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Quads and Friends

Quads and Friends. Kenshi Takayama. 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

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Quads and Friends

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  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...

  11. 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

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

  13. 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:

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

  15. 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)

  16. 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 ・・・

  17. 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

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

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