Volume Parameterization Reporter : Lei Zhang 10\24\2007.

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Presentation transcript:

Volume Parameterization Reporter : Lei Zhang 10\24\2007

What is volume Surface model 2D manifold Solid\Volume model 3D manifold

Discrete Representation triangletetrahedron quadranglehexahedron

Surface Parameterization Parameterization can be viewed as a one-to-one mapping from a suitable domain to the surface. 1997~ 2007 Huge amount of papers

Surface Parameterization Application Re-meshing Morphing Texture mapping

Volume Parameterization Parameterization can be viewed as a one-to-one mapping from a suitable domain to the volume. Not proposed formally!

Texture Mapping

Tetrahedral\Hexhedral Meshing (a)(b)(c)(d)

Rendering Acceleration (a) (d) (c) (b)

Volume Parameterization Difficulties  Math ground  High dimensional geometry  3 manifold is special  Data structure  Enormous datas  Complex connectivity  Tiny amount of papers ? ? ?

Problems Surface:  conformal mapping  harmonic mapping  … … Volume:  ?  … …

Harmonic Volume Mapping Harmonic equation

Yalin Wang, Xianfeng Gu, and Shing-Tung Yau. Volumetric harmonic map. Communications in Information and Systems, Yalin Wang, Xianfeng Gu, Tony F. Chan, Paul M. Thompson, and Shing-Tung Yau. Volumetric harmonic brain mapping. IEEE International Symposium on Biomedical Imaging, 2004.

Harmonic mapping from 3 manifold to 3D solid sphere Surface conformal mapping

Steepest descent method

Xin Li, Xiaohu Guo, Hongyu Wang, Ying He, Xianfeng Gu, and Hong Qin. Harmonic volumetric mapping for solid modeling application. SPM, 2007.

Problem Formulation

Solving Equation self-adjoint Green function Fundamental Solution Method (MFS) G. Fairweather and A. Karageorghis. The method of fundamental solutions for elliptic boundary value problems. Advances in Computational Mathematics, 1998.

MFS

Electric field Intuitive Explanation e

Algorithm

Discussion

Continuous Volume Mapping M. S. Floater. Mean Value Coordinates. CAGD, 2003,20, T. Ju, S. Schaefer and J. Warren. Mean Value Coordinates for Closed Triangular Meshes. Siggraph2005.

Mean Value Interpolation

Summary  Research Status  Not hot  Parameterization  2 manifold  3 manifold  Application  Maybe not much desirous ?

Thanks for your attention!