1. Mesh Coarsening zhenyu shu 2007.5.12

2. Mesh Coarsening Large meshes are commonly used in numerous application area Modern range scanning devices are used High resolution mesh model need more time and more space to handle Large meshes need simplification to improve speed and reduce memory storage

3. Mesh Coarsening Size, quality and speed Mesh optimization Many simplification methods now

4. QEM Garland M, Heckbert P. Surface simplification using quadric error metrics. In: Proceedings of the Computer Graphics, Annual Conference Series. Los Angeles: ACM Press, 1997. 209~216

5. QEM Quadric Error Metric method Using Pair Contraction to simplify the mesh Minimize Quadric function when contracting Define Quadric

6. Quadric Define Quadric of each vertex

7. Pair Contraction

8. Pair Selection Condition is an edge or, where t is a threshold When performing, Choose position of minimizing If A is not invertible, choose among two endpoints and midpoint of two endpoints

9. Algorithm Summary Compute the Q matrices for all the initial vertices. Select all valid pairs. Compute the optimal contraction target for each valid pair Place all the pairs in a heap keyed on cost with the minimum cost pair at the top. Iteratively remove the pair of least cost from the heap, contract this pair, and update the costs of all valid pairs involving v1.

10. Advantage Efficiency, local, extremely fast Quality, maintain high fidelity to the original mesh Generality, can join unconnected regions of original mesh together

11. Result Original model An approximation with 69451 triangles with 1000 triangles

12. Topology manipulation Hattangady N V. A fast, topology manipulation algorithm for compaction of mesh/faceted models[J]. Computer-Aided Design. 1998, 30(10): 835-843.

13. Edge collapsing

14. Edge swapping

15. Edge smoothing let N be the average of all Ci

16. Data Structure of mesh model A type of data structure to present mesh model for reference

17. Remeshing Surazhsky V, Gotsman C. Explicit surface remeshing[C]. Aachen, Germany: Eurographics Association, 2003 Improve mesh quality by a series of local modification of the mesh geometry and connectivity

18. Vertex Relocation with neighbors Find new location of to satisfy some constraints, e.g. improving the angles of the triangles incident on

19. Vertex Relocation Map these vertices into a plane, is mapped to the origin, satisfy The angles of all triangles at are proportional to the corresponding angles and sum to

20. Vertex Relocation Let new position of be the average of to improve the angles of the adjacent faces Bring new position of back to the original surface by maintain same barycentric coordinate

21. Detail (c) is original mesh, (b) is new mesh, (d) is 2D mesh which defines a parameterization of (c) Use the same barycentric coordinates in (a) and (d)

22. Area-based Remeshing Area equalization is done iteratively by relocating every vertex such that the areas of the triangles incident on the vertex are as equal as possible Extending method above to relocating vertices such that the ratios between the areas are as close as possible to some specified values

23. Area-based Remeshing Here is the area of triangle, is the area of polygon

24. Area-based Remeshing

25. Curvature sensitive remeshing More curved region contain small triangles and a dense vertex sampling, while almost flat regions have large triangles Define density function as here K and H are approximated discrete Gaussian and mean curvatures Meyer M, Desbrun M, Schroder P, et al. Discrete differential geometry operator for triangulated 2-manifolds [A]. In: Proceedings of Visual Mathematics'02, Berlin, 2002. 35 ～ 57

26. Result

28. CVD Valette S, Chassery J M. Approximated Centroidal Voronoi Diagrams for Uniform Polygonal Mesh Coarsening[J]. Computer Graphics Forum. 2004, 23(3): 381-389

29. Voronoi Diagram Given an open set  of R m, and n different points z i ; i=0,...,n-1, the Voronoi Diagram can be defined as n different regions V i such that: where d is a function of distance.

30. Centroidal Voronoi Diagram A Centroidal Voronoi Diagram is a Voronoi Diagram where each Voronoi site z i is also the mass centroid of its Voronoi Region: here is a density function of

31. Centroidal Voronoi Diagram Centroidal Voronoi Diagrams minimize the Energy given as: On mesh, Energy above becomes to

32. Here Construct CVD based on global minimization of the Energy term E 2 Construct CVD

33. Algorithm Summary Randomly choose n different cells in mesh and these cells form n regions Cluster all cells in mesh by extending these regions and choosing correct cells’ owner to minimize the energy term E 2 Now calculate each center of these regions and replace each region with it’s center Triangulate and get new mesh

34. Clustering

35. Triangulate

36. Sample

38. Result Quality and Speed

39. Pros and Cons Pros High quality of result Optimization of original mesh Cons Slow Global

40. Thanks