Surface Parametrization

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Surface Parametrization

• Jingyu Yan

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Topics

• Definition
• Properties
• Applications (Overview)
• Approaches
• Applications (Detailed)

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Definition

• A parametrization of a surface is a mapping U
  (x,y,z)-gt(u,v) from 3D space to 2D space
• In a discrete case, a triangulation S(G,X) is any
  planar triangulation P which is isomorphic to G.

A surface triangulation and its parameterization
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Properties

• Inevitable distortion
• Conformality X (u,v) -gt (x,y,z) is conformal if
  the tangent vectors to the iso-u and iso-v curves
  passing through X(u,v) are orthogonal and have
  the same norm
• N(u,v) x (dX/du) dX/dv

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Other Properties

• Angle-preservingpreserve aspect ratios of
  triangles
• Shape-preserving
• Area-preserving, etc.

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Application (Overview)

• Texture-mapping
• ? (x,y,z) -gt (u,v) from 3D to 2D
• Remeshing
• ?-1 (u,v) -gt (x,y,z) from 2D to 3D

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Approaches of Parametrization

• Harmonic Maps 1 Conformal, Angle-preserving
• Floaters 2 Shape-preserving
• Parametrization with constrains 3
• Least squares conformal maps 4
• A lot more.

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Approaches of Parametrization

• Harmonic Maps 1 Conformal, Angle-preserving
• Floaters 2 Shape-preserving
• Parametrization with constrains 3
• Least squares conformal maps 4
• A lot more.

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Harmonic Maps

• Theorem
• Fix a homeomorphism g between the boundary of D
  and the boundary of the polygonal region P then
  there is a unique harmonic map hD-gtP that agrees
  with g on the boundary of D and minimizes metric
  dispersion 1
• Metric dispersion
• a measure of the extent to which a map stretches
  regions of small diameter in D
• a measure of metric distortion

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Visualize Harmonic Maps

• Stretch the boundary of D over the boundary of
  the polygon P. The harmonic map minimizes the
  total energy Eharmh of this configuration.

D
P
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Harmonic Maps a piecewise linear approximation

• Fix the boundary
• Find h that minimize Eharm

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Advantages

• small distortion
• U. Pinkall and K.Polthier, Computing discrete
  minimal surfaces and their conjugates, 1993,
  shows a link between the minimization of metric
  dispersion and conformality

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Approaches of Parametrization

• Harmonic Maps 1 Conformal, Angle-preserving
• Floaters 2 Shape-preserving
• Parametrization with constrains 3
• Least squares conformal maps 4
• A lot more.

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Shape-preserving Parametrization

• Floater, Parametrization and smooth approximation
  of surface triangulations2

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Barycentric combination

• p ??i pi , i1,,n
• ??i 1

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Local shape preserving parametrization

• flatten the local mesh

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Local shape preserving parametrication

• Find the barycentric combination
• p ??i pi , i1,,5
• ??i 1

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The Algorithm

• Step 1 initialize the boundary condition
• x1,, xn are the internal nodes and xn1 ,, xN
  are the boundary nodes in any anticlockwise
  sequence
• Choose pn1 ,, pN to be the vertices of a
  polygon D in 2D. pn1 ,, pN are the
  parametrizaion of xn1 ,, xN

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The Algorithm

• Step 2
• Each interior node
• pi ??ij pj , j 1,,N, i1,,n
• ?ij gt 0 when pj is adjencent to pi
• ?ij 0 when pj is not adjencent to pi
• pi - ??ik pk ??ij pj , k 1,,n j n1,,N,
  i1,,n
• Ap1 ,,pnT b

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Advantages

• Preserve chord length
• Preserve barycentricity

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Approaches of Parametrization

• Harmonic Maps 1 Conformal, Angle-preserving
• Floaters 2 Shape-preserving
• Parametrization with constrains 3
• Least squares conformal maps 4
• A lot more.

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Parametrization with Constrains

• Levy, Siggraph 2001 3
• Facilitate texture-mapping
• Constrains specified by pairs of a 3D point and a
  2D point.
• (Mj, Uj), Mj ?R3 , Uj ?R2

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Parametrization with Constrains

• Minimize the objective function
• matching features
  enforcing smoothness

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Matching Features

• The equation

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Constraining the Gradient

• The gradient in a triangle T p1,p2,p3
• Local base (p1, X,Y)
• The gradient is

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Constraining the Gradient

• For a matching feature (Mj, Uj), Mj ?R3 , Uj ?R2
  , given a vector Uj to constrain the gradient of
  u

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Enforcing smoothness

• Approximate it by minimize the variation of the
  gradients of two adjacent triangles T, T

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Parametrization with Constrains

• Objective function is decomposed into three
  linear terms

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Parametrization with Constrains
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Approaches of Parametrization

• Harmonic Maps 1 Conformal, Angle-preserving
• Floaters 2 Shape-preserving
• Parametrization with constrains 3
• Least squares conformal maps 4
• A lot more.

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Least Squares Conformal Maps

• Levy, Siggraph 2002, Least Squares Conformal Maps
  for Automatic Texture Atlas Generation4
• Satisfy conformality in the least squares sense
• Optimization based

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Conformal Maps

• X (u,v) -gt (x,y,z) is conformal if the tangent
  vectors to the iso-u and iso-v curves passing
  through X(u,v) are orthogonal and have the same
  norm
• N(u,v) x (dX/du) dX/dv

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Conformality in a triangulation

• Suppose each triangle is provided a local
  orthonormal basis
• X uiv-gtxiy, U xiy-gtuiv
• dX/du i(dX/dv)0 gt
• dU/dx i(dU/dy)0 (the derivative of the inverse
  function)

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Least Squares Conformal Maps

• dU/dx i(dU/dy)0 gt
• minimize ?(dU/dx i(dU/dy))2 over the
  triangulation

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Applications (Detailed)

• Interactive Geometry Remeshing, Siggraph 2002 5
• Partitioning
• Parametrization
• Generate the control map
• Geometry resampling
• Mesh creation

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Partitioning

• split the surface into disk-like patches

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Parametrization

• Conformal mapping

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Generate Control Map

• Area distortion map
• Mean curvature map
• To denote the sampling density

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Geometry Resampling

• Generate sampling points in parametric space by
  halftoning the control map

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Mesh creation

• Delaunay triangulation over sampling points
• Connect sampling points

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References

• 1  M. Eck, T. DeRose, T.Duchamp, H. Hoppes, M.
  Lounsbery and W. Stuetzle. Multiresolution
  Analysis of Arbitrary Meshes. In SIGGRAPH 95
  Conference Proceedings, pages 173182. ACM,
  August 1995.
• 2  M. Floater. Parametrization and smooth
  approximation of surface triangulations. Computer
  Aided Geometric Design, 14(3)231250, April
  1997.
• 3  Levy, B. Constrained Texture Mapping for
  Polygonal Meshes. In Proceedings of SIGGRAPH
  (2001), pp.417424.
• 4  Levy, B., Petitjean, S., Ray, N., and
  Maillot, J. Least Squares Conformal Maps for
  Automatic Texture Atlas Generation. In
  Proceedings of SIGGRAPH (2002).
• 5 Pierre Alliez, Mark Meyer and Mathieu
  Desbrun, Interactive Geometry Remeshing. In
  SIGGRAPH '2002 Conference Proceedings
• 6   B. Levy and J.-L. Mallet. Non-distorted
  texture mapping for sheared triangulated meshes.
  In SIGGRAPH 98 Conf. Proc., pages 343352.
  Addison Wesley, 1998.
• 7 Desbun, M., Meyer, M., AND Alliez, P.
  Intrinsic parameterizations of surface meshes. In
  Proceedings of Eurographics (2002).