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Mesh Parameterization: Theory and Practice. Global Parameterization and Cone Points Matthias Nieser joint with Felix Kälberer and Konrad Polthier. QuadCover. Curvature lines are intuitive parameter lines. QuadCover : given triangle mesh ) automatically generate global parameterization.

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Mesh parameterization theory and practice l.jpg

Mesh Parameterization:Theory and Practice

Global Parameterization and Cone Points

Matthias Nieser

joint with Felix Kälberer and Konrad Polthier


Quadcover l.jpg
QuadCover

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 l.jpg
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 l.jpg
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 PracticeGlobal Parameterization and Cone Points


Parameterization l.jpg
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


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Discretization

PL functions

Gradients of a PL function:

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points


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Discrete Rotation

DefinitionLet , 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 l.jpg
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 l.jpg
Hodge-Helmholtz Decomposition

Theorem The space of PL vector fields on any surface

decomposes into

potential field

integrable 

curl-componentnot integrable 

harmonic field H

locally integrable 

X

=

+

+

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points


Assure local integrability l.jpg
Assure Local Integrability

Problem:guidance frame is usually not locally integrable

Solution: (assume frame K splits into two vector fields)

  • Compute Hodge-Helmholtz Decomposition

  • Remove curl-component (non-integrable part) of

    Result: new frame is locally integrable

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points


Global integrability l.jpg
Global Integrability

Solution:mismatch of parameter lines around closed loops

  • Compute Homology generators (= basis of all closed loops)

  • 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 integrability l.jpg
Assure Global Integrability

Solution: (… cont’ed)

  • Measure mismatch along Homology generator as curve integrals of both vector fields:

  • Compute -smallest harmonic vector fields s.t.

    Result:new frame

    is globally integrable

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points


Quadcover algorithm unbranched l.jpg
QuadCover Algorithm (unbranched)

Given a simplicial surface M:

  • Generate a guiding frame field K

    (e.g. principal curvatures frames)

  • Assure local integrability of K via Hodge Decomp.

    (remove curl-component from K)

  • Assure global continuity of K along Homology gens.

    (add harmonic field to K s.t. all periods of K are integers)

  • Global integration of K on M

    gives parameterization

Mesh Parameterization: Theory and PracticeGlobal Parameterization and Cone Points


No splitting of parameter lines l.jpg
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 l.jpg
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 surface16 l.jpg

1.+2.

3.

4.

Construct a Branched Covering Surface

Step 3: Connect layers consistently with the vectors.

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 l.jpg
Fractional Index of Singularities

Branch points will occur at singularities of the field.

Index=1/4

Index=-1/4

Index=1/2

Index=-1/2

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points


Quadcover algorithm full version l.jpg
QuadCover Algorithm (full version)

  • Generate a guiding frame field K

  • Detect branch points and compute

    the branched covering surface.

    Interpret K as vector field on M*

  • Assure local integrability of K

    via Hodge Decomposition

  • Lift generators of to generators

    of the homology group

  • Assure global continuity of K

    along Homology gens.

  • Global integration of K on M gives parameterization

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points


Examples l.jpg
Examples

Minimal surfaces with isolated branch points

Index of each singularity = -1/2

Trinoid

Schwarz-P Surface

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points


Examples20 l.jpg
Examples

Minimal surfaces: Costa-Hoffman-Meeks and Scherk.

Original parameterization using Weierstrß data

… with QuadCover

Scherk Surface

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points


Examples21 l.jpg
Examples

Surfaces with large close-to-umbilic regions

QuadCover texture

Original triangle mesh

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points


Examples22 l.jpg
Examples

Different Frame Fields

Non-orthogonal frame on hyperboloid

Non-orientable Klein bottle

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points


Examples23 l.jpg
Examples

Rocker arm test model

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points


More complex examples l.jpg
More Complex Examples

Thank You!

Mesh Parameterization: Theory and Practice Global Parameterization and Cone Points