Abstract
We introduce the Symmetry Factored Embedding (SFE) and the Symmetry Factored Distance (SFD) as new tools to analyze and represent symmetries in a point set. The SFE provides new coordinates in which symmetry is "factored out," and the SFD is the Euclidean distance in that space. These constructions characterize the space of symmetric correspondences between points -- i.e., orbits. A key observation is that a set of points in the same orbit appears as a clique in a correspondence graph induced by pairwise similarities. As a result, the problem of finding approximate and partial symmetries in a point set reduces to the problem of measuring connectedness in the correspondence graph, a well-studied problem for which spectral methods provide a robust solution. We provide methods for computing the SFE and SFD for extrinsic global symmetries and then extend them to consider partial extrinsic and intrinsic cases. During experiments with difficult examples, we find that the proposed methods can characterize symmetries in inputs with noise, missing data, non-rigid deformations, and complex symmetries, without a priori knowledge of the symmetry group. As such, we believe that it provides a useful tool for automatic shape analysis in applications such as segmentation and stationary point detection.
Supplemental Material
- Belkin, M., and Niyogi, P. 2001. Laplacian eigenmaps and spectral techniques for embedding and clustering. In Advances in Neural Information Processing Systems 14, MIT Press, 585--591.Google Scholar
- Berner, A., Bokeloh, M., Wand, M., Schilling, A., and Seidel, H.-P. 2008. A graph-based approach to symmetry detection. In IEEE/EG Symposium on Volume and Point-Based Graphics. Google ScholarDigital Library
- Bokeloh, M., Berner, A., Wand, M., Seidel, H.-P., and Schilling, A. 2009. Symmetry detection using line features. Computer Graphics Forum (Eurographics) 28, 2, 697--706.Google ScholarCross Ref
- Bronstein, A., Bronstein, M., Bruckstein, A., and Kimmel, R. 2009. Partial similarity of objects, or how to compare a centaur to a horse. Int. J. Comp. Vis. 84, 2, 163--183. Google ScholarDigital Library
- Chertok, M., and Keller, Y. 2010. Spectral symmetry analysis. In IEEE Transactions on Pattern Analysis and Machine Intelligence, to appear. Google ScholarDigital Library
- Coifman, R. R., Lafon, S., Lee, A. B., Maggioni, M., Nadler, B., Warner, F., and Zucker, S. W. 2005. Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps. Proceedings of the National Academy of Sciences 102, 21 (May), 7426--7431.Google Scholar
- Gold, S., Rangarajan, A., ping Lu, C., and Mjolsness, E. 1998. New algorithms for 2d and 3d point matching: Pose estimation and correspondence. Pattern Recognition 31, 957--964.Google ScholarCross Ref
- Hays, J., Leordeanu, M., Efros, A. A., and Liu, Y. 2006. Discovering texture regularity as a higher-order correspondence problem. In 9th European Conference on Computer Vision. Google ScholarDigital Library
- Imiya, A., Ueno, T., and Fermin, I. 1999. Symmetry detection by random sampling and voting process. In CIAP99, 400--405. Google ScholarDigital Library
- Kazhdan, M., Chazelle, B., Dobkin, D., Funkhouser, T., and Rusinkiewicz, S. 2003. A reflective symmetry descriptor for 3D models. Algorithmica 38, 1 (Oct.). Google ScholarDigital Library
- Kazhdan, M., Funkhouser, T., and Rusinkiewicz, S. 2004. Symmetry descriptors and 3D shape matching. In Symposium on Geometry Processing. Google ScholarDigital Library
- Leordeanu, M., and Hebert, M. 2005. A spectral technique for correspondence problems using pairwise constraints. In ICCV '05: Proceedings of the Tenth IEEE International Conference on Computer Vision, IEEE Computer Society, Washington, DC, USA, 1482--1489. Google ScholarDigital Library
- Leung, T., and Malik, J. 1996. Detecting, localizing and grouping repeated scene elements from an image. In European Conference on Computer Vision, 546--555. Google ScholarDigital Library
- Li, W., Zhang, A., and Kleeman, L. 2005. Fast global reflectional symmetry detection for robotic grasping and visual tracking. In ACRA05.Google Scholar
- Li, M., Langbein, F., and Martin, R. 2006. Constructing regularity feature trees for solid models. Geom. Modeling Processing, 267--286. Google ScholarDigital Library
- Lipman, Y., and Funkhouser, T. 2009. Mobius voting for surface correspondence. ACM Transactions on Graphics (Proc. SIGGRAPH) 28, 3 (Aug.). Google ScholarDigital Library
- Liu, S., Martin, R., Langbein, F., and Rosin, P. 2007. Segmenting periodic reliefs on triangle meshes. In Math. of Surfaces XII, Springer, 290--306. Google ScholarDigital Library
- Mitra, N. J., Guibas, L., and Pauly, M. 2006. Partial and approximate symmetry detection for 3d geometry. In ACM Transactions on Graphics, vol. 25, 560--568. Google ScholarDigital Library
- Nadler, B., Lafon, S., Coifman, R. R., and Kevrekidis, I. G. 2005. Diffusion maps, spectral clustering and reaction coordinates of dynamical systems. ArXiv Mathematics e-prints.Google Scholar
- Ovsjanikov, M., Sun, J., and Guibas, L. 2008. Global intrinsic symmetries of shapes. Computer Graphics Forum (Symposium on Geometry Processing) 27, 5, 1341--1348. Google ScholarDigital Library
- Pauly, M., Mitra, N. J., Wallner, J., Pottmann, H., and Guibas, L. 2008. Discovering structural regularity in 3D geometry. ACM Transactions on Graphics 27, 3, #43, 1--11. Google ScholarDigital Library
- Pinkall, U., and Polthier, K. 1993. Computing discrete minimal surfaces and their conjugates. Experimental Mathematics 2, 15--36.Google ScholarCross Ref
- Podolak, J., Shilane, P., Golovinskiy, A., Rusinkiewicz, S., and Funkhouser, T. 2006. A planar-reflective symmetry transform for 3D shapes. ACM Transactions on Graphics (Proc. SIGGRAPH) 25, 3 (July). Google ScholarDigital Library
- Raviv, D., Bronstein, A., Bronstein, M., and Kimmel, R. 2007. Symmetries of non-rigid shapes. In Int. Conf. on Comp. Vis.Google Scholar
- Reisfeld, D., Wolfson, H., and Yeshurun, Y. 1995. Context-free attentional operators: The generalized symmetry transform. IJCV 14, 2, 119--130. Google ScholarDigital Library
- Rustamov, R. 2008. Augmented planar reflective symmetry transform. The Visual Computer 24, 6, 423--433. Google ScholarDigital Library
- Shi, J., and Malik, J. 1997. Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 22, 888--905. Google ScholarDigital Library
- Shikhare, D., Bhakar, S., and Mudur, S. 2001. Compression of large 3d engineering models using automatic discovery of repeating geometric features. 233--240.Google Scholar
- Xu, K., Zhang, H., Tagliasacchi, A., Liu, L., Li, G., Meng, M., and Xiong, Y. 2009. Partial intrinsic reflectional symmetry of 3d shapes. ACM Transactions on Graphics (SIGGRAPH ASIA) 28, 5. Google ScholarDigital Library
- Yip, R. 2000. A hough transform technique for the detection of reflectional symmetry and skew-symmetry. PRL 21, 2, 117--130. Google ScholarDigital Library
- Zabrodsky, H., Peleg, S., and Avnir, D. 1993. Continuous symmetry measures, ii: Symmetry groups and the tetrahedron. Journal of the American Chemical Society 115, 8278--8289.Google ScholarCross Ref
Index Terms
- Symmetry factored embedding and distance
Recommendations
Symmetry factored embedding and distance
SIGGRAPH '10: ACM SIGGRAPH 2010 papersWe introduce the Symmetry Factored Embedding (SFE) and the Symmetry Factored Distance (SFD) as new tools to analyze and represent symmetries in a point set. The SFE provides new coordinates in which symmetry is "factored out," and the SFD is the ...
Streaming algorithms for embedding and computing edit distance in the low distance regime
STOC '16: Proceedings of the forty-eighth annual ACM symposium on Theory of ComputingThe Hamming and the edit metrics are two common notions of measuring distances between pairs of strings x,y lying in the Boolean hypercube. The edit distance between x and y is defined as the minimum number of character insertion, deletion, and bit ...
Locally Linear Embedding based on Rank-order Distance
ICPRAM 2016: Proceedings of the 5th International Conference on Pattern Recognition Applications and MethodsDimension reduction has become an important tool for dealing with high dimensional data. Locally linear embedding (LLE) is a nonlinear dimension reduction method which can preserve local configurations of nearest neighbors. However, finding the nearest ...
Comments