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