Abstract
The study of 2D shapes is a central problem in the field of computer vision. In 2D shape analysis, classification and recognition of objects from their observed silhouettes are extremely crucial and yet difficult. It usually involves an efficient representation of 2D shape space with natural metric, so that its mathematical structure can be used for further analysis. Although significant progress has been made for the study of 2D simply-connected shapes, very few works have been done on the study of 2D objects with arbitrary topologies. In this work, we propose a representation of general 2D domains with arbitrary topologies using conformal geometry. A natural metric can be defined on the proposed representation space, which gives a metric to measure dissimilarities between objects. The main idea is to map the exterior and interior of the domain conformally to unit disks and circle domains, using holomorphic 1-forms. A set of diffeomorphisms from the unit circle \(\mathbb{S}^1\) to itself can be obtained, which together with the conformal modules are used to define the shape signature. We prove mathematically that our proposed signature uniquely represents shapes with arbitrary topologies. We also introduce a reconstruction algorithm to obtain shapes from their signatures. This completes our framework and allows us to move back and forth between shapes and signatures. Experiments show the efficacy of our proposed algorithm as a stable shape representation scheme.
Chapter PDF
References
Zhu, S.C., Yuille, A.L.: A flexible object recognition and modeling system. IJCV 20, 8 (1996)
Liu, T., Geiger, D.: Approximate tree matching and shape similarity. In: ICCV, pp. 456–462 (1999)
Belongie, S., Malik, J., Puzicha, J.: Shape matching and object recognition using shape contexts. IEEE Trans. on Pattern Analysis and Machine Intelligence 24, 509–522 (2002)
Mokhtarian, F., Mackworth, A.: A therory of multiscale, curvature-based shape representation for planar curves. IEEE Trans. on Pattern Analysis and Machine Intelligence 14, 789–805 (1992)
Ericsson, A., Astrom, K.: An affine invariant deformable shape representation for general curves. In: Proc. IEEE Intl. Conf. on Computer Vision, vol. 2, pp. 1142–1149 (2003)
Sebastian, T., Klein, P., Kimia, B.: Shock based indexing into large shape databases. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2352, pp. 731–746. Springer, Heidelberg (2002)
Dryden, I., Mardia, K.: Statistical shape analysis. John Wiley and Son, Chichester (1998)
Yang, Q., Ma, S.: Matching using schwarz integrals. Pattern Recognition 32, 1039–1047 (1999)
Lee, S.M., Clark, N.A., Araman, P.A.: A shape representation for planar curves by shape signature harmonic embedding. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 2006), vol. 2, pp. 1940–1947 (2006)
Lipman, Y., Funkhouser, T.: Mobius Voting for Surface Correspondence. ACM Transactions on Graphics (Proc. SIGGRAPH) (August 2009)
Zeng, W., Zeng, Y., Wang, Y., Yin, X., Gu, X., Samaras, D.: 3D non-rigid surface matching and registration based on holomorphic differentials. In: Forsyth, D., Torr, P., Zisserman, A. (eds.) ECCV 2008, Part III. LNCS, vol. 5304, pp. 1–14. Springer, Heidelberg (2008)
Zeng, W., Lui, L.M., Gu, X., Yau, S.T.: Shape Analysis by Conformal Modules. Methods Appl. Anal. 15(4), 539–556 (2008)
Sharon, E., Mumford, D.: 2d-shape analysis using conformal mapping. International Journal of Computer Vision 70, 55–75 (2006)
Gardiner, F.P., Lakic, N.: Quasiconformal Teichmüler theory. American Mathematical Society, Providence (1999)
Henrici, P.: Applied and Computational Complex Analysis. Wiley Classics Library (1974)
Zeng, W., Yin, X.T., Zhang, M., Luo, F., Gu, X.: Generalized Koebe’s method for conformal mapping multiply connected domains. In: SPM 2009: 2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling, pp. 89–100 (2009)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
1 Electronic Supplementary Material
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Lui, L.M., Zeng, W., Yau, ST., Gu, X. (2010). Shape Analysis of Planar Objects with Arbitrary Topologies Using Conformal Geometry. In: Daniilidis, K., Maragos, P., Paragios, N. (eds) Computer Vision – ECCV 2010. ECCV 2010. Lecture Notes in Computer Science, vol 6315. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15555-0_49
Download citation
DOI: https://doi.org/10.1007/978-3-642-15555-0_49
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15554-3
Online ISBN: 978-3-642-15555-0
eBook Packages: Computer ScienceComputer Science (R0)