Spectral Surface Quadrangulation

1. Spectral Surface Quadrangulation Shen Dong, Peer-Timo Bremer, Michael Garland, Valerio Pascucciand John Hart Reporter: Hong guang Zhou Math Dept. ZJU October 26

2. Quadrangulating Surfaces

3. DAZ Productions Why Quad Meshes? • Applications • PDEs for fluid, cloth, … • Catmull-Clark subdivision • NURBS patches in CAD/CAM • Demands • Few extraordinary points • High quality elements Stam 2004

4. Related Work– Semi-Regular Triangle Remeshing Multiresolution Analysis of Arbitrary Meshes [Eck et al. 95] Multiresolution Adaptive Parameterization of Surfaces [Lee et al. 98] Globally Smooth Parameterization [Khodakovsky et al. 03]

5. Related Work– Quad Remeshing Parameterization of Triangle Meshes over Quadrilateral Domains [Boier-Martin et al. 04] Periodic Global Parameterization [Ray et al. 05]

6. Our Approach • Start with a triangulated 2-manifold

7. Our Approach • Start with a triangulated 2-manifold • Construct a “good” scalar function

8. Our Approach • Start with a triangulated 2-manifold • Construct a “good” scalar function • Quadrangulate the surface using its Morse-Smale complex

9. Our Approach • Start with a triangulated 2-manifold • Construct a “good” scalar function • Quadrangulate the surface using its Morse-Smale complex • Optimize the complex geometry

10. Our Approach • Start with a triangulated 2-manifold • Construct a “good” scalar function • Quadrangulate the surface using its Morse-Smale complex • Optimize the complex geometry • Generate semi-regular quad mesh

11. Key Features of SSQ • Few extraordinary points • Pure quad, fully conforming mesh • Topological robustness • High element quality

12. Computing the Morse-Smale Complex • Given any scalar function • Contrust Morse –Smale function over a manifold

13. wij = (cot+cot) / 2 i   j Discrete Laplacian Eigenfunctions • Discretization • Smooth surface  polygon mesh of n vertices • Scalar field  real vector of size n • Laplace operator  • Vertex i: fi =  wij ( fj – fi ) • Whole mesh :  f = L · f • Eigenfunction of  : F = F  eigenvector of L : L · f =  Morse –Smale function F: v → f

14. 1 2 3 4 5 6 7 8 Our Choice – Laplacian Eigenfunctions • Equivalence of Fourier basis functions in Euclidean space • Capture progressively higher surface undulation modes

15. Computing the Morse-Smale Complex • Given any scalar function • Identify all criticalpoints • maximum • minimum • saddle Other points :regular

16. Computing the Morse-Smale Complex • Each saddle has four lines of steepest ascent /descent • Trace ascending lines from saddle to maxima • Trace descending lines from saddle to minima

17. Shape Dependence

18. Properties of the Morse-Smale Complex • Guaranteed fully conforming, purely quadrangulardecomposition for • Any surface topology • Any function

19. persistence Noise Removal • Cancel pairs of connected critical points

20. Noise Removal

21. Quasi-Dual Complexes Morse-Smale complex Quasi-dual complex

22. Quasi-Dual Complexes In each cell, calculate the easiest path that connect the minimum to the maximum.

23. Quasi-Dual Complexes • Doubles the number of available base domains • Capture different symmetry patterns of the surface Primal Quasi-dual

24. Bunny Harmonics

25. Complex Improvement • Patches may be poorly shaped • Paths can merge

26. Build 2n2n linear system   Globally Smooth Parameterization [1,1] [0,0]

27.     Globally Smooth Parameterization Build 2n2n linear system

28.    Parameterization Globally Smooth Parameterization Bake transition function into system [Tong et al. 06] use more general formulation

29. Iterative Relaxation • For any vertex i, find a patch  such that  [0,1][0,1]

30. Iterative Relaxation • For any vertex i, find a patch  such that  [0,1][0,1] • Conform patch boundaries to the in-range charts

31. Iterative Relaxation • For any vertex i, find a patch  such that  [0,1][0,1] • Conform patch boundaries to the in-range charts • Relocate nodes to adjacent paths branching points • Resolve parameterization and repeat relaxation

32. Complex Refinement

33. Mesh Generation • Lay down kk grid in each patch • Extraordinary points can only exist at complex extrema • Fully conforming

34. Picking Eigenfunctions • Two phases • Pick range of spectrum by target number of critical points • Pick best eigenfunction within range with lowest parametric distortion Spectrum 0 k

35. Primal Quasi-dual 16th 32nd 8th Results – Torus

36. MS-complex Input Optimized complex Remesh Results – Dancer

37. Results – Heptoroid Quadrangulation Input Output |EV|=175

38. [Boier-Martin et al.] |EV| = 175 [Ray et al.] |EV| = 314 SSQ |EV| = 26 Results – Bunny

39. Angle Edge Length =6.87 =7e-4 =9.63 =7.4e-4 =12.71 =9.3e-4 Results – Bunny SSQ [Ray et al.] [Boier-Martin et al.]

40. Performance

41. Conclusion • Surface quadrangulation using Morse-Smale complex of Laplacian eigenfunction • Key features • Few extraordinary points • Pure quad, fully conforming mesh • Topologically robust • High element quality

42. Future Work • Deeper understanding of the Laplacian spectrum • Full feature and boundary support • More efficient complex optimization • Select the good eigenfunction whose gradient field most closely follows any such user- specified orientation

43. Thank you Questions ?