Abstract
A new categorical approach to size functions is given. Using this point of view, it is shown that size functions of a Morse map, f: M→ℜ can be computed through the 0-dimensional homology. This result is extended to the homology of arbitrary degree in order to obtain new invariants of the shape of the graph of the given map.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adamek, J., Herrlich, H. and Strecker, G.: Abstract and Concrete Categories, Wiley Interscience, New York, 1990.
Borceux, F.: Handbook of Categorical Algebra, Encyclop. of Math. 51, Cambridge Univ. Press.
Collina, C., Ferri, M., Frosini, P. and Porcellini, E.: SketchUp: Towards qualitative shape data management, Proc. ACCV'98, Hong Kong, 8-10 Jan. 1998, vol. 1, Lecture Notes in Comput. Sci. 1351, Springer, New York, 1998, pp. 338-343.
d'Amico, M.: ? ?-reduction of size graphs as a new algorithm for computing size functions of shapes, In: Proc. Internat. Conf. on Computer Vision, Pattern Recognition and Image Processing, Atlantic City, Feb. 27-Mar. 3, 2000, vol. 2, pp. 107-110.
Ferri, M., Frosini, P., Lovato, A. and Zambelli, C.: Point selection: A new comparison scheme for size functions (With an application to monogram recognition), Proc. ACCV'98, Hong Kong, 8-10 Jan. 1998, vol. 1, Lecture Notes in Comput. Sci. 1351, Springer, New York, 1998, pp. 329-337.
Ferri, M., Gallina, S., Porcellini, E. and Serena, M.: On-line character and writer recognition by size functions and fuzzy logic, In: Proc. ACCV '95, Dec. 5-8, Singapore, vol. 3, 1995, pp. 622-626.
Ferri, M., Lombardini, S. and Pallotti, C.: Leukocyte classification by size functions, In: Proc. 2nd IEEE Workshop on Applications of Computer Vision, Sarasota, 1994 Dec. 5-7, 1994, pp. 223-229.
Frosini, P.: Discrete computation of size functions, J. Combin. Inform. System Sci. 17 (1992), 232-250.
Frosini, P.: Connections between size functions and critical points, Math. Meth. Appl. Sci. 19 (1996), 555-569.
Frosini, P. and Landi, C.: Size functions and morphological transformations, Acta Appl. Math. 49 (1997), 85-104.
Frosini, P. and Landi, C.: Size functions and formal series, Appl. Algebra Engrg. Comm. Comput. (to appear).
Frosini, P. and Landi, C.: Size theory as a topological tool for computer vision, Pattern Recogn. Image Anal. 9 (1999), 596-603.
Greenberg, M. J. and Harper, J. R.: Algebraic Topology, a First Course, Benjamin-Cummings.
Milnor, J.: Morse Theory, Ann. of Math. Stud. 51, Princeton Univ. Press, 1963.
Uras, C. and Verri, A.: On the recognition of the alphabet of the sign language through size functions, In: Proc. XII IAPR Int. Conf. on Pattern Recognition, Jerusalem (Israel) II, IEEE Comp. Soc. Press, Los Alamitos, CA, 1994, pp. 334-338.
Verri, A., Uras, C., Frosini, P. and Ferri, M.: On the use of size functions for shape analysis, Biol. Cybern. 70 (1993), 99-107.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cagliari, F., Ferri, M. & Pozzi, P. Size Functions from a Categorical Viewpoint. Acta Applicandae Mathematicae 67, 225–235 (2001). https://doi.org/10.1023/A:1011923819754
Issue Date:
DOI: https://doi.org/10.1023/A:1011923819754