Multiresolution Analysis of Arbitrary Meshes

1. Multiresolution Analysis of Arbitrary Meshes • Matthias Eck • joint with • Tony DeRose, Tom Duchamp, Hugues Hoppe, • Michael Lounsbery and Werner Stuetzle U. of Darmstadt , U. of Washington , Microsoft , Alias

2. Overview • 1. Motivation and applications • 2. Our contribution • 3. Results • 4. Summary and future work

3. Motivation problem: complex shapes = complex meshes I have 70,000faces !

4. Difficulties: • Storage • Transmission • Rendering • Editing • Multiresolution analysis

5. multiresolution representation of mesh M • = • base shape M 0 • + • sum of local correction terms • (wavelet terms)

6. base shape M 0 mesh M

7. Applications 1. Compression 2. Multiresolution editing 3. Level-of-detail control 4. Progressive transmission and rendering

8. e < 0.8% ~70,000 faces ~11,000 faces tight error bounds

9. Applications 1. Compression 2.Multiresolution editing 3. Level-of-detail control 4. Progressive transmission and rendering

11. Applications 1. Compression 2. Multiresolution editing 3. Level-of-detail control 4. Progressive transmission and rendering

12. no visual discontinuties

13. Applications 1. Compression 2. Multiresolution editing 3. Level-of-detail control 4.Progressive transmission and rendering

14. base shape M 0 mesh M

15. 2. Our contribution

16. 2. Our contribution

17. Previous work • Lounsbery, DeRose, Warren 1993 • provides general framework for MRA • extends wavelet analysis to surfaces of arbitrary topology • Schroeder, Sweldens 1995 • similar work on sphere

18. However ... • input surface must be parametrized over a simple domain mesh • r(x) • x r

19. The problem ... • Meshes are typically given as collection of triangles, thus • MRA algorithms cannot directly be applied

20. I’m not parametrized ! M

21. ... and our solution • step 1: construct a simple domain mesh K K M

22. ... and our solution • step 1: construct a simple domain mesh K • step 2: construct a parametrization rofMover K MRA !!! r K M

23. step1:Construction of domain mesh • Main idea: • partition M into triangular regions • domain mesh K

24. mesh M partition domain mesh K

25. How to get partition ? • Our requirements: • topological type of K = topological type of M • small number of triangular regions • smooth and straight boundaries • fully automatic procedure

26. construct Voronoi-like diagram on M construct Delaunay-like triangulation mesh M

27. step 2:Construction of parametrization • map each face of domain mesh to corresponding triangular region • local maps agree on boundaries: parametrization r

28. local map

29. How to map locally? • Requirements: • fixed boundary conditions • small distortion • Best choice: harmonic maps • well-known from differential geometry • minimizing the metric distortion

30. local map planar triangle triangular region

31. 4. Results

32. 4. Results

33. 34 min. , 70,000 faces 4,600 faces , e < 1.2 % 162 faces 2,000 faces , e < 2.0 %

34. 40 min. , 100,000 faces 4,700 faces , e < 1.5 % 229 faces 2,000 faces , e < 2.0 %

35. Summary • Given: An arbitrary mesh M • We construct: a simple domain mesh and an exact parametrization for M • Allows MRA to be applied • tight error bounds • Useful in other applications

37. 5. Future work Other potential applications of parametrization: • texture mapping • finite element analysis • surface morphing • B-spline fitting

38. B - spline fitting B - spline control mesh approximating surface