Quads and Friends

1. Quads and Friends Kenshi Takayama

2. General Planar Quadrilateral Mesh Design Using Conjugate Direction Field Liu, Xu, Wang, Zhu, Guo, Chen, Wang MSRA, Univ. Sci. Tech. China, Peking Univ. • goal: planar quad (PQ) mesh design • application: architectural glass structures • key challenge: Conjugate Direction Field (CDF) design • (warning: very hard to read for non-experts)

3. Existing PQ meshing methods deal with special cases of CDF: • principal curvature directions [Liu06] • restricted singularity indices [Zadravec10] • definition of CDF: • Two tangent vectors and at a surface point are conjugate iff • & :principal curvatures • & : principal curvature directions • and need not be orthogonal

4. CDF of -th triangle is represented using angles & : • To measure smoothness of CDF, we need to determine correspondence of vectors between adjacent faces  nonlinear mixed-integer optimization = difficult! • key idea: signed-permutation operation • (read paper...)

5. : user constraints on mesh directions • Workflow input triangular mesh output PQ mesh initial CDF optimized CDF

6. Comparison conjugate direction field (proposed) 4-RoSy field [Bommes09] input mesh & direction constraint quad mesh from direction fields (non-planar) after planarization

7. Comparison cannot preserve original shape! proposed [Bommes09] + planarization

8. Connectivity Editing for Quadrilateral Meshes Peng, Zhang, Kobayashi, Wonka Arizona State Univ., Oregon State Univ., KAUST • goal: improve the quality of quad meshes by manual editing of connectivity before after

9. Basic Operations • Atomic Semantic Operations • combination of Basic Operations inverse 3-5 pair movement generation/removal of 3-5 pair

10. Pair-wise Movement Operations • combination of Atomic Semantic Operations • 5 pages of theoretical analysis • quite complicated, actually...

13. Simple Quad Domains for Field Aligned Mesh Parametrization Tarini, Puppo, Panozzo, Pietroni, Cignoni ISTI-CNR, Univ. dell’Insubria, Univ. di Genova • goal: modify input quad mesh such that its separatrices become simple • separatrix := consecutive edges connecting two irregular vertices  resulting in simpler domains original modified

14. applications: • regular remeshing • texture mapping • GPU computation (e.g., Poisson solver) • workflow: • step 1: changing the separatrices’ connectivity • step 2: remeshing and smoothing

15. step 1: changing the separatrices’ connectivity • cost function for a separatrix: • : degree of deviation from the (original) streamline • : length along the streamline • : weighting parameter • iteratively replace separatrices s.t. total energy is reduced • depth-first search in the search tree example:

16. step 2: remeshing and smoothing • basically use one of the existing methods • (unpolished writing...) • results

18. Boundary Aligned Smooth 3D Cross-Frame Field Huang, Tong, Wei, Bao State Key Lab of CAD&CG (Zhejiang Univ.), Michigan State Univ. • extension of surface symmetry field to volumes • applications: • hexahedral meshing • (texture synthesis?) Surface [Palacios11] Volume (proposed)

19. issue: ambiguity of directions • All 24 frames should be considered as equivalent • key idea: • Define a set of frames (matrices) equivalent to as • Measure the distance between two matrices using spherical functions: • where • is zero iff and are equivalent ・・・

20. Explicitly integrating on sphere is expensive! use spherical harmonics • can be analytically decomposed as: • Only 4th frequency band coefficients need to be considered Each is represented as a dim vector

21. technical details: • alignment with surface normals • nonlinear optimization on tetrahedral mesh • results