A planar-reflective symmetry transform for 3D shapes

Abstract

Symmetry is an important cue for many applications, including object alignment, recognition, and segmentation. In this paper, we describe a planar reflective symmetry transform (PRST) that captures a continuous measure of the reflectional symmetry of a shape with respect to all possible planes. This transform combines and extends previous work that has focused on global symmetries with respect to the center of mass in 3D meshes and local symmetries with respect to points in 2D images. We provide an efficient Monte Carlo sampling algorithm for computing the transform for surfaces and show that it is stable under common transformations. We also provide an iterative refinement algorithm to find local maxima of the transform precisely. We use the transform to define two new geometric properties, center of symmetry and principal symmetry axes, and show that they are useful for aligning objects in a canonical coordinate system. Finally, we demonstrate that the symmetry transform is useful for several applications in computer graphics, including shape matching, segmentation of meshes into parts, and automatic viewpoint selection.

References

1. Abbasi, S., and Mokhtarian, F. 2000. Automatic view selection in multi-view object recognition. In Proc. ICPR, vol. 1, 1013.
2. Atallah, M. 1985. On symmetry detection. IEEE Trans. on Computers 34, 663--666.
3. Besl, P. J., and Mckay, N. D. 1992. A method for registration of 3-D shapes. IEEE Trans. PAMI 14, 2, 239--256.
4. Bigun, J. 1997. Pattern recognition in images by symmetries and coordinate transformations. Computer Vision and Image Understanding 68, 3, 290--307.
5. Blanz, V., Tarr, M., Buelthoff, H., and Vetter, T. 1999. What object attributes determine canonical views. Perception 28.
6. Blum, H. 1967. A transformation for extracting new descriptors of shape. In Models for the Perception of Speech and Visual Form, MIT Press, W. Whaten-Dunn, Ed., 362--380.
7. Bonneh, Y., Reisfeld, D., and Yeshurun, Y. 1994. Quantification of local symmetry: application to texture discrimination. Spatial Vision 8, 4, 515--530.
8. Chazelle, B., Dobkin, D. P., Shouraboura, N., and Tal, A. 1995. Strategies for polyhedral surface decomposition: an experimental study. In SCG '95: Proceedings of the eleventh annual symposium on Computational geometry, ACM Press, New York, NY, USA, 297--305.
9. Chetverikov, D. 1995. Pattern orientation and texture symmetry. Computer Analysis of Images and Patterns 970, 222--229.
10. Choi, I., and Chien, S. 2004. A generalized symmetry transform with selective attention capability for specific corner angles. IEEE Signal Processing Letters 11, 2 (Feb.), 255--257.
11. Di Gesù, V., Valenti, C., and Strinati, L. 1997. Local operators to detect regions of interest. Pattern Recognition Letters 18, 11--13 (Nov.), 1088--1081.
12. Duda, R., Hart, P., and Stork, D. 2001. Pattern Classification, Second Edition. John Wiley & Sons, New York.
13. Ferguson, R. W. 2000. Modeling orientation effects in symmetry detection: The role of visual structure. In Proc. Conf. Cognitive Science Society.
14. Funkhouser, T., Min, P., Kazhdan, M., Chen, J., Halderman, A., Dobkin, D., and Jacobs, D. 2003. A search engine for 3D models. ACM Trans. Graph. 22, 1, 83--105.
15. Garland, M., Willmott, A., and Heckbert, P. S. 2001. Hierarchical face clustering on polygonal surfaces. In SI3D '01: Proceedings of the 2001 Symposium on Interactive 3D graphics, ACM Press, New York, NY, USA, 49--58.
16. Kamada, T., and Kawai, S. 1988. A simple method for computing general position in displaying three-dimensional objects. Comput. Vision Graph. Image Process. 41, 1, 43--56.
17. Katz, S., and Tal, A. 2003. Hierarchical mesh decomposition using fuzzy clustering and cuts. Proceedings of ACM SIGGRAPH 22, 3, 954--961.
18. Kazhdan, M., Chazelle, B., Dobkin, D., Funkhouser, T., and Rusinkiewicz, S. 2003. A reflective symmetry descriptor for 3D models. Algorithmica 38, 1 (Oct.).
19. Kazhdan, M., Funkhouser, T., and Rusinkiewicz, S. 2003. Rotation invariant spherical harmonic representation of 3D shape descriptors. In Symposium on Geometry Processing.
20. Kazhdan, M., Funkhouser, T., and Rusinkiewicz, S. 2004. Symmetry descriptors and 3D shape matching. In Proc. Symposium on Geometry Processing.
21. Kelly, M. F., and Levine, M. D. 1995. Annular symmetry operators: A method for locating and describing objects. In Proc. ICCV, 1016--1021.
22. Lee, J., Moghaddam, B., Pfister, H., and Machiraju, R. 2004. Finding optimal views for 3d face shape modeling. In FGR, IEEE Computer Society, 31--36.
23. Lee, C. H., Varshney, A., and Jacobs, D. W. 2005. Mesh saliency. Proceedings of ACM SIGGRAPH 24, 3, 659--666.
24. Li, X., Toon, T., Tan, T., and Huang, Z. 2001. Decomposing polygon meshes for interactive applications. In Proceedings of the 2001 Symposium on Interactive 3D graphics, 35--42.
25. Loy, G., and Zelinsky, A. 2002. A fast radial symmetry transform for detecting points of interest. In Proc. ECCV, 358--368.
26. Mangan, A. P., and Whitaker, R. T. 1999. Partitioning 3d surface meshes using watershed segmentation. IEEE Transactions on Visualization and Computer Graphics 5, 4, 308--321.
27. Manmatha, R., and Sawhney, H. 1997. Finding symmetry in intensity images. Tech. Rep. UM-CS-1997-007, University of Massachusetts, Jan.
28. Martinet, A., Soler, C., Holzschuch, N., and Sillion, F. 2005. Accurately detecting symmetries of 3D shapes. Tech. Rep. RR-5692, INRIA, September.
29. Minovic, P., Ishikawa, S., and Kato, K. 1993. Symmetry identification of a 3D object represented by octree. IEEE Transactions on Pattern Analysis and Machine Intelligence 15, 5 (May), 507--514.
30. Reisfeld, D., and Yeshurun, Y. 1992. Robust detection of facial features by generalized symmetry. In Proc. ICPR, 117.
31. Reisfeld, D., Wolfson, H., and Yeshurun, Y. 1995. Context-free attentional operators: The generalized symmetry transform. IJCV 14, 2, 119--130.
32. Shah, M. I., and Sorensen, D. C. 2005. A symmetry preserving singular value decomposition. SIAM Journal of Matrix Analysis and it's Application (October).
33. Shan, Y., Matei, B., Sawhney, H. S., Kumar, R., Huber, D., and Hebert, M. 2004. Linear model hashing and batch ransac for rapid and accurate object recognition. IEEE International Conference on Computer Vision and Pattern Recognition.
34. Shilane, P., Min, P., Kazhdan, M., and Funkhouser, T. 2004. The Princeton Shape Benchmark. In Proc. Shape Modeling International.
35. Sun, C., and Sherrah, J. 1997. 3D symmetry detection using the extended Gaussian image. IEEE Transactions on Pattern Analysis and Machine Intelligence 2, 2 (February), 164--168.
36. Thrun, S., and Wegbreit, B. 2005. Shape from symmetry. In Proceedings of the International Conference on Computer Vision (ICCV), IEEE, Bejing, China.
37. Vázquez, P.-P., Feixas, M., Sbert, M., and Heidrich, W. 2001. Viewpoint selection using viewpoint entropy. In VMV '01: Proceedings of the Vision Modeling and Visualization Conference 2001, Aka GmbH, 273--280.
38. Wolter, J. D., Woo, T. C., and Volz, R. A. 1985. Optimal algorithms for symmetry detection in two and three dimensions. The Visual Computer 1, 37--48.
39. Zabrodsky, H., Peleg, S., and Avnir, D. 1993. Completion of occluded shapes using symmetry. In Proc. CVPR, 678--679.
40. Zabrodsky, H., Peleg, S., and Avnir, D. 1995. Symmetry as a continuous feature. Trans. PAMI 17, 12, 1154--1166.
41. Zhang, J., and Huebner, K. 2002. Using symmetry as a feature in panoramic images for mobile robot applications. In Proc. Robotik, vol. 1679 of VDI-Berichte, 263--268.

Index Terms

• Computing methodologies

  • Computer graphics

    • Shape modeling

      • Parametric curve and surface models

      • Volumetric models

• Mathematics of computing

  • Probability and statistics

    • Probabilistic algorithms

    • Probabilistic reasoning algorithms

      • Markov-chain Monte Carlo methods

      • Sequential Monte Carlo methods

• Theory of computation

  • Randomness, geometry and discrete structures