1. Designing Quadrangulations with Discrete Harmonic Forms
Speaker: Zhang Bo

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.
: Ph.D. in Computer Science at the Unversity
of Southern California (USC).
Thesis title: “Towards Applied Geometry in Graphics”
Advisor: Professor Mathieu Desbrun.
: M.S. in Computer Science at Zhejiang University
Thesis Title: “Topics on Image-based Rendering”
: 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. because the 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. Singularity graph
reverse
regular

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

18. Vertex with no singularities ?
Discrete Laplace Equation:
wij = cot aij + cot bij
Can generate smooth fields even on irregular meshes!

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

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 & emax everywhere, 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. As a special line in the singularity graph
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. Summary Extended Laplace operator along singularity lines
Only three types:
regular, reverse, switch
Provide control over
singularity:
type
locations
sizing
REGULAR
REVERSE
SWITCH

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!