Multidimensional Size Functions for Shape Comparison

  • S. Biasotti
  • A. Cerri
  • P. Frosini
  • D. Giorgi
  • C. Landi
Article

Abstract

Size Theory has proven to be a useful framework for shape analysis in the context of pattern recognition. Its main tool is a shape descriptor called size function. Size Theory has been mostly developed in the 1-dimensional setting, meaning that shapes are studied with respect to functions, defined on the studied objects, with values in ℝ. The potentialities of the k-dimensional setting, that is using functions with values in ℝk, were not explored until now for lack of an efficient computational approach. In this paper we provide the theoretical results leading to a concise and complete shape descriptor also in the multidimensional case. This is possible because we prove that in Size Theory the comparison of multidimensional size functions can be reduced to the 1-dimensional case by a suitable change of variables. Indeed, a foliation in half-planes can be given, such that the restriction of a multidimensional size function to each of these half-planes turns out to be a classical size function in two scalar variables. This leads to the definition of a new distance between multidimensional size functions, and to the proof of their stability with respect to that distance. Experiments are carried out to show the feasibility of the method.

Keywords

Multidimensional size function Multidimensional measuring function Natural pseudo-distance 

References

  1. 1.
    Ackermann, W.: Zum Hilbertschen Aufbau der reellen Zahlen. Math. Ann. 99, 118–133 (1928) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Allili, M., Corriveau, D., Ziou, D.: Morse homology descriptor for shape characterization. ICPR’04 4, 27–30 (2004) Google Scholar
  3. 3.
    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. (2008, to appear) Google Scholar
  4. 4.
    Blum, H.: A transformation for extracting new descriptors of shape. In: Wathen-Dunn, W. (ed.) Models for the Perception of Speech and Visual Form, Boston, November 1964, pp. 362–380. The MIT Press, Cambridge (1967) Google Scholar
  5. 5.
    Cagliari, F., Ferri, M., Pozzi, P.: Size functions from a categorical viewpoint. Acta Appl. Math. 67, 225–235 (2001) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Carlsson, G., Zomorodian, A.: The theory of multidimensional persistence. In: Proc. of SCG’07, Gyeongju, South Korea, 6–8 June 2007 Google Scholar
  7. 7.
    Cerri, A., Ferri, M., Giorgi, D.: Retrieval of trademark images by means of size functions. Graph. Models 68, 451–471 (2006) CrossRefGoogle Scholar
  8. 8.
    Cerri, A., Frosini, P., Landi, C.: A global reduction method for multidimensional size graphs. Electron. Not. Discrete Math. 26, 21–28 (2006) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Cerri, A., Giorgi, D., Muse, P., Sur, F., Tomassini, F.: Shape recognition via an a contrario model for size functions. In: Springer Lecture Notes in Computer Science, vol. 4141, pp. 410–421. Springer, Berlin (2006) Google Scholar
  10. 10.
    Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. In: Proc. 21st Symp. Comput. Geom., pp. 263–271 (2005) Google Scholar
  11. 11.
    Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete Comput. Geom. 37, 103–120 (2007) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Cohen-Steiner, D., Edelsbrunner, H., Morozov, D.: Vines and vineyards by updating persistence in linear time. In: SCG’06, Sedona, Arizona, USA, 5–7 June 2006 Google Scholar
  13. 13.
    d’Amico, M.: A new optimal algorithm for computing size function of shapes. In: Proc. CVPRIP Algorithms III, International Conference on Computer Vision, Pattern Recognition and Image Processing, pp. 107–110 (2000) Google Scholar
  14. 14.
    d’Amico, M., Frosini, P., Landi, C.: Natural pseudo-distance and optimal matching between reduced size functions. Tech. Rep. 66, DISMI, Univ. degli Studi di Modena e Reggio Emilia, Italy (2005), see also arXiv:0804.3500v1
  15. 15.
    d’Amico, M., Frosini, P., Landi, C.: Using matching distance in size theory: a survey. Int. J. Imaging Syst. Technol. 16(5), 154–161 (2006) CrossRefGoogle Scholar
  16. 16.
    Dibos, F., Frosini, P., Pasquignon, D.: The use of size functions for comparison of shapes through differential invariants. J. Math. Imaging Vis. 21(2), 107–118 (2004) CrossRefMathSciNetGoogle Scholar
  17. 17.
    Donatini, P., Frosini, P.: Natural pseudodistances between closed manifolds. Forum Math. 16(5), 695–715 (2004) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Donatini, P., Frosini, P.: Lower bounds for natural pseudodistances via size functions. Arch. Inequal. Appl. 2(1), 1–12 (2004) MATHMathSciNetGoogle Scholar
  19. 19.
    Donatini, P., Frosini, P.: Natural pseudodistances between closed surfaces. J. Eur. Math. Society 9, 231–253 (2007) MathSciNetGoogle Scholar
  20. 20.
    Donatini, P., Frosini, P.: Natural pseudodistances between closed curves, Forum Math. (2008, to appear) Google Scholar
  21. 21.
    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
  22. 22.
    Eckhardt, U., Latecki, L.J.: Topologies for the digital spaces ℤ2 and ℤ3. Comput. Vis. Image Underst. 90, 295–312 (2003) MATHCrossRefGoogle Scholar
  23. 23.
    Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. In: Proc. 41st Ann. IEEE Symp. Found. Comput. Sci., pp. 454–463 (2000) Google Scholar
  24. 24.
    Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28, 511–533 (2002) MATHMathSciNetGoogle Scholar
  25. 25.
    Freeman, H., Morse, S.P.: On searching a contour map for a given terrain profile. J. Frankl. Inst. 248, 1–25 (1967) CrossRefGoogle Scholar
  26. 26.
    Frosini, P.: A distance for similarity classes of submanifolds of a Euclidean space. Bull. Aust. Math. Soc. 42(3), 407–416 (1990) MATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Frosini, P.: Measuring shapes by size functions. In: Intelligent Robots and Computer Vision X: Algorithms and Techniques, Boston, MA. Proc. of SPIE, vol. 1607, pp. 122–133 (1991) Google Scholar
  28. 28.
    Frosini, P.: Connections between size functions and critical points. Math. Methods Appl. Sci. 19, 555–569 (1996) MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Frosini, P., Landi, C.: Size theory as a topological tool for computer vision. Pattern Recognit. Image Anal. 9, 596–603 (1999) Google Scholar
  30. 30.
    Frosini, P., Landi, C.: Size functions and formal series. Appl. Algebra Eng. Commun. Comput. 12, 327–349 (2001) MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Frosini, P., Mulazzani, M.: Size homotopy groups for computation of natural size distances. Bull. Belg. Math. Soc. 6, 455–464 (1999) MATHMathSciNetGoogle Scholar
  32. 32.
    Frosini, P., Pittore, M.: New methods for reducing size graphs. Int. J. Comput. Math. 70, 505–517 (1999) MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Handouyaya, M., Ziou, D., Wang, S.: Sign language recognition using moment-based size functions. In: Proc. of Vision Interface 99, Trois-Rivières, Canada, 19–21 May 1999, pp. 210–216 (1999) Google Scholar
  34. 34.
    Kaczynski, T., Mischaikow, K., Mrozek, M.: Computational Homology, Applied Mathematical Sciences, vol. 157. Springer, New York (2004) Google Scholar
  35. 35.
    Landi, C., Frosini, P.: 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
  36. 36.
    Landi, C., Frosini, P.: Size functions as complete invariants for image recognition, In: Latecki, L.J., Mount, D.M., Wu, A.Y. (eds.) Vision Geometry XI. Proc. SPIE, vol. 4794, pp. 101–109 (2002) Google Scholar
  37. 37.
    Motzkin, T.: Sur quelques propriétés caractéristiques des ensembles convexes. Atti R. Accad. Naz. Lince. Ser. Rend. Classe Sci. Fis. Mat. Nat. 21, 562–567 (1935) MATHGoogle Scholar
  38. 38.
    Motzkin, T.: Sur quelques propriétés caractéristiques des ensembles bornés non convexes. Atti R. Accad. Naz. Linc. Ser. Rend. Classe Sci. Fis. Mat. Nat. 21, 773–779 (1935) MATHGoogle Scholar
  39. 39.
    Patanè, G., Spagnuolo, M., Falcidieno, B.: Families of cut-graphs for bordered meshes with arbitrary genus. Graph. Models 692, 119–138 (2007) CrossRefGoogle Scholar
  40. 40.
    Stanganelli, I., Brucale, A., Calori, L., Gori, R., Lovato, A., Magi, S., Kopf, B., Bacchilega, R., Rapisarda, V., Testori, A., Ascierto, P.A., Simeone, E., Ferri, M.: Computer-aided diagnosis of melanocytic lesions. Anticancer Res. 25, 4577–4582 (2005) Google Scholar
  41. 41.
    Uras, C., Verri, A.: Computing size functions from edge maps. Internat. J. Comput. Vis. 23(2), 169–183 (1997) CrossRefGoogle Scholar
  42. 42.
    Verri, A., Uras, C.: Invariant size functions. In: Applications of Invariance in Computer Vision. Lecture Notes in Comput. Sci., vol. 825, pp. 215–234. Springer, Berlin (1993) Google Scholar
  43. 43.
    Verri, A., Uras, C.: Metric-topological approach to shape representation and recognition. Image Vis. Comput. 14, 189–207 (1996) CrossRefGoogle Scholar
  44. 44.
    Verri, A., Uras, C., Frosini, P., Ferri, M.: On the use of size functions for shape analysis. Biol. Cybern. 70, 99–107 (1993) MATHCrossRefGoogle Scholar
  45. 45.
    Willard, S.: General Topology. Addison-Wesley, Reading (1970) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • S. Biasotti
    • 1
  • A. Cerri
    • 2
    • 3
  • P. Frosini
    • 2
    • 3
  • D. Giorgi
    • 1
  • C. Landi
    • 4
  1. 1.IMATIConsiglio Nazionale delle RicercheGenovaItalia
  2. 2.ARCESUniversità di BolognaBolognaItalia
  3. 3.Dipartimento di MatematicaUniversità di BolognaBolognaItalia
  4. 4.Dipartimento di Scienze e Metodi dell’IngegneriaUniversità di Modena e Reggio EmiliaReggio EmiliaItalia

Personalised recommendations