mesh parameterizations lizheng lu lulz_zju@yahoo.com.cn oct. 19, 2005

Mesh Parameterizations

Lizheng LuLulz_zju@yahoo.com.cn

Oct. 19, 2005

Overview

Introduction Planar Methods Non-Planar Methods

Mean Value Methods Spherical Methods

Summary

Motivation(1) Analysis on surfaces is usually performed in Eucli- dean plane, using appropriate (local) coordinates. ⇒One has to assign to

every surface point a parameter value in the plane The result of the analysis often depends on the choice of the parameterization.

Example: B-Spline Interpolation

Motivation(2)

Q: What is a good parameterization ?A: One that preserve all the (basic) geometry length, angles, area, ... ⇒ isometric parameterizationbut : possible only for developable surfaces e.g., there will always be distortion !

Try to keep the distortion as small as possible (change of length, area, angles,... )

Motivation(3): Applications

Many operations, manipulations on/with surfaces require a parameterization as a preliminary step.

e.g.: Texture mapping

Surface fitting Hierarchical representations

Mesh conversion Morphing & Deformation

Motivation: Applications of Parameterizations

Motivation: Applications of Parameterizations

Motivation: Applications of Parameterizations

Motivation: Applications of Parameterizations

Morphing

Problem Description

For a triangulated set of data points

find a parameteration

with minimal distortion

Classifications

Conformal mapping No distortion in angles

Equiareal mapping No distortion in areas

Isometric mappings No distortion, but usually impossible

Desirable Properties With minimal distortion

So how to measure and minimize it? Guaranteeing one-to-one mapping

Avoid overlapping, degenerating, flipping Most difficult and critical!

Robustness Time and space efficiency Process meshes with genus, if possible

Previous Methods:Classifications

Planar methods Early works

Cube/Polycube methods Spherical methods Partition methods …Goal: Minimizing distortion for diverse meshes

Overview

Introduction Planar Methods Non-Planar Methods

Mean Value Methods Spherical Methods

Summary

Theory Foundation

Given: A planar 3-connected graph Boundary fixed to a convex shape in R2

Result: Interior vertices form a planar triangular

Each vertex is some convex combination of its neighbors

Main Challenge Measure of distortion

Ratio of singular values (Hormann & Greiner 1998) Conformal (Levy, 2002) Dirichlet energy (Guskov, 2002) Mean value (Floater, 2003,2005 & Tao Ju,2005)

Boundary fixing Choose the shape, e.g.. Circle, square, etc. Choose the distribution

Seamless merging Partition and cutting

Main References M. S. Floater. Parameterization and smooth approximation of surface triangulations. CAGD , 1997, 14(3):231-250. Hormann, Greiner: MIPS: An efficient global parametrization method, in: Curve and Surface Design: Saint−Malo1999,153−162

M. S. Floater and M. Reimers. Meshless parameterization and surface reconstruction. CAGD , 2001, 18(2):77-92.

M. S. Floater, Mean value coordinates, CAGD , 2003,20(1), 19-27. M. S. Floater, One-to-one piecewise linear mappings over triangulations,

Math. Comp. 2003,72(242), 685-696. M. S. Floater and K. Hormann, Surface Parameterization: a Tutorial and Survey,

in Advances in Multiresolution for Geometric Modelling, N. A. Dodgson, M. S.

Floater, and M. A. Sabin (eds.), Springer-Verlag, Heidelberg, 2004, 157-186.

Linear Methods: Idea

Fixing the boundary of the mesh onto

an unit circle an unit square

Linear Methods: Idea

For interior mesh points:

⇒ Forming a linear system.

Choices of the Weights Uniform:

Chord length:

Centripetal:

Mean value:

1 1

i

ijk N k ij i

x xx x

1/ , wherei i i iw d d N

Shortcomings

Severe distortion Topology limiting

Can't process non genus-zero meshes Introduce other artifacts

Such as cutting seams

Non-linear Methods [Hormann et al. 1999] MIPS [Piponi et al. 2000] Seamless texture ma

pping of subdivision surfaces by model pelting and texture blending. SIGGRAPH

[Sander et al. 2001] Texture mapping progressive meshed. SIGGRAPH

[Zigelman et al. 2001] Texture mapping using surface flattening via multi-dimensional scaling. TVCG, 8(2), 198-207

Overview

Introduction Planar Methods Non-Planar Methods

Mean Value Methods Spherical Methods

Summary

How to Obtain Good Parameterization

Mesh independence? Very difficult

Less distortion? Maybe, defining better measure function

Possible method for minimizing distortion Choosing possible mapping domains!

Sphere, Cube/polycub, Simplified domains ...

Main References(1)Spherical Domain

Sheffer, A., Gotsman, C., Dyn, N. 2004. Robust Spherical Parameterization of Triangular Meshes. Computing, 72(1-2), 185–19

3. Praun, E., Hoppe, H. Spherical Parametrization and Remeshing. SIG

GRAPH2003. Gotsman, C., Gu, X., Sheffer, A. Fundamentals of Spherical Paramet

erization for 3D Meshes. SIGGRAPH 2003. Alexa, M. Recent advances in mesh morphing. 2002. Computer Gra

phics Forum, 21(2), 173-196. Grimm, C. Simple manifolds for surface modeling and parametrizat

ion. Shape Modeling International 2002. Haker, S., Angenent, S., et al. Conformal surface parameterization f

or texture mapping. 2000. TVCG, 6(2), 181-189. Kobbelt, L.P., Vorsatz, J., Labisk, U., Seidel, J.-p.. A shrink-wrapping

approach to remeshing polygonal surfaces. 1999. CGF. 18(3), 119-129.

Kent, J., Carlson, W., Parent, R. 1992. Shape transformation for polyhedral objects. SIGGRAPH 1992, 47-54.

Main References(2)Cube/Polycube Domain

Tarini, M., Hormann, K., Cigononi, P., Montani, C. PolyCube-Maps. SIGGRAPH 2004.

Main References(3) Simplified Domains

Schreiner, J., Asirvatham, A, Praun, E., Hoppe, H. Inter-Surface Mapping. SIGGRAPH 2004.

Khodakovsky, A., Litke, N., Schröder, P. Globally Smooth Parameterizations with Low Distortion. SIGGRAPH 2003.

Gu, X., Gortler, J., Hoppe, H. Geometry images. SIGGRAPH 2002. Sorkine, O., Cohen-or, D., et al. Bounded-distortion piecewise mesh para

metrization. 2002. IEEE Visualization, 355-362. Praun, E. Sweldens, W. Schröder, P. Consistent mesh parametrizations. SI

GGRAPH 2001. Guskov, I., Vidimce, K., Sweldens, W., Schröder, P. Normal meshes. SIGGR

APH 2000. Lee, A., Dobkin, D., Sweldens, W., Schröder, P. Multiresolution mesh morp

hing. SIGGRAPH 1999. Hoppe, H. Progressive meshes. SIGGRAPH 1996, 99-108.

Example Spherical Methods

ExamplePolycube Methods

ExampleSimplification/Cutting Methods

Overview

Introduction Planar Methods Non-Planar Methods

Mean Value Methods Spherical Methods

Summary

Mean Value Coordinates for Closed Triangular Mesh

Tao Ju, Scott Schaefer, Joe WarrenRice University

SIGGRAPH2005

Mean Value MethodsReferences

M. S. Floater. Mean value coordinates. CAGD, 14(3):19–27, 2003.

M. S. Floater. Mean value coordinates in 3D. CAGD, 22(7):623–631, 2005.

Tao, Ju Scott Schaefer, Joe Warren. Mean Value Coordinates for Closed Triangular Meshes. SIGGRAPH 2005.

Barycentric Coordinates Give , find weights such that

with barycentric coordinates

iwv

i ii

ii

w pv

w

i

ii

w

w

Boundary Value Interpolation

Given , compute such that

Given values at , construct a function

Good properties: Interpolates values at vertices Linear on boundary Smooth on interior

ipiw

ipif

[ ] i ii

ii

w ff v

w

i ii

ii

w pv

w

Continuous Barycentric Coordinates Discrete Continuous

[ ] i ii

ii

w ff v

w

[ , ] [ ]

[ ][ , ]

x

x

w x v f x dxf v

w x v dx

Mean Value Interpolation

Properties: Interpolates boundary Generates smooth function Reproduces linear function

s

[ ][ ]

[ ]1

[ ]

vx

vx

f xdS

p x vf v

dSp x v

Relation to Discrete Coordinates

MV coordinates ⇒ Closed-form solution of continuous interpolant for piecewise linear shap

es

Discrete Continuous

3D Mean Value Coordinates

( ) 0i iii ii

ii

w pv w p v

w

Project surface onto sphere centered at v

m = mean vector (integral of unit normal over spherical triangle)

Stokes’ Theorem:

3

1

( )k kk

m w p v

0jj

m ( ) 0i ii

w p v

Computing the Mean Vector

Given spherical triangle, compute mean vector (integral of unit normal) Build wedge with face normals Apply Stokes’ Theorem,

m

3

1

10

2 k kk

n m

kn

Interpolant Computation Compute mean vector:

Calculate weights

By

Sum over all triangles

3

1

10

2 k kk

n m

( )k

kk k

n mw

n p v

3

1

3

1

[ ]j j

k kj k

jkj k

w ff v

w

3

1

( )k kk

m w p v

Implementation Considerations

Special cases On boundary (co-planar)

Numerical stability Small spherical triangles Large meshes

Pseudo-code provided in paper

ApplicationsBoundary Value Problems

ApplicationsSolid Textures

Applications Surface Deformations

Real-time!

Control Mesh Surface Comp. Weights

Deformation

216 Triangles 30k Triangles

1.9 Sec. 0.03 Sec.

Initial mesh

Summary

Integral formulation for closed surfaces Closed-form solution for triangle meshes

Numerically stable evaluation Applications

Boundary Value Interpolation Volumetric textures Surface Deformation

Overview

Introduction Planar Methods Non-Planar Methods

Mean Value Methods Spherical Methods

Summary

Challenges on Spherical Domain Robustness

Non-overlapping -->> Difficult and critical 1-to-1 spherical map -->> Required

Less distortion Diverse meshes -->> Highly deformed Oversampling/downsamping -->>Irregular

So, how to miminize it? Visually pleasing, regular, …

Previous Spherical Methods(1)

[Kent et al. 92]Shape Transformation for Polyhedral Objects. SIGGRAPH.

Projections Methods: Convex and Star-Shaped Objects Methods using model knowledge Physically-Based Simulation

Simulating balloon inflation process Hybrid methods

Unsolved problem, 1-to-1 map?…

Previous Spherical Methods(2)

[Shapiro & Tal 98] Polygon Realization for Shape Transformation. The Visu. Comp. 8-9,429-444.

Limitations Difficult to optimize, due to greedy nature Lack desirable mathematical properties So simple, inefficient to large mesh

Previous Spherical Methods(3)

[Kobbelt et al. 99] A Shrink-wrapping Approach to Remeshing Polygonal Surfaces. CGF, 18(3),119-129.

[Alexa 00] Merging Polyhedral Shapes with Scattered Features. The Visu. Comp., 16(1), 26-37.

[Alexa 02] Recent Advances in Mesh Morphing. CGF, 21(2), 173-196.

Heuristic iterative No guarantee to converge Sometimes invalid embedding

[Alexa 02] Several heuristics to converge validness

Previous Spherical Methods(4)

[Haker 00]Conformal Surface Parameterization for Texture Mapping. TVCG, 6(2),181-189.

Conformal mapping Remove a single point, harmonic map: remain surface an infinite plane Stereographic projection: plane sphere

Limitations: No guarantee to embedding despite bijective and conformal map

Previous Spherical Methods(5)

[Grimm 02] Simple manifolds for surface modeling and parameterization. Shape Modeling International.

Remark: A priori chart partitions constrain spherical parameterization

Previous Spherical Methods(6)

[Sheffer 04] Robust Spherical Parameterization of Triangular Meshes. Computing, 72(1-2), 185–193.

Angle-based method Constrained nonlinear system Valid embedding guaranteeing Limitations

Highly no-linear optimization Lack efficient numerical computation procedure

Spherical Methods(1)

Spherical Parameterization and Remeshing

Emil Praun Hugues HoppeUniversity of Utah Microsoft Research

SIGGRAPH2003

Main References GU, X., GORTLER, S., AND HOPPE, H. 2002. Geometry images. ACM SIG

GRAPH 2002, pp. 355-361. SANDER, P., GORTLER, S., SNYDER, J., HOPPE, H. Signal-specialized parameterization. Eurographics Workshop on Rendering 2002, pp. 87-10

0. SANDER, P., SNYDER, J., GORTLER, S., AND HOPPE, H. 2001. Texture ma

pping progressive meshes. ACM SIGGRAPH 2001, pp. 409-416. PRAUN, E., SWELDENS, W., AND SCHRÖDER, P. 2001. Consistent mesh p

arametrizations. ACM SIGGRAPH 2001, pp. 179-184. HAKER, S., ANGENENT, S., TANNENBAUM, S., KIKINIS, R., SAPIRO, G., AN

D HALLE, M. 2000. Conformal surface parametrization for texture mapping. IEEE TVCG, 6(2), pp. 181-189.

LOSASSO, F., HOPPE, H., SCHAEFER, S., AND WARREN, J. 2003. Smooth geometry images. Submitted for publication.

HOPPE, H. 1996. Progressive meshes. ACM SIGGRAPH 96, pp. 99-108. DAVIS, G. 1996. Wavelet image compression construction kit. http://

www.geoffdavis.net/dartmouth/wavelet/wavelet.html.

Scope

Assumed meshes Geneus-0 Manifold Closed, can handle open ones also

-->> Topology equal to a sphere!!

Motivation: Geometry Images [Gu et al. ’02][Gu et al. ’02]

Completely regular Completely regular samplingsampling

Geometry imageGeometry image 257 x 257; 12 bits/channel 257 x 257; 12 bits/channel

3D Geometry3D Geometry

Motivation: Geometry Images

Geometry Images [Gu et al. ’02] No connectivity to store Render without memory gather operation

s No vertex indices No texture coordinates

Regularity allows use of image processing tools

Spherical Parametrization Genus-0 models: no Genus-0 models: no a prioria priori cuts cuts

Geometry imageGeometry image257 x 257; 12 bits/channel257 x 257; 12 bits/channel

Contribution Genus-0 no constraining cuts Less distortion in map; Better compression New applications:

Morphing GPU splines DSP

Overview

OriginalOriginal SphericalSphericalparametrizationparametrization

GeometryGeometryimageimage

RemeshRemesh

Process overview

Outline Spherical parametrization

Spherical remeshing

Results & applications

Spherical Parametrization

Mesh Mesh MM Sphere Sphere SS

[Kent et al. ’92][Kent et al. ’92][Haker et al. 2000][Haker et al. 2000][Alexa 2002][Alexa 2002][Grimm 2002][Grimm 2002][Sheffer et al. 2003][Sheffer et al. 2003][Gotsman et al. 2003][Gotsman et al. 2003]

Goals: Robustness Good

sampling

[Hoppe 1996][Hoppe 1996]

[Sander et al. 2001][Sander et al. 2001]

[Hormann et al. 1999][Hormann et al. 1999]

[Sander et al. 2002][Sander et al. 2002]

Coarse-to-fine Stretch metric

Coarse-to-Fine Algorithm Convert to progressive mesh [Convert to progressive mesh [Hoppe 96Hoppe 96]]

Parametrize coarse-to-fineParametrize coarse-to-fineMaintain embedding & minimize stretchMaintain embedding & minimize stretch

Before V-split: No

degenerate/flipped 1-ring kernel

Apply V-split:No flips if V inside

kernel

VV

Coarse-to-Fine Algorithm

Before V-split: No degenerate/flipped

1-ring kernel

Apply V-split:No flips if V inside

kernel Optimize stretch:

No degenerate (they have stretch)

VV

Coarse-to-Fine Algorithm

Traditional Conformal Metric

Preserve angles but “area compression”

Bad for sampling using regular grids

[Sander et al. 2001][Sander et al. 2001]

[Sander et al. 2002][Sander et al. 2002]

Penalizes undersampling Better samples the surface

Stretch Metric

Regularized Stretch

Stretch alone is unstable Add small fraction of inverse stretch

withoutwithout withwith

Outline Spherical parametrization

Spherical remeshing

Results & applications

Domains and Their Sphere Maps

TetrahedronTetrahedron

OctahedronOctahedron

CubeCube

Domain Unfoldings

Boundary Constraints

Spherical Image Topology

Spherical Image Topology

Spherical Image Topology

Outline Spherical parametrization

Spherical remeshing

Results & applications

Example Results

Results

Results

Results

DavidDavidDavidDavid

Model courtesy of Model courtesy of Stanford UniversityStanford University

Timing Results

Model # Faces InitialInitial OptimizeOptimizedd

Cow 23,216 7 min. 65 sec.

David 60,000 8 min. 80 sec.

Bunny 69,630 10 min. 1.5 min.

Horse 96,948 15 min. 2.5 min.

Gargoyle 200,000 23 min. 4 min.

Tyrannosaurus

200,000 25 min. 4 min.Pentium IV 3GHz, optimized codePentium IV 3GHz, optimized code

Rendering

interpretinterpretdomaindomain

renderrendertessellationtessellation

Level-of-Detail Control

nn=1=1 nn=2=2 nn=4=4 nn=8=8 nn=16=16 nn=32=32 nn=64=64

Align meshes & Interpolate Geometry Images

Geometry Compression

Image wavelets Boundary extension

rules Spherical topology Infinite C1 lattice*

Globally smooth parametrization*

*(except edge midpoints)

Compression Results

12 KB 3 KB 1.5 KB

Compression Results

50

55

60

65

70

75

80

85

90

1000 10000 100000File Size (bytes)

PSNR

Image Wavelets

PGC [Khodakovsky et al. '00]

Gu et al. max(n=257,n=513)

50

55

60

65

70

75

80

85

90

1000 10000 100000File Size (bytes)

PSNR

Image Wavelets

PGC [Khodakovsky et al. '00]

Gu et al. max(n=257,n=513)

Smooth Geometry Images

33x33 geometry image33x33 geometry image CC11 surface surface

GPUGPU

3.17 ms

[Losasso et al. 2003][Losasso et al. 2003]

Ordinary Uniform Bicubic B-splineOrdinary Uniform Bicubic B-spline

Conclusions

Spherical parametrization Guaranteed one-to-one

New construction for geometry images Specialized to genus-0 No a priori cuts better performance New boundary extension rules

Effective compression, DSP, GPU splines, …

Future Work

Explore DSP on unfolded octahedron 4 singular points at image edge midpoints

Fine-to-coarse integrated metric tensors Faster parametrization; signal-specialized

map

Direct DSM optimization Consistent inter-model parametrization

Spherical Methods(2)

Fundamentals of Spherical Parameterization

for 3D Meshes

Craig Gotsman1, Xianfeng Gu2, Alla Sheffer1

1.Technion – Israel Inst. of Tech. 2.Harvard University.

SIGGRAPH2003

Main References GU, X., AND YAU, S.-T. 2002. Computing Conformal Structures of Surfaces.

Communications in Information and Systems, 2,2, 121-146. LEVY, B., PETITJEAN, S., RAY, N., AND MAILLOT, J. 2002. Least Squares Conformal Maps for Automatic Texture Atlas Generation.TOG,21,3,362-67

1 ALEXA, M. 2000. Merging Polyhedral Shapes with Scattered Features. The Visual Computer 16, 1, 26-37. HAKER, S., ANGENENT, S., TANNENBAUM, A., et al. 2000. Conformal Surfa

ce Parameterization for Texture Mapping. IEEE TVCG, 6, 2, 1-9. LOVASZ, L., AND SCHRIJVER, A. 1999. On the Nullspace of a Colin de

Verdiere Matrix. Annales de l'Institute Fourier 49, 1017-1026. COLEMAN, T.F., LI, Y. 1996. An Interior Trust Region Approach for

Nonlinear Minimization Subject to Bounds. SIAM J. on Optimi.,6, 418-445. On a New Graph Invariant and a Criterion for Planarity. In Graph Structur

e Theory. 1993. (N. Robertson, P. Seymour,Eds.) Contemporary Mathematics, AMS, 137-147.

TUTTE. W.T. 1963. How to Draw a Graph. Proc. London Math. Soc. 13, 3, 743-768.

Scope

Assumed meshes Geneus-0 Manifold Closed

-->> Topology equal to a sphere!!

Main Idea Overview

Nonlinear extension of the linear theory barycentric coordinates 2D: General Normalized Laplacian operator 3D: Laplace-Beltrami operator [Gu&Yau02]

Spectral Graph Theory CdV(Colin de Verdiere) number CdV eigenvalue, eigenvector CdV nullspace

Spectral Graph Theory:Basic Theorem Given:

Planar 3-connected graph in Result:

Each vetex is some convex combination of its neighbors, projected the on sphere

Valid embedding

3

Barycentric CoordinatesPlanar Case Interior edge e=(i,j), assign weight , such that All other entries (i,j), let Embed boundary vertex to a closed conve

x region Solve linear systems

Laplace equation

0ijw ( )

1ijj N i

w

ijw

( ) , ( )x yI W x b I W y b

wL x b

Barycentric CoordinatesSpherical Case Define Laplace operator

But, restrict to be symmetric

( , )

( , ) ( , )

0 ( , )

w wk i

negative number i j E

L i j L i k i j

i j E

Barycentric CoordinatesExtension [Gu & Yau 02] Laplace-Beltrami operator [Gu & Yau 0

2] Inspired by class differential geometry

Nonlinear system

Bijective embedding a continuous Riemann surface on the sphere But, how is the discrete case, e.g., mesh?

0 s.t. 1, 1, ,w iL x x i n

Spectral Graph Theory:CdV Number Given n-vretex graph G=<V,E>

M(G) is a symmetric matrix Spectrum of M

CdV number Maximal integer such that

0 1( ) , , nM 0 1( ) , , nM

( )r G

1 2 .. r

( , )

0 ( , )ij

negative number i j E

M anything i j

i j E

Spectral Graph Theory:Nullspace Embedding[Lovasz & Schrijver 99] Supposed

CdV eigenvalue=0 w.l.o.g CdV eigenvectors be coordinates vector

s Result

G describes the edges of a convex polyhedron in R3 containing the origin

( ) 3r G

i

Spherical Nullspace Embeddings System

4n unknowns , ( , , )i i i ix y z

Geometric Interpretation of the Embeddings

System Analysis Properties

Quadratic non-linear Solution non-unique Degenerate always

Solving the system fsolve procedure of MATLAB ---- a subspace trust region procedure [Coleman & Li 1996]

Example

Conclusion

Non-linear Large mesh

Degenerate Robustness

Solution with degree of freedom How to control it?

Generate to higher genus? Need further improved…

Summary of Spherical Methods

Good Properties Equivalent to sphere Most meshes Less distortion Remedy metric Though, difficult to control! Needn’t prior cutting/partition Mesh independence Better for application Morphing

Summary of Spherical Methods

Future Works Generalize to non genus-0 meshes? Associate with partition? Associate with better distortion metric? Consistent spherical parameterizations

among several models (feature correspondence)

Implement acceleration?

Q&A

Thank You!