1 Mesh Parametrization and Its Applications 동의대학교 멀티미디어공학과 김형석 포항공과대학교 ( 이윤진, 이승용 )

2 Computer Graphics Definition : all technologies related to producing pictures or images using a computer Computer animation, VR(virtual reality), … Goal : Reality and Real time Reality Mapping(texture) / Rendering(light) v

3 Computer Graphics v

4 v

5 Polygonal Objects(Mesh)

6 Parametrization Embedding 3D mesh to 2D parameter space Requirements distortion minimization one-to-one mapping s t x y z v (s,t) triangular mesh in 3D parametrization in 2D

7 Parameterization:[Levy]

8 Previous Work Energy functional minimization Green-Lagrange tensor [Maillot93] orthogonality and homogeneous spacing [Lévy98] Dirichlet energy [Hormann99] Convex combination approach shape-preserving parametrization [Floater97] harmonic embedding [Eck95]

9 Convex Combination Approach (1) Convex combination and boundary condition determine shape of parameter space map boundary vertices onto a convex polygon determine coefficients for the inner vertices solve a linear system Ax = b 1-ring neighborhood in parametric space 3D mesh parameter space

10 Convex Combination Approach (2) Benefit simple and fast, one-to-one embedding Drawback high distortions near the boundary parameterization with fixed boundary 3D mesh

11 Reducing Distortion near Boundary Floating boundary for the parameter space non-linear system [Maillot93] [Lévy98] [Hormann99] linear system [Lévy01] heavy computation and/or non-one-to-one mapping parameterization with floating boundary 3D mesh

12 Motivation Extension of convex combination approach distortion minimization near the boundary simple and fast one-to-one mapping floating boundary 3D mesh fixed boundary

13 Our Approach (1) Virtual boundary virtual vertices attached to the real boundary virtual boundary is fixed but real boundary can move to reduce the distortion in parameterization virtual boundary parametrization with virtual boundary 3D mesh

14 Our Approach (2) Parametrization process Compute coefficients i,j (inner vertices + boundary vertices) Determine shape of parameter space (convex polygon) Map virtual vertices to the polygon Solve linear system making virtual boundary parametrization

15 Virtual Boundary Virtual vertices # of virtual vertices = 2  # of real boundary vertices boundary vertex is adjacent to three virtual vertices no 3D positions are required for virtual vertices real boundary connectivity of virtual vertices virtual boundary

16 Coefficient Computation (1) Shape-preserving parametrization [Floater97] conformal mapping of 1-ring neighborhood average of barycentric coordinates conformal mapping onto 2D 1-ring neighborhood in 3D averaging barycentric coord.

17 Coefficient Computation (2) Coefficients of real boundary vertices 1-ring neighborhood in 3D 1-ring neighborhood in 2D 1-ring neighborhood + virtual vertices in 2D map to 2D while preserving angles and lengths place virtual vertices in 2D

18 2D Positions of Virtual Vertices Mapping virtual vertices onto convex polygon using edge lengths between real boundary vertices real boundary mapping virtual boundary virtual boundary relation of real and virtual boundary

19 Shape of Parameter Space Strong influence on the parameterization simple choices such as circle and rectangle? Convex hull of the projection of real boundary circle rectangle convex polygon from projected boundary

20 Extended Virtual Boundary (1) More virtual vertices in multi-layered structure to reduce distortions near the real boundary 3D meshparametrization region far from the boundary

21 Extended Virtual Boundary (2) Structure each layer has the same # of virtual vertices Coefficients for virtual vertices real boundary 1 st virtual layer 2 nd virtual layer … connectivity an coefficients of virtual vertices

22 Extended Virtual Boundary (3) Effect for concave real boundary 3D mesh one layer no virtual vertices two layers three layers four layers

23 Results (1) rectanglecircle projected polygon 3D mesh map the boundary map virtual boundary

24 Results (2) Texture mapping rectangle circle, virtual boundary projected polygon, virtual boundary 3D mesh

25 Applications(Texture):[Levy]

26 Applications(Texture) ):[Levy]

27 Applications(Texture) ):[Levy]

28 Conclusion and Future Work Extension of convex combination approach distortion minimization near the boundary Virtual boundary fixed instead of the real boundary multi-layered structure Future work connectivity and coefficients of virtual vertices speed up with multilevel approach