Nonlinear Analysis 71 (2009) e624–e629

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Nonlinear Analysis

journal homepage: www.elsevier.com/locate/na

A novel computational paradigm for creating a Triangular IrregularNetwork (TIN) from LiDAR dataTarig Ali ∗, Ali MehrabianCollege of Engineering and Computer Science, University of Central Florida, Orlando, FL, 32816, United States

a r t i c l e i n f o

Keywords:TriangulationTINLiDARDEMTerrain surface

a b s t r a c t

The Triangular Irregular Network (TIN) model is an alternative to the grid-basedrepresentation of the terrain surface adopted in numerous digital mapping and geographicinformation systems. In TIN, irregularly spaced sample points model the terrain in sucha way that more points represent the areas with rough terrain and there are fewer insmooth terrain. Despite its simplicity, creating a TIN model for an area involves (a) pickingsample points, (b) connecting points into triangles, and (c) modeling the surface withineach triangle. Most of the available algorithms for picking sample points use either adense Digital Elevation Model (DEM) or a set of digitized contours as input. Now, theavailability and popularity of Light Detection And Ranging (LiDAR) devices, which produceclouds of points, require the development of a new computational paradigm for pickingsample points from the new data. This article presents a novel computational frameworkfor picking sample points to create a TIN model from LiDAR data.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction and background

Digital terrain modeling is very important procedure as it provides a sound representation of topography in digitalformat, which can easily be incorporated into both digital mapping systems and GIS. Digital Terrain Models (DTM’s) aremeans of representing the surface of the Earth in a digital data structure of some type. The most common DTM’s arethe Digital Elevation Models (DEM’s), which are regularly grid surfaces, and the Triangulated Irregular Networks (TIN’s).Commonly DTM’s are produced using photogrammetry either manually or automatically; however in both cases samplepoints are not randomly selected. The question is, to what extent can we represent the terrain surface by a regularly gridedmodel such as a DEM? Also what if we’d like to update the DEM? TIN’s, unlike DEM’s, are continuous representation ofthe surface of the terrain. In a Geographic Information System (GIS), digital terrain representation can generally be usedfor analysis and visualization purposes [1]. Different representations can be used to provide a digital model of the terrainsurface including a contour map, Digital ElevationModel (DEM), and Triangular Irregular Network (TIN) model. The contourmap uses contour lines to join points of equal elevation (height) and hence depict the surface of the terrain. Contourlines are stored in an ‘‘ordered list’’ file. Although this data structure is simple, long contour lines could have thousandsof points, resulting in a big ‘‘ordered list’’ file. The digital elevation model (DEM) is a grid-based representation of thespot heights of the terrain surface. The grid-based model is the simplest and most common form of digital representationof the terrain. The cell size or the resolution of a DEM is a very critical parameter in identifying the faithfulness ofdepicting the terrain surface. High resolution DEM is always desirable, but is more crucial when a rugged terrain ismodeled.

∗ Corresponding author.E-mail addresses: taali@mail.ucf.edu (T. Ali), mehrabia@mail.ucf.edu (A. Mehrabian).

0362-546X/$ – see front matter© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2008.11.081

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Fig. 1. The study area and the LiDAR points subset used in this experiment.

The irregularly spaced points of the TINmodel can provide amore faithful representation of the terrain surfacewithmorepoints in rugged terrain areas and fewer points in relatively flat areas. The TIN model suits visualization purposes becauseof the continuous nature that the triangular facets of the model add to the digital representation. Furthermore, not muchinformation can be derived from TIN models because unlike the case for DEM’s, a comprehensive analysis framework fortriangulated models does not yet exist. In a TIN model, the sample points are simply connected by lines to form triangles,which are represented by planes, which give a continuous representation of the terrain surface. Creating a TIN, despite itssimplicity, requires decisions about how to pick the sample points from the original data set, and further how to triangulatethem. When it comes to triangulating the sample points, a few triangulation methods are available for producing a TIN.Among the existing triangulation methods that are in use, the Delaunay Triangulation (DT) is very common and popular forits rigorous structure although it produces triangles that are not hierarchical [2,3]. As a first step in creating a TIN model,sample points can be selected frommany types of data including a cloud of points, a DEM, or a contour layer. Then the pickedor selected sample points are connected to form ‘‘commonly’’ a Delaunay triangulation. This article presents a novel methodfor picking sample points to create a TIN model from a LiDAR data set.LiDAR is a topographic data acquisition technology that uses pulses of laser to map the terrain surface [4]. The LiDAR

scanner is commonly carried in an aircraft along with units of an Inertial Navigation System (INS) and Global PositioningSystem (GPS) and it works by processing the return time for each returned laser pulse to calculate the distances andfurther produce three-dimensional positional information for a cloud of points in the mapped area. The LiDAR system hasdemonstrated its ability tomap different types of terrain surfaces including bare ground surface, urban areas, rural areas, andcanopy. LiDAR systems can also be used to capture reflectance data, in addition to their ability to collect three-dimensionalpoint data. The LiDAR data set used in the study and presented in this paper is for a study area one mile square in the LakeErie coast (Fig. 1). This LiDAR data set has been collected under the Airborne LiDAR Assessment of Coastal Erosion (ALACE)project (USGS, NOAA and NASA). The density of points on the ground is highly variable, but densities are typically 30 pointsper 100 squaremeters. The variability is due to the greater number of points along the edges of a pass, andwhere two passesoverlap. The average horizontal accuracy is 75 cm due to uncertainties related to the altitude of the aircraft and the verticalaccuracy is about 50 cm [1].

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Fig. 2. Delaunay triangulation of a set of six points.

2. Triangulation of a set of points

There are several triangulation schemes available for building a TIN from a set of n points, but the most rigorous is theDelaunay Triangulation (DT). The main property of DT is that any triangle contains no point of the set inside it (Fig. 2).If no four points of the set are co-circular, the triangulation in then unique. The DT can be built in O(n log n) time and theunderlying data structure can be searched and updated in O(log n) time. Also, the DT has an easy linear rule for interpolationbased on the convex coordinates. If we have a collection of random points with some attribute associated with it, and weneed to produce the surface that best fits the data points and reflect the true behavior of the surface, what is the best surfacemodel to use in this case, a DEM or a TIN? The best way is to triangulate the points producing a TIN (not necessary a DT). Thequestion is, why do people like to produce grided surfaces more often than producing triangulated ones in such cases? Onepossible answer to this question is that there are well-developed tools for modeling and analyzing grids. These are availablethrough Map Algebra operation [5]. On the other hand, there are no such well-developed tools available for triangulatedterrain surfaces. There are several triangulations available for a given collection of random points (DT, Greedy, Optimal, andarbitrary); therefore a unique data structure doesn’t exist. Depending on the type of the proximal order (Natural Neighbor,Greedy, Optimal, and arbitrary), a specific data structure for the triangulation could be defined. An operational environmentfor spatial modeling of TIN’s similar to that available in Map Algebra for grided terrain surfaces is certainly needed.The advancement in data acquisition techniques has resulted in more available data in point format collected randomly

by systems with reasonable cost such as LiDAR. Also, representing the terrain surface by triangulation is a more efficientrepresentation than regular gridding since it closely represents the reality. Before such a modeling environment for TIN’scan be developed, there are some issues that need to be addressed including (a) identification of a unique data structure forall possible triangulations of a given point set, (b) identification of the boundary ‘‘we don’t mean the convex hull since weknow that the exterior face of the DT is the convex hull of the point set’’ for a given TIN model, and (c) identification of thetype of a given triangulation.There are three data structures for triangulations including a simple list of triangles, indexed data structure, and indexed

topological data structure. In the list of triangle structure, for each triangle, the geometrical information associated withits three vertices (position in space, surface normal, etc.) is stored. The disadvantage of this method is that each vertex isrepeated for all triangles incident in it, resulting in a linear storage cost. The indexed data structure includes two lists ofthe vertices and triangles. For each vertex, its geometrical information is stored and for each triangle, references to its threevertices are recorded. This can be completed in O(n log n) time since a vertex’s reference for a triangle requires log n bits.The indexed topological data structure is similar to the indexed data structure, but it furthermore includes references to thethree adjacent triangles to every triangle in this data structure. The cost is inO(n log n+n) time since each triangle referencerequires (log n+ 1) bits.

3. Delaunay triangulation (DT) characteristics

The Delaunay triangulation is the dual graph of the Voronoi diagram. The Voronoi diagram, also called Thiessen polygons,subdivides the space into a set of convex polygons whose boundaries are the perpendicular bisectors between adjacentdata points, therefore exemplifying the concept of adjacency. On the basis of this characteristic, the Voronoi diagrams canhelp detect clusters. The dual relationship between DT and its Voronoi diagram provides a direct solution to the all nearestneighbors problem for a set of points in such a way that each triangle vertex is connected to its nearest neighbors. One ofthe common Voronoi diagram algorithms inserts the points one at a time into the diagram. Whenever a new point comes

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in, we need to do three things: (a) figure out which of the existing Voronoi polygons contains the new site, (b) walk aroundthe boundary of the new site’s Voronoi region, insert new edges into the diagram, and (c) delete all the old edges stickinginto the new region. Although this algorithm takes a total time of O(n2), it has been proven that if you insert the points inrandom order, the expected time is only O(n log n), regardless of which set of points are given.Another algorithm is a divide and conquer algorithm that splits the points into two halves, the leftmost n/2 points, which

we will color blue, and the rightmost n/2 points, which we will color red. Recursively compute the Voronoi diagram of thetwo halves, and thenmerge the two diagrams by finding the edges that separate the blue points from the red points. The laststep can be done in linear time by the ‘‘walking ant’’ method. An ant starts down at infinity, walking upward along the pathhalfway between some blue point and some red point. The ant wants to walk all the way up to infinity, staying as far awayfrom the points as possible. Whenever the ant gets to a red Voronoi edge, it turns away from the new red point. Wheneverit hits a blue edge, it turns away from the new blue point. Also, an interesting property of the DT is its relationship to theminimum spanning tree (MST). Given a set of n points in the plane, we can think of the points as defining a Euclidean graphwhose edges are all undirected pairs of distinct points, and an edge that has weight equal to the Euclidean distance betweenthe set of n points. AMST is a set of (n−1) edges that connect the points (into a free tree) such that the total weight of edgesis minimized. A popular algorithm for computing the MST is Kruskal’s, which sorts the edges at first, then insert them oneat a time in a total time of O(n2 log n).The DT creation algorithm that we utilize in this experiment is a randomized incremental algorithm, which inserts points

in random order, one at a time. With each insertion the triangulation is updated appropriately. As with other incrementalalgorithms,we need someway of keeping track ofwhere newly inserted points are to be placed in the diagram.Whenwe addthe next point, the problem is to convert the current DT into a new one containing this point. This will be done by creatinga non-DT containing the new point, and then incrementally fixing this triangulation to restore the Delaunay properties. Thetwomajor components of this algorithm are (a) insert point: here the triangle that contains the new point will be identified,then edgeswill be inserted in the triangulation, and fix the surrounding edges, and (b) test whether the DT in-circle propertyis satisfied, and fix the edges. This algorithm running time is O(n log n).

4. New paradigm for creating TIN’s from LiDAR data

The use of LiDAR data as input for creating a TIN model provides a dense cloud of points that over-represents the areaof interest most of the time. For example, if the area is relatively flat, there is no need to have tens of triangles that haveapproximately the same values of slope and aspect to represent it. Therefore, a novel computational framework for pickingpoints from the initial LiDAR points cloud to use in the triangulation of an area is important. In this section, we presenta novel computational framework for creating TIN’s LiDAR data. Our approach is based on the graph-theoretical dualitybetween the DT and Voronoi diagrams. Listed below is our computational framework for creating the TIN from LiDAR data:

(a) Construct the Voronoi diagram using the ‘‘divide and conquer’’ algorithm, which has been introduced in the previoussection.

(b) Use the Voronoi diagram to evaluate the local density of the LiDAR points and identify clusters above a user definedlimit. Finding clusters is actually equivalent to determining a partition of the given set of points into subsets whosewithin-class members are similar and cross-class members are different according to a predefined similarity measure.For example, dense subsets produce Voronoi polygons with small areas, and polygons in a homogeneous cluster willhave similar geometric shape [6].

(c) From the clusters identified in step (b) above, select the points in the same proximitywithin the Voronoi diagram regionsthat have elevationswithin a specific ‘‘user defined’’ threshold. This threshold can be estimated using different strategies,for example, by constructing a trend surface or using a moving average window after gridding the LiDAR points cloudinto a DEM.

(d) Then, following [7], use the ‘‘Voronoi tree concept’’ to delete the points selected in step (c) and update the Voronoidiagram. The Voronoi tree records the history of the ‘‘divide and conquer’’ construction of the Voronoi diagram. The treeleaves restrict the points to lexicographical order.

(e) Then, use the randomized incremental algorithm presented in the previous section to create the final DT.

The LiDAR data set used in the study (Fig. 1) has typical densities of 30 points per 100 square meters. This variabilityis due to the greater number of points along the edges of a pass, and where two passes overlap. The average horizontalaccuracy is 75 cm and the vertical accuracy is about 50 cm. Our new paradigm for creating TIN’s has been used on the subsetof the LiDAR data set shown in Fig. 1 above. The subject LiDAR data subset has 590 total points with 589 Voronoi polygons,with an average points elevation of 571.26 ft, maximum elevation of 572.38 ft, and minimum elevation of 570.35 ft, andwhen directly triangulated, resulted in 1157 Delaunay triangles (Fig. 3(a)). Clusters have been identified on the basis of thecriteria that we ‘‘do not allowmore than 10 Voronoi polygons per 150 square feet of the area’’. As we have stated previously,the clustering threshold is a user defined and application-driven process. The elevation threshold used here was 0.15 ft andthis value was obtained based on the elevation statistics of the original LiDAR data subset. The vertical threshold is userand application dependent as well. Using our method, 137 points were selected and deleted from the original triangulationshown in Fig. 3(a), resulting in 884 triangles. The resulting triangulation is shown in Fig. 3(b), which has a total point countof 453 and resulted in 452 Voronoi polygons. Fig. 4 depicts the overlay of the two triangulations shown in Fig. 3(a) and (b)

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Fig. 3. The test LiDAR subset (a) triangulated directly into DT, and (b) triangulated into DT after selecting, deleting, and updating the triangulation.

Fig. 4. The overlay of the two triangulations shown in Fig. 3(a) and (b).

and illustrates the difference between the two TIN’s. Note that the triangles shown in blue in Fig. 4 are part of the directtriangulation of the original LiDAR point subset, but are not part of the triangulation completed using the new methodpresented in this article on this data subset.

5. Conclusions

The computational framework presented in this article for picking sample points to create a TIN model from LiDAR datais novel in many respects. It helps to create a more realistic TIN model for a given area especially if the area is relatively flat.This is important because over-triangulating a flat area neither adds value to the terrain modeling with triangulation norimproves the visualization of the topography. Also, the computational complexity of the algorithms used in thismethod is inO(n log n) time; however, in this day and age this is not a critical algorithm design issue because of the huge computationalpower that is available at reasonable cost. This computational framework is still under development and testing, but hasshown great advantage when triangulating a LiDAR points set.

References

[1] T. Ali, On the selection of appropriate interpolation method for creating coastal terrain models from LiDAR data, in: Proc. The American Congress onSurveying and Mapping (ACSM) Conference 2004, Nashville TN, USA, April, 2004, pp. 16–21.

[2] R. Fowler, J. Little, Automatic extraction of irregular network digital terrain models, Computer Graphics 13 (1979) 199–207.[3] T. Peucker, R. Fowler, J. Little, D. Mark, The triangulated irregular network, in: Proc. The American Society of Photogrammetry: Digital Terrain Models(DTM) Symposium, St. Louis, Missouri, May 9–11, 1978, pp. 516–540.

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[4] A. Wehr, U. Lohr, Airborne laser scanning - an introduction and overview, Journal of Photogrammetry and Remote Sensing 54 (1999) 68–82.[5] M. DeMers, GIS Modeling in Raster, J. Wiley, New York, 2002, 203 p.[6] F. Aurenhammer, Voronoi diagrams - a survey of a fundamental geometric data structure, ACM Computing Surveys 23 (3) (1991) 345–405.[7] I. Gowda, D. Kirkpatrick, D. Lee, A. Naamad, Dynamic Voronoi diagrams, IEEE Transaction of Infinite Theory IT 29 (1983) 724–731.