Project Outline

• Interested in using various subdivision algorithms to smoothen meshes for better processing and visuals.

• Curious as to how different existing algorithms compare with one another in terms of geometrical properties, runtime, memory space, etc.

• Understanding different subdivision categories and pros/cons

• Existing Databases:

  • RWTT:
    • Uses photo reconstruction methods.
    • Subdivision algorithms can exacerbate problems due to too many vertices.
  • Digital Michelangelo Project:
    • 3D scanning of large figures.
    • Digitized 10 statues by Michelangelo.
  • Stanford 3D Scanning Repository:
    • Provides raw 3D scans.
    • Their zippering and volumetric range image merging methods produce smooth manifolds and already reduce noise.
      • Range Imaging:
        • Captures 3D shapes by measuring the distance from sensor to object.
        • Reconstructs a 3D model from partial scans.
        • The first step is to align each range image in the same coordinate system, but gaps may remain; hence, zippering them together is used, which is not always ideal for surface reconstruction.
      • Additional Note:
        • Some 3D scanning techniques capture connectivity, so treating them as point clouds (i.e., just x, y, z coordinates) discards useful information.
  • Alternatively, use standard methodologies like quadric edge collapse decimation and voxel-based remeshing.
  • Polytechnique
    • Multiple datasets of triangular meshes (from aim@shape and Thingi10k repo)
      • genus 0, so surface closed, topologically equiv to spheres but triangulations
      • https://www.lix.polytechnique.fr/~amturing/Software/Datasets/Datasets.html
  • Computer Graphics Group at MIT
    • https://people.csail.mit.edu/sumner/research/deftransfer/data.html

• Testing Subdivision Algorithms:

  • Either find a database with irregular meshes or decompose a mesh.
  • Curvature Metrics:
    • Gaussian Curvature:
      • Product of the two principal curvatures.
      • Gives insight into local geometric properties (e.g., ellipse, saddle-like shapes).
      • Example: Pre-div (-95870), post-div (-1.6e+8).
    • Mean Curvature:
      • Average of the two principal curvatures.
      • Example: Pre-div (0.2), post-div (0).
  • To test smoothness, consider other metrics such as normal variation.
  • Subdivision either interpolation (includes original control pts like butterfly) or approximating (doesn't include the original points)
  • Convergent analysis on the refinement meshes

References

• Research Paper on Subdivision Algorithms
• [Butterfly scheme] http://www.math.tau.ac.il/~niradyn/papers/butterfly.pdf
• [Interpolatory sqrt(3) subdivision] https://people.eecs.berkeley.edu/~sequin/CS284/PAPERS/root3subdiv.pdf
• [Loop subdivision] https://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/thesis-10.pdf
• [Original sqrt(3) subdivision] https://dl.acm.org/doi/pdf/10.1145/344779.344835
• https://cs.engr.uky.edu/~cheng/PUBL/Book_CCSS.pdf
• [General info] https://people.computing.clemson.edu/~dhouse/courses/405/notes/Catmull-Clark.pdf
• [Alg comparison] https://pubs.aip.org/aip/acp/article/1487/1/343/855266/A-comparison-of-surface-subdivision-algorithms-for?pdfCoverIconEvent=cite