Variable Resolution Spatial Interpolation Using the Simple Recursive Point Voronoi Diagram

Transcription

1 Geographical Analysis ISSN Using the Simple Recursive Point Voronoi Diagram Robert Feick, 1 Barry Boots 2 1 Department of Geography, School of Planning, University of Waterloo, Waterloo, ON, Canada, 2 Department of Geography and Environmental Studies, Wilfrid Laurier University, Waterloo, ON, Canada This article introduces a procedure for progressively increasing the density of an initial point set that can be used as a basis for interpolating surfaces of variable resolution from sparse samples of data sites. The procedure uses the Simple Recursive Point Voronoi Diagram in which Voronoi concepts are used to tessellate space with respect to a given set of generator points. The construction is repeated every time with a new generator set, which comprises members selected from the previous generator set plus features of the current tessellation. We show how this procedure can be implemented in Arc/Info and present an illustration of its application using three known surfaces and alternative generator point configurations. Initial results suggest that the procedure has considerable potential and we discuss further methods for evaluating and extending it. Introduction Spatial data consist of measurements (data values) of an attribute taken at specific locations (data sites) in a geographic space (study region). In much of the data collected in the environmental sciences, the attribute is assumed to be spatially continuous (possibly piecewise) so that the data values can be considered as a sampling of the attribute at the data sites. Suppose that our primary aim is to use the data to interpolate the values of the attribute at locations other than the data sites, thus enabling us to create a visual representation of the underlying surface (Watson 1992). This task can be considered as a specific instance of the more general problem of approximating surfaces that arises in a variety of applications in computeraided design, computer graphics, computer vision, and finite element methods. In some applications, where the number of data sites is very large, it may not be necessary to use all the data sites in order to capture the fundamental features of the Correspondence: Robert Feick, Department of Geography, University of Waterloo, Waterloo, Canada ON N2L 3G1 rdfeick@fes.uwaterloo.ca Submitted: November 11, Revised version accepted: June 6, Geographical Analysis 37 (2005) r 2005 The Ohio State University 225

2 Geographical Analysis surface. Instead, a representative subset of data sites, which enables the surface to be represented with a prespecified level of accuracy, can be selected. This can be achieved either by a process of simplification (also referred to as decimation or coarsening) in which we start with the complete set of data sites and remove some of them or by a process of refinement when we start with an insufficient subset of data sites and add additional ones (see Garland [1999] and De Floriani and Magillo [2002] for reviews). However, there are also situations where the data sites are sparse and there is no possibility of supplementing their number by increasing the size of the sample. Typically, this applies to most historical data. Further, notwithstanding advances in remote sensing and global positioning system data collection methods, in some instances it can still be impractical or too costly to collect appropriately detailed data. In addition, with historical data, we may also lack knowledge of the sampling design, although some characteristics may be inferred from the spatial distribution of data sites. Much less attention has been applied to this situation, which is the focus of this article. We propose a procedure for progressively increasing the density of an initial point set that provides a basis for interpolating surfaces of variable resolution from sparse samples of data sites. The procedure makes use of the simple recursive point Voronoi diagram (SRPVD). Our proposal is motivated by both practical and conceptual considerations. Practically, the procedure can be fully automated and implemented within existing commercial off-the-shelf (COTS) software. Conceptually, it is supported by recognition of the dual relationships between Voronoi and Delaunay constructions. The latter have a long history of use in terrain modeling and visualization, with the Delaunay triangulation being the most popular method of constructing a triangulated irregular network (TIN; Hutchinson and Gallant 1999). In part, this is because it is the only triangulation that satisfies both local and global max min angle criteria (Sibson 1978; Okabe et al. 2000, p. 93). Wang et al. (2001) also demonstrate that the basic Delaunay triangulation even outperforms data-dependent triangulations in modeling terrain surfaces. Further, several recursive forms of Delaunay triangulations have already been used as top-down, multiresolution TIN models in the refinement process described above (see De Floriani, Marzano, and Puppo 1996; De Floriani et al for reviews). These include strictly hierarchical TINs (HTINs), which involve the recursive subdivision of the initial triangle(s) into a set of nested triangles (De Floriani and Puppo 1992, 1995), and nonnested, pyramidal TINs (PTINs) in which a new structure is computed every time new points are inserted (De Floriani, Falcidieno, and Pienovi 1985; De Floriani 1989; Voightmann, Becker, and Hinrichs 1994). It will become apparent below that, in essence, the SRPVD is a dual form of PTIN. We begin our presentation by describing recursive Voronoi diagrams in general terms and providing a specific definition of the SRPVD. We then show how these constructions can be used to create surfaces of varying spatial resolution. In the third main section we describe how these procedures can be implemented in Arc/ 226

3 Robert Feick and Barry Boots Info. This is followed in the fourth section by an application of the procedure to point samples drawn from three known topographic surfaces using three different sampling designs. We conclude by discussing outstanding issues and directions for future work. Recursive Voronoi diagrams The basic Voronoi concept involves tessellating an m-dimensional space with respect to a finite set of objects by assigning all locations in the space to the closest member of the object set. This concept can also be applied recursively by tessellating the space with respect to a given set of generators and then repeating the construction every time with a new generator set consisting of objects selected from the previous generator set plus features of the current tessellation. More formally, consider a finite set of n distinct generators, G. First, construct the Voronoi diagram V(G) ofg. Next, extract a set of features Q from V(G) and create a new set of generators G 0 that comprises Q plus selected members of G. Then construct the Voronoi diagram V(G 0 )ofg 0. This step is then repeated a number of times. At each step, the number of generators that are retained may range from none to all. Boots and Shiode (2003) show that such recursive constructions provide an integrative conceptual framework for a number of disparate procedures in spatial analysis and modeling. Simple recursive point Voronoi diagram In this article, we consider a specific form of recursive Voronoi diagram, the SRPVD, in which the initial set of generators consists of n distinct points G (0) 5 {g (0)1, g (0)2,...,g (0)n }. The construction of this diagram involves the following steps: 0. define the initial set of generators G (0) ; 1. generate the ordinary Voronoi diagram V(G (0) )ofg (0) ; 2. extract all m(0) Voronoi vertices Q (0) 5 {q (0)1, q (0)2,...,q (0)m(0) }ofv(g (0) ); 3. create a new set of generator points G (1) 5 G (0) 1Q (0) ;and 4. repeat steps 1 through 3. We call the result of the kth construction, V(G (k) ), the kth generation of the SRPVD. Similarly, we call its generator set G (k) and its vertices Q (k) the kth generation of the point set and the kth generation of the vertices, respectively. The results of applying these steps for five recursions to an initial generator set consisting of five points are shown in Fig. 1. Note that the Voronoi polygons at generation k are either entirely contained within a Voronoi polygon at generation (k 1), that is, V(G (k)i ) V(G (k 1)i ), or are composed of pieces of three polygons from generation (k 1), that is, V(G (k)i ) 5 (V(G (k)i ) \ V(G (k 1)l )) [ (V(G (k)i ) \ V(G (k 1)m )) [ (V(G (k)i ) \ V(G (k 1)n )) (see Fig. 2). As noted above, a recursive 227

4 Geographical Analysis Figure 1. The first five recursions of a simple recursive point Voronoi diagram. Voronoi structure will have a dual recursive Delaunay structure. The recursive Delaunay construction that is equivalent to the SRPVD is defined as follows: 0. define the initial set of generators G (0) ; 1. generate the Delaunay triangulation D (0) of G (0) ; 2. identify the circumcenters C (0) 5 {c (0)1,...,c (0)m } of all the triangles of D (0) ; 3. create a new set of generator points G (1) 5 G (0) 1C (0) ; and 4. repeat steps 1 and 3. Figure 2. Simple recursive point Voronoi diagram generations

5 Robert Feick and Barry Boots From this description, it will be seen that our procedure is similar to the refinement strategies that have been used in some of the other application areas noted in the Introduction. A common feature of these strategies is that they start with a triangulation of the data sites, insert one or more points in each triangle, and then build another triangulation. While our procedure inserts new points at the circumcenters of the triangles, others insert points at other locations within the triangles or on their edges (e.g., Scarlatos and Pavlidis 1992; De Floriani and Puppo 1995; Cignoni, Puppo, and Scopigno 1997; Klein and Straer 1997). Using the SRPVD to construct variable resolution surfaces Using the SRPVD, the following procedure can be used to produce topographic surfaces that have increasing spatial resolution: Let G (0) be an initial set of generators, each with an associated data value, and label the set of data values D (0). Generate V(G (0) ). Extract the set Q (0) consisting of the m (0) vertices of V(G (0) ). Extrapolate data values for Q (0) using Sibson s natural neighbor interpolation (Sibson 1981) and D (0). Label extrapolated values D (1). If desired, create surface representation from D (0) 1D (1). Let G (1) 5 G (0) 1Q (0). Generate V(G (1) ). Extract the set Q (1) consisting of the m (1) vertices of V(G (1) ). Extrapolate data values for Q (1) using Sibson s natural neighbor interpolation and D (0) 1D (1). Label extrapolated values D (2). If desired, create surface representation from D (0) 1D (1) 1D (2).. Let G (k) 5 G (k 1) 1Q (k 1). Generate V(G (k) ). Extract the set Q (k) consisting of the m (k) vertices of V(G (k) ). Extrapolate data values for Q (k) using Sibson s natural neighbor interpolation and D ð0þ þ D ð1þ þþd ðkþ. Label extrapolated values D (k11). If desired, create surface representation from D ð0þ þ D ð1þ þþd ðkþ1þ. We use Sibson s interpolation procedure because the input it requires is readily obtained from two successive generations of the SRPVD. For generators in the general quadratic position, each interpolated value in D (k11) will be interpolated from three values in D ð0þ þ D ð1þ þþd ðkþ.ifd (k11)i is the interpolated value at Q (k)i and V(Q (k)i ) is the Voronoi polygon of Q (k)i, and V(Q (k)i ) is the area of V(Q (k)i ), 229

6 Geographical Analysis then Note that: D ðkþ1þi ¼ X3 p¼1 jvðq ðkþi Þj ¼ X3 jvðq ðkþi Þ\VðQ ðk 1Þp Þj D ðkþp jvðq ðkþi Þj p¼1 jvðq ðkþi Þ\VðQ ðk 1Þp Þj ð1þ By selecting vertices exclusively, an entire set of interpolated values can be generated in one pass. Further, vertices may be considered as locally optimal sites for new data interpolation locations because they maximize the distance from triples of existing interpolation locations. Operationalizing SRPVD construction in Arc/Info The SRPVD constructions described in this article were built with the Arc/Info 8.3 geographical information system platform. The input consists of a set of generator seed points, the G (0) generator set described in the previous section, and an associated set of attribute values. These data, which we refer to as Gen_i_Pts, were used as the basis for each of five Voronoi recursions. The general procedure for generating the SRPVD construction is as follows with specific Arc/Info commands identified for reference in Courier font: 1. Create a Voronoi polygon coverage for Generation i (Gen_i_VD) Gen_i_Pts using Thiessen. 2. Build node and line topology for Gen_i_VD. 3. Add a Gen field to the node attribute table of the Gen_i_VD coverage to track the age of each point. The value of the Gen field is set to i for all nodes. 4. Convert all nodes in Gen_i_VD to a point coverage (VD_i_Pts) using Node- Point. 5. Create Gen_i11_Pts coverage by Appending Gen_i_Pts and the VD_i_Pts. 6. Repeat steps 1 5 based on the updated Gen_i11_Pts coverage for k generations. Fig. 2 illustrates this procedure for generations 0 2. In practice, although the Voronoi structure is not bounded in a conceptual sense, a spatial limit needs to be established to minimize the impact of edge effects on subsequent analyses. This is discussed further in the next section. Next, attribute values for the new data points created using the process described above are interpolated on a generation-by-generation basis. The interpolation procedure is summarized below and is also illustrated in Fig. 3: 1. Union Gen_i_VD and Gen_i11_VD Voronoi polygon coverages to create a Union_i_i11 polygon coverage. 230

7 Robert Feick and Barry Boots Figure 3. Interpolation procedure example. 2. Establish relates between Union_i_i11, Gen_i_VD, and Gen_i11_VD. Calculate the ratio of each Union_i_i11 polygon s area relative to the area of the corresponding parent VD_i polygon (see shaded polygons in Fig. 3). Store the value in a PercentArea field. 3. Calculate the attribute value being interpolated for each Union_i_i11 polygon as PercentArea the attribute value recorded for the parent Gen_i_VD polygon. Store the value in a CalcAttribute field. 4. Use statistics on the Union_i_i11 coverage to create a summary table (Gen_i_stats) that lists the sum of the CalcAttribute field for each Gen_ i11_vd polygon. 5. Reselect the Gen_i11_VD and Gen_i11_Pts features for generation i11 and transfer the corresponding interpolated values from the Gen_i_stats table using relates. Illustrations To demonstrate our procedure, we applied it to known second-, third-, and fourthorder surfaces shown in Figs For convenience, these surfaces are referred to as Surfaces A, B, and C. The elevation ranges of these surfaces are 4185, 3194, and 2414 units, respectively. Three initial generator sets (sets of data sites) were considered for each surface in order to investigate the impact of generator set configuration on the results. The 121 points in the generator sets were arranged according to a randomly jittered square (JS), a stratified random (SR), and a triangular grid (TG) sampling design. Figs. 4 6 each illustrate one of these generator point configurations. 231

8 Geographical Analysis Figure 4. Surface A with jittered square generator set. Figure 5. Surface B with stratified random generator set. 232

9 Robert Feick and Barry Boots Figure 6. Surface C with triangular grid generator set. We used the procedure described in the section on Using the SRPVD to construct variable resolution surfaces to generate additional data sites and data values up to the 5th generation of the SRPVD. Column 2 in Table 2 ( All points ) lists the number of the data sites by generation from the initial point set (generation 0) to generation 5. The corresponding data for Surfaces B and C are provided in Tables 3 and 4. We undertook a series of error analyses to assess the performance of our procedure by comparing the elevation values calculated for new data points with values generated using the trend surface coefficients listed in Table 1. We recognize that our procedure, like all interpolation procedures, is subject to edge effects. Consequently, although we allowed new data points to be generated beyond the boundary of the convex hull of the sample points, we did not consider associated with such points. Columns 3, 6, and 9 ( Points examined ) in Tables 2 4 list the number of data points that remain in the analysis. As the JS, SR, and TG generator sets produce unique convex hulls, the number of points examined differs slightly across the three sample configurations. Note that the remaining content of Tables 2 4 (i.e., Extreme error columns) is discussed later in this section. To examine overall performance, we calculated both the mean absolute error (MAE) and the root-mean-squared error (RMSE) by generation (see Figs. 7 9). MAE is less sensitive to large than RMSE and so gives a better picture of the overall performance of the interpolation. Figs. 7 9 show MAE and RMSE by generation for 233

10 Geographical Analysis Table 1 Coefficients of Trend Surfaces Coefficient Surface A Surface B Surface C Constant 12, x y x E E 006 xy E E E 006 y E E E 005 x E E E 011 x 2 y E E E 011 xy E E E 011 y E E E 011 x E E 017 x 3 y E E 017 x 2 y E E 017 xy E E 017 y E E 016 x E 023 x 4 y E 023 x 3 y E 023 x 2 y E 023 xy E 022 y E 023 Surfaces A C. Note that as the elevation ranges differ across the three surfaces, MAE and RMSE are expressed as percentages to permit comparisons of error. Figs. 7 9 show that the average MAE over all generations is relatively small at less than 1.5 percent of the elevation ranges. For all three surfaces, there is a noticeable reduction in MAE between the first two generations, after which MAE stabilizes. The MAE values for the TG sample design prove to be an exception to this general statement as they increase from generation 2 to 3 before stabilizing. The behavior of RMSE is more variable, in part, reflecting its sensitivity to extreme. Nevertheless, the average RMSE for the JS and SR samples are still relatively low at approximately 2 percent of their elevation ranges. The RMSE (and, to a lesser extent, the MAE) values for the SR generator set were affected by three generation 5 data points that were assigned very low elevation values in the interpolation procedure. Some 15 generation 5 data points in the TG generator runs were also affected by this problem which caused the TG RMSE values on all three surfaces to increase noticeably at generation 5. The cause of these particular is the subject of ongoing investigation. Given the large number of data points associated with the later generations of the SRPVD construction, we focused further analyses on absolute of at least 5 percent of the elevation range of a surface. We label these extreme. The 234

11 Robert Feick and Barry Boots Table 2 Simple Recursive Point Voronoi Diagram (SRPVD) Generated Data Points and Extreme Errors by Generation (Gen.) Surface A Gen. All points Jittered square Stratified Random Triangular Grid Points # examined Extreme % Extreme Points # examined Extreme % Extreme Points # examined Extreme % Extreme ,764 16, , , Totals 29,645 24, , , With Surface A, an SRPVD data point is classed as an extreme error if its interpolated value deviates (1 or ) by 209 m from the corresponding trend surface value. number of extreme associated with Surfaces A C are listed in Tables 2 4 by generation and by generator set (i.e., JS, SR, TG). As the number of data points differ across generations and across sampling designs, the tables also report extreme as percentages of the number of points examined. The last rows in Tables 2 4 show that when an entire set of points (generations 1 5) is considered, extreme account for between 1.8 and 8.1 percent of the data points. Interestingly, the RS sampling design appeared to perform the best across all three surfaces, followed closely by the JS generator set. In contrast, the TG sample proved to have the highest number of extreme. We anticipated that would be most evident in the earliest generations because, on average, these points would be furthest from the initial generator points. When the proportion of extreme are considered, this assumption appears to be valid for the JS and RS samples on Surfaces A and B as the highest Table 3 Simple Recursive Point Voronoi Diagram (SRPVD) Generated Data Points and Extreme Errors by Generation (Gen.) Surface B Gen. All points Jittered square Stratified random Triangular grid Points # examined Extreme % Extreme Points # examined Extreme % Extreme Points # examined Extreme % Extreme ,764 16, , , Totals 29,645 24, , , With Surface B, an SRPVD data point is classed as an extreme error if its interpolated value deviates (1 or ) by 160 m from the corresponding trend surface value. 235

12 Geographical Analysis Table 4 Simple Recursive Point Voronoi Diagram (SRPVD) Generated Data Points and Extreme Errors by Generation (Gen.) Surface C Gen. All points Jittered square Stratified random Triangular grid Points # Extreme % examined Extreme Points # Extreme % examined Extreme Points # Extreme % examined Extreme ,764 16, , , Totals 29,645 24, , , With Surface C, an SRPVD data point is classed as an extreme error if its interpolated value deviates (1 or ) by 121 m from the corresponding trend surface value. percent of extreme is found in generation 1. Tables 2 4 indicate that this claim cannot be made for the TG sample as only one extreme error was recorded across all three surfaces up to generation 2, after which the proportion of extreme increased substantially. We also expected that the magnitude of would be highest in the earlier generations. However, the greatest occurred in generation 5 in all cases. This suggests that are propagated with advancing generations. This effect is most evident with increasing proximity to the edge of the study area (i.e., the convex hull). By generation 5, the SRPVD procedure produces relatively dense Figure 7. Mean absolute error (MAE) and root-mean-squared error (RMSE) Surface A. 236

13 Robert Feick and Barry Boots Figure 8. Mean absolute error (MAE) and root-mean-squared error (RMSE) Surface B. concentrations of new data points in this area; hence, edge effects could be more pronounced at this level. In addition to examining the overall level of interpolation error, we were interested in the spatial distribution of error. In Figs , the extreme associated with selected surface and generator set combinations are displayed in Figure 9. Mean absolute error (MAE) and root-mean-squared error (RMSE) Surface C. 237

14 Geographical Analysis Figure 10. Surface A extreme jittered square generators. three dimensions (see for the complete set of figures). Positive (overestimates) are shown as dark-colored extrusions above the surface in these figures, while negative (underestimates) are shown as light extrusions that extend below the surface. No vertical exaggeration was applied to the surfaces; however, the magnitude of the was multiplied by 3 to produce Z-values that could be more easily visualized. For convenience, Surfaces A C are shown extending somewhat beyond the convex hulls associated with the JS, RS, and TG generator sets. Figs show that negative are both more common and generally of a greater magnitude than positive. As underestimation is more evident as surface complexity increases, we suspect that at least part of this tendency is related to the generator sets being insufficient in size to capture topographic variations adequately. The figures also show that most of the extreme are concentrated around the edges of the convex hulls formed by the initial generator points. This is particularly apparent for the TG generator set (see Fig. 13) as all of the extreme for all three surfaces are found in proximity to the periphery of the study area. This suggests that the associated with the TG sample are at least partially an artifact of the original point configuration given that the generators were located on a nonequilateral triangular grid. The presence of the 15 most pronounced extreme Figure 11. Surface A extreme stratified random generators. 238

15 Robert Feick and Barry Boots Figure 12. Surface B extreme stratified random generators. does account for the high RMSE values evident for generations 3 5 of the TG generator set as well. The interior in the JS and RS figures show no obvious spatial distribution although there is a tendency for them to be located in small local clusters. While some of these clusters appear to be associated with initial generator points, this does not appear to be a general tendency. Indeed, we had anticipated that there might be a positive relationship between error magnitude at a generated location and the distance of that location to the nearest initial generator point. However, with the exception of the 1st generation of some surface and sample design combinations, this relationship was not found. As an example, Fig. 16 shows the situation for Surface A and the JS generator set. In this figure, the distance of each extreme error point to the nearest generator point (generation 0) is plotted. Based on the arrangement of the JS generator points, the furthest a new data point could be from a generator point would be approximately 25,000 m. In contrast, the largest concentration of extreme occurs at distances between 7000 and 12,000 m. Discussion and conclusions The primary aim of this article was to introduce an automated method, based on the SRPVD, for progressively increasing the density of an initial point set, thus providing a basis for interpolating surfaces of variable resolution from sparse samples of Figure 13. Surface B extreme Triangular grid generators. 239

16 Geographical Analysis Figure 14. Surface C extreme jittered square generators. data sites. As we show, an attractive feature of this procedure is that it can be readily implemented in Arc/Info. In our illustrative examples, absolute in excess of 5 percent of the variable ranges generally occurred at less than 5 percent of the generated data sites. Clearly, the global error measurements (e.g., MAE, RMSE, extreme error counts) discussed were affected by the general problem of interpolating data values near a study area boundary. However, the lack of systematic relationships between error values and characteristics of either the underlying surface or the initial point set suggests that the procedure may be robust with respect to these features. Further, the observation that, after an initial change, the average level of error remains relatively constant may be a desirable feature for those who require a multiresolution hierarchical data set with little variation in accuracy at different levels. For example, in hydrological modelling, one may wish to use a finer spatial resolution for detailed model runs while maintaining consistency with coarser resolutions used for cruder models. However, in order to confirm our initial findings much more testing needs to be carried out. For example, we need to further explore the sensitivity of the results for a given surface to different spatial sampling designs for selecting the generator points and to different sizes of generator point sets. We also need to consider a range of surfaces with different characteristics, especially more complex surfaces. Also, of course, we need to compare the performance of the procedure relative to other COTS interpolation procedures. Figure 15. Surface C extreme stratified random generators. 240

17 Robert Feick and Barry Boots Figure 16. Surface A extreme and generator proximity jittered square generators. Notwithstanding the need for further testing, there are several ways the procedure may be refined and extended. For example, it is possible, and probably desirable, not to add all the Voronoi vertices of a given generation to the generator set for the next generation. In the present context, there are at least two circumstances where this would be appropriate. First, we may wish to constrain the new generators to those within a specific distance range of existing generators. If the potential new generators are very close to existing ones, we can expect that they may add little new information and thus may be discarded. Similarly, if a potential new generator is more than some specified distance from an existing generator it may be unwise to extend interpolation to that generator. One means of identifying such an upper limit would be to use the characteristics of the semi-variogram of the initial data points, perhaps limiting new generators to those whose distance to existing generators is less than half of the range of the semi-variogram. A similar strategy has been used successfully in the choice of control points in classification of spatial imagery into classes (Shine and Wakefield 1999) and in two other articles in this issue (Goovaerts 2005; Kyriakidis and Yoo 2005). We may also wish to relate the addition of new generators to the nature of local changes in attribute values. While it is useful to have increased resolution in areas where there are marked changes in attribute values, little is gained by increasing the resolution in areas where there is little variation. There has already been some initial consideration of such a generator constrained recursive point Voronoi diagram (Boots et al. 2002). Although we illustrated our procedure using quantitative data, it is also applicable to categorical data. In this case, instead of a single value being interpolated at each generated point, a vector of fuzzy membership values (FMVs) for classes can be interpolated (Lowell 1994) and equation (1) will need to be adjusted accord- 241

18 Geographical Analysis ingly. FMV surfaces can then be generated for each class. Alternatively, at each recursion, each polygon can be labeled with the value of its generator and these values mapped to create a piecewise continuous surface (choropleth map). Acknowledgements We are grateful for the helpful comments of three anonymous reviewers and the guest editor, which led to improvements in both the article s content and presentation. References Boots, B., R. D. Feick, N. Shiode, and S. Roberts. (2002). Investigating Recursive Point Voronoi Diagrams. In Geographic Information Science: Second International Conference, GIScience 2002, Boulder, CO, USA, September Lecture Notes in Computer Science, Vol. 2478: 1 21, edited by M. J. Egenhofer and D. M. Mark. Berlin, Germany: Springer-Verlag. Boots, B., and N. Shiode. (2003). Recursive Voronoi Diagrams. Environment and Planning B: Planning and Design 30, Cignoni, P., E. Puppo, and R. Scopigno. (1997). Representation and Visualization of Terrain Surfaces at Variable Resolutions. The Visual Computer 13, De Floriani, L. (1989). A Pyramidal Data Structure for Triangle-Based Surface Description. IEEE Computer Graphics and Applications 9, De Floriani, L., B. Falcidieno, and C. Pienovi. (1985). Delaunay-Based Representation of Surfaces Defined over Arbitrarily Shaped Domains. Computer Vision, Graphics, and Image Processing 32, De Floriani, L., and P. Magillo. (2002). Multiresolution Mesh Representation: Models and Data Structures. In Multiresolution in Geometric Modelling, , edited by M. Floater, A. Iske, and E. Quak. Berlin, Germany: Springer-Verlag. De Floriani, L., P. Magillo, S. Bussi, and E. Bailey. (2000). Triangle-Based Surface Models. In Intelligent Systems and Robotics, , edited by G. W. Zobrist and C. Y. Ho. Australia: Gordon and Breach Scientific Publishers. De Floriani, L., P. Marzano, and E. Puppo. (1996). Multiresolution Models for Terrain Surface Description. The Visual Computer 12, De Floriani, L., and E. Puppo. (1992). A Hierarchical Triangle-Based Model for Terrain Description. In Theories and Methods of Spatio-Temporal Reasoning in Geographic Space. Lecture Notes in Computer Science, Vol. 639: , edited by A. Frank, I. Campari, and U. Formentini. Berlin, Germany: Springer-Verlag. De Floriani, L., and E. Puppo. (1995). Hierarchical Triangulation for Multi-Resolution Surface Description. ACM Transactions on Graphics 14, Garland, M. (1999). Multiresolution Modeling: Survey & Future Opportunities. Eurographics 99, STAR State of the Art Reports, Goovaerts, P. (2005). Exploring Scale-Dependent Correlations Between Cancer Mortality Rates Using Factorial Kriging and Population Weighted Semivariograms: A Simulation Study. Geographical Analysis 37,

19 Robert Feick and Barry Boots Hutchinson, M. F., and J. C. Gallant. (1999). Representation of Terrain. In Geographical Information Systems: Principles, Techniques, Applications, and Management, Vol. 1, 2nd ed , edited by P. A. Longley, M. F. Goodchild, D. J. Maguire, and D. W. Rhind. New York: Wiley. Klein, R., and W. Straer. (1997). Generation of Multiresolution Models from CAD Data for Real Time Rendering. In Theory and Practice of Geometric Modeling (Blaueuren II), edited by R. Klein, W. Straer, and R. Rau. Berlin, Germany: Springer-Verlag. Kyriakidis, P. C., and E.-H. Yoo. (2005). Geostatistical Prediction/Simulation of Point Values from Areal Data. Geographical Analysis 37, Lowell, K. E. (1994). A Fuzzy Surface Cartographic Representation for Forestry Based on Voronoi Diagram Area Stealing. Canadian Journal of Forest Research 24, Okabe, A., B. Boots, K. Sugihara, and S. N. Chiu. (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd ed. Chichester, UK: Wiley. Scallatos, L. L., and T. Pavlidis. (1992). Hierarchical Triangulation Using Cartographic Coherence. CVGIP: Graphical Models and Image Processing 54(2), Shine, J. A., and G. I. Wakefield. (1999). A Comparison of Supervised Imagery Classification Using Analyst-Chosen and Geostatistically-Chosen Training Sets. Geocomputation 99 (CD-ROM). Sibson, R. (1978). Locally Equiangular Triangulations. The Computer Journal 21, Sibson, R. (1981). A Brief Description of Natural Neighbour Interpolation. In Interpreting Multivariate Data, edited by V. Barnett. New York: Wiley. Voightmann, A., L. Becker, and K. Hinrichs. (1994). Hierarchical Surface Representations Using Constrained DELAUNEY (sic) Triangulations. In Advances in GIS Research: International Symposium on Spatial Data Handling, Vols 1 & 2: Wang, K., C. P. Lo, G. A. Brook, and H. A. Arabnia. (2001). Comparison of Existing Triangulation Methods for Regularly and Irregularly Spaced Height Fields. International Journal of Geographical Information Science 15, Watson, D. F. (1992). Contouring: A Guide to the Analysis and Display of Spatial Data. Oxford, UK: Pergamon Press. 243