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Multidimensional Size Functions for Shape Comparison

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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.

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References

  1. Ackermann, W.: Zum Hilbertschen Aufbau der reellen Zahlen. Math. Ann. 99, 118–133 (1928)

    Article  MATH  MathSciNet  Google Scholar 

  2. Allili, M., Corriveau, D., Ziou, D.: Morse homology descriptor for shape characterization. ICPR’04 4, 27–30 (2004)

    Google Scholar 

  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)

  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. Cagliari, F., Ferri, M., Pozzi, P.: Size functions from a categorical viewpoint. Acta Appl. Math. 67, 225–235 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Carlsson, G., Zomorodian, A.: The theory of multidimensional persistence. In: Proc. of SCG’07, Gyeongju, South Korea, 6–8 June 2007

  7. Cerri, A., Ferri, M., Giorgi, D.: Retrieval of trademark images by means of size functions. Graph. Models 68, 451–471 (2006)

    Article  Google Scholar 

  8. Cerri, A., Frosini, P., Landi, C.: A global reduction method for multidimensional size graphs. Electron. Not. Discrete Math. 26, 21–28 (2006)

    Article  MathSciNet  Google Scholar 

  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. Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. In: Proc. 21st Symp. Comput. Geom., pp. 263–271 (2005)

  11. Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete Comput. Geom. 37, 103–120 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  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

  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)

  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. 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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  17. Donatini, P., Frosini, P.: Natural pseudodistances between closed manifolds. Forum Math. 16(5), 695–715 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. Donatini, P., Frosini, P.: Lower bounds for natural pseudodistances via size functions. Arch. Inequal. Appl. 2(1), 1–12 (2004)

    MATH  MathSciNet  Google Scholar 

  19. Donatini, P., Frosini, P.: Natural pseudodistances between closed surfaces. J. Eur. Math. Society 9, 231–253 (2007)

    MathSciNet  Google Scholar 

  20. Donatini, P., Frosini, P.: Natural pseudodistances between closed curves, Forum Math. (2008, to appear)

  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. Eckhardt, U., Latecki, L.J.: Topologies for the digital spaces ℤ2 and ℤ3. Comput. Vis. Image Underst. 90, 295–312 (2003)

    Article  MATH  Google Scholar 

  23. Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. In: Proc. 41st Ann. IEEE Symp. Found. Comput. Sci., pp. 454–463 (2000)

  24. Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28, 511–533 (2002)

    MATH  MathSciNet  Google Scholar 

  25. Freeman, H., Morse, S.P.: On searching a contour map for a given terrain profile. J. Frankl. Inst. 248, 1–25 (1967)

    Article  Google Scholar 

  26. Frosini, P.: A distance for similarity classes of submanifolds of a Euclidean space. Bull. Aust. Math. Soc. 42(3), 407–416 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  28. Frosini, P.: Connections between size functions and critical points. Math. Methods Appl. Sci. 19, 555–569 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  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. Frosini, P., Landi, C.: Size functions and formal series. Appl. Algebra Eng. Commun. Comput. 12, 327–349 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  31. Frosini, P., Mulazzani, M.: Size homotopy groups for computation of natural size distances. Bull. Belg. Math. Soc. 6, 455–464 (1999)

    MATH  MathSciNet  Google Scholar 

  32. Frosini, P., Pittore, M.: New methods for reducing size graphs. Int. J. Comput. Math. 70, 505–517 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  34. Kaczynski, T., Mischaikow, K., Mrozek, M.: Computational Homology, Applied Mathematical Sciences, vol. 157. Springer, New York (2004)

    Google Scholar 

  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)

  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)

  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)

    MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  39. Patanè, G., Spagnuolo, M., Falcidieno, B.: Families of cut-graphs for bordered meshes with arbitrary genus. Graph. Models 692, 119–138 (2007)

    Article  Google Scholar 

  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. Uras, C., Verri, A.: Computing size functions from edge maps. Internat. J. Comput. Vis. 23(2), 169–183 (1997)

    Article  Google Scholar 

  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. Verri, A., Uras, C.: Metric-topological approach to shape representation and recognition. Image Vis. Comput. 14, 189–207 (1996)

    Article  Google Scholar 

  44. Verri, A., Uras, C., Frosini, P., Ferri, M.: On the use of size functions for shape analysis. Biol. Cybern. 70, 99–107 (1993)

    Article  MATH  Google Scholar 

  45. Willard, S.: General Topology. Addison-Wesley, Reading (1970)

    MATH  Google Scholar 

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Authors and Affiliations

  1. IMATI, Consiglio Nazionale delle Ricerche, Via De Marini 6, 16149, Genova, Italia

    S. Biasotti & D. Giorgi

  2. ARCES, Università di Bologna, via Toffano 2/2, 40135, Bologna, Italia

    A. Cerri & P. Frosini

  3. Dipartimento di Matematica, Università di Bologna, P.zza di Porta S. Donato 5, 40126, Bologna, Italia

    A. Cerri & P. Frosini

  4. Dipartimento di Scienze e Metodi dell’Ingegneria, Università di Modena e Reggio Emilia, via Amendola 2, Pad. Morselli, 42100, Reggio Emilia, Italia

    C. Landi

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  1. S. Biasotti
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  2. A. Cerri
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  3. P. Frosini
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  4. D. Giorgi
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  5. C. Landi
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Correspondence to P. Frosini.

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Biasotti, S., Cerri, A., Frosini, P. et al. Multidimensional Size Functions for Shape Comparison. J Math Imaging Vis 32, 161–179 (2008). https://doi.org/10.1007/s10851-008-0096-z

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