2
Parameterization
• Mapping from a domain (plane, sphere, simplicial complex) to surface
• Motivation: Texture mapping, surface reconstruction, remeshing …
3
Desirable Properties
• One-to-one
• Minimize some measure of distortion– Length preserving– Angle preserving– Area preserving– Stretch minimizing
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Outline
• Background– Commonly used Domains
• Plane, Simplicial Complex, Sphere
– Constrained Parameterizations– Consistent Parameterizations
• Consistent Spherical Parameterizations
• Inter-Surface Mapping
• Summary and future work
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Planar Parameterizations• Convex combination maps
– p = i pi , i=1,…,n i =1
• Stretch preserving maps
• Conformal Maps
[Tutte 63][Floater 97][Floater et al 03]
[Sheffer et al 01][Levy et al 02][Desbrun et al 02]
[Sander et al 01]
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Simplicial Parameterizations
• Planar parameterization techniques cut surface into disk like charts
• Use domain of same topology
• Work for arbitrary genus• Discontinuity along base domain edges[Eck et al 95, Lee et al 00, Guskov et al 00, Praun et al 01,
Khodakovsky et al 03]
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Spherical Parameterization
• No cuts less distortion
• Restricted to genus zero meshes
[Shapiro et al 98][Alexa et al 00][Sheffer et al 00][Haker et al 00][Gu et al 03][Gotsman et al 03][Praun et al 03]
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Constrained Parameterizations
• Texture mapping
[Levy et al 01, Eckstein et al 01, Kraevoy et al 03]
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Consistent Parameterizations
Input Meshes
with Features
Semi-Regular Meshes
Base Domain
DGP Applications
• Motivation– Digital geometry processing– Morphing– Attribute transfer– Principal component analysis
[Alexa 00, Levy et al 99, Praun et al 01]
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Stretch Minimizing Spherical Parameterization [Praun & Hoppe 03]
• Use multiresolution– Convert model to progressive mesh format– Map base tetrahedron to sphere– Add vertices one by one, maintaining valid
embedding and minimizing stretch
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Stretch Metric [Sander et al. 2001]
2D texture domain2D texture domain surface in 3Dsurface in 3D
linear maplinear map
singular values: singular values: γγ , , ΓΓ
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Conformal vs StretchConformal metric: can lead to undersampling
Stretch metric encourages feature correspondence
Conformal Stretch
Conformal
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Approach
• Find “good” spherical locations– Use spherical parameterization of one model
• Assymetric
– Obtain spherical locations using all models
• Constrained spherical parameterization– Create base mesh containing only feature
vertices– Refine coarse-to-fine– Fix spherical locations of features
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1. Find initial spherical locations using 1 model2. Parameterize all models using those locations3. Use spherical parameterizations to obtain remeshes4. Concatenate to single mesh5. Find good feature locations using all models6. Compute final parameterizations using these locations
step 1
step 2 step 3 step 6
Algorithm
+ step 4
step 5
UCSP
UCSPCSP
CSP
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Consistent Partitioning
• Compute shortest paths (possibly introducing Steiner vertices)
• Add paths not violating legality conditions– Paths (and arcs) don’t intersect– Consistent neighbor ordering
– Cycles don’t enclose unconnected vertices
• First build spanning tree
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Heuristics to avoid swirls
• Insert paths in increasing order of length
• Link extreme vertices first
• Disallow spherical triangles with any angle < 10o
• Sidedness test
• Unswirl operator
• Edge flips
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Timing
# models
#tris 1 2 5 6 Total (mins)
2 71k-200k
10 5 5 17 37
4 24k-200k
2 23 7 24 56
8 12k-363k
19 81 8 95 203
• 2.4 GHz Pentinum 4 PC, 512 MB RAM
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Comparison to CSP
• No intermediate domain
• Arbitrary genus
• Limited to 2 models
• Applications
– Morphing– Digital geometry processing– Transfer of surface attributes– Deformation transfer
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Contributions
• Directly create inter-surface map– Symmetric coarse-to-fine optimization– Symmetric stretch metric
Automatic geometric feature alignment
• Robust– Very little user input– Arbitrary genus– Hard constraints
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1. Consistent mesh partitioning2. Constrained Simplification3. Trivial map between base meshes4. Coarse-to-fine optimization
Algorithm Overview
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Consistent Mesh Partitioning
• Compute matching shortest paths (possibly introducing Steiner vertices)
• Add paths not violating legality conditions
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Legality Conditions
• Paths don’t intersect
• Consistent neighbor ordering
• Cycles don’t enclose unconnected vertices
• First build maximal graph without sep cycles
• genus 0: spanning tree
• genus > 0: spanning tree + 2g non-sep cycles
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Separating/Non-separating cycles
• Separating cycle breaks surface into 2 disjoint components
Separating cycle Non separating cycle
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Non-separating cycle test
• Grow 2 fronts starting on both sides of AB
• Non-separating if fronts meet
A
B
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Tracing non separating cycle
• Grow contour around AC
• Contour wraps around and meets itself at O
A CO
B
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Contributions
• Consistent Spherical Parameterizations for several genus-zero surfaces– Robust method for Constrained Spherical
Parameterization
• Robust partitioning of two meshes of arbitrary genus
• Methods to avoid swirls and to correct them when they arise
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Future Work
• Improve overall exectution time– Multiresolution path tracing algorithm– Linear stretch optimization
• Construct maps between surfaces of different genus
• Handle point cloud and volumetric data
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Publications
Consistent Spherical Parameterizations, Arul Asirvatham, Emil Praun, Hugues Hoppe, Computer Graphics and Geometric Modelling, 2005.
Inter-Surface Mapping, John Schreiner, Arul Asirvatham, Emil Praun, Hugues Hoppe, ACM SIGGRAPH 2004.