Acta Applicandae Mathematicae

, Volume 109, Issue 2, pp 527–554 | Cite as

Natural Pseudo-Distance and Optimal Matching between Reduced Size Functions

  • Michele d’Amico
  • Patrizio Frosini
  • Claudia Landi


This paper studies the properties of a new lower bound for the natural pseudo-distance. The natural pseudo-distance is a dissimilarity measure between shapes, where a shape is viewed as a topological space endowed with a real-valued continuous function. Measuring dissimilarity amounts to minimizing the change in the functions due to the application of homeomorphisms between topological spaces, with respect to the L -norm. In order to obtain the lower bound, a suitable metric between size functions, called matching distance, is introduced. It compares size functions by solving an optimal matching problem between countable point sets. The matching distance is shown to be resistant to perturbations, implying that it is always smaller than the natural pseudo-distance. We also prove that the lower bound so obtained is sharp and cannot be improved by any other distance between size functions.


Shape comparison Shape representation Reduced size function Natural pseudo-distance 

Mathematics Subject Classification (2000)

68T10 58C05 49Q10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Biasotti, S., De Floriani, L., Falcidieno, B., Frosini, P., Giorgi, D., Landi, C., Papaleo, L., Spagnuolo, M.: Describing shapes by geometrical-topological properties of real functions. ACM Comput. Surv. 40(4) (2008, in press) Google Scholar
  2. 2.
    Brucale, A., d’Amico, M., Ferri, M., Gualandri, L., Lovato, A.: Size functions for image retrieval: a demonstrator on randomly generated curves. In: Lew, M., Sebe, N., Eakins, J. (eds.) Proc. CIVR02, London. Lecture Notes in Computer Science, vol. 2383, pp. 235–244. Springer, Berlin (2002) Google Scholar
  3. 3.
    Cagliari, F., Fabio, B.D., Ferri, M.: One-dimensional reduction of multidimensional persistent homology (2007). arXiv:math/0702713v1
  4. 4.
    Cerri, A., Ferri, M., Giorgi, D.: Retrieval of trademark images by means of size functions. Graph. Models 68, 451–471 (2006) CrossRefGoogle Scholar
  5. 5.
    Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete Comput. Geom. 37, 103–120 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    d’Amico, M.: A new optimal algorithm for computing size function of shapes. In: CVPRIP Algorithms III, Proceedings International Conference on Computer Vision, Pattern Recognition and Image Processing, pp. 107–110 (2000) Google Scholar
  7. 7.
    Dibos, F., Frosini, P., Pasquignon, D.: The use of size functions for comparison of shapes through differential invariants. J. Math. Imaging Vis. 21, 107–118 (2004) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Donatini, P., Frosini, P.: Lower bounds for natural pseudodistances via size functions. Arch. Inequal. Appl. 2, 1–12 (2004) zbMATHMathSciNetGoogle Scholar
  9. 9.
    Donatini, P., Frosini, P.: Natural pseudodistances between closed manifolds. Forum Math. 16, 695–715 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Donatini, P., Frosini, P.: Natural pseudodistances between closed surfaces. J. Eur. Math. Soc. 9, 231–253 (2007) MathSciNetGoogle Scholar
  11. 11.
    Donatini, P., Frosini, P.: Natural pseudodistances between closed curves. Forum Math. (to appear) Google Scholar
  12. 12.
    Donatini, P., Frosini, P., Landi, C.: Deformation energy for size functions. In: Hancock, E.R., Pelillo, M. (eds.) Proceedings Second International Workshop EMMCVPR’99. Lecture Notes in Computer Science, vol. 1654, pp. 44–53. Springer, Berlin (1999) Google Scholar
  13. 13.
    Efrat, A., Itai, A., Katz, M.: Geometry helps in bottleneck matching and related problems. Algorithmica 31, 1–28 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Frosini, P.: Connections between size functions and critical points. Math. Meth. Appl. Sci. 19, 555–596 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Frosini, P., Landi, C.: New pseudodistances for the size function space. In: Melter, R.A., Wu, A.Y., Latecki, L.J. (eds.) Vision Geometry VI. Proc. SPIE, vol. 3168, pp. 52–60 (1997) Google Scholar
  16. 16.
    Frosini, P., Landi, C.: Size functions and morphological transformations. Acta Appl. Math. 49, 85–104 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Frosini, P., Landi, C.: Size theory as a topological tool for computer vision. Pattern Recogn. Image Anal. 9, 596–603 (1999) Google Scholar
  18. 18.
    Frosini, P., Landi, C.: Size functions and formal series. Appl. Algebra Eng. Commun. Comput. 12, 327–349 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Frosini, P., Landi, C.: Reparametrization invariant norms. Trans. Am. Math. Soc. 361, 407–452 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Frosini, P., Mulazzani, M.: Size homotopy groups for computation of natural size distances. Bull. Belg. Math. Soc. 6, 455–464 (1999) zbMATHMathSciNetGoogle Scholar
  21. 21.
    Frosini, P., Pittore, M.: New methods for reducing size graphs. Int. J. Comput. Math. 70, 505–517 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Garfinkel, R., Rao, M.: The bottleneck transportation problem. Nav. Res. Logist. Q. 18, 465–472 (1971) CrossRefMathSciNetGoogle Scholar
  23. 23.
    Handouyaya, M., Ziou, D., Wang, S.: Sign language recognition using moment-based size functions. In: Vision Interface 99, Trois-Rivières (1999) Google Scholar
  24. 24.
    Hirsch, M.W.: Differential Topology. Springer, Berlin (1976) zbMATHGoogle Scholar
  25. 25.
    Kuratowski, K., Mostowski, A.: Set Theory. North-Holland, Amsterdam (1968) zbMATHGoogle Scholar
  26. 26.
    Latecki, L.J., Melter, R., Gross, A. (eds.): Special issue: shape representation and similarity for image databases. Pattern Recogn. 35, 1–297 (2002) Google Scholar
  27. 27.
    Veltkamp, R., Hagedoorn, M.: State-of-the-art in shape matching. In: Lew, M. (ed.) Principles of Visual Information Retrieval, pp. 87–119. Springer, Berlin (2001) Google Scholar
  28. 28.
    Willard, S.: General Topology. Addison-Wesley, Reading (1970) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  • Michele d’Amico
    • 1
  • Patrizio Frosini
    • 1
    • 2
  • Claudia Landi
    • 3
  1. 1.ARCESUniversità di BolognaBolognaItaly
  2. 2.Dipartimento di MatematicaUniversità di BolognaBolognaItaly
  3. 3.DISMIUniversità di Modena e Reggio EmiliaReggio EmiliaItaly

Personalised recommendations