1. Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points Matthias Nieser joint with Felix Kälberer and Konrad Polthier Global Parameterization and Cone Points Matthias Nieser joint with Felix Kälberer and Konrad Polthier

2. Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points QuadCover Curvature lines are intuitive parameter lines QuadCover: given triangle mesh ) automatically generate global parameterization

3. Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points Related Work (tiny excerpt) X. Gu and S.T. Yau (2003) “Global Conformal Surface Parameterization” Y. Tong, P. Alliez, D. Cohen-Steiner, M. Desbrun (2006) “Designing Quadrangulations with Discrete Harmonic Forms” N. Ray, W.C. Li, B. Levy, A. Sheffer, P. Alliez (2005) “Periodic Global Parameterization” B. Springborn, P. Schröder, U. Pinkall (2008) “Conformal Equivalence of Triangle Meshes” and many more: Bobenko, Gotsman, Rumpf, Stephenson, …

4. Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points Input: Guidance Field Goal: Find triangle parameter lines which align with a given frame field (e.g. principal curvatures, unit length)

5. Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points Parameterization The parameter function … … has two gradient vector fields: ) Integration of input fields yields parameterization.

6. Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points Discretization PL functions Gradients of a PL function:

7. Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points Discrete Rotation Definition Let, p a vertex and m an edge midpoint. Then, the total discrete curl is

8. Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points Local Integrability of Discrete Vector Fields Theorem : is locally integrable in, i.e.

9. Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points Hodge-Helmholtz Decomposition Theorem The space of PL vector fields on any surface decomposes into + + = potential field integrable curl-component not integrable  harmonic field H locally integrable X

10. Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points Assure Local Integrability Problem: guidance frame is usually not locally integrable Solution: (assume frame K splits into two vector fields) 1.Compute Hodge-Helmholtz Decomposition 2.Remove curl-component (non-integrable part) of Result: new frame is locally integrable

11. Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points Global Integrability Solution: mismatch of parameter lines around closed loops 1.Compute Homology generators (= basis of all closed loops) 2.Measure mismatch along Homology generator (next slide…) Problem: mismatch of parameter lines around closed loops

12. Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points Assure Global Integrability Solution: (… cont’ed) 2.Measure mismatch along Homology generator as curve integrals of both vector fields: 3.Compute -smallest harmonic vector fields s.t. Result: new frame is globally integrable

13. Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points QuadCover Algorithm (unbranched) Given a simplicial surface M: 1.Generate a guiding frame field K (e.g. principal curvatures frames) 2.Assure local integrability of K via Hodge Decomp. (remove curl-component from K) 3.Assure global continuity of K along Homology gens. (add harmonic field to K s.t. all periods of K are integers) 4.Global integration of K on M gives parameterization

14. Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points No Splitting of Parameter Lines Warning: parameter lines do not split into red and blue lines !!! Consequence: a frame field does not globally split into four vector fields.

15. Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points Construct a Branched Covering Surface Step 1: Make four layers (copies) of the surface. Step 2: Lift frame field to a vector field on each layer.

16. Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points Construct a Branched Covering Surface Step 3: Connect layers consistently with the vectors. 1.+2. 3. 4. Result: The frame field simplifies to a vector field on the covering surface.

17. Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points Fractional Index of Singularities Branch points will occur at singularities of the field. Index=-1/2Index=1/2Index=1/4Index=-1/4

18. Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points QuadCover Algorithm (full version) 1.Generate a guiding frame field K 2.Detect branch points and compute the branched covering surface. Interpret K as vector field on M* 3.Assure local integrability of K via Hodge Decomposition 4.Lift generators of to generators of the homology group 5.Assure global continuity of K along Homology gens. 6.Global integration of K on M gives parameterization

19. Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points Examples Minimal surfaces with isolated branch points Index of each singularity = -1/2 Trinoid Schwarz-P Surface

20. Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points Examples Minimal surfaces: Costa-Hoffman-Meeks and Scherk. Original parameterization using Weierstrß data Scherk Surface … with QuadCover

21. Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points Examples Surfaces with large close-to-umbilic regions QuadCover texture Original triangle mesh

22. Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points Examples Different Frame Fields Non-orthogonal frame on hyperboloid Non-orientable Klein bottle

23. Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points Examples Rocker arm test model

24. Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points More Complex Examples Thank You!