Connectivity Editing for Quad-Dominant Meshes

1. Connectivity Editing for Quad-Dominant Meshes Chi-Han Peng1 and Peter Wonka1,2 1Arizona State University2King Abdullah University of Science and Technology (KAUST)

2. Research statement • To analyze methods for the connectivity control* of the irregular elements (vertices and faces) in quad-dominant (QD) surface meshes, i.e. meshes that comprise mostly 4-sided faces? * For example, controlling the topological locations, types (vertices/faces), and numbers of irregular elements in a surface QD mesh. (Not just their 3D positions).

3. Irregular elements in QD meshes • A face is regular if it is degree 4 and irregular otherwise. A face with degree n is called fn. • A vertex is regular if it is degree 4 and irregular otherwise. A vertex with degree n is called vn. f3 v5 v3 f3 v3 v6 f6 Irregular elements are color-coded with blue colors if degree<4andorange colors if degree>4 f3 v3 f5

4. Irregular elements in QD meshes • “Atomic” irregular elements: degree 3 or 5, that is, v3, v5, f3, and f5. • Irregular elements with other degrees can be broken down into a combination of atomic ones. v2 f6 v6 Two v3 Two v5 Two f5

5. Related work

6. Field-based pure-quad or QD remeshing • The irregular elements in the resulting meshes can be indirectly controlled by editing the singularities in the guiding field. Mixed-Integer Quadrangulation, Bommes et al., Siggraph 2009 Designing Quad-dominant Meshes with Planar Faces, Zaravec et al., SGP 2010.

7. Direct control of irregular vertices in polygonal meshes • Connectivity editing for triangle meshes: • Connectivity editing for pure-quad meshes: Editing Operations for Irregular Vertices in Triangle Meshes, Li et al., Siggraph Asia 2010 Connectivity Editing for Quadrilateral Meshes, Peng et al., Siggraph Asia 2011

8. Finding a suitable domain toanalyze the problem

9. Primal QD mesh domain Pure-quad meshes: QD meshes: v3 v5 v3 v5 f3 f5 Pairs: 3 possibilities Pairs: 10 possibilities … v3-v5 v3-v3 v5-v5 f3-v5 v3-f5 f5-f5 f3-f3 Triples: 4 possibilities Triples: 20 possibilities v3-v3-v5 v3-v3-v5 v3-f3-f3 v3-v3-v3 v3-v3-v3 … v3-v3-f5 v3-v5-v5 v5-v5-v5 v3-v3-f3 f3-f3-f3

10. Alternative domain • We need to analyze the problem in an alternative domain. Desired criteria are: • Pure-quad. • Primal irregular elements (vertices and faces) are mapped to irregular vertices of the same degrees. • Forward convertibility: every (QD) primal domain mesh can be mapped to an (pure-quad) alternative domain mesh. • Backward convertibility: every (edited) alternative domain mesh can still be inversely-mapped to a primal domain mesh.

11. Alternative domain • Dual, Doo-Sabin, and -subdivision domains: • Pure quad. • Catmull-Clark (CC) subdivision domain: • Pure quad. • Primal irregular elements are mapped to irregular vertices of the same indices. • Forward convertibility. • Backward convertibility.     

12. Backward convertibility of CC domain • A necessary condition for a pure-quad mesh to be inversely CC-subdivisible: irregular vertices must have even graph distances in between. Moving the v3-v5 pair CC domain: InverseCC-subdiv. Primal domain:

13. Catmull-Clark square-root () domain [Kobbelt96] • A “half-step” to achieve the connectivity of a full CC subdivision: -subdiv. -subdiv. domain (pure-quad mesh) Primal domain(QD mesh) CC domain (pure-quad mesh)

14. Catmull-Clark square-root () domain [Kobbelt96] • Pure quad. • Primal irregular elements are mapped to irregular vertices of the same indices. • Forward convertibility. • Backward convertibility.    

15. Backward convertibility of domain • Proposition: A pure-quad mesh with a sphere-like or disk-like topology can be inversely-subdivided in exactly two ways. domain: Moving the v3-v5 pair Inverse-subdiv. Primaldomain:

16. Backward convertibility of domain • Backward convertibility is guaranteed as long as an edit can be contained within a disk-like region. Move the v5-v5 pair domain: Inverse -subdiv. Primal domain:

17. Editing framework subdiv. Edits Inverse subdiv. Primal domain mesh (QD) domain mesh (pure-quad)

18. Editing operations for irregular elements in QD meshes

19. Editing a single irregular element? • Proposition: It is impossible to create, delete, move, or type-change (vertex / face) a single irregular element* in an otherwise regular QD disk-like region. * Atomic (degree 3 or 5) irregular elements.

20. Editing irregular elements in pairs • The simplest possible way to edit irregular elements is move / type-change them in pairs. • We distinguish three cases: • 3-5 pairs (e.g. v3-v5, v3-f5,…). • 3-3 pairs. • 5-5 pairs.

21. Moving a 3-5 pair It can be moved in exactly four directions: left, right, up, down.

22. Moving a 3-5 pair Since they always move in the same directions, a 3-5 pair cannot be merged (and cancelled) by themselves.

23. Moving a 3-5 pair Note that a single step would type-change both irregular elements.

24. Moving a 3-3 or 5-5 pair They can be moved in exactly four directions: getting closer, farther apart, rotating clockwise, rotating counter-clockwise.

25. Merging a 3-3 or 5-5 pair • Proposition: A 3-3 or 5-5 pair can be merged into a single (non-atomic) irregular element iff their types are the same. Primaldomain: Two f5 One v6 domain:

26. Cancelling irregular elements • To cancel irregular elements, a minimal of three (two of them with opposite degrees) are required: domain Primal domain

27. T-junction (adjacent v3-f5 pair) editing • Proposition: A T-junction can be moved in exactly four directions.

28. T-junction cancellation • Two T-junctions can be completely cancelled or reduced to an irregular element pair, depending on their relative orientations.

29. T-junction cancellation Fully cancellable case 1

30. T-junction cancellation Fully cancellable case 2

31. T-junction cancellation Reduces to an irregular element pair

32. Editing field singularities and irregular elements • A field singularity pair with the opposite indices can be canceled.*Impossible for a pair of irregular elements with the opposite degrees (e.g. v3-v5). • A single field singularity can be moved.* Impossible for a single irregular vertex. 1. 2. Rotational Symmetry Field Design on Surfaces, Palacios et al., Siggraph 2007

33. Applications and results

34. Exploring QD mesh designs • Irregular vertices are necessary to maintain sharp features (corners and edges). • Irregular faces can lead to smoother mesh lines. Pure-quad QD Dual of pure-quad

35. QD mesh designs for geometric optimization Four f3 and eight v5 Four v5 Four f5

36. Exploring QD mesh designsfor geometric optimization Equiangular faces optimization(fairness term of [Liu et al. 2006])

37. Exploring QD mesh designsfor geometric optimization Planarity optimization ([Liu et al. 2006])

38. QD mesh connectivity improvement

39. Future work • Global connectivity optimization by automating the editing operations, driven by topological and geometric heuristics (e.g. curvature). • QD meshing for finite element methods – versus triangle meshing and pure-quad meshing?

40. Acknowledgements • Helmut Pottmann for reformulating the derivation of the discrete Gauss Bonnet theorem. • Xiang Sun and Caigui Jiang for helping with the geometric optimizations. • Alexander Schiftner, Yu-Kun Lai, Martin Marinov, and Dong-ming Yan for providing datasets. • Yoshihiro Kobayashi and Christopher Grasso for the renderings. • Virginia Unkefer for the proofreading. • The anonymous reviewers for their insightful comments. • This work was supported by the National Science Foundation and KAUST. Thank you for listening!