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Bicubic G 1 interpolation of arbitrary quad meshes using a 4-split. Geometric Modeling and Processing 2008. S. Hahmann G.P. Bonneau B. Caramiaux. CAI Hongjie Mar. 20, 2008. Authors. Stefanie Hahmann Main Posts Professor at Institut National Polytechnique

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bicubic g 1 interpolation of arbitrary quad meshes using a 4 split

Bicubic G1 interpolation of arbitrary quad meshes using a 4-split

Geometric Modeling and Processing 2008

S. Hahmann G.P. Bonneau B. Caramiaux

CAI Hongjie

Mar. 20, 2008

authors
Authors

Stefanie Hahmann

  • Main Posts

Professor at Institut National Polytechnique

de Grenoble (INPG), France

Researcher at Laboratorie Jean Kuntzmann (LJK)

  • Research

CAGD

Geometry Processing

Scientific Visualization

authors1
Authors

Georges-Pierre Bonneau

  • Main Posts

Professor at Université

Joseph Fourier

Researcher at LJK

  • Research

CAGD

Visualization

outline
Outline
  • Applications of surface modeling
  • Background
    • Subdivision surface
    • Global tensor product surface
    • Locally constructed surface
  • Circulant Matrices
  • Vertex Consistency Problem
  • Surface Construction by Steps
applications of surface modeling
Applications of Surface Modeling
  • Medical imaging
  • Geological modeling
  • Scientific visualization
  • 3D computer graphic animation
a peep of hd 3d animation
A peep of HD 3D Animation

From Appleseed EX Machina (2007)

subdivision surface

Doo-Sabin 细分方法

Catmull-Clark 细分方法

Loop 细分方法

Butterfly 细分方法

Subdivision Surface

From PhD thesis of Zhang Jinqiao

locally constructed surface
Locally Constructed Surface

From S. Hahmann, G.P. Bonneau. Triangular G1 interpolation by 4-splitting domain triangles

circulant matrices
Circulant Matrices
  • Definition: A circulant matrix M is of the form
  • Remark: Circulant matrix is a special case of Toeplitz matrix
circulant matrices1
Circulant Matrices
  • Property: Let f(x)=a0+a1x +…+ an-1xn-1,

then eigenvalues, eigenvectors and determinant of M are

    • Eigenvalues:
    • Eigenvectors:
    • Determinant:
examples of circulant matrices
Examples of Circulant Matrices
  • Determine the singularity of

Solution: f(x)=0.5+0.5xn-1,

examples of circulant matrices1
Examples of Circulant Matrices
  • Compute the determinant of
  • Compute the rank of
vertex consistency problem
Vertex Consistency Problem
  • For C2 surface assembling

If G1 continuity at boundary is satisfied,

then

vertex consistency problem1
Vertex Consistency Problem
  • Twist compatibility for C2 surface

then

vertex consistency problem2
Vertex Consistency Problem
  • Matrix form

It is generally unsolvable when n is even

sketch of the algorithm
Sketch of the Algorithm
  • Given a

quad mesh

  • To find 4

interpolated bi-cubic

tensor surfaces for

each patch with

G1 continuity at

boundary

preparation simplification
Preparation: Simplification
  • Simplification of G1 continuity condition
choice of
Choice of
  • Let be constant, depended only on n (the order of vertex v)
  • Specialize G1 continuity condition at ui=0, then
  • Non-trivial solution require
choice of1
Choice of
  • Determine

ni is the order of vi

step 1 determine boundary curve
Step 1:Determine Boundary Curve
  • Differentiate G1 continuity equation and specialize at ui=0, then
  • Matrix form
examples of circulant matrices2
Examples of Circulant Matrices
  • Determine the singularity of

Solution: f(x)=0.5+0.5xn-1,

step 1 determine boundary curve1
Step 1:Determine Boundary Curve
  • Differentiate G1 continuity equation and specialize at ui=0, then
  • Matrix form
step 1 determine boundary curve2
Step 1:Determine Boundary Curve
  • Notations
  • Selection of d1,d2
step 2 twist computations
Step 2:Twist Computations
  • d1,d2 is in the image of T
  • Determine the twist
  • Determine
step 3 edge computations
Step 3: Edge Computations
  • Determine
  • Determine Vi(ui)

where

V0,V1 are two n×n matrices determined by G1 condition

step 4 face computations
Step 4: Face Computations
  • C1 continuity between inner micro faces
  • We choose A1,A2,A3,A4 as dof.
conclusions
Conclusions
  • Suited to arbitrary topological quad mesh
  • Preserved G1 continuity at boundary
  • Given explicit formulas
  • Low degrees (bi-cubic)
  • Shape parameters control is available
reference
Reference
  • S. Hahmann, G.P. Bonneau, B. Caramiaux

Bicubic G1 interpolation of arbitrary quad meshes using a 4-split

  • S. Hahmann, G.P. Bonneau

Triangular G1 interpolation by 4-splitting domain triangles

  • Charles Loop

A G1 triangular spline surface of arbitrary topological type

  • S. Mann, C. Loop, M. Lounsbery, et al

A survey of parametric scattered data fitting using triangular interpolants