Siggraph ’ 06 Paper Reading Seminar: Meshes Section

1. Siggraph’06 Paper Reading Seminar: Meshes Section Chen Zhonggui 2006.5.29

2. Spectral Surface Quadrangulation Shen Dong* Peer-Timo Bremer* Michael Garland* Valerio Pascucci† John C. Hart* *University of Illinois at Urbana-Champaign †Lawrence Livermore National Laboratory

3. Overview

4. Morse Theory

5. Morse-Smale Complex • Connecting the saddles and extrema via gradient flow quadrangulates the surface. maximum minimum saddle

6. Quasi-Dual Complex

7. Quasi-Dual Complex

8. Laplacian Eigenfunction • Discrete Laplacian operator • Rewriting in matrix form

9. Spectral Surface Analysis

10. Properties of Laplacian Eigenfunctions • Minima and maxima are interleaved in such a way that high valence nodes are extremely rare. • Multisaddles almost never arise, thus practically guaranteeing that extraordinary points can only occur at extrema. • The number of nodal domains of the eigenfunction with eigenvalue is at most k.

11. Properties of Laplacian Eigenfunctions

12. Properties of Laplacian Eigenfunctions

13. Multiresolution Spectral Analysis

14. Quadrangular Base Complex

15. Global Smooth Parameterization

16. Transition Function

17. Parameterization • Convex combination method • Nodes of the complex are always constrained to lie at the corners of D

18. Adjusting Patch Boundaries IterativeRelaxation • Adjusting patch boundaries

19. Iterative Relaxation • Relocating Nodes of the Complex

20. MeshGeneration

21. Conclusions • Less extraordinary points. • “feature sensitive” eigenfunctions?

22. Modified Subdivision Surfaces with continuous curvature Adi Levin Cadent Ltd.

23. Overview

24. Catmull-Clark Surface Step1. Linear subdivision Step2. Weighted averaging Averaging mask for regular vertex

25. Catmull-Clark Surface • continuous on the regular quad regions. • continuous at extraordinary vertices.

26. Surface Blending around Extraordinary Vertices • The modified surface :

27. Local Parameterization

28. Characteristic Map V One-ring neighboring vertices of extraordinary vertex V M: local subdivision matrix

29. Characteristic Map • The parametric coordinates are : the two sub-dominant eigenvectors of local subdivision matrix M.

30. Computing Polynomial p(u,v) • For valences n > 4 we use cubic polynomials, and in case n = 3, we use quadratics. • Using the (u,v) parameter values at the points, we calculate the coefficients of p(u,v) by a least-squares fit.

31. Surface Blending • The modified surface • Blending weight function

32. Conclusions • Can be used to modify the limit surface of other subdivision schemes. • The increased area of influence and the violation of the convex-hull property.

33. Edge Subdivision Schemes and The Construction of Smooth Vector Fields Ke Wang WeiWei Yiying Tong Mathieu Desbrun Peter Schröher Caltech

34. Overview

35. Discrete Exterior Calculus http://ddg.cs.columbia.edu/

36. Discrete Differential Forms Constructing higher regularity bases for discretedifferential forms Subdivision Process bases for discretedifferential 0-forms Loop[87] Scheme bases for discretedifferential 2-forms Half-box splines bases for discretedifferential 1-forms New scheme

37. Delaunay Triangulations Streaming Computation of Martin IsenburgUC Berkeley Yuanxin LiuUNC Chapel Hill Jonathan ShewchukUC Berkeley Jack SnoeyinkUNC Chapel Hill

38. Algorithms for large data sets • Divide-and-conquer algorithms • Cut a problem into small subproblems that can be solved independently • Cache-efficient algorithms • Cooperate with the hardware’s memory hierarchy • External memory algorithms • Control over where, when, and how data structures are stored on disk • Streaming algorithms • Sequentially read a stream of data and retain only a small portion of the information in memory.

39. Spatial Coherence

40. Our Approach • spatialfinalization !!! • enhance inputwith tags thatsay: “there are no more points in this area”

41. Streaming Delaunay Pipeline enhances points withspatial finalization uses spatial finalizationto certify triangles as final

42. Spatial Finalization of Points  compute bounding box

43. 4 5 4 9 5 8 3 6 5 6 1 6 1 7 4 7 2 2 3 7 1 9 7 4 5 1 7 7 7 5 8 7 8 9 2 1 7 7 8 7 3 1 7 8 7 6 8 7 7 4 8 8 9 7 9 8 2 6 9 6 5 8 2 Spatial Finalization of Points compute bounding box  create finalization grid • count points • store sprinkles

44. 5 4 9 8 6 1 6 7 4 7 3 7 9 7 7 5 8 7 8 9 2 7 7 8 7 3 7 8 7 6 8 7 7 4 8 8 9 7 9 8 2 6 9 6 5 8 2 Spatial Finalization of Points compute bounding box create finalization grid • count points • store sprinkles  output finalized points • release chunks • add finalization tags

45. 1 4 9 7 6 0 3 5 4 7 3 7 9 7 7 5 8 7 8 9 2 7 7 8 7 3 7 8 7 6 8 7 7 4 8 8 9 7 9 8 2 6 9 6 5 8 2 Spatial Finalization of Points compute bounding box create finalization grid • count points • store sprinkles  output finalized points • release chunks • add finalization tags

46. 0 4 9 2 6 1 0 5 4 5 3 6 9 7 7 5 8 7 8 9 2 7 7 8 7 3 7 8 7 6 8 7 7 4 8 8 9 7 9 8 2 6 9 6 5 8 2 Spatial Finalization of Points compute bounding box create finalization grid • count points • store sprinkles  output finalized points • release chunks • add finalization tags

47. 0 7 1 0 4 3 1 2 7 1 1 7 7 9 2 7 7 8 7 3 7 8 7 6 8 7 7 4 8 8 9 7 9 8 2 6 9 6 5 8 2 Spatial Finalization of Points compute bounding box create finalization grid • count points • store sprinkles  output finalized points • release chunks • add finalization tags

48. 0 2 1 0 4 1 6 3 2 5 1 7 6 4 5 8 6 7 4 8 8 8 7 9 8 2 6 9 6 5 8 2 Spatial Finalization of Points compute bounding box create finalization grid • count points • store sprinkles  output finalized points • release chunks • add finalization tags

49. 0 2 1 0 4 1 1 3 2 3 2 3 7 2 5 9 5 4 7 2 Spatial Finalization of Points compute bounding box create finalization grid • count points • store sprinkles  output finalized points • release chunks • add finalization tags

50. 0 2 1 0 4 Spatial Finalization of Points compute bounding box create finalization grid • count points • store sprinkles  output finalized points • release chunks • add finalization tags