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Layer-Constrained Triangulated Irregular Network Algorithm Based on Ground Penetrating Radar Data and Its Application

更新时间:2016-07-05

With the rapid development of visualization technology in scientific computing, three-dimensional (3D) geological modelling technology has increasingly attracted more and more attention, and has been widely used in the fields of digital geology, petroleum exploration and geotechnical Engineering. The 3D geological modelling is the abstract reconstruction and reproduction for the geological bodies through 3D visualization technology[1]. The complexity of geological spatial relation increases the complicacy of data structure, topological relation and corresponding algorithms, which makes geological modelling very difficult[2], and the 3D modelling and visualization of geological data is the hotspot in the domain of geosciences. Surface element model[3] is widely applied to represent the digital geological surface, including regular rectangular grid and triangular irregular network (TIN).Compared with regular rectangular grid, the TIN method can change the size and number of triangular patches according to the complexity of the geological surface, which eliminates the data redundancy in visualization step, and maintain a high fitting accuracy[4]. The current research work[4-16] of TIN construction assumes that the discrete data point set is static and does not discuss how to handle dynamic data points in different levels. From the application view, the research of TIN application in geology field focused on terrain and stratum information, and almost all of them do not involve the 3D modelling of geological disease information according to our knowledge. Geological disease is a disaster phenomenen in geological environment, such as underground cavity, stagnant water and loose soil, and it has a great destructive effect on railway subgrade[17], highway tunnel lining[18] and cultural relics[19-20].

According to the hierarchical division of the discrete data points, the paper proposes a layer-constrained TIN (named LC-TIN) algorithm, which can construct a closed or non-closed 2D/3D surface model, and the construction procedure is dynamic from layer to layer, which can realize high precision and arbitrary shape. In addition, a detection platform of geological disease has been developed which adopted the LC-TIN algorithm, and the practice proved its validity.

The rest of this paper is organized as follows. Section 1 presents the related work about TIN algorithm and geological disease detection based on GPR data, some preliminary definitions are given in Section 2 and Section 3 discusses the details of the LC-TIN method. In Section 4, the experimental results are analyzed, and the application of LC-TIN in the detection platform of geological disease is discussed in Section 5, Section 6 concludes this work.

1 Related Work

The principle of TIN algorithm is to construct scattered and non-duplication points to form a continuous and non-overlapping irregular triangle network according to certain rules (such as Delaunay criterion) in order to simulate the 2D/3D object surface, and many scholars studied the TIN algorithm. Aiming at dealing with the inefficiency of network construction, Miao[8] rapidly constructed TIN in linear time by making virtual grid, and then adopted a local optimization method to handle the redundancy problem; Zheng[13] proposed a constrained lines embedding method in the TIN model, which first checked the influence domain of every constrained line and then extracted the influence domain boundary; Xuan[14] gave a new method(the sum of the quadratic distance from three vertices to the gravity center) to determine the skinny triangle for LiDAR data, in which the edge points cannot be detected only by the triangle shape variable; Longtin[15] gave a representation method for density elevation data, which transforms digital evaluation data to a TIN expression in order to reduce the quantity of terrain data; Chen[16] proposed a high precision model of digital elevation, which is based on 3D TIN method.

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The TIN method has been widely used to model terrain and stratum information[4-7,9].Xiong[3] used TIN to realize 3D modelling for the fault information, and adopted VTK (Visualization Toolkit) to implement the prototype system; Tan[5] proposed an automatic triangulation algorithm based on the triangular topological relation, and applied this algorithm to develop a 3D TIN platform; Wang[6] used the improved TIN algorithm to merge the adjacent convex shell blocks, which can optimize the sub-TIN data to form a complete topographic data for underwater navigation. Huang[7] emphatically discussed the robustness of Delaunay mesh generation based on the Bowyer-Watson incremental point insertion technology,and also addressed the stripping algorithm of TINs; Mao[9] proposed a new method of morphological analysis for geological interfaces based on TIN model, which used 3D TIN model to simulate geological interfaces and calculate the general geometry parameters. From the view of geological disease, most of research work[17-20] focused on the detailed diagnosis and preventive measures, and didn’t concern how to model them in 3D platform.

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From the above discussion it is obvious that the above algorithms do not constrain discrete data points in TIN model, and do not illustrate how to use TIN algorithm to model geological disease data. This paper gives a layer-constrained TIN method, which can construct TIN layer by layer dynamically, and the proposed algorithm has been applied to detect geological disease information, the details of LC-TIN are described in Section 2.

2 Preliminaries

In order to describe the LC-TIN algorithm, some basic conceptions have been given as follows.

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The geological data point δ is a tuple, ζ represents a 3D coordinate value, and ζ=<x,y,z>, γ is the attribute value of δ, for example the reflection value of electromagnetic wave.

Def 2 Geological data slice ζ

(1)

ζ is an m×n matrix, and the pth (p∈[1,u]) slice is described as ζp which follows any direction in 3D space, represents the geological data point, which exists in the ith row and the jth column on the pth slice, 1≤im,1≤jn.

Def 3 Pickup line l

l is the bc pickup line on ζp following any direction in 3D space, where b∈{1,2,…t} represents the number of the bth geological disease body, and bc is the corresponding cth line of the bth geological disease body, c∈{1,2,…Nb} and Nb is the number of pickup lines for the bth geological disease body. l may be closed or non-closed forms, which is used to pick up the interesting information on ζp. The amount of point δ in l is defined as

Def 4 Pickup line set ψ

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Input: Ψ

Def 5 Adjacent pickup line pair η

ηbc describes two adjacent pickup lines in ψb, and where c∈{1,2,…Nb-1}, p,q∈[1,u] and pq.

Def 6 Base line ϑ

The base line of ηbc is defined as ϑbc, it represents the l where contains the start edge of the first triangle in ηbc according to TIN rules (such as the sum of circumcircle areas is minimum).In order to ensure there are enough points in non-base line to form triangles, the less points line in and is selected as ϑbc. is the ith geological data point on ϑbc, and i∈{1,2,…Nηbc}, here Nηb is the δ number of ϑbc. Correspondingly,the non-base line in ηbc is defined as

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Def 7 Base edge σ

σ is defined as the line which is formed by two adjacent points on certain l, for example, is the line formed by two adjacent points and on ϑbc, i∈{1,2,…Nηbc-1}. describes the start triangle edge on ηbc according to TIN rules, and σ needs dynamically change during the triangle construction.

Def 1 Geological data point δ=<ζ,γ>

Def 8 Pickup triangle T

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The ith T on ηbc along any direction is defined as or where j∈{1,2,3}) and δj cannot be on the same l.

Def 9 Pickup triangle set Φ

Φ is the set of T which is generated by η, for example, the set of T generated by ηbc is named as Φbc, and according to Delaunay rules.

Def 10 Pickup 3D body Ω

The 3D body constructed by Φb={Φbc|c=1,2,…,Nb} is defined as Ωb, and Ω={Ωb|b=1,2,…,t}.

Def 11 Cosine value and circumcircle radius

Two points of current σ are and i∈[1,Nηbc-1], the third point is on (that is ), so the edge length of is described as follows

(2)

The lengths of the other two edges (A and B) are defined as follows

(3)

(4)

The Cosine value of the angle facing edge constructed by the third point is described as follows

(5)

The circumcircle radius of triangle constructed by the third point is defined as

(6)

The above conceptions can be summarized in Tab.1.

Tab.1 Basic conceptions in LC-TIN algorithm

NotationDescriptionδGeologicaldatapointδ=<ζ,γ>,andζ=ζGeologicaldataslice,ζ=δ11…δ1n︙⋱︙δm1…δmnæèççöø÷÷=δ1…δmæèççöø÷÷1≤i≤m,1≤j≤nlPickupline,andlbcpisthebcpickuplineonζpψPickuplinesetψb={lbcp|p=1,2,…u}andψ={ψb|b=1,2,…t}ηAdjacentpickuplinepair,andηbc=,c∈{1,2,…Nb},p,q∈[1,u]andp≠qϑBaseline,baselineofηbcisdefinedasϑbc,non-baselineiϑbcσBaseedge,σiηbcisthelineformedbytwoadjacentpointsδiϑbcandδi+1ϑbconϑbc,i∈[1,Nηbc-1]TPickuptriangle,andTiηbc={δj|δj∈lbcporlbc+1q},j∈[1,3],δjcannotbeonthesamel.ΦPickuptriangleset,andthesetofTgeneratedbyηbcisnamedasΦbc,andΦbc={Tiηbc|i∈[1,Nbcp+Nbc+1-2]},Φb={Φbc|c=1,2,…,Nb}ΩPickup3Dbody,the3DbodyconstructedbyΦbisnamedasΩb,Ω={Ωb|b=1,2,…,t}.cosCosinevalue,andCosinevalueoftheanglefacingedgeσiηbcconstructedbythethirdpointδjϑ^bciscos(δjϑbc)=|A|2+|B|2+|σiηbc|22|A||B|τCircumcircleradius,andthecircumcircleradiusoftriangleconstructedbythethirdpointδjϑ^bcisτ(δjϑ^bc)=1-cos2σiηb2|σiηbc|

3 Principles of LC-TIN Algorithm

In order to triangulate the finite point sets in three-dimensional space, the Delaunay criterion should satisfy the following rules.

① At least one δj(j=1, 2, 3) in falls into or of ηbc and the three δj cannot on the same l.

and (ij) cannot overlap, that is to say, any edge in ηbc can only belong to one T.

③ The corresponding edges which connect and can only belong to adjacent T.

According to closeness, 3D bodies can be classified into closed and non-closed types, and corresponding to η there are 4 cases, which are described in Fig. 1.

Fig.1 Closeness of ηbc

The LC-TIN algorithm is the greedy algorithm based on Delaunay rules, which is also a local optimum algorithm. The algorithm input is ψ, the output is Ω, step (2) represents traversing all the ζ, steps (3) and (4) indicate to traverse all the geological disease bodies and their l, and steps (5)(6) can generate the current η and select ϑ, step (7) traverses all the points on the ϑ, steps (8)-(10) select σ and construct T through choosing the third point following some rules (such as the sum of circumcircle areas is minimum or the maximum angle), at last Φ is generated and formed Ω. The LC-TIN algorithm, which constructs t geological bodies, is described as follows.

Algorithm: LC-TIN

The ψ of the bth geological disease body is named as ψb which is the set of all the l s on the bth geological disease body, ψb={l|p=1,2,…u} and ψ={ψb|b=1,2,…t} indicates the whole set of pickup line sets.

Output: Ω

(1) if ΨNULL{

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(8) ϑbc

(3) for(c=1;Nb-1;c+){

(4) for (p=1;p<u;p++) {

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(5)

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(6) select ϑbc from ηbc

(14) }∥end for (3)

(2) for(b=1;b<t;b++){

(9) select the 3th point from which satisfies or

(10) construct by and add it to Φbc

(11) Handle other special points to form T

(12) } ∥end for (7)

(13) }∥ end for(4)

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(7) for (i=1;Nηbc;i++) {

(15) add Φbc to Φb

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(16) }∥end for (2)

(17) generate the whole Φ and form the whole Ω

(18) } ∥ end if

4 Experimental Results Analysis

4.1 Comparison of 3D modelling effects

Fig.2 Application of LC-TIN

Described in Fig. 2, the LC-TIN algorithm can construct a triangle network under the constraint of triangle non-overlap and satisfy the Delaunay criterion. Fig.2b and Fig.2c show the details of 3D body 1 and 2 in Fig. 2a respectively, and they compare with the corresponding 3D bodies (described by Fig. 2d and Fig.2e) which adopt the method of connecting triangles sequentially and layer by layer.Obviously,the LC-TIN algorithm can draw the 3D body more smoothly and naturally than the compared method.

4.2 Comparison of time complexity

Different experiments have been conducted according to different ζ amounts, different σ numbers, no matter whether data points are allocated equally on different ζ or not, in order to test the runtime of the LC-TIN algorithm. The hardware and software running environments are specified as follows: CPU is Intel(R) Core(TM) i5-4210H 2.90 GHz, memory is RAM 4.00 GB, the operating system is Windows 8 64 bit, the development platform is VS.NET 2013, programming language is C++, and the rendering method is OpenGL.

For the construction of one geological body, the amount of total data points is supposed to be N, the number of total layers is M, the amount of data points allocating to every layer is mi, i∈[1,m]. Shown in Fig. 3, this paper gave the running time of LC-TIN algorithm when data points are allocated equally on every layer, and N=1 000, 3 000, 5 000,10 000 and M=10, 20,30,40,50 respectively (m=mi=N/M). From Fig. 3 it can be noted that the data point number is reduced and running time is decreased with the increasing layer amount under the constraint of the same data point amount. Different N describes the same trend and bigger M, more running time. The running time deceases with the layer increasing on the premise that the total data point amount does not change, the reason is that the time complexity of the LC-TIN is O(Mm2) and M<m, this characteristic is useful to construct 3D body. The triangle number is shown in Tab. 2, when data points are allocated equally on every layer.

Fig.3 Runtime of LC-TIN (points average distribution)

Tab.2 Triangles number under average distribution

LayeramountTotalpointamount100030005000100001021166135102012025420210961131019320213302097614110255203044021576201103122032750224762521032620357

Shown as Tab.2, with the increasing of total points, the generated triangles rise continuously under the same layer amount, with the increasing of total layers, the generated triangles have little changes and the running time decreased under the same total point amount, described as Fig. 3, the decreased point number on each layer leads to the reduced running time, and this characteristic is useful to construct a complex 3D body.

If the data points are not allocated on each layer equally, the base number is supposed to be n, and the allocation rule satisfies mi=i×n and As shown in Fig. 4,compared with equal allocation of data points, the time change is little which shows that it has little influence on running time no matter whether data points are equal allocated or not.

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The amount of triangles is described in Tab. 3 under the inequality allocation situation. Similar to Tab. 2, with the increasing of total points amount, the generated triangles rise gradually at the same layer number, and for the same total point amount, with the increasing of total layer number, the generated triangles little changed, but if the total layer amount increases, the running time reduces at the same data points, described as Fig. 4.

Fig.4 Runtime of LC-TIN (points non-average distribution)

Tab.3 Triangles number under non-average distribution

TotallayerTotaldatapoints100030005000100001021146145102212023120211661251017920230302127616810273203194021376232103012023950219062401031720336

5 Application in Geological Disease Recognition

For geological engineering application, it has important significance that how to quickly identify the geological diseases (such as underground water or empty) and underground landfill (such as underground pipeline), which can prevent and diagnize the public disasters. The paper verified the validity of LC-TIN algorithm based on GPR data, and the GPR instrument is MTGR-4F vehicle borne geological radar which is independently developed by our research team. Its detailed parameters are shown as follows: the detection depth is 5 m, the time window is 100 ns, the sampling number is 512, and the antenna frequency is 200 MHz. The experimental data is obtained from Middle East Third Ring Road, Chaoyang District, Beijing City. The survey targets include: ① the layered structure of the underground; ② whether there are underground pipelines or not; and ③ whether there are empty cavities or not. The value ranges of experimental data is in [-23 679, 15 421], the number of measured lines is 3, the number of sampling paths is 18 632, 18 088 and 18 838 respectively, the experimental data collection work is shown in Fig.5.

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Fig.5 Acquirement of experiment data

C++ programming language and OpenGL have been adopted to develop the 3D visualization platform based on GPR data, which used LC-TIN algorithm to construct 3D model for geological diseases.

Shown in Fig. 6a, the GPR data should be organized firstly, which are the original survey line data generated by our MTGR-4F GPR instrument, then the Kriging algorithm has been adopted to interpolate the GPR data, which can generate the spatial data covered all the detection area and is described by Fig. 6b,the paper utilized the disease information on the 1-5 slices in Fig. 6c, which based on geological expert experience through human-computer interaction model, and the LC-TIN algorithm has been used to construct the 3D model of geological diseases in Fig. 6d, this method can not only compute the total or partial volume of disease body accuracy and flexibly, but also can project the 3D disease object onto any 3D plane, which can provide a reliable basis for the prevention and diagnosis of the geological disaster problems. As described in Fig.6d, the geological diseases can be displayed in the detection area totally or partially, which is generated by visualizing different parts between two slices.

Fig.6 3D visualization platform based on LC-TIN

6 Conclusion

The TIN algorithm is widely used in digital geology, petroleum exploration, geotechnical engineering and other fields, which solves a series of problems, such as the formation of terrain modelling, the drainage network extraction, the routes design, and so on. At present, the automatic extraction of geological disease information is very difficult, while it is necessary for geological experts to recognize and extract the information in the process of human-computer interaction. In this process, discrete data points are dynamically changing at different levels. To solve this problem, the paper proposed a layered-constrained TIN algorithm for 3D modelling , and the algorithm is adopted to construct a geological disease model, which can calculate geological diseases (such as underground cavity or water) information conveniently, and provide a reliable basis for accurate treatment of geological disasters. In the future, how to deeply analyze the inner information of 3D geological bodies is an important research topic, which needs to design proper volume element models. In addition GPU-based rendering technology is another research hotpot which can enhance the graphics processing efficiency.

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《Journal of Beijing Institute of Technology》2018年第1期文献

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