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Microtiles : Extracting Building Blocks from Correspondences

Microtiles : Extracting Building Blocks from Correspondences. Javor Kalojanov , Martin Bokeloh , Michael Wand, Leonidas Guibas , Hans-Peter Seidel, Philipp Slusallek. Overview. Partial Symmetries. Inverse Procedural M odeling. Part I. Structuring Partial Symmetries. Goal.

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Microtiles : Extracting Building Blocks from Correspondences

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  1. Microtiles: Extracting Building Blocks from Correspondences JavorKalojanov, Martin Bokeloh, Michael Wand, Leonidas Guibas, Hans-Peter Seidel, Philipp Slusallek

  2. Overview Partial Symmetries Inverse Procedural Modeling

  3. Part I Structuring Partial Symmetries

  4. Goal • Exchangeable building blocks

  5. Pairwise matches

  6. Transformation Groups • Global symmetries • Closed groups of s • Partial symmetries

  7. Related Work • Mitra et al. • Transformation space clustering • Pauly et al. • Regular patterns in transformation space • Lipman et al. • Point-wise symmetry aware distance

  8. Problem • Input • A geometric shape • Point-wise partial correspondences • Partial symmetries • Output • Characteristic building blocks

  9. Assumptions • Point-wise equivalence relation • Reflexive: • Symmetric: • Transitive: • Connectivity is preserved • Homeomorphisms

  10. Correspondence Graph Properties • Set of disconnected cliques (orbits) • Infinitely many cliques • Point-wise correspondence sets • Set of edge labels

  11. Microtiles – Instances and Classes • Microtile (definition) • Connected set of points • Same set of mappings • Microtile classes • Equivalent microtiles

  12. Properties • Existence and uniqueness • Disjoint • No partial correspondences between tiles • Encode all symmetries • Algebraic model: permutation groups

  13. Part II Understanding Inverse Procedural Modeling

  14. Overview • Compute rules, describing a class of similar models • Bokeloh et al. [2010] Output Input

  15. -Similarity • Local neighborhoods match exemplar radius radius Output radius Input

  16. -Symmetry • Correspondence set • -Symmetry w.r.t. rigid transformations • if the r-neighborhoods match

  17. Continuous Symmetries • -Slippable points • Infinitely many equivalent points • Gelfand and Guibas [2004]

  18. Microtiles 1-slippable 2-slippable

  19. The Space of r-Similar Shapes • Theorem: Given a shape S, all -similar shapes to S can be constructed out of the -microtiles of S • Unique construction

  20. Proof Outline • Show that tile boundaries remain invariant • - similarity implies 𝑟-similarity at tile boundaries

  21. Towards a Shape Grammar • Constraint for construction of shapes -similar to S • Microtile adjacency has to be present in S • Necessary for -similarity • Sufficiency not yet shown

  22. Results

  23. Conclusion • Structure for partial symmetries • Cannonical • Encodes all symmetries • Tiles do not match partially • Maximal permutation groups • Connection to inverse procedural modeling • Describes all r-similar shapes • Pairwise assembly • Unique construction

  24. Limitations & Future Work • Extraction • Robust and scalable • Ill-posed problems e.g. noisy scans • Applications • Shape understanding • Instatiation patterns • Generating new shapes • Constructive rules (grammar)

  25. Thank You!

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