Mesh Parameterization:Theory and Practice

Mesh Parameterization:Theory and Practice

Global Parameterization and Cone Points

Matthias Nieserjoint with Felix Kälberer and Konrad Polthier

Global Parameterization and Cone Points

Matthias Nieserjoint with Felix Kälberer and Konrad Polthier

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points

QuadCoverQuadCover

Curvature lines are intuitive parameter lines

QuadCover: given triangle mesh) automatically generate global parameterization

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points

Related Work (tiny excerpt)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 PracticeGlobal Parameterization and Cone Points

Input: Guidance FieldInput: Guidance Field

Goal: Find triangle parameter lines which alignwith a given frame field (e.g. principal curvatures, unit length)

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points

ParameterizationParameterization

The parameter function …

… has two gradient vector fields:

) Integration of input fields yields parameterization.

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points

DiscretizationDiscretization

PL functions

Gradients of a PL function:

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points

Discrete RotationDiscrete 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 FieldsLocal Integrability of Discrete Vector Fields

Theorem: is locally integrable in ,i.e.

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points

Hodge-Helmholtz DecompositionHodge-Helmholtz Decomposition

Theorem The space of PL vector fields on any surfacedecomposes into

+ +=

potential field integrable

curl-component not integrable

harmonic field Hlocally integrable

X

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points

Assure Local IntegrabilityAssure Local Integrability

Problem: guidance frame is usually not locally integrable

Solution: (assume frame K splits into two vector fields)1. Compute Hodge-Helmholtz Decomposition2. Remove curl-component (non-integrable part) of

Result: new frame is locally integrable

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points

Global IntegrabilityGlobal Integrability

Solution: mismatch of parameter lines around closed loops1. Compute Homology generators (= basis of all closed loops)

2. Measure mismatch along Homology generator (next slide…)

Problem: mismatch of parameter lines around closed loops

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points

Assure Global IntegrabilityAssure 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

Mesh Parameterization: Theory and PracticeGlobal Parameterization and Cone Points

QuadCover Algorithm (unbranched)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 Mgives parameterization

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points

No Splitting of Parameter LinesNo 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 SurfaceConstruct 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 SurfaceConstruct 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 SingularitiesFractional Index of Singularities

Branch points will occur at singularities of the field.

Index=-1/2Index=1/2Index=1/4 Index=-1/4

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points

QuadCover Algorithm (full version)QuadCover Algorithm (full version)

1. Generate a guiding frame field K2. Detect branch points and compute

the branched covering surface.Interpret K as vector field on M*

3. Assure local integrability of Kvia Hodge Decomposition

4. Lift generators of to generatorsof the homology group

5. Assure global continuity of Kalong Homology gens.

6. Global integration of K on M gives parameterization

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points

ExamplesExamples

Minimal surfaces with isolated branch pointsIndex of each singularity =

-1/2

Trinoid Schwarz-P Surface

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points

ExamplesExamples

Minimal surfaces: Costa-Hoffman-Meeks and Scherk.

Original parameterization using

Weierstrß dataScherk Surface

… with QuadCover

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points

ExamplesExamples

Surfaces with large close-to-umbilic regions

QuadCover texture

Original triangle mesh

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points

ExamplesExamples

Different Frame Fields

Non-orthogonal frame on

hyperboloid

Non-orientable Klein bottle

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points

ExamplesExamples

Rocker arm test model

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points

More Complex ExamplesMore Complex Examples

Thank You!