research article a novel model of conforming delaunay ...delaunay re nement is a technique for...

Research ArticleA Novel Model of Conforming Delaunay Triangulation forSensor Network Configuration

Yan Ma, Yan-ling Hao, and Feng-min Tian

College of Automation, Harbin Engineering University, Harbin 150001, China

Correspondence should be addressed to Yan Ma; [email protected]

Received 19 August 2014; Accepted 10 September 2014

Academic Editor: Zheng-Guang Wu

Copyright © 2015 Yan Ma et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Delaunay refinement is a technique for generating unstructured meshes of triangles for sensor network configuration engineeringpractice. A new method for solving Delaunay triangulation problem is proposed in this paper, which is called endpoint triangle’scircumcircle model (ETCM). As compared with the original fractional node refinement algorithms, the proposed algorithm canget well refinement stability with least time cost. Simulations are performed under five aspects including refinement stability, thenumber of additional nodes, time cost, mesh quality after intruding additional nodes, and the aspect ratio improved by singleadditional node. All experimental results show the advantages of the proposed algorithm as compared with the existing algorithmsand confirm the algorithm analysis sufficiently.

1. Introduction

Recently, the concept of intelligent network system is verypopular in the world. Actually, how can we deploy andoptimize the sensor? It is still a difficult issue to scientists thataffects both cost and detection capability, which are requiredconsiderations of both coverage and connectivity. A sensornode may perform the dual function of sensing the environ-ment and acting as a relay node. In a real sensor networksystem, all sensor nodes distribute as a discrete data set, whichwill form a mesh network to provide monitoring of the envi-ronment. The terms mesh network will be used throughoutthis paper to describe a sensor network configuration [1].

Delaunay triangulation (DT) is an effective method tocarve up a discrete data region, which is especially widelyused in sensor network configuration engineering field [2–5]. In most cases, there is a constricting relationship amongthe discrete data.The discrete datamay comprise some vectorlines and close polygons, which must be included in theresult of partition. In general, the Delaunay triangulationwill not contain all edges of the graph. So far there are twotypes of DT algorithm: constrained Delaunay triangulation[6, 7] and conforming Delaunay triangulation [8, 9]. Theformer method is a best approximation of the Delaunay

triangulation, given that it must contain all features in thegraph. Generally, the DT property cannot be preserved andthe quality of the mesh declines in constrained Delaunaytriangulation, which will influence the stability and con-vergence of finite element numerical calculation. In themeanwhile, the conforming DT method can be consideredas a degenerate Delaunay triangulation, whose relationshipsto the constrained graph are that each vertex of the graphis a vertex of the triangulation and each edge of the graphis a union of edges of the triangulation and also satisfiesthe DT property. But any introduced new node will lead theoriginal graph to change. Therefore, the method should beused restrainedly depending on the actual situation.

Usually, constructing conforming DT is more difficultthan constructing constrained DT, as it requires a numberof points to achieve conformity. The core technique of theconforming DT is to subdivide the constraints. This paperpresents a novel node refinement algorithm, which has bettertriangulation quality and fewer additional nodes than otheralgorithms.

The rest of this paper is organized as follows. In Section 2,the basic Delaunay triangulation problem is briefly intro-duced and the main idea of endpoint triangle’s circumcirclemodel (ECTM) is described in detail in Section 3. Then

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2015, Article ID 720249, 7 pageshttp://dx.doi.org/10.1155/2015/720249

2 Abstract and Applied Analysis

the convergence and complexity of ECTM are analyzed inSections 4 and 5. Section 6 gives the simulation experiments’results of the new ECTM with other methods. Finally, aconclusion is drawn in Section 7.

2. Problem Description

Suppose Ω(𝑃, 𝐵) is a planer straight line graph, where 𝐵 =𝐵0, 𝐵1, . . . , 𝐵

𝑀, 𝐵0is exterior feature constraint, 𝐵

𝑗> 0 are

interior feature constraints, and 𝑃 = 𝑃1, 𝑃2, . . . , 𝑃

𝑁is the

collection of all discrete points and endpoints of feature lines.If𝐵𝑖is a close constraint, 𝑟

𝑖describes the single connected

domain of𝐵𝑖;Ω(𝑃, 𝐵) should satisfy the following conditions.

All defined regions by interior constraints are in thedefined region by exterior constraints; namely, 𝑟

𝑖⊂ 𝑟0.

The mutual parts of feature constraints are the finitepoints in collection 𝑃; namely, 𝐵

𝑖∩ 𝐵𝑗

⊂ 𝑃, where 0 ≤ 𝑖 ̸=𝑚 ≤ 𝑗.

For any Ω(𝑃, 𝐵) with the above conditions, insertingsome additional points on the features, how can we get ageometrical equivalent DT graph by using the whole discretedata and additional points corresponding to former graph?Two types of refinement algorithms for conforming DT areconsidered for solving the problem. One is refining featurelines at first and then executing DT, such as in the literatureof [10–12]. Others have the different ideas exactly. Firstlythey execute DT and then refine the feature lines. References[10, 13–17] are the representative algorithms.

Inspired by the algorithms of [14, 16, 17] an improvedfeature refinement algorithm named endpoint triangle’s cir-cumcircle model (ETCM) is proposed in this paper.

3. Endpoint Triangle’s Circumcircle Model

3.1. Basic Idea of ETCM Definition. Endpoint’s triangle con-taining a feature is such a triangle which uses one of theendpoints as a vertex and intersects one edge of the trianglewith the feature simultaneously.

Suppose 𝑠[𝑃ℎ𝑃𝑒] is a feature that is not contained in

DT meshes, Δℎis the endpoint’s triangle of 𝑃

ℎ, and Δ

𝑒is

the endpoint’s triangle of 𝑃𝑒; Cir(Δ

ℎ) and Cir(Δ

𝑒) are the

circumcircles of Δℎand Δ

𝑒, respectively. The basic idea of

ETCM is as follows.Let 𝐽ℎ

= Cir(Δℎ) ∩ 𝑠[𝑃

ℎ𝑃𝑒], 𝐽𝑒

= Cir(Δ𝑒) ∩ 𝑠[𝑃

ℎ𝑃𝑒]. If

𝐽ℎ

∩ 𝐽𝑒

= ⌀, choose the longer one between 𝐽ℎand 𝐽𝑒as the

line to be inserted; the corresponding point of intersection isas additional point.Then let the shorter one be the remainderfeature line; execute the approach to the shorter one asmentioned above. It will stop execution until 𝐽

ℎ ∩𝐽𝑒 ̸= ⌀ and

take the midpoint of 𝐽ℎ ∩ 𝐽𝑒 as additional point at that time.

In particular if the feature line being treated influences thefeature line which has been inserted, the influenced featureline segment should be transacted again.

3.2. Description of ETCM. For designing the ECTM algo-rithm, we must confirm a data structure at first. Thereare mainly four structures being considered as shown inAlgorithm 1.

Triangleint index;intidx v[3];bool cross;Triangle ∗ neighbourtri[3];Triangle ∗ front;Triangle ∗ next;

FeatureSegmentint indexint idx vb;int idx ve;FeatureSegment ∗ frontFeatureSegment ∗ next;

TriIndexTriangle ∗ pTri;TriIndex ∗ front;TriIndex ∗ next;

Vertexint index;Int 𝑥;int 𝑦;TriIndex ∗ neighbourtri;

Algorithm 1: Data structure of ETCM.

The structure Triangle records the information of trian-gles, including the index number, the index of three vertexes,the tag of intersection, the pointers of three neighboringtriangles, and the pointers indicating front or next in thelinked list.The structure TriIndex is a doubly linked list of tri-angle. The structure FeatureSegment records the informationof constraint feature, including the index number, the indexof start point and endpoint, and the pointers indicating frontor next in the linked list. The structure Vertex records theinformation of vertex, including the index number, the vertexcoordinate, and the head pointer of neighboring triangles list.The data structure definitions are shown in Algorithm 1.

The overview of the ECTM algorithm is as shown inAlgorithm 2.

The function InsertNewNode(crosstri, 𝑃𝑎) is described as

shown in Algorithm 3.For the ETCM algorithm, it can ensure the distribution of

additional points is unique. Due to the fact that the additionalpoint is in the influence polygons of feature segment, we cansearch the additional points in the influence polygons insteadof in the whole domain. So it has a high-level efficiencycompared to other algorithms.

4. Convergence of the Number ofAdditional Points

In the process of dealing with the feature line, take the pointcorresponding to the longer one in the two interceptive seg-ments, which is generated by the two endpoints’ circumcirclesand feature line as the adding point. It is the point that makesthe ratio largest between the interceptive segment and thewhole feature line and constructs two triangles which sharethe interceptive segments and satisfy DT property. By dealing

Abstract and Applied Analysis 3

Input parameters: DT meshes and feature lines.Step 1. Construct a stack 𝐹𝑆 𝑟𝑒𝑚𝑎𝑖𝑛 for storing the features waiting to be treated, and a doubly

linked list 𝐹𝑆 𝑓𝑖𝑛𝑖𝑠ℎ𝑒𝑑 for storing the treated features.Step 2.While 𝐹𝑆 𝑟𝑒𝑚𝑎𝑖𝑛 is not empty, popup a feature segment 𝑠[𝑃

𝑖𝑃𝑗].

Step 3. If 𝑃𝑗belongs to the vertexes of the neighbor triangle of 𝑃

𝑖, continue.

ElseStep 4. Find the triangles intersected with 𝑠[𝑃

𝑖𝑃𝑗] by the topology relationship of meshes, and

construct the influence polygon of 𝑃𝑖𝑃𝑗named 𝑐𝑟𝑜𝑠𝑠𝑡𝑟𝑖.

Assign the value to the endpoints of feature segment: 𝑃ℎ

= 𝑃𝑖, 𝑃𝑒

= 𝑃𝑗, and Cir(Δ

ℎ)

intersects 𝑠[𝑃ℎ𝑃𝑒] at 𝑄

1, Cir(Δ

𝑒) intersects 𝑠[𝑃

ℎ𝑃𝑒] at 𝑄

2.

Step 5.While (𝑃ℎ𝑄1 +

𝑄2𝑃𝑒 <

𝑃ℎ𝑃𝑒)

Step 5-1. If (𝑃ℎ𝑄1 ≥

𝑄2𝑃𝑒), take 𝑄1 as adding point, i.e. 𝑃𝑎 = 𝑄1.

link 𝑠[𝑃ℎ𝑄1] into 𝐹𝑆 𝑓𝑖𝑛𝑖𝑠ℎ𝑒𝑑.

Take 𝑃𝑎as the head of the remainder, 𝑃

𝑒as the tail of the remainder.

ElseStep 5-2. Take 𝑄

2as the adding point, i.e. 𝑃

𝑎= 𝑄2.

link 𝑠[𝑄2𝑃𝑒] into 𝐹𝑆 𝑓𝑖𝑛𝑖𝑠ℎ𝑒𝑑.

Take 𝑃𝑎as the tail of the remainder, 𝑃

ℎas the head of the remainder.

Step 5-3. InsertNewNode(𝑐𝑟𝑜𝑠𝑠𝑡𝑟𝑖, 𝑃𝑎)

Step 5-4. Update 𝑄1, 𝑄2

Cir(Δℎ) intersects 𝑠[𝑃


1, Cir(Δ

𝑒) intersects 𝑠[𝑃


2.

Step 6. Take the midpoint of 𝑄1𝑄2as additional point 𝑃

𝑎, link 𝑠[𝑃

ℎ𝑃𝑎] and 𝑠[𝑃

𝑎𝑃𝑒] into

𝐹𝑆 𝑓𝑖𝑛𝑖𝑠ℎ𝑒𝑑, and the remainder of feature segment is empty.Step 7. InsertNewNode(𝑐𝑟𝑜𝑠𝑠𝑡𝑟𝑖, 𝑃

𝑎) and return to Step 4.

Algorithm 2

Step 1. Find the triangle Δ𝑃𝑎or quadrangle ◻𝑃

𝑎which includes 𝑃

𝑎in the triangle marked by 𝑐𝑟𝑜𝑠𝑠𝑡𝑟𝑖.

Step 2. Search the triangles whose circumcircle contains 𝑃𝑎iteratively, construct the influence

polygons of 𝑃𝑎named 𝑒𝑓𝑓𝑡𝑟𝑖, and judge whether 𝑃

𝑎is contained in a triangle by the sign of area coordinates.

Step 3. If the sharing edge of two triangles in 𝑒𝑓𝑓𝑡𝑟𝑖 is the feature segment having been treated,take this edge out of 𝐹𝑆 𝑓𝑖𝑛𝑖𝑠ℎ𝑒𝑑, and push it into 𝐹𝑆 𝑟𝑒𝑚𝑎𝑖𝑛.

Step 4. Connect 𝑃𝑎with the vertexes of the influence polygons for constructing new triangles.

Replace the triangle recorded in 𝑒𝑓𝑓𝑡𝑟𝑖, then add the remainder of new triangles to the tail of the triangle array.Step 5. Update 𝑐𝑟𝑜𝑠𝑠𝑡𝑟𝑖Step 5-1. 𝑐𝑟𝑜𝑠𝑠𝑡𝑟𝑖 = 𝑐𝑟𝑜𝑠𝑠𝑡𝑟𝑖 − (𝑐𝑟𝑜𝑠𝑠𝑡𝑟𝑖 ∩ 𝑒𝑓𝑓𝑡𝑟𝑖);Step 5-2. If (𝑠[𝑃

ℎ𝑃𝑒] ∩ 𝑠[𝐶

𝑘𝐶𝑘+1

] ̸= ⌀, where 𝐶𝑘and 𝐶

𝑘+1are two neighbor vertexes of influence polygons of 𝑃

𝑎)

Step 5-3. If (𝑃𝑎

= 𝑄1) Δℎ

= Δ𝑃𝑎𝐶𝑘𝐶𝑘+1

, add Δℎto the head of 𝑐𝑟𝑜𝑠𝑠𝑡𝑟𝑖.

ElseStep 5-4. Δ

𝑒= Δ𝑃𝑎𝐶𝑘𝐶𝑘+1

, add Δ𝑒to the tail of 𝑐𝑟𝑜𝑠𝑠𝑡𝑟𝑖.

Step 6. Update topology relationship of triangles by the method in [18].

Algorithm 3

with the remainder feature line as the above operation, untilthe intersection of two interceptive segments is not empty,there is no doubt that the quantity of additional points foreach single feature line would be least compared to any othermethod. But for the whole planer straight line graph, it isnot true in fact. Because of some triangle taking the treatedfeature segment as its edge, that may become a part of theinfluence polygon of the feature segment being processed. Sothe influenced feature segment should be treated again, whichwould produce some new additional points.

Assume that, in a planer straight line graph with 𝑁points and 𝑀 edges, 𝐴𝐵 is a feature line, Circle(𝐴) is thelargest circle comprising no other points and features except𝐴, and Circle(𝐵) has the same definition as Circle(𝐴). If

Circle(𝐴) = Circle(𝐵), stop refining 𝐴𝐵. If Circle(𝐴) ̸=Circle(𝐵), Circle(𝐴) ∩ 𝐴𝐵 = 𝐴, Circle(𝐵) ∩ 𝐴𝐵 = 𝐵, thelength of 𝐴𝐵 is 𝐷, and 𝑑 is the smallest distance between𝐴𝐵 and other points or edges, divide 𝐴𝐵 into [𝐷/2𝑑]

segments; each segment length is no longer than 2𝑑. Sothe circle which takes each segment as its diameter has thecharacter of containing no other points inside. Including𝐴𝐴and 𝐵𝐵, [𝐷/2𝑑] + 1 new points would be added on 𝐴𝐵, thatis,𝑂(1). Because the number of additional points for a featureline does not influence the embedded feature line, the numberof all additional points is 𝑂(𝑁) at best.

Actually, it has redundancy in refining 𝐴𝐵 by usingthe method mentioned above. In ECTM algorithm, it onlydeals with the influenced polygons. So the average radius of


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DT: constraints number = 9 AAR = 0.3066

(a) DT

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(c) Algorithm Tsai

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(d) Algorithm Lu

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(e) Algorithm Sapidis

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(f) Algorithm Mei (1)

Figure 1: Continued.


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(g) Algorithm Mei (2)

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(h) ETCM

Figure 1: Triangulation results of node refinement algorithms with 60 nodes and 9 feature segments.

the gained Circle(𝑃𝑖) is larger, and the number of additional

points is smaller. Thus the number of additional points byusing ETCM algorithm is convergent.

5. Time Complexity of ETCM

Suppose that𝑇𝑐𝑗is the time cost for calculating the position of

the 𝑗th additional point and 𝑇𝑖𝑗is the time cost for inserting

the 𝑗th additional point. It was pointed out in literature [19]that the average neighboring triangle number with a point isabout six in a DT mesh. So for finding an endpoint’s triangle,averaged six intersection judgments would be needed, andthen the second triangle intersected with the feature line canbe confirmed in the meantime. If a feature line crosses with𝑛 triangles, averaged 6 + (𝑛 − 2) × 2 intersection judgmentsare required to the influence range of the feature line, andthe time cost is a constant for calculating the intersectionpoint between a circumcircle and a feature line; that is, 𝑇𝑐

𝑗=

𝑂(1). So for finding the triangle or quadranglewhich containsadditional point, 𝑛 − 1 times of judgments whether a point isincluded in a triangle are needed, and the time cost of eachjudgment is also a constant. In literature [9], it was shown thatthat is a linear process, so𝑇𝑖

𝑗= 𝑂(1).The time complexity can

be calculated as follows:

𝑇𝑐

=

𝑁

∑

𝑖=1

𝑇𝑐

𝑗≤ 𝑁 × max {𝑇𝑐

𝑗} ⇒ 𝑇

𝑐= 𝑂 (𝑁)

𝑇𝑖=

𝑁

∑

𝑖=1

𝑇𝑖

𝑗≤ 𝑁 × max {𝑇𝑖

𝑗} ⇒ 𝑇

𝑖= 𝑂 (𝑁) .

(1)

So

𝑇ECTM = 𝑇𝑖+ 𝑇𝑐

= 𝑂 (𝑁) , (2)

where 𝑇𝑐 is the time cost for calculating the positions of alladditional points and 𝑇𝑖 is the time cost for inserting alladditional points.

6. Experiments Results and Analysis

6.1. Assessment Criteria. There are no criteria for evaluatingthe performance of refinement embedding algorithms in theexisting references. So we give five assessment criteria asfollows:

(1) stability of refinement;(2) number of additional nodes, which means that, by

using as small number of additional points as possi-ble, we can change the original datum set as little aspossible;

(3) time cost;(4) the quality of the mesh after intruding additional

nodes: when AAR is 0.5 approximately, it indicates aresult of fine quality;

(5) the average aspect ratio (AAR) improved by singleadditional point. The model is given by the followingequation:

AAR𝑝

=(AARrefine − AARCDT)

𝑛, (3)

where AAR is the average of all aspect ratios of meshes,AARrefine is the AAR after refinement, AARCDT is the AARafter constrained DT, and 𝑛 is the number of additionalpoints.

6.2. Performance Test. The performance tests were per-formed for all refinement algorithms previously described


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ew n

odes

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ber

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Figure 2: Number of new nodes.

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t (𝜇

s)

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Figure 3: Time cost of inserting feature segment.

including Tsai [15], Lu et al. [13], Sapidis and Perucchio [14],Wei et al. [17], and ETCM.

Take 15 planer straight line graphs as testing datum,whichcomprises a group uniformity distribution random points ofeach.The coordinates of the points are all in [0, 1000], and thenumber of feature lines is 15% of the number of the points.The testing environment is on a computer with P4 2.8G, 256× 2M RAM, Windows XP, and Visual C++ 6.0, and the timecost testing tool is CodeTest4.0.

Figure 1 shows a group of resulting mesh of a planerstraight line graph, where the thick dashed lines are feature

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Figure 4: AAR of node refinement algorithms.

lines and square points are additional points. Figures 2 to 5show the performance evaluating results. For testing stabilityof refinement, we adopt the method that exchanges the twoendpoints of each feature line and then judge whether the DTresults are conformable.

In the refinement stability test result as Figure 1, all algo-rithms have unique result except algorithm Mei. It indicatesthat, except Mei, all the other algorithms are stable. Figure 2shows that method in which Tsai requires the largest numberof additional nodes, methods Lu and Sapidis take the secondplace, and methods Mei and ETCM need nearly the sameleast number. From Figure 3, we can conclude that methodTsai’s time cost is the biggest and ETCM cost the leasttime. In particular, ECTM’s time cost curve is with the leastundulation, which shows a nearly linear relationship with thenumber of points. It indicates that the time cost of ETCMis mainly affected by points’ number and hardly affected bythe distribution of the points. The robustness of ETCM isbest. The AAR results are shown in Figure 4; all refinementalgorithms can make the AAR of the meshes resume to or bebetter than the AAR of meshes before inserting feature lines.Method Lu has the most obvious improvement, Tsai’s effect isthe worst, and the other threemethods are almost at the samelevel. Figure 5 shows the AAR improved by single additionalpoint.MethodTsai has theworst capability, Lu is a little better,and the others are pretty much the same thing.

7. Conclusions

Based on the performance analysis of the DT algorithm,a new refinement algorithm named endpoint triangle’s cir-cumcircle model (ETCM) is proposed in this paper. Theperformance analysis is given and confirmed by simulations.Simulation results show that ETCM achieves as well as


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Figure 5: Improved AAR of each new node.

the best in refinement stability, additional point number, andmesh quality. In particular, the time cost of ETCM isminimal,and it is influenced by the data distribution least. Hence, itcan be concluded that the proposed refinement techniqueECTM can be used to solve some practical problems withfaster requirement by maintaining good quality of solution.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (no. 51379049), the FundamentalResearch Funds for the Central Universities of China (no.HEUCFX41302), and the Scientific Research Foundationfor the Returned Overseas Chinese Scholars, HeilongjiangProvince (no. LC2013C21).

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