Mesh Parameterization

1
Mesh Parameterization

• Tan-Chi Ho
• (???)

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Outline

• Introduction to mesh parameterization
• Introduction
• Selected methods for mesh parameterization
• Floaters method
• Angle based flattening
• Texture mapping progressive meshes
• My research project
• A parameterization approach for papercraft design

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Introduction

• Whats mesh parameterization
• Find a one-to-one mapping from a suitable domain
  to the surface.
• In general, we map the 3D surface into 2D domain.

f
(u1,u2) (u1,u2)
x
x
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Types of mapping

• Isometric mapping
• Length-preserving mapping
• Conformal mapping
• Angle-preserving mapping
• Equiareal mapping
• Preserve area at every part of surface.

Isometric Conformal Equiareal
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General work for mesh parameterization

• To minimize the distortion between mapping
• Angle distortion
• Area distortion
• Approaches
• Numerical methods
• Non-linear iterative methods based on the
  measured mapping error.

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Floaters parameterization

• Parameterization and smooth approximation of
  surface triangluations. Computer Aided Geometric
  Design, 14(3), 1997.
• Convex combination map
• Each mapped internal vertex f(vi) will be a
  convex combination of its neighbors f(vj), and so
  must lie in their convex hull.
• The weight wi,j is assigned using barycentric
  mapping.
• Formulate as a linear equation system.

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Floaters parameterization

• Steps for Floaters parameterization
• Fix the boundary mapping.
• Solve the linear equation system for internal
  vertices mapping.

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Angle Based Flattening

• Parameterization of faceted surfaces for meshing
  using angle based flattening. Engineering with
  Computers, 17(3), 2001.
• To minimize the angle difference of the parameter
  triangles.
• The energy function
• The constraints

a42
a51
a41
a43
a53
a32
a33
a52
a23
a13
a31
a11
a12
a21
a22
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Angle Based Flattening

• Its a constraint minimization problem.
• Reformulate the problem using Lagrange multiplier
  formulation.
• Solve the non-linear system using Newtons method.

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Texture Mapping Progressive Meshes

• Texture Mapping Progressive Meshes, SIGGRAPH,
  2001.
• To minimize the stretch error after mapping.
• The mapping error is measured by the proposed
  texture stretch metric.

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Texture Mapping Progressive Meshes

• Steps for parameterization

Initial Param.
Initial param.
Measure L(M)
Re-param.
If L(M) exceed threshold
yes
After optimization
no
Result
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A parameterization approach for papercraft design
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Problem statement

• Its possible to unfold a 3D triangulated mesh
  without distortion.
• The shape of the unfolded patterns are too
  complex to be reconstructed.

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Goal

• Our goal
• To develop a parameterization framework suited
  for papercraft design.
• The error after parameterization should be
  bounded.
• The length of edges on the mesh boundary should
  preserved.

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Related work

• Bounded distortion piecewise mesh
  parameterization. IEEE Visualization, 2002.

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Related work

• Making Papercraft Toys from Meshes using
  Strip-based Approximate Unfolding. SIGGRAPH, 2004.

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Related work

• D-Charts Quasi-Developable Mesh Segmentation.
  Eurographics, 2005.

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Our approach
Initial Cutting
Boundary Parameterization
Add seams
Internal Parameterization
Result
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Boundary parameterization

• To flatten the 3D mesh boundary into 2D domain
• Minimize the angle difference.
• The length of all edge should be the same.

? Constraint optimization problem
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Boundary parameterization

• Modify the ABF method
• Objective function
• Where
• Constraints

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Boundary parameterization

• Rewrite the problem using Lagrange multiplier
  formulation
• Solve the following non-linear equation system
  for aij

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Boundary parameterization

• Construct the boundary loop on the 2D domain
  using aij and ei.

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Boundary parameterization

• Problems
• For boundary loop which is not star shaped.
• Self intersection of the boundary loop.

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Internal parameterization

• Parameterize the internal vertices iteratively
  from the boundary.
• Place the internal vertex.
• Measure the triangle stretch.
• Add seams from vertices with higher stretch to
  the boundary.

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Internal parameterization
Boundary Parameterization
Internal Param.
Add seams
Place internal vertex
Measure the texture stretch metric
Does the stretch error exceed threshold
yes
No
Have all vertices been parameterized
No
yes
Result
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Internal parameterization

• Vertex placement
• The optimal vertex placement should consider its
  neighboring triangles.
• Modify the method proposed by Sorkin et al.
  (Bounded-distortion Piecewise Mesh
  Parameterization. IEEE Visualization, 2002)

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Internal parameterization

• Triangle stretch measurement
• Use the stretch metric proposed by Sander et al.
  (Texture Mapping Progressive Meshes. SIGGRAPH,
  2001)
• Once the triangle stretch exceed the threshold,
  add seams from that triangle to the boundary and
  re-parameterize the mesh.

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Internal parameterization

• Problem
• The texture stretch metric proposed by Sander et
  al. is not sufficient to describe the mapping
  error.
• Lack of anisotropic information
• Lack of developability information

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Mesh cutting

• Purpose
• To make the mesh able to parameterization.
• To reduce the distortion during parameterization.
• Goal
• The seams should be
• as short and invisible
• as possible.

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Mesh cutting

• Minimum spanning tree
• The seams are generated using MST algorithm
  proposed by Sheffer. (Spanning Tree Seams for
  Reducing Parameterization Distortion of
  Triangulated. SMI, 2002)

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Mesh cutting

• Problems
• The geometry features should be considered during
  seam generation.
• There should be some common rules for adding
  seams in the papercraft design.

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Current status

• Current status
• The boundary parameterization method is done.
• Future work
• A better stretch metric for measuring the mapping
  error.
• A new mesh cutting scheme taking the geometry
  features into account.
• General cutting principle for papercraft design.