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Mesh%20Parameterization:%20Theory%20and%20Practice

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Title: Mesh%20Parameterization:%20Theory%20and%20Practice


1
Mesh ParameterizationTheory and Practice
  • Differential Geometry Primer

2
Parameterization
  • surface
  • parameter domain
  • mapping and

3
Example Cylindrical Coordinates

4
Example Orthographic Projection

5
Example Stereographic Projection

6
Example Mappings of the Earth
  • usually, surface properties get distorted

orthographic 500 B.C.
stereographic 150 B.C.
Mercator 1569
Lambert 1772
conformal (angle-preserving)
equiareal (area-preserving)
7
Distortion is (almost) Inevitable
  • Theorema Egregium (C. F. Gauß)
  • A general surface cannot be parameterized
    without distortion.
  • no distortion conformal equiareal isometric
  • requires surface to be developable
  • planes
  • cones
  • cylinders

8
What is Distortion?
  • parameter point
  • surface point
  • small disk around
  • image of under
  • shape of

9
Linearization
  • Jacobian of
  • tangent plane at
  • Taylor expansion of
  • first order approximation of

10
Infinitesimal Dis(k)tortion
  • small disk around
  • image of under
  • shape of
  • ellipse
  • semiaxes and
  • behavior in the limit

11
Linear Map Surgery
  • Singular Value Decomposition (SVD) of
  • with rotations and
  • and scale factors (singular values)

12
Notion of Distortion
  • isometric or length-preserving
  • conformal or angle-preserving
  • equiareal or area-preserving
  • everything defined pointwise on

13
Example Cylindrical Coordinates
  • ? isometric

14
Example Orthographic Projection
  • with
  • ?

neither conformal nor equiareal
15
Example Stereographic Projection
  • with
  • ? conformal

16
Computing the Stretch Factors
  • first fundamental form
  • eigenvalues of
  • singular values of
  • and

17
Measuring Distortion
  • local distortion measure
  • has minimum at
  • isometric measure
  • conformal measure
  • overall distortion

18
Piecewise Linear Parameterizations
  • piecewise linear atomic maps
  • distortion constant per triangle
  • overall distortion

19
Linear Methods
  • the terms and
    are quadratic in the parameter points
  • Dirichlet energy
  • Conformal energy
  • minimization yields linear problem

Pinkall Polthier 1993Eck et al. 1995
Lévy et al. 2002Desbrun et al. 2002
20
Linear Methods
  • both result in barycentric mappings with
    discrete harmonic weights for interior vertices
  • Dirichlet maps require to fix all boundary
    vertices
  • Conformal maps only two
  • result depends on this choice
  • best choice ? Mullen et al. 2008
  • both maps not necessarily bijective

21
Non-linear Methods
  • MIPS energy
  • Area-preserving MIPS

Hormann Greiner 2000
Degener et al. 2003
22
Non-linear Methods
  • Green-Lagrange deformation tensor
  • Stretch energies ( , , and symmetric
    stretch)

Maillot et al. 1993
Sander et al. 2001 Sorkine et al. 2002
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