Gaussian Bayes Classifier for 2D Shapes in Kendall Space

  • Hibat Allah RouahiEmail author
  • Riadh MtibaaEmail author
  • Ezzeddine ZagroubaEmail author
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 684)


We propose a 2D-shape Gaussian Bayes classifier based upon Kendall’s representations that help to quotient out the effects of non-altering shape geometric transformations. The Kendall space is a non linear space that coincides with the unit sphere modulo an isometry group. The proposed Riemannian metric is more apt in the case where shapes are different only in translation, scale and rotation. In addition to that, the manifold structure of this space renders the multivariate statistical analysis implementation unfeasible in practice. Consequently, tools such as learning and classification models are non trivial and not frequently available. To overcome these issues, we adapt the Gaussian Bayes classifier to this space. We computed the likelihood parameters through appropriate projections onto Kendall tangent space that provides a good linear approximation. In order to validate the robustness of our classifier, we proceeded to computer simulations using several benchmarks.


Kendall space Gaussian Bayes classifier Likelihood parameters 


  1. 1.
    Srivastava, A., Liu, X., Mio, W., Klassen, E.: A computational geometric approach to shape analysis in images. In: Advances in Neural Information Processing Systems, pp. 1579–1586 (2003)Google Scholar
  2. 2.
    Siddiqi, K., Pizer, S. (eds.): Medial Representations: Mathematics, Algorithms and Applications, vol. 37. Springer Science & Business Media, Netherlands (2008)zbMATHGoogle Scholar
  3. 3.
    Cootes, T.F., Taylor, C.J., Cooper, D.H., Graham, J.: Active shape models-their training and application. Comput. Vis. Image Underst. 61(1), 38–59 (1995)CrossRefGoogle Scholar
  4. 4.
    Subrahmonia, J., Cooper, D.B., Keren, D.: Practical reliable Bayesian recognition of 2D and 3D objects using implicit polynomials and algebraic invariants. IEEE Trans. Pattern Anal. Mach. Intell. 18(5), 505–519 (1996)CrossRefGoogle Scholar
  5. 5.
    Zhang, J., Zhang, X., Krim, H., Walter, G.G.: Object representation and recognition in shape spaces. Pattern Recogn. 36(5), 1143–1154 (2003)CrossRefGoogle Scholar
  6. 6.
    Kendall, D.G., Barden, D., Carne, T.K., Le, H.: Shape and Shape Theory, vol. 500. Wiley, Chichester (1999)CrossRefzbMATHGoogle Scholar
  7. 7.
    Kendall, D.G.: Shape manifolds, procrustean metrics and complex projective spaces. Bull. Lond. Math. Soc. 16(2), 81–121 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bookstein, F.L.: Landmark methods for forms without landmarks: morphometrics of group differences in outline shape. Med. Image Anal. 1(3), 225–243 (1997)CrossRefGoogle Scholar
  9. 9.
    Dryden, I.L., Mardia, K.V.: Statistical Shape Analysis, vol. 4. Wiley, Chichester (1998)zbMATHGoogle Scholar
  10. 10.
    Amor, B.B., Su, J., Srivastava, A.: Action recognition using rate-invariant analysis of skeletal shape trajectories. IEEE Trans. Pattern Anal. Mach. Intell. 38(1), 1–13 (2016)CrossRefGoogle Scholar
  11. 11.
    Giebel, S.M., Schiltz, J., Schenk, J.P.: Statistical shape analysis for the classification of renal tumors affecting children. Pak. J. Statist. 29(1), 129–138 (2013)MathSciNetGoogle Scholar
  12. 12.
    Jayasumana, S., Salzmann, M., Li, H., Harandi, M.: A framework for shape analysis via Hilbert space embedding. In: Proceedings of the IEEE International Conference on Computer Vision, pp. 1249–1256 (2013)Google Scholar
  13. 13.
    Vinu, G., Sim, A., Alemany, S.: The k-means algorithm for 3D shapes with an application to apparel design. Adv. Data Anal. Classif. 10(1), 103–132 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Soderkvist, O.: Computer vision classification of leaves from swedish trees. Masters thesis, Linkoping University (2001)Google Scholar
  15. 15.
    Dryden, I.L.: Shapes package. R Foundation for Statistical Computing, Vienna, Austria, Contributed package (2015).

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Laboratoire LIMTIC, Institut Supérieur d’InformatiqueUniversité de Tunis El ManarArianaTunisie
  2. 2.Institut Supérieur des sciences appliquées et de technologie de Sousse (ISSAT)Université de SousseSousseTunisie

Personalised recommendations