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Wiener Subdivision: Implementing Marc Alexa's Filtering Methodology

Derive and implement a subdivision scheme based on Wiener filtering of meshes. Create refined mesh by relocating vertices for a smooth surface. Explore Wiener filtering theory, m-ring neighborhoods, and linear system solutions. Using C++ in Wiener Subdivision development with GTS mesh format on Windows XP. Utilize TNT for numerical toolkit and Jama/C++ for linear system solutions. Enhance mesh data structure and refinement with Wiener Subdivision filtering. Parameters include neighborhood size, smoothness control, and vertex location fraction. Results show different parameter combinations.

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Wiener Subdivision: Implementing Marc Alexa's Filtering Methodology

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  1. Wiener Subdivision Presented by Koray KAVUKCUOGLU Geometric Modeling Spring 2004

  2. Outline • Introduction • Concepts • Wiener Filtering • Theory • Wiener Subdivision • Midpoint Subdivision • Application of Filter • Parameters • Results

  3. Introduction aim • Derive and Implement a subdivison scheme Based on Marc Alexa’s Wiener Filtering of Meshes methodology • Midpoint Linear Subdivision • Create refined mesh • Wiener Filtering • Relocate vertices to obtain a smooth surface

  4. Wiener Filtering Filtering of Irregular Meshes using Wiener Filter Recovering original smooth geometry from noisy data

  5. Wiener Filtering - Theory Mesh • Triangular domain  (K,V) connectivity info vertices in R3 Topological Distance ()

  6. Wiener Filtering - Theory • Neighborhood Definition m-ring neighborhood Collection of rings, with radius up to m • Expectation linear operator Correlation Distance between two vertices

  7. Wiener Filtering - Theory Representation of Vertex Locations vertex position in noisy mesh random noise contribution true vertex position Estimate each point as a linear sum of given noisy points Find coefficients that minimize square of discrepancy

  8. Wiener Filtering - Theory Linear System Solution of this system gives, coefficients aij Need to define distance and correlation functions 1 d i d 2 d

  9. Wiener Subdivision development environment • Language C++ • Mesh format GTS • Windows XP • Cygwin external libs / tools • TNT (template numerical toolkit) Supersedes Lapack++ • Jama/C++ (uses TNT - linear system solution) • Mesh Viewer for visualization

  10. Wiener Subdivision mesh data structure • Tree  each triangle divided into 4 childs Triangles Edge Sharing

  11. Wiener Subdivision mesh refinement • Linear midpoint subdivision

  12. Wiener Subdivision filtering • computing Topology • compute m-ring neighborhood BFS over vertices • compute distance and correlation x is parameterized for smoothness control

  13. Wiener Subdivision filtering • solve linear system • LU decomposition method • Jama/C++

  14. Wiener Subdivision parameters • size of m-ring neighborhood (1, 2, …) <-m> • smoothness parameter <-sp> • fraction of old vertex location in new location <-p>

  15. results -m1 / -n3 / -sp2

  16. results -m1 / -n3 / -sp0 -m2 / -n3 / -sp2

  17. results -m1 / -n3 / -sp0 -m1 / -n3 / -sp2 -m2 / -n3 / -sp2

  18. results -m1 / -n3 / -sp2 -m1 / -n3 / -sp2 / -p0.3

  19. results -m2 / -n3 / -sp2 -m1 / -n3 / -sp2 / -p0.3

  20. results

  21. results

  22. Questions?

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