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.
Chapter PDF
Similar content being viewed by others
References
Attene, M., Falcidieno, B.: Remesh: An interactive environment to edit and repair triangle meshes. In: Shape Modeling and Applications (2006)
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)
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)
Berner, A., Bokeloh, M., Wand, M., Schilling, A., Seidel, H.P.: Generalized intrinsic symmetry detection. Tech. rep, Max-Planck Institute for Informatics (2009)
Bronstein, A., Bronstein, M., Kimmel, R.: Numerical Geometry of Non-Rigid Shapes. Springer (2008)
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)
Gelfand, N., Guibas, L.: Shape segmentation using local slippage analysis. In: Symposium on Geometry Processing (2004)
Grushko, C., Raviv, D., Kimmel, R.: Intrinsic local symmetries: A computational framework. In: Eurographics Workshop on 3D Object Retrieval (2012)
Jiang, W., Xu, K., Chang, Z.Q., Zhang, H.: Skeleton-based intrinsic symmetry detection on point clouds. Graphical Models 75, 177–188 (2013)
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)
Kim, V.G., Lipman, Y., Chen, X., Funkhouser, T.: Möbius transformations for global intrinsic symmetry analysis. Computer Graphics Forum 29, 1689–1700 (2010)
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)
Lipman, Y., Chen, X., Daubechies, I., Funkhouser, T.A.: Symmetry factored embedding and distance. ACM Transactions on Graphics 29(103), 1–12 (2010)
Meyer, M., Desbrun, M., Schröder, P., Barr, A.H.: Mathematics and Visualization 3, chap. Discrete Differential-Geometry Operators for Triangulated 2-Manifolds. Springer (2002)
Mitra, N., Bronstein, A., Bronstein, M.: Intrinsic regularity detection in 3d geometry. In: European Conference on Computer Vision (2010)
Mitra, N., Pauly, M., Wand, M., Ceylan, D.: Symmetry in 3d geometry: Extraction and applications. In: Eurographics State of the Art Report (2012)
Mount, D., Arya, S.: ANN: A library for approximate nearest neighbor searching (2010). http://www.cs.umd.edu/mount/ANN/
Mukhopadhyay, A., Bhandarkar, S., Porikli, F.: Detection and characterization of intrinsic symmetry. Tech. rep., arXiv 1309.7472 (2013)
Ovsjanikov, M., Mérigot, Q., Mmoli, F., Guibas, L.: One point isometric matching with the heat kernel. Computer Graphics Forum 29, 1555–1564 (2010)
Ovsjanikov, M., Mrigot, Q., Patraucean, V., Guibas, L.: Shape matching via quotient spaces. Computer Graphics Forum 32, 1–11 (2013)
Ovsjanikov, M., Sun, J., Guibas, L.: Global intrinsic symmetries of shapes. In: Symposium on Geometry Processing (2008)
Raviv, D., Bronstein, A., Bronstein, M., Kimmel, R.: Symmetries of non-rigid shapes. In: International Conference on Computer Vision (2007)
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)
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)
Rinow, W.: Über Zusammenhänge zwischen der Differentialgeometrie im Großen und im Kleinen. Mathematische Zeitschrift 35, 512–528 (1932)
Solomon, J., Ben-Chen, M., Butscher, A., Guibas, L.: Discovery of intrinsic primitives on triangle meshes. In: Eurographics (2011)
Wang, H., Simari, P., Su, Z., Zhang, H.: Spectral global intrinsic symmetry invariant functions. Graphics Interface (2014)
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)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Shehu, A., Brunton, A., Wuhrer, S., Wand, M. (2015). Characterization of Partial Intrinsic Symmetries. In: Agapito, L., Bronstein, M., Rother, C. (eds) Computer Vision - ECCV 2014 Workshops. ECCV 2014. Lecture Notes in Computer Science(), vol 8928. Springer, Cham. https://doi.org/10.1007/978-3-319-16220-1_19
Download citation
DOI: https://doi.org/10.1007/978-3-319-16220-1_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-16219-5
Online ISBN: 978-3-319-16220-1
eBook Packages: Computer ScienceComputer Science (R0)