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This article was downloaded by: [University of New Hampshire]On: 08 October 2014, At: 19:47Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number:1072954 Registered office: Mortimer House, 37-41 Mortimer Street,London W1T 3JH, UK
International Journal ofGeographical InformationSciencePublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/tgis20
Representation of 3-D elevation in terraindatabases using hierarchicaltriangulated irregularnetworks: a comparativeanalysisMAHDI ABDELGUERFI , CHRIS WYNNE , EDGARCOOPER & LADNER ROYPublished online: 06 Aug 2010.
To cite this article: MAHDI ABDELGUERFI , CHRIS WYNNE , EDGAR COOPER& LADNER ROY (1998) Representation of 3-D elevation in terrain databasesusing hierarchical triangulated irregular networks: a comparative analysis,International Journal of Geographical Information Science, 12:8, 853-873, DOI:10.1080/136588198241536
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int. j. geographical information science, 1998, vol. 12, no. 8, 853 ± 873
Research Article
Representation of 3-D elevation in terrain databases using hierarchical
triangulated irregular networks: a comparative analysis
MAHDI ABDELGUERFI, CHRIS WYNNE, EDGAR COOPER,LADNER ROYComputer Science Department, University of New Orleans, USAemail: {mahdi,cwynne,eccooper,rladner}@cs.uno.edu
and KEVIN SHAWNaval Research Laboratory, Stennis Space Center, Mississippi, USAemail: [email protected]
(Received 18 June 1997; accepted 17 May 1998 )
Abstract. 3-D terrain representation plays an important role in a number ofterrain database applications. Hierarchical Triangulated Irregular Networks(TINs) provide a variable-resolution terrain representation that is based on anested triangulation of the terrain. This paper compares and analyzes existinghierarchical triangulation techniques. The comparative analysis takes into accounthow aesthetically appealing and accurate the resulting terrain representation is.Parameters, such as adjacency, slivers, and streaks, are used to provide a measureon how aesthetically appealing the terrain representation is. Slivers occur whenthe triangulation produces thin and slivery triangles. Streaks appear when thereare too many triangulations done at a given vertex. Simple mathematical expres-sions are derived for these parameters, thereby providing a fairer and a moreeasily duplicated comparison. In addition to meeting the adjacency requirement,an aesthetically pleasant hierarchical TINs generation algorithm is expected toreduce both slivers and streaks while maintaining accuracy. A comparative ana-lysis of a number of existing approaches shows that a variant of a methodoriginally proposed by Scarlatos exhibits better overall performance.
1 . Introduction
Terrain Databases (TDB’s) generally include terrain elevation data, ground fea-tures, 3-D objects, and texture. Three methods are commonly used to representelevation: contours, grids, and Triangulated Irregular Networks (TINs). In contours,lines of constant elevations are speci® ed at constant intervals. One advantage of thismethod is that contour maps for most of the world are readily available. DigitalElevation Model (DEM) grids are matrices of equally spaced elevation points. DEMgrids, such as National Imagery and Mapping Agency’s (NIMA’s), Digital ElevationData (DTED) (DMA 1993), are available from several sources. The TIN modelapproximates a topographic surface using a network of planar, non-overlapping,and irregularly shaped triangle faces (DeFloriani 1987). The irregular shape of thetriangles allows TINs to easily adapt to the roughness of the terrain, thus providinga surface representation using a limited amount of data. For instance, in Abdelguer®et al. (1996a), a DTED rectangular grid composed of 90 939 nodes is reduced to
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only 1737 nodes using TINs. The ability of the TIN model to adjust its resolutionbased on the complexity of the terrain being modeled makes it more e� cient in awide range of applications, including real-time display and automated terrain analysis(Polis et al. 1992 and 1995) . Another important advantage of the TIN model is thatit can incorporate surface-speci® c constraints such as prespeci® ed linear and areaground features (DeFloriani 1987 and 1989).
Ground features can be further categorized into two classes: (a) natural groundfeatures such as valleys, ridges, rivers and lakes; (b) man-made features such as atransportation network. 3-D objects represent cultural features such as buildings. Ingeneral, texture data represents naturally occurring features (Hardis andSureshchandran 1996) such as soil, grass, and trees.
Many applications of TDBs require the use of terrain data at varying levels ofdetails. An obvious, but not e� cient, solution is to store the terrain and associatedfeatures at the highest possible resolution. Coarse views can then be generated fromthe stored representation as needed. Take the case of the process of generating asatellite view of an area that has been stored at the highest level of resolution. Togenerate the satellite view, the system will have to perform a number of di� cult andcomputationally expensive tasks such as the thining of features, and the combiningand/or thining of TINs. As a result, this solution is generally not appropriate forreal-time simulation.
Hierarchical TINs provide an e� cient way of representing terrain at variouslevels of resolution. A node in the hierarchy represents a TIN surface approximationthat is a re® nement of some parent node (of course the latter property does notapply to the root node). In general, a gridded version of the terrain (such as DTEDlevel II) is processed ² into TIN data which represent the terrain at various levels ofdetail ranging from very coarse to very ® ne. The generated TINs are then organizedin a hierarchical fashion. The rendering software can then select an appropriatesubset of the TIN data for display according to a particular eye-point position. Ananalytical model that determines the appropriate subset of the TIN data for real-time simulation has been reported in Devarajan et al. (1993).
This paper compares and analyses existing hierarchical triangulation techniques.The comparative analysis takes into account how aesthetically appealing and accur-ate the resulting terrain representation is. Parameters, such as adjacency, slivers, andstreaks, are used to provide a measure on how aesthetically appealing the resultingterrain representation is. Slivers occur when the triangulation produces thin andslivery triangles. Streaks appear when there are too many triangulations done at agiven vertex. Simple mathematical expressions are derived for these parameters,thereby providing a fairer and a more easily duplicated comparison. In addition tomeeting the adjacency requirement, an aesthetically pleasant hierarchical TINs gen-eration algorithm is expected to reduce both sliveriness and streakiness while main-taining accuracy. A comparative analysis of a number of existing approaches showsthat a variant of a method originally proposed by Scarlatos exhibits better overallperformance.
The remainder of the paper is organized as follows: in §2, criteria for an aesthetic-ally pleasant TIN representation are introduced. Existing hierarchical algorithms arereviewed in §3. Using the introduced criteria, these algorithms are then analyzed andcompared in §4. Concluding remarks are given in §5.
² TIN can also be generated from other sources such as elevation contours and raster data.
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A comparative analysis of terrain database applications 855
2 . Criteria for an aesthetically pleasing TINs triangulation
A number of hierarchical TINs generation algorithms has been suggested (see§3 ), each with its own characteristics, be® tting the particular needs of the user. Foran aesthetically pleasing hierarchical TIN triangulation algorithm, we identify fourmajor considerations:
E adjacency propertyE nesting propertyE streakiness propertyE sliveriness property
The ® rst criterion is the adjacency property, which requires that each edge, orany part of it, in a triangle can be shared by no more than two triangles. TriangulationB in ® gure 1, for example, does not adhere to this requirement. Triangle T2 has edgeV1 ± V2 that has parts in T1 and T3 also. Additionally, if an edge is shared by twotriangles, it must be shared in its entirety and not shared in parts, such as inTriangulation D in ® gure 1. This requirement guarantees a continuous surface, asopposed to non-adjacent triangulations which might produce gaps between triangles(Triangulation C in ® gure 1 ). Moreover, with adjacency, relationships from oneprimitive (edge, vertex, and triangle) to a neighbour (horizontal navigation) are moredirect, thereby decreasing the amount of data needed to store these relationshipsand facilitating the navigation from triangle to triangle within a hierarchical level.
The second desirable property is the decomposition of triangles into smallertriangles in moving from one level to the next level of detail. The importance of thisproperty is that it provides for the easy navigation from one level of hierarchy tothe next (vertical navigation). For example, given any triangle, it should be e� ortlessto retrieve the data to zoom in or out. This property, henceforth, is labelled thenesting property between a parent and its children ( ® gure 2 ) .
Of course this feature also provides for an easy method of providing a relationshipbetween triangles of di� erent levels. This is of utmost importance when attributes
Figure 1. Adjacency property.
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Figure 2. Nested parent5children relationship.
are attached to triangle faces. Additionally, this property makes the multiscaletriangulation more manageable. It is noteworthy that when this property does nothold there is, in fact, no hierarchy as the representation becomes a Directed AcyclicGraph (DAG).
It is noted that the adjacency and nested parent5child relationship requirementsare especially important in the context of real-time terrain simulation (Devarajan etal. 1993), as they allow for a simpler data storage and ease of navigation (bothhorizontally and vertically) through the di� erent precision levels.
The third desirable trait shall be referred to as the streakiness e� ect, which occurswhen there are too many triangulations done at a given vertex, resulting in a terrainwith many line streaks from that vertex. In ® gure 3, the vertex that is repeatedlydivided ended up with 9 edges stemming from it after 3 triangulations. Consider thecase where the same vertex is part of 4 or 5 triangles in the original hierarchicallevel. In such a case, it could have ended up being associated with 40 edges after thethird triangulation. This number translates to an unappealing display when theterrain is being viewed.
The ® nal property to be considered is the sliveriness property, which occurs whentriangulations produce thin and slivery triangles, as opposed to more equilateralones ( ® gure 4 ). In addition to its negative e� ect on the aesthetics of the area beingviewed, sliveriness causes numerical interpolation problems (DeFloriani 1987, 1989).
It should be noted that sliveriness and streakiness are sometimes related. Forinstance, high streakiness usually (but not always) leads to sliveriness. Conversely,
Figure 3. Example of streakiness.
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A comparative analysis of terrain database applications 857
sliveriness does not necessarily imply streakiness. It is for these reasons that bothmeasures are utilized in this paper.
The four properties mentioned above form the main basis for ® nding an aesthetic-ally pleasing triangulation algorithm, though of course there are other criteria to beconsidered as well. They include:
E Reduction of unnecessary triangulations (possibly leading to more storage cost,and therefore, retrieval time).
E Amount of new elevation information gained from one hierarchy level to thenext (the higher the better).
3 . Methods of hierarchical triangulation
T ernary hierarchical triangulation is the simplest method, and thereby one of themost common forms of dividing triangles (DeFloriani 1989). In a ternary triangula-tion, a triangle can be split into three parts by ® nding an internal point P. Thetriangle is then divided by connecting each of its vertices to P. Point P now fallsunder the category of signi® cant break point , meaning a point that is used to breaka triangle down to smaller triangles. Signi® cant implies that it is inserted because itdoes meet or exceed a certain elevation change. Conversely, an insigni® cant breakpoint would be de® ned as one that is used only to break a triangle.
The advantage of the ternary method is that it meets the adjacency requirement.At this point, it is imperative to note that adjacency is meant for triangles at thesame hierarchical level, i.e., a Ternary triangulation produces a surface that is continu-ous at each level. One major ¯ aw of ternary division is that the algorithm inherentlyintroduces arti® cially elongated triangles. The other major ¯ aw of this approach isthat edges in the original triangles are never brokenÐ leading to inaccuracies sincethese edges might need to be divided at a lower level.
In actual viewing applications, the elongated triangles produced many streaks( lines) extending from various vertices after a few triangulations. In ® gure 5, pointA is already part of 5 edges. One more triangulation could give it a maximum of 9
edges. Data storage with a Ternary triangulation is also simple. Every triangle eitherhas three children or none. Since the adjacency property exists for this method,navigation with access primitives is reasonably e� ortless.
A Quarternary triangulation partitions a triangle into four children. This divisioncan easily be achieved by connecting the three insigni® cant break points which aremid-points of the three edges of the original triangle. Remember, they are insigni® cantsince they are used to decompose the triangle even if the elevations of the breakpoints and the original triangle vertices are identical. The problem with theQuarternary triangulation is that continuity might be lost if adjacent triangles arenot uniformly split, i.e., if one triangle is split, but its neighbour is not, then the
Figure 4. Sliveriness property.
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Figure 5. Ternary triangulation.
elevations where the split occurred would not be consistent with the elevation of theunsplit neighbour (e.g., the adjacency requirement is not met). This side-e� ect canbe resolved by dividing all triangles of a level, instead of only the ones that requiredivision. Of course, this defeats the purpose of hierarchical TINs in that it wouldintroduce many unncessary triangles ( like the uniform rectangular grid, and therebynot be irregular).
As seen in ® gure 6 of a Quarternary triangulation, point A would have di� erentelevations at the third level of the hierarchy. Additionally, Quarternary triangulationhas the drawback of requiring additional information to support the relationship ofadjacent triangles. Triangles that contain point A in ® gure 6 would require moreinformation to know all of its neighbours at the same level of resolution. The datarequired to store the information to traverse from one level to the next is the sameas the Ternary triangulation, except that a parent has four children or none.
On the plus side, the advantage of the Quarternary method are its simplicity inimplementation and the fact that it leads to an aesthetically pleasant terrain (apart
Figure 6. Quarternary triangulation (DeFloriani 1987 ).
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A comparative analysis of terrain database applications 859
from vertical gap). It should be noted that Multigen ³ ’s, Continuous Adaptive Terrain(CAT) (Willis 1996) modeling technique for hierarchical terrain representation isbased on a variant of the Quarternary method. CAT allows for morphing, i.e. asmooth and gradual transition, between di� erent levels of details. Other advantagesof the CAT can be found in Willis (1996).
Two interesting algorithms that have been devised are the Delaunay Pyramid andits derivative, the Constrained Delaunay Pyramid. Both algorithms are based on theDelaunay triangulation, considered the method of choice to triangulate arbitrarilydistributed points (DeFloriani 1989). However, the Delaunay triangulation cannotbe directly applied to create a hierarchical structure in that the insertion of a newpoint will cause a modi® cation that may involve the entire TIN network(DeFloriani 1989).
The disadvantage of the Delaunay Pyramid is its complexity. Roughly speaking,the Delaunay Pyramid takes an existing group of triangles and inserts a new pointof interest. The inserted point along with the points in the original group of trianglesare used to separate a new set of triangles that replaces the original group. Thoughthe algorithm itself will not be discussed in more detail here, it is important to knowwhat it produces and to understand its bene® ts as well as its weaknesses. An exampleof the result of the Delaunay Pyramid is shown in ® gure 7.
In the original terrain, [a, b, c] form a group of triangles. A new point is theninserted. The vertices of [a, b, c] are then combined with the new point to formtriangles [g, h, i, j, k]. The same method would be applied to triangles [d, e, f ] toproduce triangles [ l, m, n, o, p]. One drawback with this method is that the informa-tion gained from one level to the next is that for each pyramid, say of 3 triangles,only one extra signi® cant’ break point is introduced. The amount of elevation data
Figure 7. An example of Delaunay pyramid (DeFloriani 1989 ).
³ Multigen Inc. is a leading commercial provider of software CAD tools for virtual worldreconstruction.
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gained is very low, whereas ternary gains one signi® cant’ point for each triangle andQuarternary gains three insigni® cant’ points.
As is easily seen in ® gure 7, this method does not satisfy the nesting propertyand as a result, triangles at di� erent levels have no direct relationships with eachother. Therefore, one cannot zoom-in on one triangle but must zoom-in on the areamaking up the pyramid containing the triangle. As for storage, this algorithm requirescomplex data structures to navigate through the pyramid and to encode the morecomplex relationship that exists between parents and children. Take triangle a’ inthe original scheme, it would be the parent of both triangles g’ and h’ afterapplying the Delaunay Pyramid. Though this in itself is not catastrophic, it does createan extra inconvenience. If one were to take a triangle and zoom in on it, such astriangle `b’, the result would be all triangles g’ to k’. The changes in the elevationcontours would not be smooth but disjointed.
Of course through the years many esoteric algorithms have been devised, againeach with its own appeal. Scarlatos ’ algorithm, for example, uses ® ve di� erentstrategies for dividing a triangle according to various rules. The strategies rangefrom centre split, to centre split with a one-edge split, two-edge split, and three-edgesplit (Scarlatos and Pavlidis 1992) , with the result shown in ® gure 8. Though thismethod merits some consideration, it does not provide continuity from triangle totriangle due to the fact that an edge might not break on all triangles containing theedge. The division in this algorithm is based on each triangle separately from itsneighbours, each one adapting to the terrain features within itself. Furthermore, apotential drawback of this method is that it, as its title implies, is a hierarchicaltriangulation using terrain features’ (Scarlatos and Pavlidis 1992) , as shown in® gure 9. In general, vector-based systems, such as ESRI’s ARC/INFO (Zeiler 1994)or DMA’s Vector Product Format (VPF) (DoD 1993) and its derivative ExtendedVPF (EVPF) (Abdelguer® et al. 1996b), are designed to provide disjoint layers(coverages) thereby allowing the user to only process needed information. A hierarch-ical triangulation using terrain features would create a large and redundant data setas features (such as roads) would be duplicated in the elevation and transportationcoverages.
Triangulation is not the only approach to representing hierarchical terrain data.There are various grid-based approaches that generally perform well. The advantageof grid-based methods is the simplicity due to the regularity. The disadvantage,however, is the inaccuracy of irregular terrains due to the same regularity. Important
Figure 8. Bisections for Scarlatos’ algorithm (Scarlatos 1992 ).
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A comparative analysis of terrain database applications 861
Figure 9. An example of Scarlatos’ algorithm (Scarlatos 1992 ).
points might be missed if they are not on the regular grid (or require many divisionsto reach). These grid-based methods are easy to visualize: divide any terrain intoevenly spaced square grids. Subsequently, a hierarchy is attained by various ways ofsubdividing squares as shown in ® gure 10.
Of course there are not too many methods simpler than hierarchical quadrilat-erals. However, recognizing the superiority of TINs, many have tried to integrateTINs into the hierarchical quadrilateral. One method was introduced by Herzen andBarr which more or less takes hierarchical quadrilaterals and, with some restrictions,divides the quadrilaterals into triangles (Herzen and Barr 1987), with the followingresults shown in ® gure 11. The good news is that adjacency is retained. However,there is much too much excess of unnecessary triangles. Therefore, it is more e� cientto triangulate in the ® rst place instead of triangulating from the grids. Other methodsof TIN integration are variants of dividing squares into triangles, many of whichyield the same ine� ciency in data storage.
A hierarchical triangulation algorithm has been recently described in Wynne(1996). To some extent, it can be considered as a derivative of Scarlatos’ method.The algorithm can be described using the rules given below:
E Rule 1 . If there is one edge break, then the break point will connect to thetriangle vertex that does not contain the edge with the break ( ® gure 12) .
Figure 10. Grid decomposition.
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Figure 11. Grid-based triangulation (Herzen et al. 1987 ).
Figure 12. One edge with break point.
E Rule 2 . If there are two edge breaks, then the two break points will connectto form one triangle. The remaining quadrilateral will be divided by connectingone of the break points with a non-adjacent vertex ( ® gure 13).
Note that it does not matter whether division of the quadrilateral [1AB3] isdone by connecting B’ to 1’ or from A’ to 3’. Adjacency is still maintained. In fact,both ways are recommended and should be randomly chosen to easily guaranteediversi® cation of triangle alignment, i.e., not all triangles with two break points willbe broken the same way.
E Rule 3 . If all edges have break points, then a Quarternary division is performed,as shown in ® gure 14.
It should be emphasized that unlike the Quarternary triangulation, here, thethree break points are all signi® cant.
Figure 13. Two edges with break points.
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A comparative analysis of terrain database applications 863
Figure 14. Two edges with break pionts.
E Rule 4 . Rule 4 is only applied if no edge contains a break point. Acknowledgingthe possibility that internal (within a triangle) elevation changes might neverbe found since the inside of a triangle has not been checked, it is then necessaryto check the inside for a signi® cant break point. If an internal break point isfound, then a Ternary triangulation is performed. Since Ternary contributes tothe streakiness and sliveriness factors, it must be restressed that rule 4 is onlyapplied if there is no edge break point.
One important aspect of all hierarchical triangulations is the concept of breakpoints. Obviously the more break points (especially signi® cant ones) gained fromone level to the next, the better the triangulation due to the amount of elevationinformation gained. And if the break points are signi® cant, then the accuracy of theterrain representation is usually higher. An experiment involving this method isdescribed as an appendix.
It is worth noting that a potential problem with some of the above algorithmsis that an internal break point may be too close to an edge or vertex. Should thisoccur, there would be triangular slivers that, though maintaining continuity, createa less aesthetically appealing terrain. This predicament can be relieved ( ® gure 15) byconsidering only internal points with distance to both vertices of the edge longerthan a prede® ned xy-tolerance. A more accurate method is to have a variabletolerance based on the ratio of the triangle size.
4 . Performance analysis
In comparing the aforementioned techniques, it would appear that a multi-resolution terrain would look best under the Quarternary method. Hence, a goodtriangulation algorithm should come as close to the Quarternary method as possiblein terms of the appearance of the terrain. However, a good triangulation should notcontain the discontinuity that may be induced by the Quarternary method norshould it lead to excessive unnecessary triangles of the Quarternary method.
Before analysing the proposed algorithm, we summarize the particular featuresof each method. It is recalled that the term signi® cant point’ here refers to the points
Figure 15. Points to be checked and xy-tolerance.
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inserted into a triangle (or triangle group) to form new children triangles. Thesesigni® cant points are those which are inserted because of the change in elevationand not just for the purpose of dividing the triangle. The comparison is summarizedin table 1 and is explained thereafter.
E TernaryÐ Low accuracy since the new vertex in the trisection is the only added
signi® cant point.Ð Excessive sliveriness.Ð Line streaks can reach an unacceptable level.
E QuarternaryÐ Low accuracy since the new vertices in the division are the mid-points
of the edges, which might not be signi® cant.Ð Non-adjacency occurs without total division of all triangles in a level,
which would produce too much unnecessary divisions.Ð Good equilateral triangles ( low sliveriness).Ð No line streaks.
E DelaunayÐ Low accuracy since multiple triangles may only have one signi® cant
point of new elevation inserted. Hence the amount of elevation datagained from level to level is not high.
Ð Low sliveriness.Ð Streaky line still may exist.Ð Parents and children do not have a direct relationship.
E ScarlatosÐ Has medium accuracy since one triangle may have multiple new vertices
to be considered for the next level of hierarchy.Ð Uses other features as constraints.Ð Streakiness is medium to high since triangles are mostly divided from
the vertices.E Grid-based
Ð Accuracy can vary depending on the size of the grid; the smaller themore accurate.
Table 1. Comparison of methods.
NestedParent:
Algorithm Accuracy UT Adjacency Sliveriness child Streaks
Ternary Low (1SP) None Yes High Yes HighQuarternary Very low (3IP) High No No Yes NoDelaunay Low (=1SP) None Yes No No PossibleScarlatos Medium (1 ± 3SP) None Yes Medium Yes MediumGrid-based Variable High Yes No Yes NoVariant of Medium (1 ± 3SP) None Yes Low Yes Low
Scarlatosmethod
UT=Unnecessary Triangulations.IP=Insigni® cant breakpoint.SP=Signi® cant breakpoint.
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A comparative analysis of terrain database applications 865
Ð The disadvantage is the amount of data that must be stored since thereare too many unnecessary triangles, especially if small grids are used toensure accuracy.
The advantage of the variant of Scarlatos’ algorithm is that it is a good comprom-ise among the other methods. Accuracy is acceptable in that three signi® cant pointscan be found for each triangle. In fact, going from one level to the next, as many asthree signi® cant points can be gained from each triangle. Furthermore, sliverinessand streakiness are greatly reduced. Also, there is no unnecessary triangulations.
Because the argument for the variant of Scarlatos’ algorithm is based on thereduction of streaks and slivers, it is necessary to examine and compare theseproperties in detail. For this comparison, the grid-based and Delaunay Pyramidmethods have been excluded (grid-based due to its unnecessary triangulations, andDelaunay Pyramid due to its lack of a nesting relationship between a parent and itschildren).
For the streakiness comparison, a new measurement index is proposed. This newindex is based on the number of new triangulations stemming from a vertex.Generally, if a triangulation bisects a triangle from a vertex, then streakiness occurs.Since all algorithms triangulate the same way from level to level, successive bisectionsfrom a vertex will propagate many lines from one vertex. For all algorithms, themeasure of streakiness is measured by the number of vertex bisections (see ® gure 16 )relative to the number of possible bisections. The streakiness measurement index isexpressed as:
Streakiness=total vertex bisections
total bisections(1 )
The total bisections in the above expression refers to the total number of bisectionswithin an algorithm. Without loss of generality, we will assume that the bisectionsare equally probable. The streakiness measurement index ranges from 0 to 1. A smallvalue indicates less streakiness (with 0 being the ideal case). For Scarlatos’ variant( ® gure 17), the total number of bisections is 11, 5 of which are vertex based. Thenumbers give a streaky measurement of 0.45. Scarlatos’ method gives a result of 0.53
since 8 out of the 15 bisections are vertex based.As for the Ternary and Quarternary methods, they are both straightforward. The
Ternary method has a streakiness measurement of 1.0, since all of its bisections arevertex based. Conversely, the Quarternary method gives a measurement of 0.0 sinceits bisections are edge-based.
As for comparing the sliveriness property of the algorithms, it will be assumed
Figure 16. Types of bisection.
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Figure 17. Bisections for the variant of Scarlatos’ algorithm.
that all algorithms start from one equilateral triangle and that all bisections areequally probable. With this in mind, the proposed method is one used by Scarlatosand Pavlidis (1992) who expressed sliveriness as a proportion of the perimeter tothe area:
Sliveriness=Perimeter2
Area(2 )
The sliveriness measurement gives an amount of 20.78 for an equilateral triangle(Scarlatos and Pavlidis 1992). Additionally, a small ® gure indicates a more equilateraltriangle, whereas a large ® gure indicates a more slivery triangle (see ® gure 18 ). UnlikeScarlatos, however, the triangulation is from a given equilateral triangle, which willshow a fairer and more easily duplicated comparison.
For example, for the Scarlatos’ variant, the result for each triangulation averagesto be 26.26. Table 2 details the results of all algorithms. Note that in most cases, thegrowth of sliveriness is steep. In Ternary method, once a sliver is found, it can neverbe eliminated. So the sliveriness gets worse as additional triangulations are performed.
As table 2 indicates, Quarternary is the best, which was expected. Notice how
Figure 18. Sliveriness example.
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Table 2. Streakiness and sliveriness comparisons.
Average Maximum MinimumAlgorithm Streakiness sliveriness sliveriness sliveriness
Ternary 1.0 12.37 12.37 12.37Quarternary 0.0 1 1 1Scarlatos 0.53 7.02 12.37 1Scarlatos’ Variant 0.45 6.48 12.37 1
Streakiness=0.0 is best, 1.0 is worst.Sliveriness=lower is better, 1 is best.Results have been normalized so that Quarternary method produces a result of 1 (result=
result Õ 19.78 ).
well the Scarlatos’ variant emerged. It is the second best, but holds signi® cantadvantages over the Quarternary in other areas (see table 1 ) .
5 . Conclusion and future work
This paper de® nes parameters to measure how aesthetically appealing a TIN-based hierarchical terrain representation is. Thereafter, mathematical expressions tomeasure these parameters are derived. Based on these introduced parameters, acomparative analysis of existing hierarchical algorithms is performed. A variant ofScarlatos’ algorithm is shown to exhibit better overall performance.
Currently there is no standard format for the distribution and exchange ofhierarchical TIN data. Additionally, the hierarchical representation of terrain raisesthe issue of hierarchical representation of ground features (such as roads, rivers,ridges and valleys) and 3-D objects. The hierarchical representation of terrain andspatial objects places considerable demand on the system’s capabilities. As a result,it is not surprising that existing vector-based data exchange formats such as NIMA’sVPF have no built-in mechanism to store and access hierarchical terrain data andmultiscale features. These are much needed extensions to VPF. Such extensions areunder way as our current work includes the enhancement of VPF to handle thee� cient storage of hierarchical TINs as well as the storage and access of features atvarying levels of resolution. Some preliminary results can be found in Wynne (1996).
Acknowledgments
The authors are very grateful to Dr Paola Magillo (Department of Computerand Information Sciences (DISI) of the University of Genova, Italy) and Dr LoriScarlatos (Department of Computer Science and Visual Media, Hampshire College,Amherst, Massachusetts) for their careful reading of a previous version of this paper.Their valuable comments and suggestions have considerably improved the paper.
This work was sponsored by the National Imagery and Mapping Agency’s(NIMAÐ formerly known as the Defense Mapping Agency) Terrain ModelingProgram O� ce (TMPO) and the Defense Modeling and Simulation O� ce, underProgram Element 0603832D, with Jerry Lenczowski as program manager. The viewsand conclusions contained in this paper are those of the authors and should not beconsidered as representing those of the NIMA.
References
Abdelguerfi, M ., Cooper, E., Wynne, C., and Shaw, K ., 1996a, Towards virtual wordreconstruction using the modeling and simulation extended vector product (MSEVP).
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In IAST ED International Conference on Modeling, Simulation and Optimization, paper242-091 (also available as an NRL Formal Report 7441-96-9654 ) (Goldcoast, Australia,N RL).
Abdelguerfi, M ., Cooper, E., Wynne, C., and Shaw, K ., 1996b, A terrain database repres-entation using an extended vector product format (EVPF). In Fifth InternationalConference on Information and Knowledge Management of Data, Maryland, pp. 27 ± 33.
DeFloriani, L., 1987, Surface representations based on triangular grids. T he V isual Computer,3, 27 ± 50.
DeFloriani, L., 1989, A pyramidal data structure for triangle-based surface description. IEEEComputer Graphics and Applications, 19, 67 ± 78.
DMA , 1993, Digitizing the Future (Fairfax, VA: Defense Mapping Agency), pp. 2031 ± 2137.DoD , 1993, Vector Product FormatÐ Military Standard (??????: Department of Defense, Mil-
STD), 2407.Devaraian, V., 1993, Terrain modeling for real-time simulation. In Proceedings of the American
Congress on Surveying and Mapping /American Society for Photogrammetry and RemoteSensing (ACSM/ASPRS) Annual Convention & Exposition, New Orleans, L ouisiana(Los Angeles: ACSM (ASPRS), pp. 129 ± 138.
Hardis, K . C., and Sureshchandran, S., 1996, Terrain database correlation testing withindatabase generation systems for DIS. 13th DIS Workshop, Florida, pp. 367 ± 380.
Herzen, B. V., and Barr, A. H ., 1987, Accurate triangulation of deformed intersectingsurfaces. Computer Graphics, 21, 103 ± 110.
Polis, M . F., and McKeown, J., 1992, Iterative TIN generation from Digital elevation models.IEEE Computer Society Conference on Computer V ision and Pattern Recognition,787 ± 790.
Polis, M . F., G ifford, J. G ., and McKeown, J., 1995, Automating the construction of large-scale virtual worlds. IEEE Computer, 28, 58 ± 64.
Scarlatos, L., and Pavlidis, T., 1992, Hierarchical triangulation using catographic coherence.Computer V ision, Graphics, and Image Processing, 54, 147 ± 161.
W illis, L., 1996, CAT terrain support coming soon to a MultiGen near you!. T ake Flight:T he MultiGen Newsletter, 3, 8.
Wynne, C., 1996, The integration of hierarchical TINs into the vector product format, MSThesis, Computer Science Department, University of New Orleans.
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Appendix: Implementation
This appendix describes an implementation of the variant of Scarlatos’s algorithm.In order to demonstrate the capabilities of the algorithm a region over Killeen, TX,has been chosen as a prototype. The selected area (see ® gure A1 ) spans a 5 Ö 5 section(Õ 97 ß 30 ¾ to Õ 97 ß 35 ¾ longitude and 31 ß 5 ¾ to 31 ß 10 ¾ latitude) of Kelleen, TX and hassu� cient variances in elevation as well as multiple bodies of water to demonstratethe capabilities of our algorithm. In addition, several DMA products exist for theregion providing an excellent source of data.
Data in the form of level II DTED (Digital Topographic Elevation Data) andDTOP (Digital Topographic Data) data exists for the prototype region of Killeen,TX, and have been used in the construction of the Hierarchical TIN. More speci® c-ally, 90 300 nodes extracted from level 2 DTED and 639 nodes from ® ve shorelineedges extracted from the hydrography coverage in DTOP have been used in theTIN generation.
In this set of examples ( ® gures A2 to A5 ), Killeen, Texas is examined. In thestarting point, the entire area is divided into a 9 Ö 9 grid of squares. Then eachsquare is divided into two triangles of the same size, giving an initial triangularnetwork. From this starting point, the network is re® ned for higher details threetimes. In ® gures A3, A4 and A5, (a) represents an overhead from the most generalizedview in ® gure A3 level 1 to the most accurate in A5. The wireframe shown in (b),
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Figure A1.
and the shaded wireframe of (c) exhibit the variation in elevation from one level tothe next.
Another way to use the algorithm is not as the primary triangulation, but as asecondary. For instance, a Delaunay triangulation can be used to generate theoriginal set of TINs from which hierarchical TIN can be derived. Hence, the toplevel of TINs will consist of the most signi® cant points in the grid, and subsequentlevels can be derived from there.
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Figure A2.
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Figure A3.
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Figure A4.
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Figure A5.
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