Survey Review, 31, 245 (July 1992)

ACCURACY AND PRECISION OF VOLUMES COMPUTEDFROM TRIANGULATED IRREGULAR NETWORKS

M. R. Shortis and P. G. Joyce

Department of Surveying and Land Information, University of Melbourne,Australia

ABSTRACT

A common task within the discipline of surveying and mapping is the measurement of coordinatesto define surfaces. Both natural and artificial surfaces can be modelled using Triangulated IrregularNetworks (TINs) in order to compute volumes, extract sections or interpolate contours. The accuracyand precision of the derived information is dependent upon many factors. First, the overallmathematical accuracy of using TINs to compute volumes is discussed. Second, the mathematicalrelationship between the precision of the surface coordinates and the precision of the derived volumesis investigated using empirical testing. Several different types of data are used to analyse the influenceof precision.

INTRODUCTION

A Triangulated Irregular Network (TIN) is a series of connected, non-overlapping triangular facets which model a natural or artificial surface [11].TINsare in widespread use by· surveyors, cartographers and the mapping industry tomodel the topography of the surface of the Earth [10]. The triangulated networkapproach is just one of a number of alternatives, such as grids, profiles and B-spline patches [1].However, the triangle is the two dimensional simplex, that is, itis the simplest area shape in two dimensions. The use of such simplices bestowscertain inherent advantages on the model, such as honouring the original datapoints, optimisation of the accuracy of the surface model and simplification of theinterpolation and calculation of sections, contours and volumes.

Perhaps the most important advantage of TINs is the ability to model surfacesfrom scattered data points. Modern surveying total station and analyticalphotogrammetric instruments and techniques allow almost unlimited versatility indata gathering [13]. Surface measurement can be judiciously varied in density orplacement of points to realise the most accurate surface with the greatest efficiency.A pseudo-random distribution of data points may further enhance efficiency andaccuracy because it accords with the collection of planimetric detail and lines ofchange in surface grade.

This paper is concerned only with two and a half dimensional (2·5D) TINs. A2·5D TIN is constrained so that any point can only have a single value in the thirddimension. Where the third dimension is height, a model of a topographic surfacemust therefore preclude vertical cliffs, overhangs or caves. The advantage of the2·5D TIN is that the triangulated network is formed in plan only, and the thirddimension is used only for interpolation. The third dimension is of course notlimited to height, and can be any measure of interest. Whilst not a justification forthe constraint to 2·5D, it is relevant that overhangs are meaningless to contourmaps of, for example, population density.

Indeed, 2·5D TINs are used by many disciplines to model surface or intensity

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TRIANGULATED IRREGULAR NETWORKS

z

y

IIIIII I I

~

xFig. 1. Triangular prism used for volume calculation.

maps based on scattered data points. Groups such as geographers andmathematicians tend to use their own terminology, but by far the most commonvariant across all disciplines, regardless of the nature of the third dimension, is theDelaunay triangulation. Characterised by the formation algorithm, Delaunay isknown by a variety of other names such as the Voronoi diagram dual, the Thiessenpolygon dual, Digital Terrain Model (DTM) and Digital Elevation Model (DEM).Voronoi diagrams and Thiessen polygons are typically used by other disciplines toderive areas of interest. Whilst DTMs and TINs are widely seen as synonymous,DEMs are most often associated with gridded or raster data. However, it is oftenthe case that for surface modelling the DEM points are interpolated onto a gridfrom an initial TIN, frequently for the purpose of effective visualisation of theshape of the surface.

The formation of the network of triangles is not an end in itself. The finalproduct of the modelling process is derived information. Typically this informationcomprises contours, sections, areas and volumes. The precision and accuracydiscussion in this paper will concentrate on volumes, because verification ofvolumes is always open to the most uncertainty and disputation. Whereas contoursand sections can be independently checked and areas can be tested against othercomputation techniques, volume verification is limited to 'counting trucks' orsimilar techniques which may be contaminated by compaction or other effects.

To compute a volume from a TIN, each triangle of the TIN must be projectedonto a horizontal datum plane. The volume of the resultant triangular prism iscalculated by simple geometry (see Fig. 1). By accumulating the volumes of allindividual prisms, the total volume enclosed by the TIN and the datum plane canbe calculated. Many software packages such as GEOCOMP, CiviICAD,Intergraph software, ARC/INFO and others use TINs and this method tocompute volumes. The precision of volumes computed in this way is generally notknown.

Little research has been reported on the precision of results interpolated fromDTMs, and even less about TINs specifically. One investigator used a series of

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M. R. SHORTIS AND P. G. JOYCE

check points to estimate a quantity that would represent the overall accuracy of theDTM [8]. Other research has looked at factors including data source, adjacencyand topology accuracy [12]. This paper aims to establish the relationship betweenprecision of measurements and precision of volumes computed using TINs, as wellas the overall accuracy of volumes computed from TINs.

ALTERNATIVE METHODS OF SURFACE MODELLING

As mentioned in the introduction, the three main methods for surface modellingare TINs, grids and directly fitting mathematical surfaces to the data.

B-spline patches are the mathematical algorithms most often used for surfacemodelling. Whilst appropriate for artificial or designed surfaces, for naturalsurfaces a least squares fit is necessary since more data than degrees of freedomexist. If the fitted surface does not honour the data points as measured, errors willbe introduced into subsequent interpolations. In surveying, this approach isgenerally not acceptable and this technique will not be discussed further in thispaper.

Gathering surface data in a gridded fashion is time consuming and requiresconsiderable effort if good accuracy is desirable. Motorised total stations· arecapable of efficiently pointing to locations on a regular grid, but the procedurecould not be justified with respect to the time required to set out and then collecteach data point. Perhaps the only exception to this rule is an analyticalphotogrammetric plotter, which can drive the measuring mark to each gridlocation where the operator (or digital image correlator) can observe the height ona stereopair of aerial or satellite images.

If gathering gridded data is not feasible, then a process of interpolation ofheights for the grid locations must occur. Many methods exist to interpolate theheights, but there is no universally accepted method. All methods of interpolationsuffer from deficiencies under particular circumstances or with particular data.Indeed, the grid interpolation process must introduce errors, as to estimate thevalue of a surface at a point based on the values around it will not always give anaccurate result [14]. However, a fundamental problem with -gridded data willalways exist. The basic structure of the data does not facilitate variation in densitywith roughness of topography nor the delineation of string features such as ridgelines or other changes in surface grade. Such features can be incorporated into agrid structure, but at a cost which is unacceptable when compared to the inherentflexibility of TINs.

In general, TINs are the preferred surface model since they do not suffer fromany of the above deficiencies. One criticism which can be levelled at TINs is thedata storage requirement, as TIN files must hold the connectivity between thetriangle apexes as well as the coordinate data. However it has been shown thatTINs can use less storage than grid based methods [9].

There are alternatives and variations to the Delaunay criterion for the formationof TINs. Local search routines can be used to minimize the computing andtherefore maximise the rate of the formation of the surface [2].Direct manipulationof the Voronoi polygons, rather than the Voronoi dual, conveniently allows theinsertion of new data points into an existing network and the ability to processdynamic data [4].

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TRIANGULA TED IRREGULAR NETWORKS

A secondary category of surface modelling exists where a mathematical surface(or surface patches) are fitted to the grid or TIN based model. Using a grid modelas a base would compound the interpolation uncertainty cited above and probablyintroduce propagation of systematic errors. The use of smooth surface patches ontop of a TIN is reasonable in certain applications, for instance the modelling of asmooth, natural surface like a sand dune.

A more detailed discussion of alternative modelling techniques is given elsewhere[3].

THE INFLUENCE OF TRIANGLE FORMATION ALGORITHMS ON ACCURACY

The universal recognition of the Delaunay as the optimal triangulationalgorithm is because of several properties. The most important is that for any givendata only one Delaunay triangulation exists. Most other methods would arrive atdifferent triangulations given different starting conditions. Indeed, many possibletriangulations exist for any given data set, but all networks for a particular dataset have the same number of triangles. Another important property of theDelaunay triangulation is that the triangles comprising the network are wellconditioned. Long, narrow triangles may produce inaccurate interpolation and aresaid to be poorly conditioned. Well conditioned triangles are therefore as close toequilateral as possible [7].

On the other hand, there are a number of known problems associated with TINsand all have an influence on accuracy. One potential problem is where trianglesformed using the Delaunay criteria overlap features known to be connectedlinearly. Natural or artificial changes in surface grade, such as a ridge line or thekerb inverts of a road section, are the commonly cited examples where anunconstrained triangulation can be grossly inaccurate. Any triangle side whichdoes not honour the ridge orinvert line will misrepresent the surface. This problemcan be overcome by using break lines. The name originated by using these lines to'break' the triangulation to obtain a more accurate surface representation.Modern algorithms account for the break lines at the time of initial triangulation,simply by ensuring that each break line becomes a triangle side. Traditionaltechniques use post processing of the Delaunay network, applying a sequence ofside swaps until the break lines are honoured [5]. Thus the known linearity ispreserved. The resultant triangulation is known as a constrained Delaunaytriangulation.

The assumption of linearity between data points is both a strength and aweakness of TINs. Linear interpolation is a strength because it is the lowest orderof interpolation and because of the simplicity of the computations. The use of ahigh order interpolation implies a higher order of surface shape, which may notexist. The direct relationship between data density and DTM accuracy, combinedwith the ability to use random and pseudo-random (break line) data, allows theuser to optimise the data collection to the needs of the survey. Linear interpolationis a weakness when the data is sparse, since the surface representation will begeneralised and data interpolated from the TIN will reflect this generalisation. Thereadily apparent effect of this weakness is the discernible triangle shapes which areoften seen in contour plots interpolated from sparse data.

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M. R. SHORTIS AND P. G. JOYCE

Fig. 2. A Delaunay triangulation with the circumcircle shown for one triangle in the network.

3a 3b 3,c

Fig. 3. Triangulation of a convex quadrilateral.

The most serious potential problem with the Delaunay criteria is the existenceof a neutral case. The actual criteria is that for each triangle, the circumcircledefined by the three vertices must not contain another data point (see Fig. 2).

For any four points that form a convex quadrilateral, two possible triangulationsexist (see Fig. 3). Normally the Delaunay criteria can be used to automaticallyselect the optimum triangulation, defined as the case where the two triangles areas close to equilateral as possible. When four points are concentric, the twoalternative triangulations are identical using the Delaunay criteria (see Fig. 3c).However, results interpolated from the two triangulations are not identical. Takefor example four points at the apexes of a unit square. The points are concentric,so either pair of triangles would satisfy the Delaunay criteria.

The volume computed using triangles in Fig. 4a using a datum of 0·0 m is4·483 m3• The volume computed using the triangles of Fig. 4b and the same datumis 3·867 m3• Both answers are mathematically correct, but one is certainly a moreaccurate representation of the actual surface.

Break lines can often be used to resolve this problem, but not always. If an areaof curved surface is covered by data that is gridded in plan, the ambiguity of Fig.4 exists for each grid cell. The two dashed lines in Fig. 5 are both valid break lines.

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TRIANGULATED IRREGULAR NETWORKS

9.6

2.04a

4.5

0.6

9.6

2.04b

4.5

0.6

Fig. 4. Triangulation of a unit square.

Fig. 5. Selection of diagonals for the triangulation of gridded data.

However it is not obvious which diagonal should become the break line across thegrid square indicated, without recourse to information other than coordinate dataand simple linear interpolation. In this situation the triangle formation algorithmmay arbitrarily select one diagonal in preference to the other because there is noother available criterion. Yet the accumulated effects of this selection can have asignificant impact on the volume calculated. For the concave surface case shownin Fig. 5, selection of the inappropriate diagonal would lead to an over-estimationof the volume, while for a convex surface an under-estimation would result. Theonly guaranteed remedies for this problem are to increase the data density or toforce appropriate break lines across every grid cell. The former may not be cost-effective and the latter may not be practical.

In the absence of any other criteria, the triangle formation process could bepreset to select diagonals based on volume bias. This leads to the slightly ludicroussituation that the software or the user could opt for a maximised or a minimisedvolume when concentric points were encountered during a triangulation.Nevertheless, this is not a trivial problem since concentric points occur by chanceas well as in the obvious case of gridded data sets.

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Fig. 6. Poorly conditioned triangles near the extremities of a data set.

Precise measurement of a data set may realise a precise volume computed fromthe TIN, but precision does not imply that the computed volume is accurate. Thevolume is effected (possible by more than through data precision) by data density,which is directly related to how well the data represents the surface. Adequate datadensity will ensure any occurrences of the neutral case will have an insignificanteffect, and the assumption of linearity between points will be more valid than forsparse data.

Another issue worthy of consideration is boundary specification. A Delaunaytriangulation is by definition bounded by a convex shape (called the convex hull).The usual result of this is long, narrow triangles around the edge of thetriangulation (see Fig. 6). These poorly conditioned triangles may result in, forexample, unacceptable or illogical contour patterns near the periphery of the dataset. If any of the edge triangles are included in the volume computation, erroneousresults will be obtained because extra areas will be enclosed which are not part ofthe actual data set. A boundary string can be defined which excludes these edgetriangles and therefore removes such edge effects.

If the specified boundary points are not included in the triangulation data set,the bounded area may cover some of the unwanted edge triangles. These trianglesmust then be broken along the boundary via post-processing, which again mayintroduce poorly conditioned triangles. Careful boundary specification, usingexisting data points, will avoid this problem.

THE INFLUENCE OF COORDINATE DATA PRECISION

Initial research carried out as part of a student project [6] indicated that therewas an essentially linear relationship between the precision of the coordinate datamaking up a TIN, and the precision of the volume (contained by the TIN surface,a boundary string and a horizontal datum surface). The main thrust of the testingwas purely empirical and very straightforward. The coordinates in the data setwere perturbed by random normal deviates and the volume recomputed a numberof times. The mean and the standard deviation of the volume could therefore becomputed from the multiple estimates.

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TRIANGULA TED IRREGULAR NETWORKS

This approach of empirical testing isolates the influence of precision of the dataset coordinates. The separation of factors in an error analysis of any kind isimportant because once a functional or statistical relationship is established, thecontribution of the factor under consideration can be eliminated. Other factors canthen be isolated, analysed and eliminated.

The initial research was extended in two aspects in order to increase theconfidence in the conclusions. First, the number of repetitions was increased from16 to 100.The calculations are therefore based on a much larger sample of randomnumbers which provides a much better approximation of a true normaldistribution. Small samples of random numbers tend to be biased in the mean, thestandard deviation, the shape of the probability distribution or all three. Second,four different types of data sets were processed. The variation in topography anddata distribution increases the confidence with which a universal relationshipbetween coordinate precision and volume can be adopted as a hypothesis.

Software was written to form a Delaunay TIN of a surface and then computethe volume enclosed by the surface, a boundary defined by a separate string of plancoordinates, and a constant datum plane. The boundary applied was inside theconvex hull of the triangulation and selected so as to eliminate the problem ofboundary edge effects, as discussed above. However this strategy does notguarantee that there are no poorly conditioned triangles near the periphery of thedata set, as boundary string points can not be part of a data set undergoingperturbation. This would lead to inflated variations in the volume caused byvariations in the total area covered by the data set. It might be argued that anapproach using a consistent boundary is unrealistic, but a boundary would beimposed on the data set in the vast majority of practical circumstances.

Software was also written to perturb the data set with random normal deviates.The deviates were computed using sequences of computer generated randomnumbers and the central limit theorem. In effect this enabled a random error of aspecified precision to be applied to the data set coordinates. An entire populationof deviates for any data set very accurately represents the probability function ofa normal distribution.

A typical sequence of processing for a single data set and a single precision levelwould be: (1) make 100 copies of the coordinate data set, (2) perturb each copy,(3) form a Delaunay TIN for each copy, (4) compute a volume for each copy.

The mean and standard deviation of the 100 volumes was then computed. Thusa precision of measurement could be associated with a standard deviation of theresultant volume. The above sequence of copying and perturbing simulates re-measurement of the data.

Note that the above sequence has the consequence that the connectivity of thetriangle network is not preserved for all copies. This may lead to different networksif there are instances of the neutral case, or indeed instances which are marginallyclose to neutral, due to the random re-positioning of points. This strategy wasadopted to maintain realism, as re-measurement of a data set may result in adifferent triangulation network for real data.

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M. R. SHORTIS AND P. G. JOYCE

THE DATA SETS

Data sets from hydrographic, photogrammetric, survey and mathematicallygenerated sources were used to embrace as wide a range of topologies and datadistributions as possible.

The survey data set comprises 500 points measured on the excavated surface ofan open cut mine. The survey was performed using a total station and was for thepurpose of volume computation. The area covered is approximately 300 m by200 m and the depth of the pit is 50 m. This data set is a mixture of random pointson the natural surface plus pseudo-random strings of points which define changesin grade.

The photogrammetric data set comprises 680 points covering the front of thehead of a bald man. The data were observed from a stereo pair of ZeissOberkochen SMK photographs using a Zeiss lena Stecometer stereocomparator.The data set was acquired as part of ongoing research into biostereometrics andCAD modelling of the human body. This data is essentially random in nature.

The hydrographic data set comprises approximately 1000points extracted froma much larger data set of a shipping channel and the approach to a wharf basin.The selected area is part of the shipping channel being dredged to a rectangularsection. The data was acquired during dredging operations using a depth sounderand radio frequency positioning system. This data set is pseudo-random as theaverage separation of the sounding lines is considerably larger than the averageseparation of the soundings within the lines.

The three real data sets described above provide a reasonable range of practicalexamples of TIN surfaces. To supplement these sets, approximately 1000 randompoints were mathematically generated over the surface of a northern hemisphereof radius 1000 m. This data type was included as the true volume of the enclosedsurface could be calculated exactly, allowing comparison between measured andactual results.

Five different methods of generating the coordinates on the hemisphere wereused to produce a range of data distributions and because there is no valid reasonfor excluding any particular Method. method (a) computed 970 points along linesof longitude and latitude spaced at approximately 3.50 over the surface. The nextthree methods are random in plan only and require corresponding heights to thesurface to be computed. Method (b) superimposed a regular 30 m grid over thearea of the hemisphere. Method (c) generated random points inside a square witha side length equal to the hemisphere diameter and accepted only those inside theinscribed circle. Method (d) used random polar coordinates (radius and angle).Finally, a method (e) used a fixed radius and random latitude and longitude togenerate random points on the surface of the hemisphere.

Notice that in methods (d) and (e) the generated points will exhibit a clusteringdefect at the zenith. In both cases this is due to the convergence of the meridiansnear the pole, the former in plan and the latter on the surface. This is not a problemfor the purposes of this investigation.

A range of measurement precisions was tested in order to establish therelationship between data set measurement precision and volume precision. Forthe survey data, precisions of 0·01, 0'02, 0·05 and 0·1 m were adopted. Thephotogrammetric data was in millimetres, so precisions of 0·01, 0·02, 0·05, 0·1, 0·2,

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TRIANGULATED IRREGULAR NETWORKS

0.10 -e- Method (a)

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-A- Method (c)u<0 -0.208~"0 ...sr Method (d)> -0.30

Coordinate Precision (m)-0.40 + Method (e)

0.001 0.01 0.1 1.0 10.0

Fig. 7. Graph of volume accuracy (percentage error of the mean volume) versus coordinateprecision for the hemisphere data sets.

0·5 and 1·0mm were used. Precisions of 0'01, 0·1, 0·2, 0·5 and 1·0m were tested forthe hydrographic data set. For the mathematically generated hemispherical data,precisions of 0'001, 0'01, 0'1, 1·0 and 10·0m were tested. This last range ofprecisions represents five orders of magnitude, which in land surveying termsmight correspond to precise differential GPS compared to a magnetic compasstraverse. It is unlikely that any single measurement technique could approach sucha variation in precision.

ANAL YSIS OF THE RESULTS

The results of the computations can be found in the graphs shown in Figs 7-11.The graph of the percentage error in the mean volume for the hemisphere isvirtually identical for all five data distributions. The percentage error is a measureof accuracy and the graph shows very little deviation from zero. Indeed, the worsterror in the entire set of all mean hemisphere volumes is less than 0.5%

• Theconsistent under estimation of the mean volume at the worst precision value is dueto the fixed boundary of the hemisphere. A precision of ± 10m allows some pointsnear the edge of the data set to be perturbed to positions outside of the boundary,but part of the increase in volume is not retained due to the cut-off at the boundaryedge. On the other hand, points which are perturbed to new positions inside of theboundary retain the full impact of volume lost, thereby decreasing the volume. Theworst case of this should be method (a) because of the many points near the edgeof the hemisphere, and this contention is supported by the results.

The relation between volume precision and surface point precision is essentiallylinear. In all cases the effect of changing the measurement precision is aproportional change in the precision of the result. The only exceptions are in thehemispherical data generated by methods (a) and (b) at the smallest precisionvalues used. The unexpected degradation in volume precision is attributable to theoccurrence of large numbers of concentric points in the data. As the precision ofthe data increases in magnitude this effect disappears. Due to the corresponding

386

M. R. SHOR TIS AND P. G. JOYCE

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Fig. 8. Graph of volume precision versus coordinate precision for the survey data set.

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0.05

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0.00 0.20 0.40 0.60 0.80 1.00 1.20

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Fig. 9. Graph of volume precision versus coordinate precision for the photogrammetric data set.

387

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0.25

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Fig. 10. Graph of volume precision versus coordinate precision for the hydrographic data set.

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-a- Method (b)

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Fig. 11. Graph of volume precision versus coordinate precision for the hemisphere data sets.

increase in the magnitude of the random deviates, the possibility of concentricpoints dramatically decreases.

CONCLUSIONS

This paper has addressed the issues of accuracy and precision of 2·5D TINs. Theaccuracy of TINs has been discussed in terms of the most appropriate surfacerepresentation and a maximised correspondence between the TIN and the actual

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M. R. SHORTIS AND P. G. JOYCE

surface. Degradation of accuracy can be avoided by judicious selection of pointdistributions, point density, break lines and boundary strings. The precision ofTINs has been investigated via empirical testing of the relationship between theprecision of the data set coordinates and the precision of the volume. Except forminor effects from the neutral case of triangle formation, the precision relationshipis linear. That is, any increase or decrease in the precision of the point coordinatesin the TIN data set will result in a directly proportional increase or decrease in theprecision of a derived volume.

It is apparent from the results of the empirical testing that the gradient of thelinear relationship between data set precision and volume precision variesconsiderably. If the gradient could be predetermined then it would be possible topredict the required data set precision to achieve a desire volume precision. Furtherresearch is required to determine whether there is a relationship between thisgradient and the topology, the data density or some other characteristic of thesurface. Cases where the planimetry and height precision are typically quitedifferent, for example hydrographic data sets, are also worthy of additionalinvestigation.

Finally, to overcome the deficiencies of 2·5D TINs, full 3D TINs are the subjectof continuing research by the authors. Current research is concentrating on theformation and optimisation of 3D TINs. It is essential that the analysis of accuracyand precision should be extended to embrace the three dimensional case.

References

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2. EUick, M. H., 1979. Contouring by use of a triangular mesh. Cartographic Journal, 16 (1): 24-29.3. Fenton, P., 1982. Digital Terrain Modelling Techniques for Volume Calculations and Road

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4. Gold, C. M., 1991. Problems with handling spatial data - the Voronoi approach. CISM JournalACSGC, 45 (1): 65-80.

5. Heller, M., 1990. Triangulation Algorithms for Adaptive Terrain Modelling. Proceedings Volume1, 4th International Symposium on Spatial Data Handling, Zurich, Switzerland, pp. 163-174.

6. Johari, A. G., 1987. Precision and accuracy of volumes. Unpublished Final Year Project Report.Department of Surveying and Land Information, The University of Melbourne, Australia. 35pages.

7. Lawson, C. L. 1977. Software for C1 Surface Interpolation. In Mathematical Software III, Ed. J. R.Rice. Academic Press.

8. Li, Z., 1988. On the Measure of Digital Terrain Model Accuracy. Photogrammetric Record, 12(72): 873-877.

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10. Petrie, G. & Kennie, T. J. M., 1990. Terrain Modelling in Surveying and Civil Engineering. Whittles,London. 351 pages.

11. Peuker, T. K., Fowler, R. J., Little, J. J. & Mark, D. M., 1978. The Triangulated IrregularNetwork. Proceedings, DTM Symposium, St Louis, U.S.A., pp. 516-540.

12. Ryan, T. & Masters, E., 1991. Spatial Accuracy as a Quality Indicator. Proceedings, Symposiumon Spatial Data Accuracy, Melbourne, Australia, pp. 175-185.

13. Shortis, M. R. & Benwell, G. L., 1984. Acquisition of spatial data by survey and photogrammetrictechniques. Proceedings, Australasian Computer Graphics Association Conference, Melbourne,Australia, 7 pages.

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