Characterization of Partial Intrinsic Symmetries

  • Aurela Shehu
  • Alan Brunton
  • Stefanie Wuhrer
  • Michael Wand
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8928)

Abstract

We present a mathematical framework and algorithm for characterizing and extracting partial intrinsic symmetries of surfaces, which is a fundamental building block for many modern geometry processing algorithms. Our goal is to compute all “significant” symmetry information of the shape, which we define as \(r\)-symmetries, i.e., we report all isometric self-maps within subsets of the shape that contain at least an intrinsic circle or radius \(r\). By specifying \(r\), the user has direct control over the scale at which symmetry should be detected. Unlike previous techniques, we do not rely on feature points, voting or probabilistic schemes. Rather than that, we bound computational efforts by splitting our algorithm into two phases. The first detects infinitesimal \(r\)-symmetries directly using a local differential analysis, and the second performs direct matching for the remaining discrete symmetries. We show that our algorithm can successfully characterize and extract intrinsic symmetries from a number of example shapes.

Keywords

Symmetry Shape analysis Shape matching Intrinsic geometry Slippability analysis 

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References

  1. 1.
    Attene, M., Falcidieno, B.: Remesh: An interactive environment to edit and repair triangle meshes. In: Shape Modeling and Applications (2006)Google Scholar
  2. 2.
    Aubry, M., Schlickewei, U., Cremers, D.: The wave kernel signature: A quantum mechanical approach to shape analysis. In: International Conference on Computer Vision Workshops (2011)Google Scholar
  3. 3.
    Ben-Chen, M., Butscher, A., Solomon, J., Guibas, L.: On discrete killing vector fields and patterns on surfaces. Computer Graphics Forum 25, 1701–1711 (2010)CrossRefGoogle Scholar
  4. 4.
    Berner, A., Bokeloh, M., Wand, M., Schilling, A., Seidel, H.P.: Generalized intrinsic symmetry detection. Tech. rep, Max-Planck Institute for Informatics (2009)Google Scholar
  5. 5.
    Bronstein, A., Bronstein, M., Kimmel, R.: Numerical Geometry of Non-Rigid Shapes. Springer (2008)Google Scholar
  6. 6.
    Brunton, A., Wand, M., Wuhrer, S., Seidel, H.P., Weinkauf, T.: A low-dimensional representation for robust partial isometric correspondences computation. Graphical Models 76, 70–85 (2014)CrossRefGoogle Scholar
  7. 7.
    Gelfand, N., Guibas, L.: Shape segmentation using local slippage analysis. In: Symposium on Geometry Processing (2004)Google Scholar
  8. 8.
    Grushko, C., Raviv, D., Kimmel, R.: Intrinsic local symmetries: A computational framework. In: Eurographics Workshop on 3D Object Retrieval (2012)Google Scholar
  9. 9.
    Jiang, W., Xu, K., Chang, Z.Q., Zhang, H.: Skeleton-based intrinsic symmetry detection on point clouds. Graphical Models 75, 177–188 (2013)CrossRefGoogle Scholar
  10. 10.
    Kalojanov, J., Bokeloh, M., Wand, M., Guibas, L., Seidel, H.P., Slusallek, P.: Microtiles: Extracting building blocks from correspondences. In: Symposium on Geometry Processing (2012)Google Scholar
  11. 11.
    Kim, V.G., Lipman, Y., Chen, X., Funkhouser, T.: Möbius transformations for global intrinsic symmetry analysis. Computer Graphics Forum 29, 1689–1700 (2010)CrossRefGoogle Scholar
  12. 12.
    Lasowski, R., Tevs, A., Seidel, H.P., Wand, M.: A probabilistic framework for partial intrinsic symmetries in geometric data. In: International Conference on Computer Vision (2009)Google Scholar
  13. 13.
    Lipman, Y., Chen, X., Daubechies, I., Funkhouser, T.A.: Symmetry factored embedding and distance. ACM Transactions on Graphics 29(103), 1–12 (2010)Google Scholar
  14. 14.
    Meyer, M., Desbrun, M., Schröder, P., Barr, A.H.: Mathematics and Visualization 3, chap. Discrete Differential-Geometry Operators for Triangulated 2-Manifolds. Springer (2002)Google Scholar
  15. 15.
    Mitra, N., Bronstein, A., Bronstein, M.: Intrinsic regularity detection in 3d geometry. In: European Conference on Computer Vision (2010)Google Scholar
  16. 16.
    Mitra, N., Pauly, M., Wand, M., Ceylan, D.: Symmetry in 3d geometry: Extraction and applications. In: Eurographics State of the Art Report (2012)Google Scholar
  17. 17.
    Mount, D., Arya, S.: ANN: A library for approximate nearest neighbor searching (2010). http://www.cs.umd.edu/mount/ANN/
  18. 18.
    Mukhopadhyay, A., Bhandarkar, S., Porikli, F.: Detection and characterization of intrinsic symmetry. Tech. rep., arXiv 1309.7472 (2013)Google Scholar
  19. 19.
    Ovsjanikov, M., Mérigot, Q., Mmoli, F., Guibas, L.: One point isometric matching with the heat kernel. Computer Graphics Forum 29, 1555–1564 (2010)CrossRefGoogle Scholar
  20. 20.
    Ovsjanikov, M., Mrigot, Q., Patraucean, V., Guibas, L.: Shape matching via quotient spaces. Computer Graphics Forum 32, 1–11 (2013)CrossRefGoogle Scholar
  21. 21.
    Ovsjanikov, M., Sun, J., Guibas, L.: Global intrinsic symmetries of shapes. In: Symposium on Geometry Processing (2008)Google Scholar
  22. 22.
    Raviv, D., Bronstein, A., Bronstein, M., Kimmel, R.: Symmetries of non-rigid shapes. In: International Conference on Computer Vision (2007)Google Scholar
  23. 23.
    Raviv, D., Bronstein, A., Bronstein, M., Kimmel, R.: Diffusion symmetries of non-rigid shapes. In: International Symposium on 3D Data Processing, Visualization and Transmission (2010)Google Scholar
  24. 24.
    Raviv, D., Bronstein, A., Bronstein, M., Kimmel, R.: Full and partial symmetries of non-rigid shapes. International Journal of Computer Vision 89, 18–39 (2010)CrossRefGoogle Scholar
  25. 25.
    Rinow, W.: Über Zusammenhänge zwischen der Differentialgeometrie im Großen und im Kleinen. Mathematische Zeitschrift 35, 512–528 (1932)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Solomon, J., Ben-Chen, M., Butscher, A., Guibas, L.: Discovery of intrinsic primitives on triangle meshes. In: Eurographics (2011)Google Scholar
  27. 27.
    Wang, H., Simari, P., Su, Z., Zhang, H.: Spectral global intrinsic symmetry invariant functions. Graphics Interface (2014)Google Scholar
  28. 28.
    Xu, K., Zhang, H., Jiang, W., Dyer, R., Cheng, Z., Liu, L., Chen, B.: Multi-scale partial intrinsic symmetry detection. ACM Transactions on Graphics 31(181), 1–11 (2012)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Aurela Shehu
    • 1
    • 2
  • Alan Brunton
    • 3
  • Stefanie Wuhrer
    • 1
  • Michael Wand
    • 4
  1. 1.Cluster of Excellence MMCISaarland UniversitySaarbrückenGermany
  2. 2.Saarbrücken Graduate School of Computer ScienceSaarbrückenGermany
  3. 3.Fraunhofer Institute for Computer Graphics Research IGDDarmstadtGermany
  4. 4.Utrecht UniversityUtrechtNetherlands

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