Ordering points for incremental TIN construction from DEMs James J. Little and Ping Shi Department of Computer Science University of British Columbia Vancouver, BC, Canada, V6T 1Z4
[email protected] 604-822-4830 (Tel) 604-822-5485 (FAX)
Abstract The standard method of building compact triangulated surface approximations to terrain surfaces (TINs) from dense digital elevation models(DEMs) adds points to an initial sparse triangulation or removes points from a dense initial mesh. Typically, in each triangle in the current TIN, the worst tting point, in terms of vertical distance, is selected. The order of insertion of the points is determined by the magnitude of the maximum vertical di erence. This measure produces triangulations that minimize the maximum vertical distance between the TIN and the source DEM. Other approximation criteria are often used, however, including the root-mean-squared error or the mean absolute error, both for the vertical di erence and normal di erence, i.e. the distance in the direction of the normal to the triangular approximation. For these approximation criteria, we still select the worst t point, but determine the insertion order by various sums of errors over the triangle. Experiments show that using these better evaluation measures signi cantly reduces the size of the TIN for a given approximation error.
1 Introduction Terrain data is often speci ed on a grid of points, a digital elevation model (DEM). For analysis and display, we can derive from the DEM a triangulation of points selected to represent the DEM with minimal error. This triangulation is often called a Triangulated Irregular Network (TIN)[PFLM78]. The early work in deriving TINs from DEMs was inspired by the curve approximation
(a) (b) Figure 1: (a) Curve simpli cation: the Douglas-Peucker method adds the farthest point from each segment on the current polygonal curve. Successive approximations are show by progressively ner dashed lines.(b) Terrain surface represented by contours: as before, successive approximations are shown with progressively ner dashed lines. procedure of Douglas and Peucker [DP73] which derives a polygonal curve that approximates a curve but recursively nding the farthest point from each segment in the current approximation (Figure 1(a)). In unpublished work, W.R. Franklin (1975) developed a procedure to approximate a terrain height eld by iteratively adding the farthest point from each face in a recursively constructed triangulated surface (Figure 1(b)). Many techniques for approximating a surface, usually a terrain height eld, begin by selecting points that are expected to be critical in the nal approximation[GH95, PM92, HG97, Kum94, Hel90, CG87, DR93, JS98]. The approximation to the surface is improved by adding points to this initial triangulation. One particular method, [FL79], nds in each triangle the point that is most poorly t by the current triangulation and adds that point to the Delaunay triangulation[Lis94] of the points. Iteratively following this process produces triangulations that eventually t the surface well, but with many fewer points than the dense source data. There is always a tradeo between the error of the approximation and the number of points used; of course the error is minimal with the full set of DEM points. This \greedy" procedure orders the points for insertion by the magnitude of the vertical error at the selected point in the triangle. The procedure is not optimal, in the sense that the resulting triangulation may not have the minimal error, however measured, for a particular number points. But the method is simple to implement and produces compact triangulations. It is clear that this method is not guaranteed to t the surface well at surface discontinuities or at slope discontinuities, both of which occur frequently in terrain and especially in range maps produced in computer vision. Several methods [FL79, SC93, LS98, LS00] exist 2
for identifying surface curves called structural lines on these discontinuities. Using these lines as the initial skeleton reduces the size of the triangulation for a given error. A signi cant reduction in the size of the triangulation can be produced simply by adopting di erent measures for ordering the selected points in greedy insertion. These measures integrate various errors over We will rst describe triangulation and introduce the new measures for evaluation points for insertion. Finally we will present experiments using these measures in TINs.
2 Triangulation The original work in this area[FL79] proposed incremental \greedy" triangulation of a TIN by inserting points based upon the error in each triangle, and preservation of structural lines found by marking ridges and channels and then generalizing these 3D lines. Incremental improvement is widely used now, together with many variations in criteria for adding points. Typically, a point is selected in each triangle; the \worst" point is the point with maximal vertical deviation from the approximating triangle. Many di erent strategies can be used for determining the order of insertion. [FL79] proceeded by inserting the worst point in a triangle if it exceeded a desired error tolerance, continuing until all points in the DEM were t within this tolerance by the triangulation. Our main innovation is to rank the points by new criteria, instead of the maximum vertical deviation. Fundamentally, these attempt to assess the change in the volume of the di erence between the DEM and the TIN. The candidate point is still the point p with maximum vertical error jVpj. One can also consider normal error|distance normal to the approximating plane, jNpj. Let At be the area of the triangle t containing p. i is the index in the DEM of points in the current triangle. We can envision an error surface that is the pointwise di erence between the DEM and TIN. In what follows, summations are assumed over points in the current triangle. The signum function returns the sign of a real value: 1 if positive, 0 if zero, and ?1 if negative. The point is ranked in the insertion schedule by various measures: 1. normal volume (NV): jNpj At approximates the error surface locally by a triangular pyramid whose apex is at the point of maximal deviation, and then computes its volume. It uses the absolute normal error at this point as the height of the pyramid. 2. absolute normal sum (ANS): jNij approximates the volume under the error surface using the sum of the absolute value 3
of the normal errors. 3. normal sum (NS): Ni sums the normal errors. 4. absolute vertical sum (AVS): jVij approximates the volume under the error surface using the sum of absolute value of the vertical errors. This is the usual approximation to the volume of the error surface. 5. vertical sum (VS): Vi totals the vertical errors. 6. vertical volume (VV): jVpj At approximates the error surface locally by a triangular pyramid whose apex is at the point of maximal deviation, and then computes its volume. It uses the absolute vertical error at this point as the height of the pyramid. 7. squared normal sum (SNS): (Ni)2 sums the squared normal errors. 8. squared vertical sum(SVS): (Vi)2 sums the squared vertical errors. 9. signed squared vertical sum(SSVS): signum(Vi)(Vi)2 sums the signed squared vertical errors. The normal sum measure (NS) and the vertical sum measure (VS) sum the errors. The absolute normal sum measure (ANS) and the absolute vertical sum measure (AVS) sum the absolute value of the errors. The squared normal sum (SNS) and squared vertical sum (SVS) integrate the squared errors; these should be suitable for reducing the root-meansquared error (RMSE). Figure 2 shows two local portions of a surface; the surface is totally positive (a) and oscillates around zero (b). Both have the same squared vertical sum; for SVS their values are 4:881. The signed squared vertical sum (SSVS) takes into account the sign of the error to behave like NS and VS. Using the signed squared vertical sum (SSVS), the surface in (a) has value 4:881 and (b) has value 0:960. Thus the worst tting point in both (the central point) would be added to (a) long before (b) using the signed measure. The RMSE decreases from 4.8 to 1.56 with one split in (a) while the error decreases only from 4.8 to 3.94 with one split in (b). Thus locally the signed measure is a good guide to improving the triangulation. More points must be added to (b) to reduce its approximation error correspondingly. 4
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DEMs name width height range avg. grad mag 5000/(w*h) RMSE range MAE range crater 336 459 950 4.12 0.032 2.8-12.2 2.1-9.0 lcrater 230 459 940 3.43 0.047 1.8-11.8 1.4-9.2 ritz1 300 300 80 1.05 0.055 0.9-3.5 0.65-2.45 yakima3 256 256 600 3.79 0.076 0.6-10.0 1.4-10.0 yakima2 256 256 900 11.76 0.076 4.2-40.0 3.4-30.0 baker 314 468 3274 12.4 0.034 5.6-30.0 3.9-30.0 Table 1: Speci cations for the DEMs.
3 Experiments These measures have been extensively testedronPa variety of DEMs. The root-mean-squared 2 =1 e , where e is the di erence between the error (RMSE) is computed at all points i as i n DEM height at i and P =1thee TIN approximation. The mean-absolute error (MAE) is computed at all points i as n . The MAE is less sensitive to large errors than the RMSE and gives a better picture of the behavior of the approximation. Table 1 shows the speci cations for the DEMs tested. All are taken from the USGS 1:24000 series DEMs. The speci cations include the size of the DEM (width and height), the range in elevation, average slope and relevant measures from the experiments, including the percent of the DEM points used in the ne approximation (5000 points) and the range of RMSE and MAE in the experiments. Experiments in producing TINs were run on DEMs, running the greedy triangulation. The triangulation stopped when 5000 points had been included. n i
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Figure 3: Number of points versus RMSE for Crater Lake left (lcrater). The rankings reported in Table 2 are the ratio over the entire range of RMSE for the number of points required by the TIN for each particular measure to the number in the TIN using the traditional maximal vertical di erence, Vp . The tables are presented in Appendix B. Table 3 shows similar ratios over the entire range of MAE for the number of points required by the TIN for each measure to the number in the TIN using maximal vertical di erence, Vp. There are twelve tables in Appendix B: three groups of four. Each group of four contains results without and with lines, and for each selection of lines or not, results are reported for RMSE and MAE. The rst group of four shows the results in terms of the relative size of the TIN. The second group shows relative performance, i.e., for a given size of TIN, the RMSE or MAE relative to using the vertical maximum measure. The nal group of four shows relative performance when the size of the TIN is relatively small; typically the results are best for compact TINs. Figure 3 plots the number of points required as a function of RMSE in the Crater Lake DEM; Figure 4 shows a blowup of that gure for small RMSE. Figures 5 and 6 show similar plots for MAE.
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Figure 6: Number of points versus MAE for Crater Lake left (lcrater), small MAE. This is a blowup of a portion of the previous gure.
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Figure 7: Two triangulations with the same RMSE (11.25) for Crater Lake left (lcrater): (a) the triangulation (700 points) for maximum vertical measure; (b) the triangulation (422 points) for SVS. Figure 7(a) shows the triangulation of the Crater Lake DEM using the maximum vertical measure using 700 points; the RMSE is 11.25. Figure 7(b) shows the triangulation of the Crater Lake DEM using measure SVS; now the triangulation requires only 422 points to achieve the same RMSE, 40% fewer points. The maximum vertical errors in (a) and (b) are 51.3 and 107.9, respectively.
3.1 With Lines The rankings reported in Tables 4 and 5 are the ratio over the entire range of RMSE and MAE for the number of points required by the TIN for this measure to the number in the TIN using the traditional maximal vertical di erence, Vp, using structural lines as an initial skeleton for the surface. 9
3.2 Comparison The ratio of the number of points required for a given quality of t with a particular measure to the number of points required with the standard method demonstrates that almost all the proposed measures improve on the standard method in some cases. To assess which is the most useful measure, we can compare the average e ect of the various measures. For each DEM, we nd the best measure and its proportional count. For example, the ratio for Crater Lake, which we will use as our example, is 0:622 for measure SVS, as shown in the following table. rank 1 2 3 4 5 6 7 8 9 ratio 0.622 0.627 0.656 0.684 0.714 0.808 1.075 1.106 1.137 measure SVS SSVS VS AVS NV VV SNS ANS NS We convert that to a reduction by subtracting it from 1.0: rank 1 2 3 4 5 6 7 8 9 reduction 0.378 0.373 0.344 0.316 0.286 0.192 -0.075 -0.106 -0.137 measure SVS SSVS VS AVS NV VV SNS ANS NS We can read the number 0:378 in the previous table as saying that measure SVS reduces the number of points needed for a particular RMSE by 37.8%, compared to using the maximum vertical error. For the six DEMs, we compute the maximum reduction over all measures, for RMSE, and record the best measure: DEM crater lcrater ritz1 yakima3 yakima2 baker max reduction 0.378 0.323 0.258 0.444 0.272 0.257 measure SVS SVS SVS SVS SVS SVS The same numbers for MAE are: DEM crater lcrater ritz1 yakima3 yakima2 baker max reduction 0.481 0.417 0.316 0.597 0.345 0.288 measure VS AVS VS VS VS SVS We divide the results for each DEM by the maximum and get the relative reduction, a number whose maximum is 1.0 and is negative when the reduction is an increase. The following table shows the relative reduction for Crater Lake: rank 1 2 3 4 5 6 7 8 9 relative reduction 1.0 0.99 0.95 0.91 0.87 0.77 0.58 0.56 0.55 measure SVS SSVS VS AVS NV VV SNS ANS NS We then take the average of these relative reductions, for each measure, over all DEMs. The order of performance of the measures is, for RMSE:
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Figure 8: Average relative reductions, RMSE, relative to the best measure: (a) no lines, where the best is SVS, (b) lines, where the best is SSVS. measure SVS SSVS VS AVS NV VV SNS NS ANS nolines 1.0 0.976 0.878 0.827 0.679 0.506 -0.00781 -0.134 -0.178 measure SSVS SVS VS AVS NV VV SNS NS ANS lines 1.0 0.978 0.919 0.863 0.696 0.672 0.145 -0.108 -0.177 The order of performance of the measures is, for MAE: nolines VS AVS SVS SSVS NV NS SNS ANS VV 0.989 0.953 0.935 0.911 0.801 0.747 0.719 0.705 0.559 lines VS SSVS AVS SVS NS NV SNS ANS VV 1.0 0.965 0.963 0.950 0.847 0.843 0.783 0.777 0.730 These results are plotted in Figures 8 and 9. One way to visualize the rankings derived from the relative reductions is to examine the plot (Fig. 10) showing them as we vary the RMSE from minimum to maximum values for Crater Lake. Measure SVS is best for RMSE; it is now shown since it is the benchmark value. If shown it would be a horizontal value at 1.0. Several of the measures are better for some RMSE range, particularly at low and high RMSE, but on average measure SVS is best. When a measure is worse than the standard method, the relative reduction value becomes negative. One may suppose that selecting points other than the worst t for insertion might improve the performance. Is the centroid of the triangle a feasible alternative? Using the centroid is at least 25% worse than using the worst t point, in the sense that the error measure for a particular number of points is 25% larger. 11
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4 Discussion These measures achieve the following relative reductions for RMSE, when structural lines are not used:
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DEMs crater lcrater ritz1 yakima3 yakima2 baker 0.378 0.323 0.258 0.444 0.272 0.257 SVS SVS SVS SVS SVS SVS and MAE: DEMs crater lcrater ritz1 yakima3 yakima2 baker 0.481 0.417 0.316 0.597 0.345 0.288 VS AVS VS VS VS SVS That is, the reductions range from 26% to 44% for RMSE and from 32% to 60% for MAE. In the previous section the measures were compared only for the case without structural lines. When structural lines are used, the relative reductions increase substantially (except for yakima3): DEMs crater lcrater ritz1 yakima3 yakima2 baker 0.422 0.392 0.345 0.333 0.329 0.344 SSVS SSVS SSVS SSVS SVS SSVS and MAE: DEMs crater lcrater ritz1 yakima3 yakima2 baker 0.519 0.475 0.380 0.318 0.384 0.376 VS VS VS VS VS VS Measures SVS and SSVS, both using the squared vertical errors, are very e ective for RMSE in both nolines and lines: both are ranked rst and second in the two cases and deliver 95% or more of the best reduction. Measures NV, ANS, NS, VV, and SNS and are much less e ective, and several actually are less e ective than the standard method. Measures VS, SSVS, AVS, and SVS (VS and AVS use vertical errors) are most e ective for MAE in both nolines and lines. VS is best in both, while SSVS and AVS are greater than 95% in lines. For MAE, most measures are e ective, i.e., they produce at least 50% of the best relative reduction. Surprisingly, both measures using the summed errors, without the absolute value, i.e., VS and SSVS, are very e ective: for RMSE SSVS is best when structural lines are used, and for MAE VS is best both with and without structural lines. Using these measures introduces no additional cost beyond the standard maximum computation, while signi cantly improving the compactness of the resulting triangulation. The proposed measures are signi cantly worse than the max vertical criterion at reducing the maximum vertical di erence between the DEM and the TIN approximation: the maximum vertical di erence is three times worse for the best of the measures. However, the maximum deviation between the DEM and the TIN in the direction normal to the TIN is smaller. For the lcrater DEM, SNS, squared normal sum, reduces the maximum normal 13
error by 68.2% relative to the error produced by the max vertical criterion. But this is unfair: using the maximum normal deviation as the measure reduces the maximum normal error by a factor of three relative to SNS. The normal measure is unbiased while the vertical measure exaggerates errors relative to a slanted plane in the following sense. The ratio of the vertical error at a point, v , to the normal error, n, is the secant of the angle between the plane and the horizontal. For a horizontal plane the errors are the same but as the plane tends toward vertical the ratio of the vertical error to the normal error tends to in nity. We also evaluated the following measure: 10. signed squared normal sum: (SSNS) signum(Ni)(Ni)2 sums the signed squared normal errors. The overall ratios for the squared normal sum (SNS) and signed squared normal sum (SSNS) relative to the max vertical criterion for the normal RMS, as opposed to the vertical RMS used above, are: DEMs crater lcrater ritz1 yakima3 yakima2 baker 0.762 0.694 0.382 0.856 0.767 0.509 SNS SSNS SSNS SSNS SSNS SNS Reductions range from 38.9% to 85.6% in the RMS normal error. Likewise the mean absolute normal error is signi cantly reduced by SSNS (signed squared normal sum), by 56.5% in the lcrater data. The measures based on normal error behave relatively poorly for the vertical MAE and RMSE but when the approximation error for the TIN is measured by the normal MAE and RMSE these measures are best. Interestingly, the results without structural lines are better when the approximation error is measured in the normal direction. As a group of measures these lead to signi cant savings in the size of TINs derived from DEMs and allow us to select the order of insertion of points during incremental TIN construction to suit the approximation error we choose.
References [CG87]
Zi-Tan Chen and J. Armando Guevara. Systematic selection of very important points (VIP) from digital terrain model for constructing triangular irregular networks. In N. Chrisman, editor, Proc. of Auto-Carto 8 (Eighth Intl. Symp. on Computer-Assisted Cartography), pages 50{56, Baltimore, MD, 1987. American Congress of Surveying and Mapping.
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[DP73]
David H. Douglas and Thomas K. Peucker. Algorithms for the reduction of the number of points required to represent a digitized line or its caricature. The Canadian Cartographer, 10(2):112{122, 1973.
[DR93]
Nira Dyn and Shmuel Rippa. Data-dependent triangulations for scattered data interpolation and nite element approximation. Applied Numer. Math., 12:89{ 105, 1993.
[FL79]
Robert J. Fowler and James J. Little. An automatic method for the construction of irregular network digital terrain models. In Proceedings of SIGGRAPH '79, pages 199{207, Chicago, Illinois, August 1979.
[GH95]
Michael Garland and Paul S. Heckbert. Fast polygonal approximation of terrains and height elds. Technical Report CMU-CS-95-181, Carnegie Mellon U., September 1995.
[Hel90]
Martin Heller. Triangulation algorithms for adaptive terrain modeling. In Proceedings of the 4th International Symposium of Spatial Data Handling, pages 163{174, 1990.
[HG97]
Paul S. Heckbert and Michael Garland. Survey of polygonal simpli cation algorithms. Technical report, CMU, 1997.
[JS98]
Bernd Junger and Jack Snoeyink. Importance measures for TIN simpli cation by parallel decimation. In International Symposium on Spatial Data Handling '98, pages 637{646, 1998.
[Kum94] Mark P. Kumler. An intensive comparison of triangulated irregular networks (TINs) and digital elevation models (DEMs). Cartographica, 31(2):1{48, 1994. [Lis94]
Dani Lischinski. Incremental Delaunay triangulation. In P. Heckbert, editor, Graphics Gems IV. Academic Press, 1994.
[LS98]
James J. Little and Ping Shi. Structural lines for triangulations of terrain. IEEE Workshop on Applications of Computer Vision, 1998.
[LS00]
James J. Little and Ping Shi. Structural lines, TINs, and DEMs. Algorithmica (to appear), 2000.
[PFLM78] T. K. Peucker, R. J. Fowler, J. J. Little, and D. M. Mark. The Triangulated Irregular Network. In Proc. of the Digital Terrain Models Symp., pages 516{532, St. Louis, MO, 1978. 15
[PM92]
M.F. Polis and D.M. McKeown. Iterative TIN generation from digital elevation models. In Proc. IEEE Conf. Computer Vision and Pattern Recognition, 1992, pages 787{790, 1992.
[SC93]
Francis Schmitt and Xin Chen. Vision-based construction of CAD models from range images. In Proc. 4th International Conference on Computer Vision, pages 129{136, 1993.
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Appendix A The following gures show the DEMs used in the experiments as grayscale images; increasing height is brighter.
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Figure 11: (a) Crater Lake (b) Left Crater
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Appendix B The following tables describe the ratios of the performance of the various measures: the rst group shows the ratio of the size of the TIN needed for comparable performance to the max vertical criterion, using either RMSE or MAE, without and with structural lines; the second group shows the ratio of the performance (RMSE or MAE) for TINs of comparable size. The measures are listed in order of best to worst, in terms of relative size of the resulting TIN (the rst four tables) and the relative performance, whether RMSE or MAE (the last four).
Size Ratios No Lines
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1 0.622 SVS lcrater 0.677 SVS ritz1 0.742 SVS yakima3 0.556 SVS yakima2 0.728 SVS baker 0.743 SVS
2 0.627 SSVS 0.692 SSVS 0.743 SSVS 0.562 SSVS 0.734 SSVS 0.754 SSVS
3 0.656 VS 0.716 AVS 0.749 VS 0.604 VS 0.753 VS 0.792 VS
4 0.684 AVS 0.728 NV 0.765 AVS 0.636 AVS 0.799 AVS 0.800 AVS
5 0.714 NV 0.749 VS 0.798 SNS 0.689 VV 0.851 NV 0.818 NV
6 0.808 VV 0.815 VV 0.833 NS 0.695 NV 0.910 VV 0.878 VV
7 1.075 SNS 1.043 SNS 0.846 ANS 1.073 SNS 1.034 NS 1.031 SNS
8 1.106 ANS 1.073 ANS 0.862 NV 1.150 NS 1.040 ANS 1.062 NS
9 1.137 NS 1.125 NS 0.884 VV 1.326 ANS 1.058 SNS 1.072 ANS
Table 2: Without structural lines: Ratio of points needed for each measure, for a given root-mean-squared error (RMSE), to the points needed for the same RMSE using the max vertical criterion; the value is the average over a broad range of RMSE. The measures are listed in order of best to worst.
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1 0.519 VS lcrater 0.583 AVS ritz1 0.684 VS yakima3 0.403 VS yakima2 0.655 VS baker 0.712 SVS
2 0.550 AVS 0.610 VS 0.697 AVS 0.414 SVS 0.682 SVS 0.712 VS
3 0.565 SSVS 0.613 SVS 0.718 SVS 0.430 AVS 0.685 SSVS 0.713 AVS
4 0.573 SVS 0.624 SSVS 0.719 SSVS 0.449 SSVS 0.699 AVS 0.731 SSVS
5 0.591 NV 0.626 NV 0.729 NS 0.458 NV 0.758 NV 0.759 NV
6 0.631 NS 0.675 ANS 0.736 SNS 0.509 SNS 0.795 NS 0.792 NS
7 0.644 SNS 0.685 NS 0.744 ANS 0.532 NS 0.806 ANS 0.808 SNS
8 0.654 ANS 0.688 SNS 0.806 NV 0.583 ANS 0.827 SNS 0.810 ANS
9 0.756 VV 0.766 VV 0.836 VV 0.589 VV 0.849 VV 0.816 VV
Table 3: Without structural lines: Ratio of points needed for each measure, for a given mean absolute error (MAE), to the points needed for the same MAE using the max vertical criterion; the value is the average over a broad range of MAE.
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1 0.596 SSVS lcrater 0.608 SSVS ritz1 0.366 SSVS yakima3 0.675 SSVS yakima2 0.671 SVS baker 0.656 SSVS
2 0.602 SVS 0.613 SVS 0.376 SVS 0.682 SVS 0.673 SSVS 0.671 SVS
3 0.611 VS 0.654 VS 0.386 VS 0.716 VS 0.702 VS 0.676 VS
4 0.637 AVS 0.672 AVS 0.405 AVS 0.738 AVS 0.702 AVS 0.706 AVS
5 0.712 VV 0.729 NV 0.413 NS 0.750 VV 0.750 NV 0.756 NV
6 0.718 NV 0.732 VV 0.415 SNS 0.793 NV 0.800 VV 0.787 VV
7 1.023 SNS 1.006 SNS 0.439 ANS 1.125 SNS 0.889 NS 0.874 NS
8 1.156 ANS 1.012 NS 0.448 NV 1.596 ANS 0.944 ANS 0.887 ANS
9 1.183 NS 1.061 ANS 0.453 VV 1.668 NS 0.958 SNS 0.887 SNS
Table 4: With structural lines: Ratio of points needed for each measure, for a given rootmean-squared error (RMSE), to the points needed for the same RMSE using the max vertical criterion; the value is the average over a broad range of RMSE.
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1 0.481 VS lcrater 0.525 VS ritz1 0.620 VS yakima3 0.697 VS yakima2 0.616 VS baker 0.624 VS
2 0.499 AVS 0.541 SSVS 0.632 SSVS 0.699 AVS 0.619 AVS 0.634 SSVS
3 0.519 SSVS 0.543 AVS 0.641 NS 0.703 SSVS 0.624 SVS 0.647 SVS
4 0.526 SVS 0.548 SVS 0.643 SVS 0.704 SVS 0.626 SSVS 0.648 AVS
5 0.558 NV 0.591 NV 0.647 AVS 0.711 NV 0.665 NV 0.685 NS
6 0.584 NS 0.615 NS 0.675 ANS 0.723 VV 0.696 NS 0.695 NV
7 0.613 SNS 0.640 SNS 0.676 SNS 0.727 SNS 0.729 ANS 0.703 ANS
8 0.626 ANS 0.649 ANS 0.731 NV 0.729 NS 0.733 VV 0.714 SNS
9 0.635 VV 0.661 VV 0.747 VV 0.742 ANS 0.740 SNS 0.743 VV
Table 5: With structural lines: Ratio of points needed for each measure, for a given mean absolute error (MAE), to the points needed for the same MAE using the max vertical criterion; the value is the average over a broad range of MAE.
23
Performance ratios No Lines
name crater
1 0.765 SSVS lcrater 0.812 SVS ritz1 0.871 SSVS yakima3 0.779 SVS yakima2 0.864 SVS baker 0.846 SVS
2 0.768 SVS 0.814 SSVS 0.871 SVS 0.784 SSVS 0.874 SSVS 0.848 SSVS
3 0.808 VS 0.849 AVS 0.882 VS 0.900 VV 0.885 VS 0.874 AVS
4 0.820 AVS 0.858 VS 0.883 AVS 0.911 VS 0.888 AVS 0.877 VS
5 0.857 NV 0.883 NV 0.914 SNS 0.948 AVS 0.926 NV 0.903 NV
6 0.869 VV 0.912 VV 0.934 NS 1.027 NV 0.993 VV 0.951 VV
7 1.125 SNS 1.146 SNS 0.935 NV 1.720 NS 1.154 NS 1.053 SNS
8 1.139 ANS 1.180 ANS 0.939 ANS 1.774 SNS 1.163 ANS 1.059 NS
9 1.145 NS 1.195 NS 0.943 VV 1.836 ANS 1.208 SNS 1.064 ANS
Table 6: Without structural lines: Ratio of RMSE for each measure, for a given number of points, to the RMSE for the same number of points using the max vertical criterion; the value is the average over a broad range of TIN sizes.
24
name crater
1 0.669 VS lcrater 0.741 AVS ritz1 0.828 VS yakima3 0.665 SVS yakima2 0.795 VS baker 0.818 VS
2 0.682 AVS 0.745 VS 0.832 AVS 0.669 SSVS 0.805 AVS 0.821 AVS
3 0.702 SSVS 0.763 SVS 0.844 SSVS 0.671 VS 0.822 SVS 0.825 SVS
4 0.710 SVS 0.763 SSVS 0.846 SVS 0.695 AVS 0.829 SSVS 0.827 SSVS
5 0.712 NV 0.774 NV 0.849 NS 0.736 NV 0.840 NV 0.858 NV
6 0.768 NS 0.856 SNS 0.852 SNS 0.758 VV 0.940 NS 0.875 NS
7 0.784 ANS 0.859 NS 0.859 ANS 0.963 NS 0.951 ANS 0.886 ANS
8 0.786 SNS 0.859 ANS 0.887 NV 1.011 SNS 0.952 VV 0.904 SNS
9 0.817 VV 0.866 VV 0.904 VV 1.036 ANS 0.981 SNS 0.915 VV
Table 7: Without structural lines: Ratio of MAE for each measure, for a given number of points, to the MAE for the same number of points using the max vertical criterion; the value is the average over a broad range of TIN sizes.
25
Lines
name crater
1 0.763 SSVS lcrater 0.829 SVS ritz1 0.849 SSVS yakima3 0.762 SSVS yakima2 0.884 SVS baker 0.840 SSVS
2 0.766 SVS 0.829 SSVS 0.852 SVS 0.775 SVS 0.897 SSVS 0.842 SVS
3 0.790 VS 0.885 VS 0.872 VS 0.869 VV 0.917 AVS 0.861 VS
4 0.809 AVS 0.885 AVS 0.881 AVS 0.900 VS 0.934 VS 0.870 AVS
5 0.853 VV 0.922 VV 0.914 NS 0.942 AVS 0.952 NV 0.906 NV
6 0.874 NV 0.930 NV 0.915 SNS 1.035 NV 1.019 VV 0.927 VV
7 1.123 SNS 1.210 NS 0.923 NV 1.703 NS 1.192 NS 1.003 NS
8 1.140 NS 1.223 SNS 0.927 ANS 1.713 SNS 1.235 ANS 1.026 ANS
9 1.199 ANS 1.236 ANS 0.927 VV 1.850 ANS 1.242 SNS 1.036 SNS
Table 8: With structural lines: Ratio of RMSE for each measure, for a given number of points, to the RMSE for the same number of points using the max vertical criterion; the value is the average over a broad range of TIN sizes.
26
name crater
1 0.650 VS lcrater 0.762 VS ritz1 0.815 VS yakima3 0.650 SSVS yakima2 0.818 AVS baker 0.799 VS
2 0.666 AVS 0.768 AVS 0.823 SSVS 0.660 SVS 0.822 VS 0.813 AVS
3 0.698 SSVS 0.777 SSVS 0.828 SVS 0.662 VS 0.834 SVS 0.816 SSVS
4 0.703 SVS 0.779 SVS 0.829 AVS 0.691 AVS 0.839 SSVS 0.820 SVS
5 0.720 NV 0.801 NV 0.836 NS 0.727 VV 0.851 NV 0.852 NS
6 0.769 NS 0.873 VV 0.854 ANS 0.734 NV 0.958 NS 0.853 NV
7 0.786 SNS 0.875 NS 0.855 SNS 0.960 NS 0.963 VV 0.869 ANS
8 0.792 6 0.890 SNS 0.874 NV 0.998 SNS 0.995 ANS 0.883 SNS
9 0.799 ANS 0.896 ANS 0.889 VV 1.039 ANS 1.005 SNS 0.895 VV
Table 9: With structural lines: Ratio of MAE for each measure, for a given number of points, to the MAE for the same number of points using the max vertical criterion; the value is the average over a broad range of TIN sizes.
27
Performance Ratios for compact TINs No Lines
name crater
1 0.722 SVS lcrater 0.756 SVS ritz1 0.823 SSVS yakima3 0.641 SVS yakima2 0.795 SSVS baker 0.779 SVS
2 0.732 SSVS 0.775 SSVS 0.829 VS 0.645 SSVS 0.798 SVS 0.808 SSVS
3 0.747 VS 0.780 NV 0.831 SVS 0.677 VS 0.813 VS 0.821 NV
4 0.770 AVS 0.794 AVS 0.848 SNS 0.705 AVS 0.856 AVS 0.823 VS
5 0.789 NV 0.823 VS 0.856 AVS 0.738 VV 0.893 NV 0.829 AVS
6 0.876 VV 0.856 VV 0.874 ANS 0.756 NV 0.934 VV 0.869 VV
7 0.964 SNS 0.904 ANS 0.881 NS 0.960 SNS 0.970 NS 0.954 SNS
8 0.993 ANS 0.914 NS 0.912 NV 1.013 NS 0.972 ANS 0.966 ANS
9 1.023 NS 0.934 SNS 0.914 VV 1.173 ANS 0.972 SNS 1.021 NS
Table 10: Without structural lines: Average ratio of RMSE for each measure, for a given number of points, to the RMSE for the same number of points using the max vertical criterion, for compact TINs.
28
name crater
1 0.673 SSVS lcrater 0.655 SSVS ritz1 0.757 SSVS yakima3 0.650 SSVS yakima2 0.719 SVS baker 0.687 SSVS
2 0.678 SVS 0.659 SVS 0.763 VS 0.658 SVS 0.721 SSVS 0.704 VS
3 0.696 VS 0.677 VS 0.765 SVS 0.699 VS 0.740 VS 0.707 SVS
4 0.717 AVS 0.702 AVS 0.774 NS 0.711 AVS 0.741 AVS 0.728 AVS
5 0.761 NV 0.740 NV 0.783 AVS 0.724 VV 0.776 NV 0.761 NV
6 0.783 VV 0.741 VV 0.784 SNS 0.761 NV 0.791 VV 0.799 SNS
7 0.891 NS 0.803 NS 0.796 ANS 0.968 SNS 0.827 NS 0.801 VV
8 0.896 SNS 0.820 SNS 0.828 NV 1.032 NS 0.843 SNS 0.805 ANS
9 0.918 ANS 0.840 ANS 0.836 VV 1.093 ANS 0.844 ANS 0.821 NS
Table 11: With structural lines: Average ratio of RMSE for each measure, for a given number of points, to the RMSE for the same number of points using the max vertical criterion, for compact TINs.
29
Lines
name crater
1 0.590 VS lcrater 0.591 VS ritz1 0.732 VS yakima3 0.457 VS yakima2 0.689 VS baker 0.674 VS
2 0.607 AVS 0.601 SSVS 0.737 SSVS 0.464 AVS 0.689 AVS 0.675 SSVS
3 0.607 SSVS 0.607 SVS 0.742 NS 0.482 SSVS 0.689 SVS 0.691 SVS
4 0.610 SVS 0.611 AVS 0.747 SVS 0.486 SVS 0.693 SSVS 0.694 AVS
5 0.648 NV 0.634 NS 0.751 AVS 0.496 NV 0.725 NV 0.718 ANS
6 0.657 NS 0.652 NV 0.761 SNS 0.547 VV 0.742 NS 0.723 NS
7 0.684 SNS 0.666 SNS 0.763 ANS 0.557 SNS 0.750 VV 0.729 NV
8 0.689 ANS 0.668 ANS 0.800 NV 0.571 NS 0.759 ANS 0.730 SNS
9 0.698 VV 0.683 VV 0.810 VV 0.607 ANS 0.764 7 0.768 VV
Table 12: With structural lines: Average ratio of MAE for each measure, for a given number of points, to the MAE for the same number of points using the max vertical criterion, for compact TINs.
30
name crater
1 0.637 VS lcrater 0.685 AVS ritz1 0.788 VS yakima3 0.456 VS yakima2 0.752 VS baker 0.758 SVS
2 0.680 AVS 0.702 SVS 0.797 SSVS 0.474 AVS 0.759 SSVS 0.762 VS
3 0.686 SVS 0.710 VS 0.808 AVS 0.503 SSVS 0.766 SVS 0.762 AVS
4 0.688 SSVS 0.712 NV 0.810 SVS 0.508 SVS 0.787 AVS 0.785 SSVS
5 0.718 SNS 0.719 SSVS 0.816 SNS 0.513 NV 0.831 NV 0.788 NV
6 0.737 NS 0.721 ANS 0.819 ANS 0.567 SNS 0.843 NS 0.818 SNS
7 0.738 ANS 0.727 NS 0.830 NS 0.593 VV 0.851 ANS 0.822 ANS
8 0.742 NV 0.744 SNS 0.881 NV 0.597 NS 0.851 SNS 0.823 NS
9 0.823 VV 0.825 VV 0.886 VV 0.623 ANS 0.890 VV 0.833 VV
Table 13: Without structural lines: Average ratio of MAE for each measure, for a given number of points, to the MAE for the same number of points using the max vertical criterion, for compact TINs.
31
