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

Geometric Modeling and Processing 2008 Bicubic G1 interpolation of arbitrary quad meshes using a 4-split S. Hahmann G.P. Bonneau B. Caramiaux CAI Hongjie Mar. 20, 2008

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**Authors Stefanie Hahmann Main Posts Research**

Professor at Institut National Polytechnique de Grenoble (INPG), France Researcher at Laboratorie Jean Kuntzmann (LJK) Research CAGD Geometry Processing Scientific Visualization

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**Authors Georges-Pierre Bonneau Main Posts Professor at Université**

Joseph Fourier Researcher at LJK Research CAGD Visualization

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**Outline Applications of surface modeling Background Circulant Matrices**

Subdivision surface Global tensor product surface Locally constructed surface Circulant Matrices Vertex Consistency Problem Surface Construction by Steps

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**Applications of Surface Modeling**

Medical imaging Geological modeling Scientific visualization 3D computer graphic animation

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A peep of HD 3D Animation From Appleseed EX Machina (2007)

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**Subdivision Surface From PhD thesis of Zhang Jinqiao Doo-Sabin 细分方法**

Catmull-Clark 细分方法 Loop 细分方法 Butterfly 细分方法 From PhD thesis of Zhang Jinqiao

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**Locally Constructed Surface**

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

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**Circulant Matrices Definition: A circulant matrix M is of the form**

Remark: Circulant matrix is a special case of Toeplitz matrix

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**Circulant Matrices Property: Let f(x)=a0+a1x +…+ an-1xn-1,**

then eigenvalues, eigenvectors and determinant of M are Eigenvalues: Eigenvectors: Determinant:

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**Examples of Circulant Matrices**

Determine the singularity of Solution: f(x)= xn-1,

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**Examples of Circulant Matrices**

Compute the determinant of Compute the rank of

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**Vertex Consistency Problem**

For C2 surface assembling If G1 continuity at boundary is satisfied, then

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**Vertex Consistency Problem**

Twist compatibility for C2 surface then

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**Vertex Consistency Problem**

Matrix form It is generally unsolvable when n is even

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**Sketch of the Algorithm**

Given a quad mesh To find 4 interpolated bi-cubic tensor surfaces for each patch with G1 continuity at boundary

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**Preparation: Simplification**

Simplification of G1 continuity condition

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**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

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Choice of Determine ni is the order of vi

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**Step 1:Determine Boundary Curve**

Differentiate G1 continuity equation and specialize at ui=0, then Matrix form

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**Examples of Circulant Matrices**

Determine the singularity of Solution: f(x)= xn-1,

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**Step 1:Determine Boundary Curve**

Differentiate G1 continuity equation and specialize at ui=0, then Matrix form

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**Step 1:Determine Boundary Curve**

Notations Selection of d1,d2

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**Step 2:Twist Computations**

d1,d2 is in the image of T Determine the twist Determine

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Change of G1 Conditions From To

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**Step 3: Edge Computations**

Determine Determine Vi(ui) where V0,V1 are two n×n matrices determined by G1 condition

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**Step 3: Edge Computations**

Determine

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**Step 4: Face Computations**

C1 continuity between inner micro faces We choose A1,A2,A3,A4 as dof.

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Results

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Results

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**Conclusions Suited to arbitrary topological quad mesh**

Preserved G1 continuity at boundary Given explicit formulas Low degrees (bi-cubic) Shape parameters control is available

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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

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Thanks! Q&A

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