，利用该椭圆DBSCAN方法对提高地面探测精度进行了研究。比较而言，除了（2）中的参数外，该算法中使用的所有参数对圆DBSCAN方法保持不变，其中a=b=0.5用于将搜索区域更改为圆。使用圆DBSCAN的地面检测精度结果，用红色绘制在图9中。在低噪声率（1兆赫左右）下，两者都达到了0.8左右的F测量值。随着噪声率的增加，采用椭圆DBSCAN方法可以显著提高地面检测精度。这定量图也表明，在太阳噪声率较高情况下，椭圆DBSCAN具有更好的性能。注意，这个结论适用于光子计数激光高度计数据，其表面回波的点密度高于背景噪声。如果表面返回率太低，无法从视觉上区分表面返回和噪声，则很难实现该算法的良好性能。
V. CONCLUSION
In this letter, a density-based algorithm has been proposed for classifying photon-counting lidar point clouds as surface or noise returns. In consideration of finding surface returns more accurately from the lidar point cloud, the area shape of a data point search for its nearest neighbors was modified to be an ellipse to match general characteristics of terrain or vegetation. This adaptive clustering method was then implemented and tested on photon-counting lidar altimetry data. Validation showed that surface and canopy can be expected to be observable using the proposed algorithm during the ICESat-2 mission. Performance measurement demonstrated that this method has better performance for smoother surfaces and lower noise rate conditions.
本文提出了一种基于密度的光子计数激光雷达点云分类算法。为了从激光雷达点云中，更准确地找到地表回波，将最近邻点数据点搜索的面积形状修改为椭圆，以匹配地形或植被的一般特征。然后对光子计数激光雷达测高数据进行了自适应聚类测试。验证结果表明，在ICESAT-2任务中，利用该算法可以观测到地表和树冠。性能测试结果表明，该方法在光滑表面和低噪声条件下具有较好的性能。
Future work will consider the following issues. First, in our current work, only objects which have high density in horizontal direction were studied. We will develop a definition to extend the approach for more complicated objects such as steep crevasses in photon-counting point cloud. Second, more realistic lidar data sets will be studied to evaluate the proposed algorithm performance for more complicated terrains and atmospheric conditions. Third, additional tests will be performed to quantify the algorithm performance in detecting both vegetation canopy and ground in dense forests.
steep crevasses [stiːp krɪˈvæsɪz]：陡峭的裂缝
quantify[ˈkwɒntɪfaɪ]：量化
今后的工作将考虑以下问题。首先，在我们目前的工作中，只研究水平方向具有高密度的物体。我们将发展一个定义来扩展对更复杂物体的方法，例如光子计数点云中的陡峭裂缝。其次，将研究更真实的激光雷达数据集，以评估所提出的算法在更复杂的地形和大气条件下的性能。第三，将进行额外的测试，量化算法在密林植被冠层和地面检测中的性能。
个人总结
一、DBSCAN算法改进
1. 原DBSCAN算法
（1）概述
某样本在eps距离内有至少MinPts个样本，则该样本可以成为核样本。通过找到一个核样本，找到其附近的核样本，再找到附近核样本的附近的核样本。递归寻找。形成一个高密度区域。
（2）核样本的邻域
半径为Eps的圆，当时，说明q在p的邻域内。
（3）Dist(p,q)定义
Fig.1
如图Fig.1所示，二维数据集，存在点p(tp, hp)与q(tq, hq)，t表示delta_time，h表示Elevation。
Dist(p,q)可以表示为：
公式1
其中，tscale和hscale用于标准化，以便测试数据集中的点在T轴和H轴上具有可比较性。
2. 改进DBSCAN算法
（1）概述
由于本实验中，绝大多数簇在水平方向有比垂直方向更高的密度，因此，Eps邻域范围由圆改进为椭圆。
（2）核样本的邻域
椭圆。同样，当时，说明q在p的邻域内。
（3）Dist(p,q)定义
Fig.2
Dist(p,q)可以表示为：
公式2
则根据，可推导出椭圆公式为：
可知，该椭圆长轴长为，短轴长为
二、改进DBSCAN算法的参数估算
1. Eps
Eps，取用常见数值2
2. MinPts
（1）总面积：
（2）一个椭圆的面积：
其中，a=0.5，b=0.2【论文中并没有给出选取值的理由】
（3）总点数：
（4）一个椭圆内的平均点数：
（5）多次计算，求取平均值得到最终
值。MinPts需满足：
3. 实验选取值
MinPts值，从大于4ρ的最小整数开始，逐一增加。模拟光子计数雷达数据，应用ρ ≈ 0.3，MinPts = 4。MABEL数据集，应用ρ ≈ 3.7，MinPts = 16。
三、定量评测改进算法的性能
1. 定义True Surface Returns
沿二维地面实况，创建上／下边界，高度为10cm。两个边界组成的窗口，可以被认为是True Surface Returns的标准。因此，如果一个光子，在轮廓“窗口”内，则可以被归类为Surface Returns。图Fig.3中，黑色虚线即为轮廓窗口。
Fig.3
2. 计算Recall、Precision、F-Measure
选取五条轨迹。针对每一个轨道，计算TP, FP和FN。如图Fig.3所示，轮廓窗口标记为黑色虚线。被预测为地面返回且落在窗口内的是TP (Hit)。被预测为地面返回但未落在窗口内的是FP (False Alarm)。同时，被预测为噪点但落在窗口内的是FN (Miss)。
根据公式计算Recall、Precision、F-Measure：
分别计算这五组轨迹的F-measure值，并选取平均数。
3. 实验结果一：F-Measure与表面粗糙度参数P的关系
F-Measure与表面粗糙度参数P的对比结果，如Fig.4所示。随着P值的增加，合成地形变得不那么粗糙，F值从0.58显著增加到0.86。因此，该算法在光滑表面上具有较好的性能。
Fig.4
4. 实验结果二：F-Measure与噪声率
噪声率根据大气和太阳条件而变化：0.5MHz模拟夜间采集，2MHz和5MHz分别代表晴空和朦胧大气的白天采集[19]。
当我们将噪声率从0.5MHz增加到5MHz时，F-Measure保持0.8的平均值（Fig.5中的蓝色曲线），因此该椭圆DBSCAN算法被认为是稳健的。除此之外，研究发现，较低的噪声率会使检测性能稍有改善。
Fig.5
5. 实验结果三：对比改进前后的DBSCAN算法
圆DBSCAN算法中，更改参数a=b=0.5，用于将搜索区域更改为圆。其余参数不变。使用圆DBSCAN的地面检测精度结果，用红色绘制在图Fig.5中。
在低噪声率（1兆赫左右）下，两者都达到了0.8左右的F-Measure。随着噪声率的增加，采用椭圆DBSCAN方法可以显著提高地面检测精度。 这定量图也表明，在太阳噪声率较高情况下，椭圆DBSCAN具有更好的性能。
注意，该结论适用于其表面回波的点密度高于背景噪声。如果表面返回率太低，无法从视觉上区分表面返回和噪声，则很难实现该算法的良好性能。
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