Points

Abstract

We have seen how dominance, the assignment of spatial locations to the closest generator, leads to a fundamental spatial partition -the Voronoi diagram. In some cases only the generator location is known, and in others some attributes, such as colour, may be assigned.In forest mapping, the Voronoi diagram clearly gives a good estimate of the context of an individual tree in a forest: its constraining neighbours are found directly if tree crowns may be detected from aerial imagery. The locations of the tree crowns may then be used as generators, and the area of each associated Voronoi cell gives a reasonable first estimate of the tree's spread, in competition with its neighbours, and tables may be constructed to compare this with the tree height and wood volume. (This applies where the trees are in close competition, and does not apply at the edge of the forest stand.) Summing these volume estimates for a particular area may be used to estimate harvestable timber. In some cases the tree type may also be estimated, and the resulting volume totals used for harvest planning.The same approach may be used for other types of natural resource assessments. Coal thickness for example, or permeable aquifer thickness, is often assessed by shallow drilling -and the total thickness observed at each location. (This may often involve several thin strata, and often only the total thickness is needed -this is difficult by modelling the geological 'tops' of each stratum. Experience suggests that stable regional estimates may be made by assuming the local thickness at any location is best estimated by the closest real observation -the Voronoi cell generator. Summing the thickness times the cell area often gives satisfactory results for minimum effort. The same holds true for total rainfall estimation based on rain gauges -as originally suggested by Mr. Thiessen.A related, and common, problem is that of point density mapping. This is usually attempted by counting circles -counting how many points fall within a given circle, and then moving it and trying again: a very small circle, or a very large one, would give implausible results. Noting that point density (points per unit area) is the reciprocal of the Voronoi cell area (area per unit point) greatly simplifies this process. In this case the local density around each generator is 1/(cell area). Mapping this directly will often give a highly variable density surface, and some traditional smoothing function is often appropriate. Nevertheless, the whole problem of using counting circles has been eliminated.A significant application of this approach may be found in the work of van de Weijgaert described earlier. A major catalogue of a significant portion of the cosmos was performed by the 2dF Galaxy Redshift Survey, completed in 2002, extending over a huge number of galaxies, and light-years, in 3D ( Figure 78). In order to speculate on the cosmic structure they needed to produce a density map rather than just galaxies/points, and so they developed the 'Delaunay Tessellation Field Esti...