Designing Quadrangulations with Discrete Harmonic Forms

1. Designing Quadrangulationswith Discrete Harmonic Forms Speaker: Zhang Bo2007.3.8

2. References Designing Quadrangulations with Discrete Harmonic Forms Y.Tong P.Alliez D.Cohen-Steiner M.Desbrun Caltech INRIA Sophia-Antipolis, France Eurographics Symposium on Geometry Processing (2006)

3. About the Author: Yiying Tong • 2005-present: Post doctoral Scholar in Computer Science Department, Calteth. • 2000-2004: Ph.D. in Computer Science at the Unversity of Southern California (USC). Thesis title: “Towards Applied Geometry in Graphics” Advisor: Professor Mathieu Desbrun. • 1997-2000: M.S. in Computer Science at Zhejiang University Thesis Title: “Topics on Image-based Rendering” • 1993-1997: B. Engineering in Computer Science at Zhejiang University Eurographics Young Researcher Award 2005 INRIA: 法国国家信息与自动化研究所 CGAL developer Siggraph Significant New Researcher Award 2003

4. Methods for Quadrangulations • Among many: • clustering/Morse [Boier-Martin et al 03, Dong et al. 06] • global conformal param [Gu/Yau 03] • curvature lines [Alliez et al. 03, Marinov/Kobbelt 05] • isocontours [Dong et al. 04] • two potentials • (much) more robust than streamlines • periodic global param(PGP) [Ray et al. 06] • PGP : nonlinear + no real control • This paper: one linear system only • This paper: discrete forms & tweaked Laplacian

5. About Discrete Forms Discrete k-form A real number to every oriented k-simplex 0-forms are discrete versions of continuous scalar fields 1-forms are discrete versions of vector fields

6. About Exterior Derivative • Associates to each k-form ω a particular (k+1)-form dω • If ω is a 0-form (valued at each node), i.e., a function on the vertices, then dω evaluated on any oriented edge v1v2 is equal to ω(v1) -ω(v2) • Potential: 0-form u is said to be the potential of w if w = du • Hodge star: maps a k-form to a complimentary (n-k)-form • On 1-forms, it is the discrete analog of applying a rotation of PI/2 to a • vector field

7. About Harmonic Form Codifferential operator: Laplacian: 满足 的微分形式称为调和形式, 特别 的函数 称为调和函数

8. One Example

9. Why Harmonic Forms ? • Suppose a small surface patch composed of locally “nice” quadrangles • Can set a local coordinate system (u, v) • du and dv are harmonic, so u and v are also harmonic. becausethe exterior derivative of a scalar field is harmonic iff this field is harmonic This property explain the popularity of harmonic functions in Euclidean space

10. Discrete Laplace Operator u = harmonic 0-form

11. Necessity of discontinuities • Harmonic function on closed genus-0 mesh? Only constants! Globally continuous harmonic scalar potentials are too restrictive for quad meshing

12. Adding singularities • Poles, line singularity du dv contouring

13. With more poles… Crate saddles

14. Why ? Poincaré–Hopf index theorem! ind(v)=(2-sc(v))/2 ind(f)=(2-sc(f))/2 sc() is the number of sign changes as traverses in order Discrete 1-forms on meshes and applications to 3D mesh parameterization StevenJ.Gortler ,Craig Gotsman ,Dylan Thurston, CAGD 23 (2006) 83–112

15. Line Singularity -> T-junctions

16. regular reverse Singularity graph

17. Singularity lines between “patches” • Special continuity of 1-forms du and dv i.e., special continuity of the gradientfields • only three different cases in order to guarantee quads

18. Vertex with nosingularities ? Discrete Laplace Equation: wij = cot aij + cot bij Can generate smooth fieldseven on irregular meshes!

19. N- N+ Handling Singularities • Vertex with regular continuity • as simple as jump in potential:

20. Handling Singularities • Vertex with reverse continuity

21. Handling Singularities • Vertex with switch continuity

22. Building a Singularity Graph • Meta-mesh consists of • Meta-vertices, meta-edges, meta-faces • Placing meta-vertices • Umbilic points of curvature tensor (for alignment) • User-input otherwise • Tagging type of meta-edges • can be done automatically or manually • Geodesic curvature along the boundary • will define types of singularities • Small linear system to solve for corner’s (Us,Vs) • “Gauss elimination”: row echelon matrix

23. Assisted Singularity Graph Generation • Two orthogonal principal curvature directions emin & emaxeverywhere, except at the so-called umbilics

24. Final Solve • Get a global linear system for the 0-forms u and v of the original mesh as discussion above • The system is created by assembling two linear equations per vertex, but none for the vertices on corners of meta-faces • This system is sparse and symmetric, Can use the supernodal multifrontal Cholesky factorization option of TAUCS, Efficient!

25. Handle Boundaries • As a special line in the singularity graph • Force the boundary values to be linearly interpolating the two corner values

26. Mesh Extraction A contouring of the u and v potentials will stitch automatically into a pure quad mesh

28. Mesh alignments control provide (soft) control over the final mesh alignments

29. Mesh size control

30. Results

31. Singularity graph

32. Harmonic Functions u,v

33. du, dv

34. Final Remesh

35. B-Spline Fitting

36. More result

38. REGULAR REVERSE SWITCH Summary • Extended Laplace operator along singularity lines • Only three types: • regular, reverse, switch • Provide control over • singularity: • type • locations • sizing

39. Summary • Sparse and symmetric linear system, average 7 non-zero elements per line, can be compute fast! • Not a fully automatic mesher Singularity graph

40. Thank you!