Mesh Parameterization: Theory and Practice

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Mesh ParameterizationTheory and Practice

• Global Parameterization and Cone Points
• Matthias Nieser
• joint with Felix Kälberer and Konrad Polthier

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QuadCover

• Curvature lines are intuitive parameter lines

QuadCover given triangle mesh ) automatically
generate global parameterization
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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,

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Input Guidance Field

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

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Parameterization

• The parameter function

has two gradient vector fields
) Integration of input fields yields
parameterization.
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Discretization

• PL functions

Gradients of a PL function
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Discrete Rotation

• Definition Let , p a vertex
  and m an edge midpoint.
• Then, the total discrete curl is

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Local Integrability of Discrete Vector Fields

• Theorem is locally integrable in ,
• i.e.

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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



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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

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Global Integrability
Problem mismatch of parameter lines around
closed loops

• Solution mismatch of parameter lines around
  closed loops
• Compute Homology generators ( basis of all
  closed loops)
• Measure mismatch along Homology generator (next
  slide)

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Assure Global Integrability

• Solution ( conted)
• 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

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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

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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.

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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.
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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.
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Fractional Index of Singularities

• Branch points will occur at singularities of the
  field.

Index-1/2
Index1/2
Index1/4
Index-1/4
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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

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Examples

• Minimal surfaces with isolated branch points

Index of each singularity -1/2
Trinoid
Schwarz-P Surface
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Examples

• Minimal surfaces Costa-Hoffman-Meeks and Scherk.

Original parameterization using Weierstrß data
with QuadCover
Scherk Surface
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Examples

• Surfaces with large close-to-umbilic regions

QuadCover texture
Original triangle mesh
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Examples

• Different Frame Fields

Non-orthogonal frame on hyperboloid
Non-orientable Klein bottle
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Examples

• Rocker arm test model

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More Complex Examples

• Thank You!