Journal of Mathematical Imaging and Vision

, Volume 52, Issue 2, pp 218–233 | Cite as

Generalized Shapes and Point Sets Correspondence and Registration

  • Bogdan M. MarisEmail author
  • Paolo Fiorini


The theory of shapes, as proposed by David Kendall, is concerned with sets of labeled points in the Euclidean space \(\mathbb {R}^d\) that define a shape regardless of translations, rotations and dilatations. We present here a method that extends the theory of shapes, where, in this case, we use the term generalized shape for structures of unlabeled points. By using the distribution of distances between the points in a set we are able to define the existence of generalized shapes and to infer the computation of the correspondences and the orthogonal transformation between two elements of the same generalized shape equivalence class. This study is oriented to solve the registration of large set of landmarks or point sets derived from medical images but may be employed in other fields such as computer vision or biological morphometry.


Generalized shapes Point sets Correspondence  Registration Procrustes analysis 


  1. 1.
    Arun, K.S., Huang, T.S., Blostein, S.D.: Least-squares fitting of two 3-d point sets. IEEE Trans. Pattern Anal. Mach. Intell. 9(5), 698–700 (1987)CrossRefGoogle Scholar
  2. 2.
    Bandeira, A.S., Charikar, M., Singer, A., Zhu, A.: Multireference alignment using semidefinite programming. In: ITCS, pp. 459–470 (2014).Google Scholar
  3. 3.
    Belongie, S., Malik, J., Puzicha, J.: Shape context: A new descriptor for shape matching and object recognition. In. In NIPS, vol. 54, pp. 831–837. Citeseer, Citeseer (2000).Google Scholar
  4. 4.
    Besl, P.J., McKay, N.D.: A method for registration of 3-D shapes. IEEE Trans Pattern Anal Mach Intell 14(2), 239–256 (1992)CrossRefGoogle Scholar
  5. 5.
    Bloom, G.S.: A counterexample to a theorem of s. piccard. J. Comb. Theory, Ser. A 22(3), 378–379 (1977)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bookstein, F.L.: Principal warps: thin-plate splines and the decomposition of deformations. IEEE Trans Pattern Anal Mach Intell 11(6), 567–585 (1989)CrossRefzbMATHGoogle Scholar
  7. 7.
    Boutin, M., Kemper, G.: On reconstructing n-point configurations from the distribution of distances or areas. arxiv:0304192 (2003).
  8. 8.
    Boutin, M., Kemper, G.: Which point configurations are determined by the distribution of their pairwise distances? Int. J. Comput. Geometry Appl. 17(1), 31–44 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Boyer, D.M., Lipman, Y., St Clair, E., Puente, J., Patel, B.A., Funkhouser, T., Jernvall, J., Daubechies, I.: Algorithms to automatically quantify the geometric similarity of anatomical surfaces. Proceedings of the National Academy of Sciences of the United States of America 108(45), 18,221–18,226 (2011).Google Scholar
  10. 10.
    Bronstein, A., Bronstein, M., Kimmel, R.: Numer Geom Non-Rigid Shapes, 1st edn. Springer Publishing Company, Incorporated (2008)Google Scholar
  11. 11.
    Castillo, R., Castillo, E., Guerra, R., Johnson, V.E., McPhail, T., Garg, A.K., Guerrero, T.: A framework for evaluation of deformable image registration spatial accuracy using large landmark point sets. Phys Med Biol 54(7), 1849 (2009)CrossRefGoogle Scholar
  12. 12.
    Chui, H.: Non-rigid point matching: Algorithms, extensions and applications. Ph.D. thesis, Yale University (2001).Google Scholar
  13. 13.
    Chui, H., Rangarajan, A.: A new point matching Algorithm for non-rigid registration. Comput Vis Image Underst 89(2–3), 114–141 (2003)Google Scholar
  14. 14.
    Ghosh, D., Sharf, A., Amenta, N.: Feature-driven deformation for dense correspondence. Medical Imaging: Visualization, Image-Guided Procedures, and Modeling (Proc. SPIE) 7261 (2009).Google Scholar
  15. 15.
    Huang, Q.X., Guibas, L.J.: Consistent shape maps via semidefinite programming. Comput. Graph. Forum 32(5), 177–186 (2013)CrossRefGoogle Scholar
  16. 16.
    Jian, B., Vemuri, B.C.: Robust Point Set Registration Using Gaussian Mixture Models (2011).Google Scholar
  17. 17.
    Kendall, D.G.: Shape manifolds, procrustean metrics, and complex projective spaces. Bull Lond Math Soc 16(2), 81–121 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Kendall, D.G., Barden, D., Carne, T.K., Le, H.: Shape and Shape Theory. Wiley, New York (1999)CrossRefzbMATHGoogle Scholar
  19. 19.
    Kim, V.G., Li, W., Mitra, N.J., DiVerdi, S., Funkhouser, T.: Exploring collections of 3D models using fuzzy correspondences . ACM Trans. Graph. 31(4), 1–11 (2012).Google Scholar
  20. 20.
    Lemke, P., Skiena, S., Smith, W.: Reconstructing sets from interpoint distances . Discrete Comput. Geom. 25, 597–631 (2003).Google Scholar
  21. 21.
    Leordeanu, M., Hebert, M.: A spectral technique for correspondence problems using pairwise constraints. In: ICCV, pp. 1482–1489 (2005).Google Scholar
  22. 22.
    Lipman, Y., Funkhouser, T.: Mobius voting for surface correspondence. ACM Transactions on Graphics (Proc. SIGGRAPH) 28(3) (2009).Google Scholar
  23. 23.
    Mémoli, F., Sapiro, G.: A theoretical and computational framework for isometry invariant recognition of point cloud data. Found Comput Math 5(3), 313–347 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Myronenko, A., Song, X.S.X.: Point Set Registration: Coherent Point Drift. IEEE Trans Pattern Anal Mach Intell 32(12), 2262–2275 (2010)CrossRefGoogle Scholar
  25. 25.
    Osada, R., Funkhouser, T.A., Chazelle, B., Dobkin, D.P.: Shape distributions. ACM Trans. Graph. 21(4), 807–832 (2002)CrossRefGoogle Scholar
  26. 26.
    Pachauri, D., Kondor, R., Singh, V.: Solving the multi-way matching problem by permutation synchronization. In: NIPS, pp. 1860–1868 (2013).Google Scholar
  27. 27.
    Rangarajan, A., Chui, H., Bookstein, F.L.: The Softassign Procrustes Matching Algorithm. Inf Process Med Imaging 1230, 29–42 (1997)CrossRefGoogle Scholar
  28. 28.
    Schönemann, P.H.: A generalized solution of the orthogonal procrustes problem. Psychometrika 31(1), 1–10 (1966)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Scott, G.L., Longuet-Higgins, H.C.: An Algorithm for associating the features of two images. Proc Royal Soc B 244(1309), 21–26 (1991)CrossRefGoogle Scholar
  30. 30.
    Shapiro, L.S., Brady, J.M.: Feature-based correspondence: an eigenvector approach. Image Vis Comput 10(5), 283–288 (1992)CrossRefGoogle Scholar
  31. 31.
    Wahba, G.: Spline Models for Observational Data, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 59. SIAM (1990).Google Scholar
  32. 32.
    Wang, L., Singer, A.: Exact and stable recovery of rotations for robust synchronization. CoRR abs/1211.2441 (2012).Google Scholar
  33. 33.
    Yaniv, Z., Cleary, K.: Image-Guided Procedures: A Review (2006).Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of VeronaVeronaItaly

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