Mesh Parameterization Theory and Practice Global Parameterization and

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

Quad. Cover Curvature lines are intuitive parameter lines Quad. Cover: given triangle mesh ) automatically generate global parameterization 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, … 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) 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. Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points

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

Local Integrability of Discrete Vector Fields Theorem: is locally integrable in i. e. 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 X curl-component integrable = Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points harmonic field H locally integrable not integrable + +

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 Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points is locally integrable

Global Integrability Problem: mismatch of parameter lines around closed loops 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…) 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 Result: new frame is globally integrable Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points s. t.

Quad. Cover 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 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. 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. 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. Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points

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

Quad. Cover 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 Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points

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

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

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

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