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Mesh Parameterizations Lizheng Lu [email protected] Oct. 19, 2005.
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Transcript of Mesh Parameterizations Lizheng Lu [email protected] 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!