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.
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