1. Triangular Mesh Decimation
Martin Franc, Václav Skala
University of West Bohemia in Plzen
Czech Republic

2. Contents Motivation Decimation Previous work Algorithm modification
Results
Conclusion

3. Motivation
fast and interactive visualization of large and complex data (CAD, 3D scanner, CT, MRI)
reduction of the number of triangles preserving important details of the model

4. Decimation W. Schroeder 1992
Simplification methods based on a specific mesh element removal
vertex decimation
edge decimation – edge contraction
patch decimation – vertex clustering
Fast and simple method
Can be generalized to 3D

5. Decimation General scheme Non-trivial triangulation in 3D
mesh element importance evaluation
element removal
arising “hole” triangulation
Non-trivial triangulation in 3D

6. Decimation Vertex decimation vertex topology assessment
importance evaluation
the least vertex removal
triangulation

7. Decimation Edge decimation Patch decimation edge importance evaluation
the least important edge removal
triangulation
Patch decimation
patch importance evaluation
the least important removal

8. Previous work Vertex and edge decimation combination Parallelization
vertices evaluation
the least important vertex selection
adjacent edges importance evaluation
triangulation
Parallelization
independent set of vertices
multithread programming (no critical sections)

9. Previous work Edge contraction criterion Independent set of vertices
minimal length
minimal surface area after triangulation
Independent set of vertices
independent set of vertices
super independent set of vertices

10. Algorithm Vertices evaluation The least important one search
topology assessment
importance evaluation
The least important one search
vertices sorting (according to their importance)
super independent set creation
Vertex removal
edges evaluation (optimal edge selection)
consistency check
triangulation

11. Data structure
Winged edge modification

12. Implementation Symmetrical multiprocessor (shared memory) Windows NT
Threads
no critical sections
as many threads as free processors
procedure
get number of free processors (P)
divide a set of vertices onto P parts
run P threads, each with its own subset of vertices
Parallel section
vertices importance evaluation
vertices removal
No load balancing

13. First results
Speedup Time ratio of various parts of the algorithm

14. Algorithm modification 1
Vertices sorting removal
remove vertices under some importance threshold only
importance of a vertex (x)
maximum vertex importance in whole set

15. Algorithm modification 1
Vertices evaluation
topology assessment
importance evaluation
Vertices removal
threshold function (bucketing instead of sorting)
independent set of vertices
edges evaluation
triangulation

16. Algorithm modification 1
+ Speedup
– Higher approximation error
– Independent set of vertices

17. Algorithm modification 2
Hash function
basic function
modification

18. Algorithm modification 2
Idea

19. Algorithm modification 2
Function coefficients
Application of the principle of indep. sets
vertices impacted by previous decimation step are moved to the end of the cluster

20. Algorithm modification 2
Preprocessing
vertices evaluation
clusters creation (according the importance)
Processing of the least important cluster
vertex removal
triangulation
new evaluation of surrounded vertices which are moved to the end of proper cluster

21. Results
Mentioned approaches comparison

22. Results
Time comparison (rough)

23. Results
Example of non-trivial data

24. 871,414 triangles 430,000 triangles 87,000 triangles
Results
Example of reduced data
871,414 triangles 430,000 triangles 87,000 triangles

25. Results Example of reduced data
137,072 triangles ,706 triangles ,854 triangles ,248 triangles

26. 58,328 triangles 29,000 triangles 6,000 triangles
Results
Example of reduced data
58,328 triangles 29,000 triangles ,000 triangles

27. Conclusion
Fast parallel algorithm for simplification of large triangular meshes
Efficient sequential algorithm
+ non-manifold meshes reduction
– simple triangulation function
Future work
triangulation method improvement
decimation controlled by the approximation error
volume decimation