geometric modeling based on triangle meshes
TRANSCRIPT
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Surface Parameterization
Christian RösslINRIA Sophia-Antipolis
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Christian Rössl, INRIA 237
Outline
•Motivation
• Objectives and Discrete Mappings• Angle Preservation• Discrete Harmonic Maps• Discrete Conformal Maps• Angle Based Flattening
• Reducing Area Distortion
• Alternative Domains
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Christian Rössl, INRIA 238
Surface Parameterization
[www.wikipedia.de]
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Christian Rössl, INRIA 239
Surface Parameterization
[www.wikipedia.de]
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Christian Rössl, INRIA 240
Surface Parameterization
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Christian Rössl, INRIA 241
Motivation
•Texture mapping
Lévy, Petitjean, Ray, and Maillot: Least squares conformal maps for automatic texture atlas generation, SIGGRAPH 2002
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Christian Rössl, INRIA 242
Motivation
•Many operations are simpler on planar domain
Lévy: Dual Domain Exrapolation, SIGGRAPH 2003
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Christian Rössl, INRIA 243
Motivation
• Exploit regular structure in domain
Gu, Gortler, Hoppe: Geometry Images, SIGGRAPH 2002
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Christian Rössl, INRIA 244
Surface Parameterization
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Christian Rössl, INRIA 245
Surface Parameterization
f
X U
Jacobian
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Christian Rössl, INRIA 246
Surface Parameterization
f
X U
dX = J dU
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Christian Rössl, INRIA 247
Surface Parameterization
f
X U
dX = J dU
||dX ||2 = dU JTJ dU{ First Fundamental Form
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Christian Rössl, INRIA 248
• By first fundamental form I– Eigenvalues λ1,2 of I
– Singular values σ1,2 of J (σi2= λi)
• Isometric
– I = Id, λ1= λ2=1
• Conformal
– I = µ Id , λ1 / λ2=1
• Equiareal
– det I = 1, λ1 λ2=1
Characterization of Mappings
angle preserving
area preserving
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Christian Rössl, INRIA 249
Piecewise Linear Maps
•Mapping = 2D mesh with same connectivity
f
X U
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Christian Rössl, INRIA 250
Objectives
• Isometric maps are rare
•Minimize distortion w.r.t. a certain measure– Validity (bijective map)
– Boundary
– Domain
– Numerical solution
triangle flip
e.g.,spherical
linear / non-linear?
fixed / free?
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Christian Rössl, INRIA 251
Discrete Harmonic Maps
• f is harmonic if
• Solve Laplace equation
• In 3D: "fix planar boundary and smooth"
u and v are harmonic
Dirichlet boundary conditions
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Christian Rössl, INRIA 252
Discrete Harmonic Maps
• f is harmonic if
• Solve Laplace equation• Yields linear system
• Convex combination maps
– Normalization
– Positivity
(again)
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Christian Rössl, INRIA 253
Convex Combination Maps
• Every (interior) planar vertex is a convex combination of its neighbors
• Guarantees validity if boundary is mapped to aconvex polygon (e.g., rectangle, circle)
•Weights– Uniform (barycentric mapping)
– Shape preserving [Floater 1997]– Mean Value Coordinates [Floater 2003]
• Use mean value property of harmonic functions
Reproduction of planar meshes
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Christian Rössl, INRIA 254
Conformal Maps
• Planar conformal mappings
satisfy the Cauchy-Riemann conditions
and
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Christian Rössl, INRIA 255
Conformal Maps
• Planar conformal mappings
satisfy the Cauchy-Riemann conditions
• Differentiating once more by x and y yields
•
and
and ⇒
and similar
conformal ⇒ harmonic
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Christian Rössl, INRIA 256
Discrete Conformal Maps
• Planar conformal mappings
satisfy the Cauchy-Riemann conditions
• In general, there are no conformal mappings for piecewise linear functions!
and
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Christian Rössl, INRIA 257
Discrete Conformal Maps
• Planar conformal mappings
satisfy the Cauchy-Riemann conditions
• Conformal energy (per triangle T)
•Minimize
and
→
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Christian Rössl, INRIA 258
Discrete Conformal Maps
• Least-squares conformal maps [Lévy et al. 2002]
• Satisfy Cauchy-Riemann conditions in least-squares sense
• Leads to solution of linear system
• Alternative formulation leads to same solution…
where→
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Christian Rössl, INRIA 259
Discrete Conformal Maps
• Same solution is obtained for
cotangent weights
Neumann boundary conditions
[Desbrun et al. 2002]Discrete Conformal Maps
+ fixed vertices
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Christian Rössl, INRIA 260
Discrete Conformal Maps
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Christian Rössl, INRIA 261
Discrete Conformal Maps
• Free boundary depends on choice of fixed vertices (>1)
ABF
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Christian Rössl, INRIA 262
Angle Based Flattening
• Perserve angles specify problem in angles– Constraints
• triangle• Internal vertex•Wheel consistency
– Objective function
ensure validity
preserve angles 2D ~3D
"optimal" angles (uniform scaling)
[Sheffer&de Sturler 2000]
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Christian Rössl, INRIA 263
Angle Based Flattening
• Free boundary
• Validity: no local self-intersections• Non-linear optimization
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Christian Rössl, INRIA 264
Angle Based Flattening
• Free boundary
• Non-linear optimization– Newton iteration– Solve linear system in every step
[Zayer et al. 2005]
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Christian Rössl, INRIA 265
And how about area distortion?
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Christian Rössl, INRIA 266
Reducing Area Distortion
• Energy minimization based on– MIPS [Hormann & Greiner 2000]
– modification [Degener et al. 2003]
– "Stretch" [Sander et al. 2001]
– modification [Sorkine et al. 2002]
or
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Christian Rössl, INRIA 267
Non-Linear Methods
• Free boundary• Direct control over distortion
• No convergence guarantees• May get stuck in local minima• May not be suitable for large problems• May need feasible point as initial guess• May require hierarchical optimization even for
moderately sized data sets
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Christian Rössl, INRIA 268
Linear Methods
• Efficient solution of a sparse linear system
• Guaranteed convergence
• Fixed convex boundary
• May suffer from area distortion for complex meshes
• An alternative approach to reducing area distortion…
– How accurately can we reproduce a surface on the plane?
– How do we characterize the mapping?
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Christian Rössl, INRIA 269
Reducing Area Distortion
isometry
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Christian Rössl, INRIA 270
Reducing Area Distortion
• Quasi-harmonic maps [Zayer et al. 2005]
• Iterate (few iterations)
– Determine tensor C from f– Solve for g
estimate from f
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Christian Rössl, INRIA 271
Examples
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Christian Rössl, INRIA 272
Examples
Stretch metric minimization
Using [Yoshizawa et. al 2004]
→
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Christian Rössl, INRIA 273
Reducing Area Distortion
• Introduce cuts area distortion vs. continuity
• Often cuts are unavoidable (e.g., open sphere)
Treatment of boundary is important!
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Christian Rössl, INRIA 274
Reducing Area Distortion
• Solve Poisson system [Zayer et al. 2005]
estimate from previous map
* Similar setting used in mesh editing
*
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Christian Rössl, INRIA 275
Spherical Parameterization
• Sphere is natural domain for genus-0 surfaces
• Additional constraint
• Naïve approach– Laplacian smoothing and back-projection– Obtain minimum for degenerate configuration
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Christian Rössl, INRIA 276
Spherical Parameterization
• (Tangential) Laplacian Smoothing and back-projection– Minimum energy is obtained for degenerate solution
• Theoretical guarantees are expensive– [Gotsman et al. 2003]
• A compromise?!– Stereographic projection– Smoothing in curvilinear coordinates
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Christian Rössl, INRIA 277
Arbitrary Topology
•Piecewise linear domains– Base mesh obtained by mesh decimation
– Piecewise maps – Smoothness
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Christian Rössl, INRIA 278
Literature
• Floater & Hormann: Surface parameterization: a tutorial and survey, Springer, 2005
• Lévy, Petitjean, Ray, and Maillot: Least squares conformal maps for automatic texture atlas generation, SIGGRAPH 2002
• Desbrun, Meyer, and Alliez: Intrinsic parameterizations of surface meshes, Eurographics 2002
• Sheffer & de Sturler: Parameterization of faceted surfaces for meshing using angle based flattening, Engineering with Computers, 2000.