Some properties of digital H-spaces
Some properties of digital H-spaces
In this paper, we study certain properties of digital H-spaces. We prove that a digital image that has the same digital homotopy type with any digital H-space is also a digital H-space. We show that the digital fundamental group of a digital H-space is abelian. We give examples that are related to a digital homotopy associative H-space and a κ-contractible digital H-space. Several important applications of digital H-spaces are given in computer vision and image processing. Finally, we deal with the importance of digital H-space in digital topology and image processing. We conclude that any κ-contractible digital image is a digital H-space.
___
- [1] Adams JF. On the non-existence of elements of Hopf invariant one. Ann Math 1960; 72: 20-104.
- [2] Arkowitz M. Introduction to Homotopy Theory. New York, NY, USA: Springer, 2011.
- [3] Ayala R, Dom´ınguez E, Franc´es AR, Quintero A. Homotopy in digital spaces. Lect Notes Comput Sc 2000; 1953: 3-14.
- [4] Boxer L. Digitally continuous functions. Pattern Recogn Lett 1994; 15: 833-839.
- [5] Boxer L. A classical construction for the digital fundamental group. J Math Imaging Vis 1999; 10: 51-62.
- [6] Boxer L. Properties of digital homotopy. J Math Imaging Vis 2005; 22: 19-26.
- [7] Boxer L. Digital products, wedges, and covering spaces. J Math Imaging Vis 2006; 25: 159-171.
- [8] Boxer L, Karaca I. Fundamental groups for digital products. Adv Appl Math Sci 2012; 11: 161-180.
- [9] Ege O, Karaca I. Lefschetz fixed point theorem for digital images. Fixed Point Theory A 2013; 253: 1-13.
- [10] Ege O, Karaca ¨ ˙I. Digital H-spaces. In: ISCSE 2013 3rd International Symposium on Computing in Science and Engineering; 2425 October 2013; Ku¸sadası, Turkey. ˙Izmir, Turkey: Gediz University Publications. pp. 133-138.
- [11] Ege O, Karaca I. Cohomology theory for digital images. Rom J Inf Sci Tech 2013; 16: 10-28.
- [12] Han SE. Non-product property of the digital fundamental group. Inform Sciences 2005; 171: 73-91.
- [13] Han SE. Digital fundamental group and Euler characteristic of a connected sum of digital closed surfaces. Inform Sciences 2007; 177: 3314-3326.
- [14] Herman GT. Oriented surfaces in digital spaces. Graph Model Im Proc 1993; 55: 381-396.
- [15] Herman GT. Finitary 1-simply connected digital spaces. Graph Model Im Proc 1998; 60: 46-56.
- [16] Kong TY. A digital fundamental group. Comput Graph 1989; 13: 159-166.
- [17] Kopperman R. On storage of topological information. Discrete Appl Math 2005; 147: 287-300.
- [18] Kovalevsky A. Finite topology as applied to image analysis. Comput Vision Graph 1989; 45: 141-161.
- [19] Malgouyres R. Homotopy in two-dimensional digital images. Theor Comput Sci 2000; 230: 221-233.
- [20] Mazo L, Passat N, Couprie M, Ronse C. Paths, homotopy and reduction in digital images. Acta Appl Math 2011; 113: 167-193.
- [21] Rosenfeld A, Kak AC. Digital Picture Processing. New York, NY, USA: Academic Press, 1976.
- [22] Rosenfeld A. Continuous functions on digital pictures. Pattern Recogn Lett 1986; 4: 177-184.
- [23] Spanier EH. Algebraic Topology. New York, NY, USA: McGraw-Hill, 1966.
- [24] Switzer R. Algebraic TopologyHomology and Homotopy. New York, NY, USA: Springer-Verlag, 1975.