1. Visibility Preserving Terrain Simplification An Experimental Study
Boaz Ben-Moshe (Ben-Gurion U.)
Matthew Katz (Ben-Gurion U.)
Joseph Mitchell (U. at Stony Brook)
Yuval Nir (Ben-Gurion U.)
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2. Project Goals
Define a visibility-based measure of quality of simplification.
Develop a visibility preserving terrain simplification method - VPTS.
Should preserve most of the visibility
Should be efficient
Experiment with VPTS, as well as with other TS methods, using the new quality measure.
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3. Motivation
What is terrain simplification (TS), and Why is it needed (especially in the context of facility location).
What are the common ways to measure quality of simplification.
What types of facility location tasks or other tasks might use VPTS.
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4. Definitions
Terrain T : A xy-monotone triangulation of a set P of points in 3-space.
Simplification T’ of T : A xy-monotone triangulation of a subset P’ of P.
Quality (error) measure : Determines how well T’ approximates T.
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5. Error measures Common error measures :
The maximum vertical distance between T and T’
The volume of the region consisting of all points above T and below T’ or vice versa
Our visibility-based quality measure :
The expected similarity of the “views” from p and p’, respectively, where p in T and p’ in T’ are an arbitrary pair of corresponding points.
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6. Visibility-based measure
Ideally, if two points in T see (do not see) each other,
then the corresponding points in T’ should also see
(not see) each other.
Let X be a set of pairs of points.
Let V be the set of pairs (a,b) in X for which T.los(a,b)=T’.los(a,b) .
The quality of T’ (with respect to X) is |V| / |X| .
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7. Visibility-based measure
Ideal measure : random set - all pairs.
Transmitter-receiver measure : receivers and potential transmitters locations – all mixed pairs in given range.
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8. Visibility-Preserving TS - Overview
Typically, the view from p is blocked by ridges
Main stages:
Compute the ridge network (a collection of chains of edges of T).
Approximate the ridge network. The ridge network induces a subdivision of the terrain into patches.
Simplify each patch (independently), using one of the standard TS methods.
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9. Defining the ridge network
Three types of edges.
Take all difluent edges.
Two edges are connected if they share a vertex &
no flow from one side of the 2-chain to the other.
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10. Defining the ridge network
Special cases:
flow-splitting triangles – take one of receiving edges
a few more special cases (e.g., lakes).
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11. Defining the ridge network
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12. Approximating the ridge network
Goal: Replace RN with an approx network of size k.
Preliminary phase: divide RN into chains and assign to each chain a level of importance.
Phase 1: collapse all chains - replace each chain c by a single-edge chain defined by the two endpoints of c.
Phase 2: repeatedly drop a leaf edge of min importance from current network, until current size is k’.
Phase 3: repeatedly refine the chain that “needs” it most, until the desired size k is achieved.
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13. Approximating the ridge network
Original ridge network
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14. Approximating the ridge network
Reducing the num of vertices from 31 to k=9 using k’=6:
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15. Approximating the ridge network
Phase 1: collapsing all chains
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16. Approximating the ridge network
Phase 2: Remove the least important leaf chain
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17. Approximating the ridge network
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18. Approximating the ridge network
Phase 2: Remove the least important leaf chain
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19. Approximating the ridge network
Phase 2: Remove the least important leaf chain
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20. Approximating the ridge network
Phase 2: Remove the least important leaf chain
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21. Approximating the ridge network
End of Phase 2
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22. Approximating the ridge network
Phase 3: Refine the max dist chain
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23. Approximating the ridge network
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24. Approximating the ridge network
Phase 3: Refine the max dist chain
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25. Approximating the ridge network
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26. Approximating the ridge network
Phase 3: Refine the max dist chain
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27. Approximating the ridge network
End of Phase 3: k =9
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28. Approximating the ridge network
Original RN  Approximate RN
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29. Approximating the ridge network
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30. Approximating the ridge network
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31. Approximating the ridge network
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32. The main TS algorithm
The (simplified) Ridge Network induces a subdivision of the terrain into regions:
For each region (map[i]) in the subdivision
If map[i] is “big” then recursively apply VPTS to map[i].
Else (map[i] is “small”) simplify map[i] using a “standard” simplification method (such as Garland’s “Terra”).
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33. Experimental Results
Input: 20 terrains, each of size 15,000-20,000, representing different geographic regions.
VPTS was implemented using CGAL.
“Regular” TS methods: Terra [Heckbert-Garland], GcTin [Silva-Mitchell], Qslim [Garlad-Heckbert].
Each input terrain was ‘compressed’ to 1000, 500, 250,125 points.
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34. Tests
Note: every test was repeated 4 times, for each of the 4*20*4 = 320 compressed terrains.
Thus in total about 320*4*4 = 5120 quality of simplification evaluations were done.
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35. Results
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36. Triangulation representation
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37. Future work Compare VPTS with other TS methods.
Implement a robust version of VPTS.
Implement a version where regular visibility is replaced by “RF visibility”.
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38. Fin
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39. Conclusion:
It is a good idea to compress terrains before applying facility location algorithms
Original: 150,000
Object1: 10,000 – 99.5%
Object2: 1,000 – 97.8 %
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