# A Unified Spectral Embedding for Shape Correspondence

### Zizhao Wu, Ruyang Shou, Xinguo Liu

#### Abstract

Spectral embedding, as one of shape representative techniques, takes hold of many researchers’ attention in field of shape correspondence. One of the biggest challenges of spectral correspondence method is that embeddings of different shapes need to be aligned in the embedding space in order to eliminate sign flip and ordering ambiguity of their eigenfunctions, before seeking for correspondence. In this paper, we introduce a spectral correspondence method by embedding shapes in a unified space simultaneously. In the unified embedding space, the sample points of the same shape with small intrinsic distances, and from different shapes with high similarity, are close to each other. Our unified embedding can be used for correspondence directly, without need of alignment. Furthermore, the unified embedding captures both the spatial arrangement and the feature similarity. Shape correspondence is achieved with such embedding by minimizing an objective function. Results show the efficiency of our method.

#### References

- Anguelov, D., Srinivasan, P., Koller, D., Thrun, S., Rodgers, J., and Davis, J. (2005). Scape: shape completion and animation of people. ACM Trans. Graph., 24(3):408- 416.
- Belkin, M. and Niyogi, P. (2003). Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput., 15(6):1373-1396.
- Bronstein, A., Bronstein, M., and Kimmel, R. (2008). Numerical geometry of non-rigid shapes. SpringerVerlag New York Inc.
- Elad, A. and Kimmel, R. (2001). Bending invariant representations for surfaces. In CVPR (1), pages 168-174.
- Gal, R., Shamir, A., and Cohen-Or, D. (2007). Poseoblivious shape signature. IEEE Trans. Vis. Comput. Graph., 13(2):261-271.
- Hilaga, M., Shinagawa, Y., Komura, T., and Kunii, T. L. (2001). Topology matching for fully automatic similarity estimation of 3d shapes. In SIGGRAPH, pages 203-212.
- Jain, V. and Zhang, H. (2006). Robust 3d shape correspondence in the spectral domain. In Proc. of Shape Modeling International, pages 118-129.
- Jiang, H. and Yu, S. X. (2009). Linear solution to scale and rotation invariant object matching. In CVPR, pages 2474-2481.
- Mateus, D., Cuzzolin, F., Horaud, R. P., and Boyer, E. (2007). Articulated shape matching by robust alignment of embedded representations. In IEEE Workshop on 3D Representation for Recognition (3DRR 2007). IEEE Computer Society Press.
- Mateus, D., Horaud, R. P., Knossow, D., Cuzzolin, F., and Boyer, E. (2008). Articulated shape matching using laplacian eigenfunctions and unsupervised point registration. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition.
- Ovsjanikov, M., Mérigot, Q., Mémoli, F., and Guibas, L. J. (2010). One point isometric matching with the heat kernel. Comput. Graph. Forum, 29(5):1555-1564.
- Roweis, S. and Saul, L. (2000). Nonlinear dimensionality reduction by locally linear embedding. Science, 290(5500):2323-2326.
- Sahilliog?lu, Y. and Yemez, Y. (2010). 3d shape correspondence by isometry-driven greedy optimization. In CVPR, pages 453-458.
- Sahilliog?lu, Y. and Yemez, Y. (2011). Coarse-to-fine combinatorial matching for dense isometric shape correspondence. Comput. Graph. Forum, 30(5):1461- 1470.
- Scott, G. L. and Longuet-Higgins, H. C. (1991). an algorithm for associating the features of two images. In Proc. Biol Sci.
- Sun, J., Ovsjanikov, M., and Guibas, L. (2009). A concise and provably informative multi-scale signature based on heat diffusion. In Eurographics Symposium on Geometry Processing (SGP).
- Torki, M. and Elgammal, A. M. (2010). One-shot multi-set non-rigid feature-spatial matching. In CVPR, pages 3058-3065.
- Vaxman, A., Ben-Chen, M., and Gotsman, C. (2010). A multi-resolution approach to heat kernels on discrete surfaces. ACM Trans. Graph., 29(4).

#### Paper Citation

#### in Harvard Style

Wu Z., Shou R. and Liu X. (2013). **A Unified Spectral Embedding for Shape Correspondence** . In *Proceedings of the International Conference on Computer Graphics Theory and Applications and International Conference on Information Visualization Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2013)* ISBN 978-989-8565-46-4, pages 94-99. DOI: 10.5220/0004278700940099

#### in Bibtex Style

@conference{grapp13,

author={Zizhao Wu and Ruyang Shou and Xinguo Liu},

title={A Unified Spectral Embedding for Shape Correspondence},

booktitle={Proceedings of the International Conference on Computer Graphics Theory and Applications and International Conference on Information Visualization Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2013)},

year={2013},

pages={94-99},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0004278700940099},

isbn={978-989-8565-46-4},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the International Conference on Computer Graphics Theory and Applications and International Conference on Information Visualization Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2013)

TI - A Unified Spectral Embedding for Shape Correspondence

SN - 978-989-8565-46-4

AU - Wu Z.

AU - Shou R.

AU - Liu X.

PY - 2013

SP - 94

EP - 99

DO - 10.5220/0004278700940099