A Graph Matching Approach to Symmetry Detection and Analysis

  • Michael Chertok
  • Yosi Keller
Part of the Studies in Computational Intelligence book series (SCI, volume 386)


Spectral relaxation was shown to provide an efficient approach for solving a gamut of computational problems, ranging from data mining to image registration. In this chapter we show that in the context of graph matching, spectral relaxation can be applied to the detection and analysis of symmetries in n-dimensions. First, we cast symmetry detection of a set of points in ℝ n as the self-alignment of the set to itself. Thus, by representing an object by a set of points S ∈ ℝ n , symmetry is manifested by multiple self-alignments. Secondly, we formulate the alignment problem as a quadratic binary optimization problem, solved efficiently via spectral relaxation. Thus, each eigenvalue corresponds to a potential self-alignment, and eigenvalues with multiplicity greater than one correspond to symmetric self-alignments. The corresponding eigenvectors reveal the point alignment and pave the way for further analysis of the recovered symmetry. We apply our approach to image analysis, by using local features to represent each image as a set of points. Last, we improve the scheme’s robustness by inducing geometrical constraints on the spectral analysis results. Our approach is verified by extensive experiments and was applied to two and three dimensional synthetic and real life images.


Feature Point Local Feature Rotational Symmetry Interest Point Zernike Moment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [BBM05]
    Berg, A.C., Berg, T.L., Malik, J.: Shape matching and object recognition using low distortion correspondences. In: Proceedings, IEEE Conference on Computer Vision and Pattern Recognition, vol. 1, pp. 26–33. IEEE Computer Society, Washington, DC, USA (2005)Google Scholar
  2. [Bio01]
    Bioid, the bioid face database (2001),
  3. [BL03]
    Brown, M., Lowe, D.G.: Recognising panoramas. In: Proc. IEEE Int. Conf. Computer Vision, vol. 2, pp. 1218–1227 (2003)Google Scholar
  4. [Che01]
    Chen, S.: Extraction of local mirror-symmetric feature by odd-even decomposition. In: Proceedings International Conference on Image Processing, vol. 3, pp. 756–759 (2001)Google Scholar
  5. [Cox69]
    Coxeter, H.S.M.: Introduction to Geometry. John Wiley & Sons, Inc. (1969)Google Scholar
  6. [CPML07]
    Cornelius, H., Perďoch, M., Matas, J., Loy, G.: Efficient Symmetry Detection Using Local Affine Frames. In: Ersbøll, B.K., Pedersen, K.S. (eds.) SCIA 2007. LNCS, vol. 4522, pp. 152–161. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  7. [CSS07]
    Cour, T., Srinivasan, P., Shi, J.: Balanced graph matching. In: Schölkopf, B., Platt, J., Hoffman, T. (eds.) Advances in Neural Information Processing Systems, vol. 19, pp. 313–320. MIT Press, Cambridge (2007)Google Scholar
  8. [DG04]
    Derrode, S., Ghorbel, F.: Shape analysis and symmetry detection in gray-level objects using the analytical Fourier-Mellin representation. Signal Processing 84(1), 25–39 (2004)CrossRefzbMATHGoogle Scholar
  9. [ELPZ97]
    Eldar, Y., Lindenbaum, M., Porat, M., Zeevi, Y.Y.: The farthest point strategy for progressive image sampling. IEEE Transactions on Image Processing 6(9), 1305–1315 (1997)CrossRefGoogle Scholar
  10. [EWC00]
    Evans, C.S., Wenderoth, P., Cheng, K.: Detection of bilateral symmetry in complex biological images. Perception 29(1), 31–42 (2000)CrossRefGoogle Scholar
  11. [FB81]
    Fischler, M.A., Bolles, R.C.: Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartography. Communications of the ACM 24(6), 381–395 (1981)CrossRefMathSciNetGoogle Scholar
  12. [HLEL06]
    Hays, J.H., Leordeanu, M., Efros, A.A., Liu, Y.: Discovering texture regularity via higher-order matching. In: 9th European Conference on Computer Vision, pp. 522–535 (May 2006)Google Scholar
  13. [HZ04]
    Hartley, R.I., Zisserman, A.: Multiple View Geometry in Computer Vision, 2nd edn. Cambridge University Press (2004); ISBN: 0521540518Google Scholar
  14. [KCD+02]
    Kazhdan, M.M., Chazelle, B., Dobkin, D.P., Finkelstein, A., Funkhouser, T.A.: A reflective symmetry descriptor. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2351, pp. 642–656. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  15. [KG98]
    Kiryati, N., Gofman, Y.: Detecting symmetry in grey level images: The global optimization approach. International Journal of Computer Vision 29(1), 29–45 (1998)CrossRefGoogle Scholar
  16. [KK99]
    Kim, W., Kim, Y.: Robust rotation angle estimator. IEEE Transactions on Pattern Analysis and Machine Intelligence 21(8), 768–773 (1999)CrossRefGoogle Scholar
  17. [KS06]
    Keller, Y., Shkolnisky, Y.: A signal processing approach to symmetry detection. IEEE Transactions on Image Processing 15(6), 2198–2207 (2006)CrossRefMathSciNetGoogle Scholar
  18. [LCL08]
    Lee, S., Collins, R., Liu, Y.: Rotation symmetry group detection via frequency analysis of frieze-expansions. In: Proceedings of CVPR (June 2008)Google Scholar
  19. [LE06]
    Loy, G., Eklundh, J.-O.: Video and Image Bayesian Demosaicing with a Two Color Image Prior. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV 2006. LNCS, vol. 3951, pp. 508–521. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  20. [LH05]
    Leordeanu, M., Hebert, M.: A spectral technique for correspondence problems using pairwise constraints. In: International Conference of Computer Vision (ICCV), vol. 2, pp. 1482–1489 (October 2005)Google Scholar
  21. [Low03]
    Lowe, D.: Distinctive image features from scale-invariant keypoints. International Journal of Computer Vision 20, 91–110 (2003)Google Scholar
  22. [Luc04]
    Lucchese, L.: Frequency domain classification of cyclic and dihedral symmetries of finite 2-D patterns. Pattern Recognition 37, 2263–2280 (2004)Google Scholar
  23. [LW99]
    Lei, Y., Wong, K.C.: Detection and localisation of reflectional and rotational symmetry under weak perspective projection. Pattern Recognition 32(2), 167–180 (1999)CrossRefGoogle Scholar
  24. [MCUP02]
    Matas, J., Chum, O., Urban, M., Pajdla, T.: Robust wide baseline stereo from maximally stable extremal regions. In: Proceedings of the British Machine Vision Conference (BMVC), London, pp. 384–393 (2002)Google Scholar
  25. [Mil72]
    Miller, W.: Symmetry Groups and their Applications. Academic Press, London (1972)zbMATHGoogle Scholar
  26. [MS04]
    Mikolajczyk, K., Schmid, C.: Scale and affine invariant interest point detectors. International Journal of Computer Vision 60(1), 63–86 (2004)CrossRefGoogle Scholar
  27. [MSHS06]
    Martinet, A., Soler, C., Holzschuch, N., Sillion, F.: Accurate detection of symmetries in 3d shapes. ACM Transactions on Graphics 25(2), 439–464 (2006)CrossRefGoogle Scholar
  28. [MTS+05]
    Mikolajczyk, K., Tuytelaars, T., Schmid, C., Zisserman, A., Matas, J., Schaffalitzky, F., Kadir, T., Van Gool, L.: A comparison of affine region detectors. Int. J. Comput. Vision 65(1-2), 43–72 (2005)CrossRefGoogle Scholar
  29. [NZ06]
    Nilsback, M.-E., Zisserman, A.: A visual vocabulary for flower classification. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2006) (to appear)Google Scholar
  30. [OPM02]
    Ojala, T., Pietikäinen, M., Mäenpää, T.: Multiresolution gray-scale and rotation invariant texture classification with local binary patterns. IEEE Trans. Pattern Anal. Mach. Intell. 24(7), 971–987 (2002)CrossRefGoogle Scholar
  31. [PLC+08]
    Park, M., Lee, S., Chen, P.-C., Kashyap, S., Butt, A.A., Liu, Y.: Performance evaluation of state-of-the-art discrete symmetry detection algorithms. In: Computer Vision and Pattern Recognition Conference, CVPR 2008 (June 2008)Google Scholar
  32. [PY04]
    Prasad, V.S.N., Yegnanarayana, B.: Finding axes of symmetry from potential fields. IEEE Transactions on Image Processing 13(12), 1559–1566 (2004)CrossRefMathSciNetGoogle Scholar
  33. [RWY95]
    Reisfeld, D., Wolfson, H., Yeshurun, Y.: Context free attentional operators: the generalized symmetry transform. International Journal of Computer Vision, 119–130 (1995)Google Scholar
  34. [SIT01]
    Shen, D., Ip, H., Teoh, E.K.: Robust detection of skewed symmetries by combining local and semi-local affine invariants. Pattern Recognition 34(7), 1417–1428 (2001)CrossRefzbMATHGoogle Scholar
  35. [SLH91]
    Scott, G.L., Longuet Higgins, H.C.: An algorithm for associating the features of two images. Royal Society London 244, 21–26 (1991)CrossRefGoogle Scholar
  36. [SM97]
    Schmid, C., Mohr, R.: Local grayvalue invariants for image retrieval. IEEE Transactions on Pattern Analysis and Machine Intelligence 19(5), 530–535 (1997)CrossRefGoogle Scholar
  37. [SM00]
    Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 22(8), 888–905 (2000)CrossRefGoogle Scholar
  38. [SNP05]
    Shiv Naga Prasad, L.S., Davis, V.: Detecting rotational symmetries. In: Tenth IEEE International Conference on Computer Vision, ICCV 2005, vol. 2, pp. 954–961 (2005)Google Scholar
  39. [TT05]
    Tang, F., Tao, H.: Object tracking with dynamic feature graph. In: PETS 2005, pp. 25–32 (2005)Google Scholar
  40. [TV98]
    Trucco, E., Verri, A.: Introductory Techniques for 3-D Computer Vision, pp. 333–334. Prentice-Hall, New Jersey (1998)Google Scholar
  41. [VP06]
    Valstar, M., Pantic, M.: Fully automatic facial action unit detection and temporal analysis. In: IEEE Int’l Conf. on Computer Vision and Pattern Recognition 2006, vol. 3 (May 2006)Google Scholar
  42. [Wey52]
    Weyl, H.: Symmetry. Princeton University Press (1952)Google Scholar
  43. [ZMLS07]
    Zhang, J., Marszalek, M., Lazebnik, S., Schmid, C.: Local features and kernels for classification of texture and object categories: A comprehensive study. International Journal of Computer Vision 73(2), 213–238 (2007)CrossRefGoogle Scholar
  44. [ZPA95]
    Zabrodsky, H., Peleg, S., Avnir, D.: Symmetry as a continuous feature. IEEE Transactions on Pattern Analysis and Machine Intelligence 17(12), 1154–1166 (1995)CrossRefGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Bar-Ilan UniversityRamat-GanIsrael

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