Hierarchical Voronoi diagrams for spatial data indexing and paging

1. Hierarchical Voronoi diagrams for spatial data indexing and paging Christopher Gold Dept. Computing & Mathematics, University of Glamorgan, Pontypridd CF37 1DL Wales UK

2. Introduction • The Voronoi diagram (VD) partitions space into cells whose interior points are closest to the generating point • The Delaunay triangulation (DT) is the dual of this – its edges connect generators that have adjacent cells • We can walk through either by using simple geometric tests and following the VD or DT edges

4. The Problem • Hierarchies are desirable for rapid searching, paging large data sets, etc. • We walk through the higher level cells to find the one containing the desired location, drop to the next level and continue at a finer resolution • BUT: The VD cannot be made directly into a hierarchy – lower level cells will not fit exactly into higher level cells

5. Two Voronoi levels

6. Spatial Indexing • Traditionally we have a coarse grid whose cells contain finer details • Quad-trees: the map is subdivided into hierarchical rectangles as points are added • Spherical trees: triangles on the sphere • Our approach: Christaller hierarchy (administrative optimization model)

7. Christaller hierarchy

8. “FastWalk” • This implies that a cell is within a higher level index cell if its generator is inside • We can “walk” from cell to cell as described before, then drop down to the next level and continue • Each index generator has a pointer down to some child generator in the index cell • This gives a logarithmic search time

9. Three-level Voronoi hierarchy

10. Three levels of Voronoi cells

11. Index Generators • Any desired method may be used to get generators at the index level • - They may be sampled from the lower level generators • - They may be on a grid • - They may be random • Special care is needed if an index cell is nearly empty • These principles are enough to give fast access to data entirely stored in memory

12. Data Structures • Edge structures are preferable for navigating through a level as they allow direct walking rather than searching within individual triangles, etc. • We use the Quad-Edge data structure as it stores both the primal and the dual links • We walk along the edges to locate points • Index generators have “down” pointers giving access to a more detailed level

13. The Quad-Edge data structure TQuad (Type of one quad) N : TQuad (N points to the next Edge/ Face) R : TQuad (R points to the next ¼ Edge (TQuad)) V : TPoint (V points to the vertices/ faces)

15. Make-Edge

17. Splice

18. Paging • Spatial indexing is straightforward; paging is harder if our structure is split into disc pages • However, the same approach may be used: • Each index cell contains one page • When searching we walk along the index level and then drop down to a page • To allow crossing from one page to the next we keep duplicates of the edges that cross to the next page, one on each side

22. Page Boundaries • When walking through the structure we locate a vertex at an index level and move down to a page • To continue, or to find its neighbours, we may leave that page: we detect this as the other end point of the edge is on another disc page • Walking to the end point on the other page (via the index level) gives us the edge connectivity • The same approach may be used for parallel processing • This method puts spatially adjacent points on the same disc page – a very important principle

23. Paged Walk

24. Moving Points • Data structures may be maintained by dynamic point insertion and deletion, and by kinetic point movement. • These operations, along with the “Walk” algorithm, may be performed on the paged structure.

25. The dynamic point VD and DT • Point insertion using incremental algorithm • Point deletion – inverse process using Devilliers algorithm

26. Kinetic moving-points

27. Applications • Christaller administration optimization • Image boundaries • Terrain modelling and generalization • Lidar data processing • LOD (Level Of Detail) of terrain • Paging/indexing of large triangulations

28. Boundary approximation

29. Voronoi cells of some generators

30. Raw Lidar data

31. Voronoi cell approximation

32. Terrain Visualization and Level of Detail (LOD) • Different LOD with distance from viewpoint • Different LOD with relief (roughness) • Similar to standard LOD with several levels precomputed

33. Terrain model – level 0

34. Level 1 generators

35. Level 2 generators

36. Overlapping level 0 and level 1

37. Overlapping level 1 and level 2

38. Conclusions and Acknowledgements • This is a general purpose hierarchy that uses the existing topology structures • It provides a coherent paging scheme – nearby points are on the same page • It follows the standard Voronoi proximal model, and fast indexing methods