consistent parameterizations
DESCRIPTION
Consistent Parameterizations. Arul Asirvatham Committee Members Emil Praun Hugues Hoppe Peter Shirley. Parameterization. Mapping from a domain (plane, sphere, simplicial complex) to surface. Motivation: Texture mapping, surface reconstruction, remeshing …. Desirable Properties. - PowerPoint PPT PresentationTRANSCRIPT
Consistent ParameterizationsArul Asirvatham
Committee MembersEmil Praun
Hugues HoppePeter Shirley
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Parameterization• Mapping from a domain (plane, sphere,
simplicial complex) to surface
• Motivation: Texture mapping, surface reconstruction, remeshing …
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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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Contributions• Consistent Spherical Parameterizations
• Inter-surface maps
Consistent Spherical Parameterizations
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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 3Dlinear 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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Finding spherical locations
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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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Constrained Spherical Parameterization
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Approach
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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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Swirls
• Unnecessarily long paths
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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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Sidedness test
AB
D
C E B
A
E
D
C
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Morphing [Praun et al 03]
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Morphing
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Morphing
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Attribute Transfer
+
Color Geometry
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Attribute Transfer
+
Color Geometry
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Face Database
=avg
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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
Inter Surface Maps
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IntroductionNo intermediate domain– Reduced distortion– Natural alignment of features
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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
• Shortest path between AC is separating
A CB
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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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Tracing non separating cycle
• Trace paths from O to A and C
A CB
O
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Automatic Insertion Of Feature Points
Add features if not enough to resolve genus
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Genus-0 example
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Genus-1 example
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Genus-2 example
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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.
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Thank You