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Diffusion equations over arbitrary triangulated surfaces for filtering and texture applications.

Chunlin Wu1, Jiansong Deng, Falai Chen

  • 1Department of Mathematics, University of Science and Technology of China, Hefei, PR China. wucl@ustc.edu.cn

IEEE Transactions on Visualization and Computer Graphics
|March 29, 2008
PubMed
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This study introduces numerical methods for solving partial differential equations (PDEs) on triangular mesh surfaces. These efficient methods are flexible for arbitrary geometries and topologies, demonstrating effectiveness in various computer graphics applications.

Area of Science:

  • Computer Graphics
  • Numerical Analysis
  • Differential Geometry

Background:

  • Triangular meshes are widely used in computer graphics due to their rendering and storage advantages.
  • Processing data on these complex surfaces using partial differential equations (PDEs) presents unique challenges.

Purpose of the Study:

  • To develop and present numerical schemes for solving diffusion equations on triangular mesh surfaces.
  • To ensure methods are applicable to meshes with arbitrary geometry and topology.

Main Methods:

  • Investigated several diffusion equations tailored for triangular mesh representations.
  • Developed corresponding numerical schemes employing implicit/semi-implicit time discretization for efficiency.
  • Validated methods on meshes with arbitrary triangle angles and arbitrary genus (open or closed).

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Main Results:

  • The proposed numerical schemes effectively solve diffusion equations on diverse triangular mesh surfaces.
  • Methods demonstrated flexibility across various geometries and topologies, including open and closed meshes.
  • Implicit/semi-implicit time discretization ensured computational efficiency.

Conclusions:

  • The developed methods offer a flexible and efficient approach for data processing on triangular mesh surfaces using PDEs.
  • Applications in image processing, texture generation, and harmonic map regularization show the methods' effectiveness.