Abstract
We propose an automatic method for finding symmetries of 3D shapes, that is, isometric transforms which leave a shape globally unchanged. These symmetries are deterministically found through the use of an intermediate quantity: the generalized moments. By examining the extrema and spherical harmonic coefficients of these moments, we recover the parameters of the symmetries of the shape. The computation for large composite models is made efficient by using this information in an incremental algorithm capable of recovering the symmetries of a whole shape using the symmetries of its subparts. Applications of this work range from coherent remeshing of geometry with respect to the symmetries of a shape to geometric compression, intelligent mesh editing, and automatic instantiation.
- Attalah, M. J. 1985. On symmetry detection. IEEE Trans. Comput. 34, 663--666.Google ScholarDigital Library
- Brass, P. and Knauer, C. 2004. Testing congruence and symmetry for general 3-dimensional objects. Comput. Geom. Theory Appl. 27, 1, 3--11. Google ScholarDigital Library
- Highnam, P. T. 1985. Optimal algorithms for finding the symmetries of a planar point set. Tech. Rep. CMU-RI-TR-85-13 (Aug). Robotics Institute, Carnegie Mellon University, Pittsburgh, PA.Google Scholar
- Hobson, E. W. 1931. The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press, Cambridge, UK.Google Scholar
- Ivanic, J. and Ruedenberg, K. 1996. Rotation matrices for real spherical harmonics, direct determination by recursion. J. Phys. Chem. A. 100, 6342--6347. (See also Additions and corrections in vol. 102, No. 45, 9099-9100).Google ScholarCross Ref
- Jiang, X.-Y. and Bunke, H. 1991. Determination of the symmetries of polyhedra and an application to object recognition. In Proceedings of the International Workshop on Computational Geometry---Methods, Algorithms and Applications (CG '91). Lecture Notes in Computer Science, vol. 553. Springer, London, UK, 113--121. Google ScholarDigital Library
- Kazhdan, M. M., Funkhouser, T. A., and Rusinkiewicz, S. 2003. Rotation invariant spherical harmonic representation of 3D shape descriptors. In Proceedings of the 2003 Eurographics/ACM Siggraph Symposium on Geometry Processing (SGP '03). Eurographics Association, Aire-la-Ville, Switzerland, 167--175. Google ScholarDigital Library
- Kazhdan, M. M., Funkhouser, T. A., and Rusinkiewicz, S. 2004. Symmetry descriptors and 3D shape matching. In Proceedings of the 2004 Eurographics/ACM Siggraph Symposium on Geometry Processing (SGP '04). Eurographics Association, Aire-la-Ville, Switzerland. Google ScholarDigital Library
- Knuth, D. E., Morris, Jr., J. H., and Pratt, V. R. 1977. Fast pattern matching in strings. SIAM J. Comput. 6, 2, 323--350.Google ScholarDigital Library
- Minovic, P., Ishikawa, S., and Kato, K. 1993. Symmetry identification of a 3-D object represented by octree. IEEE Trans. Patt. Analy. Mach. Intell. 15, 5, 507--514. Google ScholarDigital Library
- Prince, E. 2004. Mathematical Techniques in Crystallography and Materials Science, 3rd Ed. Springer, Berlin, Germany.Google Scholar
- Ramamoorthi, R. and Hanrahan, P. 2004. A signal-processing framework for reflection. ACM Trans. Graph. 23, 4, 1004--1042. Google ScholarDigital Library
- Sun, C. and Sherrah, J. 1997. 3D symmetry detection using extended Gaussian image. IEEE Trans. Patt. Analy. Mach. Intell. 19, 2 (Feb.), 164--168. Google ScholarDigital Library
- Wolter, J. D., Woo, T. C., and Volz, R. A. 1985. Optimal algorithms for symmetry detection in two and three dimensions. Visual Comput. 1, 37--48.Google ScholarCross Ref
- Zabrodsky, H., Peleg, S., and Avnir, D. 1995. Symmetry as a continuous feature. IEEE Trans. Patt. Analy. Mach. Intell. 17, 12, 1154--1166. Google ScholarDigital Library
Index Terms
- Accurate detection of symmetries in 3D shapes
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