Geodesics-Based Topographical Feature Extraction From Airborne Lidar Data For Disaster Management Zhi Wang 1,2,3 *，Huiying Li 4, Lixin Wu 1,2 1 Key Laboratory of Environment Change & Natural Disaster of MOE, Beijing Normal University, 19 Xinjiekou Wai Street, Beijing, P.R. China, 100875; 2. Academy of Disaster Reduction and Emergency Management, Ministry of Civil Affairs, Ministry of Education, Beijing Normal University, 19 Xinjiekou Wai Street, Beijing, P.R. China,100875; 3. College of Resources and Civil Engineering, Northeastern University, No.11, Lane3, WenHua Road, HePing District, Shenyang, P.R. China, 110004; 4. College of Computer Science and Technology, Jilin University, No. 2699, Qianjin Street, Changchun, P.R. China, 130012 * Corresponding author:
[email protected] Abstract—Hundreds of thousands of lives were lost in the natural disasters such as geological earthquakes, floods, landslides and mud-rock flow in every year. Nowadays, with the rapid development in airborne LiDAR techniques, extraction of the multi-scale topographical features from high-resolution topographic data acquired via airborne LiDAR would lead to fundamentally new understandings of earth essential to mapping flood, landslide and mud-rock flow hazards for decision makers. In this paper, we define topographic features in a multi-scale manner using a center-surround operator on Gaussian-weighted mean curvatures. These multi-scale topographical features would allow improved detecting, understanding and prediction of flood inundation, landslide and mud-rock flow likelihood. For example, experimental results identify that proposed method can be employed for detecting landslide. Keywords- geodesics ； discrete curvature ； airborne lidar ； topographic feature； disaster management
I.
INTRODUCTION
In recently years hundreds of thousands of lives were lost in the natural disasters such as geological earthquakes, floods, landslides and mud-rock flow. Fundamental scientific advances will be needed to mitigate losses from these disasters as well as protect human life and property. LiDAR [1] provides a threedimensional sampled representation of the surfaces of earth and terrestrial objects. The collected point cloud of geometric samples on the surface of the subject can then be used to construct digital 3D models useful for a wide variety of military and civil applications, such as surveying, architecture, engineering, virtual reality, cultural heritage and historic sites preservation [2]. Recently, with the rapid development in airborne LiDAR techniques, extraction of the multi-scale topographical features from high-resolution topographic data
Our research was supported by the National High Technology Research and Development Program of China (Grant No. 2008AA121601), the Natural Science Foundation of China (Grant No. 40901220) and the China Postdoctoral Science Foundation (Grant No. 20090450305).
acquired via airborne LiDAR would lead to fundamentally new understandings of earth essential to mapping flood, landslide and mud-rock flow hazards for decision makers. II.
ESTIMATING DISCRETE CURVATURE
To accomplish the above missions, in this paper we firstly investigate the key technology for estimating discrete curvature on surface of LiDAR data by using geodesics [3], then extracting multi-scale topographical features of digital terrain models. The curvature of terrain surface indicates the property of a terrain surface. The mean curvatures of vertices of a surface indicate whether they are feature vertex or zero-feature vertex. A lot of studies have been done for estimating discrete mean curvature of TIN based surfaces. Unfortunately, no one is widely accepted as the most accurate method or the best method for curvature estimation. Surazhsky et al. (2003) showed that the paraboloid fit method [4] is the best one for estimating the mean curvature of meshes. However, the paraboloid fit method is time-consuming and may have extreme difficulties for fitting the osculating quadric in the area of high curvature. Geodesic curves are the fundamental concept in geometry and mathematics to generalize the idea of “straight lines” to arbitrary curved surfaces and general manifolds. Smooth geodesics are locally shortest curves and have vanishing geodesic curvature. In this paper, the first and second discrete differential-geometry operators for point clouds were proposed by using geodesics-based analysis (as shown in Figure 1). These differential-geometry operators are used to estimate geometric attributes of triangular meshes, including Gaussian curvature and mean curvature.
nvi
K G (Vi ) = Where
nvi
As the red arc shown in Figure 1, the length of geodesics arc ηi , j can be measured by following equations.
ηi , j = ei , j + γ j , j +1
The angle
ki , j 24
ei3, j + ο (e3 )
γ j , j +1 = α j , j +1 +
ei , j ei , j +1ki , j ki , j +1 4 sin α j , j +1
−
ei2, j ki2, j + ei2, j +1ki2, j +1 8 tan α j , j +1
K G (Vi ) + ο (e3 ) (2)
Suppose
C = C1 ∪ C2 ∪ C3 ∪ " ∪ Cn is a closed
piecewise curve and the outer angle
θi
k g is the geodesics curvature in simply connected region D on the surface. According to Gauss-
Bonnet theory, they follow that relationship:
If
4sin α i , j
D
Ci
i =1
k g = 0 , the following relation can be obtained.
∑θ + ∫∫ i
Svi
−
8 tan α i , j
ei2, j +1 + ei2, j +1 8 tan α i , j
⎛ ei , j ei , j +1 ei2, j + ei2, j +1 ei , j ei , j +1 ei2, j + ei2, j +1 ⎞ − + − ⎜n ⎟ ⎜ vi 2sin α j , j +1 4 tan α j , j +1 4 sin α i , j 8 tan α i , j ⎟ ⎜∑ ⎟ 2 2 ⎜ j =1 + ei , j +1ei , j +1 − ei , j +1 + ei , j +1 ⎟ ⎜ ⎟ 4 sin α i , j 8 tan α i , j ⎝ ⎠ (7) According to the theory of differential geometry, the mean curvature H continuous on the piecewise surface is defined as the average of normal curvature k n :
H=
1 2π
2π
∫ k (θ )dθ n
KdA = 2π
(5)
AVi is the area of geodesic triangles, the geodesic
K G (Vi ) can be compute by following equation:
(8)
0
According to Euler formula: (9)
The formula for computing the mean curvature can be simplified as
curvature
4sin α i , j
kn (θ ) = k M ( vi ) cos 2 θ + km ( vi ) sin 2 θ
nVi
i =1
+
ei2, j + ei2, j +1
nvi ⎛ ⎞ = 4 ⎜ 2π − ∑ α j , j +1 ⎟ / ⎜ ⎟ j =1 ⎝ ⎠
(3)
C = C1 ∪ C2 ∪ C3 ∪ " ∪ Cn is geodesics arc, so
Suppose
−
4 tan α j , j +1
n
∑θi + ∑ v∫ k g ds + ∫∫ KdA = 2π i =1
+
ei , j +1ei , j +1
−
ei , j ei , j +1
at the junction of the
smooth curves,
n
2sin α j , j +1
ei2, j + ei2, j +1
We proposed a new formula for calculating the discrete Gauss curvature of a piecewise surface by using equation 1 to equation 6:
α j , j +1 from corresponding edges, the
ei , j and the discrete curvature ki , j .
length of edge
j =1
(6)
AVi
(1)
between two geodesics arc can be
calculated from the angle
AVi = ∑
i =1
AVi can be calculated by following equation: ei , j ei , j +1
Figure 1. Geodesics-based analysis.
2π ∑ θi
(
)
H = k M ( vi ) + km ( vi ) / 2
In case of the triangular surface S , the mean curvature of the point on S can be estimated using the following formula:
H = ∑ length(e ∩ S ) e∈E
β (e)
(10)
2
In this paper, we use the angle between two triangular surfaces to estimate the angle between two geodesic triangular surfaces by using following equation:
K H ( vi )
1 nvi ∑ (ηi , j ( m) β j −1( m), j ( m) ) 2 j =1 = Avi
(11)
In case of the triangular surface S , the length of geodesic arc and the corresponding edges have the following relations:
ηi , j ( m ) =
ei , j 2
+
kG ( vi ) ei , j 3 ( ) 24 2
(12)
Figure 3.
Estimating the discrete curvatures on a revolution surface.
III.
DETECTING TOPOGRAPHIC FEATURES
The most widely used set of topographic characteristics[5], which defined by Wood (1996), is the subdivision of all points on a surface into one of plane, peak, pit, ridge, channel, and pass (as illustrated in Figure 4), which are defined as terrain features. The proposed algorithm firstly detects the terrain features according to the curvature of terrain surface.
By using equation 7 to equation 12, we find a new formula for calculating the discrete mean curvature:
K H (Vi )
3 ⎛ ⎛ nvi e ⎞ K G (Vi ) ⎛ ei , j ⎞ ⎞ i, j ⎟ ⎟ β = 2⎜⎜ ∑ + ⎜ ⎟ j −1( m ), j ( m ) / ⎜ ⎜ j =1 2 ⎟ 24 ⎝ 2 ⎠ ⎟ ⎠ ⎝⎝ ⎠
⎛ ei , j ei , j +1 ei2, j + ei2, j +1 ei , j ei , j +1 ei2, j + ei2, j +1 ⎞ − + − ⎜ nv ⎟ ⎜ i 2sin α j , j +1 4 tan α j , j +1 4sin α i , j 8 tan α i , j ⎟ ⎜∑ ⎟ 2 2 ⎜ j =1 + ei , j +1ei , j +1 − ei , j +1 + ei , j +1 ⎟ ⎜ ⎟ 4sin α i , j 8 tan α i , j ⎝ ⎠ (12) To illustrate the usefulness of the algorithm, the scheme was tested on triangular meshes that represent tesselations of synthetic geometric models. As shown in Figure 2, the results are compared with the analytically computed values of the nonuniform rational B-spline (NURBs) surfaces, these meshes originated from. As shown in Figure 3, this work manifests the algorithms developed in this paper is provided the better results than tradition one and Meyer’s method [6] for curvature estimation especially for high resolution data.
(a)
(b)
(c)
Figure 2. Estimating the discrete curvatures on a revolution surface.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4. The category of terrain features: (a) peak, (b) pit, (c) ridge, (d) channel, (e) pass.
In our method, we firstly compute mean curvature for each vertex. Then, we compute the Gaussian-weighted average of the mean curvatures for each vertex within a radius 2 σ , where σ is Gaussian's standard deviation. The topographic features are determined at different scales by varying σ . The multiscale model is used to ignore local perturbations that go against the overall trend of the linear feature. Let the mean curvature map M define a mapping from each vertex of a TIN model to its mean curvature, i.e. let M (v ) denote the mean curvature of vertex v . Let the neighborhood N (v, σ ) for a vertex v , be the set of points within a distance σ . In our scheme, we use the Euclidean distance N (v, σ ) = { vi :|| vi − v ||< σ , vi is a vertex on the TIN surface}. Let G ( M (v ), σ ) denote the Gaussianweighted average of the mean curvature. G ( M (v ), σ ) can be computed by using underlying formula:
∑ G ( M (v ), σ ) =
M ( x ) exp[ − || vi − v || /(2σ )] 2
2
vi ∈ N ( v , 2 σ )
∑
exp[ − || vi − v || /(2σ )] 2
2
(13)
vi ∈ N ( v , 2 σ )
where σ is the standard deviation of the Gaussian filter. For all the results in this paper we have used seven scales σ ∈ {1ε , 2ε , 3ε , 4ε , 5ε , 6ε , 7ε } , where ε is defined as 0.3% of the length of the diagonal of the bounding box of the model (Lee et al., 2005). In our method, we assigned a weight to each vertex according to the relationship between it and the global topographic features. Let the topographic feature map W define a mapping from each vertex of a TIN model to its feature. As shown in Figure 5 (b), the mean curvature map may have far too many “bumpy” being flagged as features. However, we can promote salience maps with a small number of high values by calculating Gaussian-weighted mean curvature in large scale. One can see that the topographic features are more coherent in the large-scales. Figure 5(c)-(f) gives an overview of topographic feature map such as peak, pit, ridge, channel and pass in different scales.
(a)
(b)
(a)
(b)
(c)
(d)
(e)
Figure 6. Images show the topographic features at scales of the salient features at scales of 1 ε , 3 ε , 5 ε , and 7 ε .
IV.
CASE STUDY –DETECTING LANDSLIDE FEATURES
The 2010 Haiti earthquake was a catastrophic magnitude 7.0M earthquake. During an earthquake, hillside stability is further threatened as the ground is shaken. The combination of widespread deforestation and the recent earthquake in Haiti lead to more landslides in the already hard-hit country. In the experiment, we use the airborne LiDAR data over the affected area from RIT's Laboratory for Imaging Algorithms and Systems to show how did the proposed method help detecting landslide.
(c)
(a)
(d)
(e)
(f)
Figure 5. Topographic feature detection: Image (a) shows the “Crater” model, image (b) shows its mean curvature distribution. Images (c) - (f) show the salient features at scales of 1 ε , 3 ε , 5 ε , and 7 ε .
Figure 6 shows the detection of topographic features such as peak, pit, ridge, channel and pass using our method in different scales. We use pseudo-colors to texture the surface according to the feature weights: warmer colours (reds and yellows) show high weights, cooler colours (greens) show low weights, and blues show zero-feature. In our algorithm, we use values of Gaussian-weighted mean curvature to evaluate the point to the extent of topographic feature. The feature classification can easily be achieved according to the rules of Wood (1996) [5].
(b)
(c)
(d)
Figure 7. Image (a) shows the landslide in aerial imagery; (b) and (c) show the 3D model of affected area; (d) shows the landslide area (areas with red color) according to the discrete curvatures.
Figure 7 (a) shows the landslide in aerial imagery. As we know, it is very difficult for finding landslides in remote sensing images. Figure 7 (b) and (c) show the 3D model of affected area. Figure 7 (d) shows the results for identifying Landslide area from our new scheme from airborne LiDAR data. We distinguish the landslide area (areas with red color) and the un-affected area (areas with green color) according to the discrete curvatures. V.
long time. It should be possible to significantly speed it up by using a multi-resolution mesh hierarchy to accelerate filtering at coarser scales. We foresee the computation and use of topographic feature detection as an increasingly important area in TIN data processing. A number of tasks can benefit from a computational model of topographic feature detection, such as improved detecting, understanding and prediction of flood inundation, landslide and mud-rock flow likelihood for decision makers.
CONCLUDING REMARKS
In this paper, we developed a model of topographic feature detection using center-surround filters with Gaussian-weighted mean curvatures in a multi-scale manner. We use various models to evaluate the performance of our proposed scheme in terms of accuracy in estimating the discrete curvature and extracting topographic features. The comparisons show that our scheme is able to estimating more accurate discrete curvature so as to generate important topographic features. Moreover, in the experiment, we employed the proposed method to extracting topographic features from airborne LiDAR data to identify landslide area. The model of topographic feature detection promises to be a rich area for further research. We are currently defining topographic features using mean curvature. It should be possible to improve this by using better measures of shape, such as principal curvatures and normals. Our current method of computing topographic feature map in large scale takes a
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