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Various properties of digital covering spaces have been substantially used in studying digital homotopic properties of digital images. In particular, these are so related to the study of a digital fundamental group, a classification of digital images, an automorphism group of a digital covering space and so forth. The goal of the present paper, as a survey article, to speak out utility of digital covering theory. Besides, the present paper recalls that the papers [1, 4, 30] took their own approaches into the study of a digital fundamental group. For instance, they consider the digital fundamental group of the special digital image (X, 4), where X := $SC^{2,8}_4$ which is a simple closed 4-curve with eight elements in $Z^2$, as a group which is isomorphic to an infinite cyclic group such as (Z, +). In spite of this approach, they could not propose any digital topological tools to get the result. Namely, the papers [4, 30] consider a simple closed 4 or 8-curve to be a kind of simple closed curve from the viewpoint of a Hausdorff topological structure, i.e. a continuous analogue induced by an algebraic topological approach. However, in digital topology we need to develop a digital topological tool to calculate a digital fundamental group of a given digital space. Finally, the paper [9] firstly developed the notion of a digital covering space and further, the advanced and simplified version was proposed in [21]. Thus the present paper refers the history and the process of calculating a digital fundamental group by using various tools and some utilities of digital covering spaces. Furthermore, we deal with some parts of the preprint [11] which were not published in a journal (see Theorems 4.3 and 4.4). Finally, the paper suggests an efficient process of the calculation of digital fundamental groups of digital images.
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참고문헌 (33)
참고문헌 신청
R. Ayala / 2003 / Homotopy in digital spaces / Discrete Applied Math 125(1):3~24
C. Berge / 1976 / Graphs and Hypergraphs / North-Holland
L. Boxer / 1994 / Digitally continuous functions / Pattern Recognition Letters 15:833~839
L. Boxer / 1999 / A classical construction for the digital fundamental group / Jour. of Mathematical Imaging and Vision 10:51~62
L. Boxer / 2008 / The Classification of Digital Covering Spaces / Jour. of Mathematical Imaging and Vision 32:23~29
C. Berge / 1976 / Graphs and Hypergraphs / North-Holland
L. Boxer / 1994 / Digitally continuous functions / Pattern Recognition Letters 15:833~839
L. Boxer / 1999 / A classical construction for the digital fundamental group / Jour. of Mathematical Imaging and Vision 10:51~62
L. Boxer / 2008 / The Classification of Digital Covering Spaces / Jour. of Mathematical Imaging and Vision 32:23~29
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