Characterization of Semi-open Subcomplexes in Abstract Cellular Complex
The concept of abstract cellular complexes was introduced by Kovalevsky (Computer Vision, Graphics and Image Processing, 46:141–161, 1989) and established that the topology of cellular complex is the only possible topology of finite sets to describe the s
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Abstract The concept of abstract cellular complexes was introduced by Kovalevsky (Computer Vision, Graphics and Image Processing, 46:141–161, 1989) and established that the topology of cellular complex is the only possible topology of finite sets to describe the structure of images. Further, the topological notions of connectedness and continuity in abstract cellular complexes were introduced while using the notions of an open subcomplex, closed subcomplex, and boundary of a subcomplex, etc. In this paper, the notion of semi-open subcomplex in abstract cellular complex is introduced and some of its basic properties are studied by defining the notions of semi-closure, semi-frontier, and semi-interior. Further, a homogeneously n-dimensional complex is characterized while using the notion of semi-open subcomplexes. Introduced is also the concept of a quasi-solid in subcomplex. Finally, a new algorithm for tracing the semifrontier of an image is presented.
Keywords Semi-open subcomplex Semi-closed subcomplex Semi-frontier Semi-interior Semi-closure Quasi-solid Semi-region Semi-frontier tracing algorithm
N. Vijaya (&) Adhiyamaan College of Engineering, Hosur, India e-mail: [email protected] G. S. S. Krishnan PSG College of Technology, Coimbatore, India e-mail: [email protected]
G. S. S. Krishnan et al. (eds.), Computational Intelligence, Cyber Security and Computational Models, Advances in Intelligent Systems and Computing 246, DOI: 10.1007/978-81-322-1680-3_30, Springer India 2014
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N. Vijaya and G. S. S. Krishnan
1 Introduction Digital image processing is a rapidly growing discipline with a broad range of applications in medicine, environmental sciences, and in many other fields. The field of digital image processing refers to processing two-dimensional pictures by a digital computer. Rosenfeld [2, 3] represented a digital image by a graph whose nodes are pixels and whose edges are linking adjacent pixels to each other. He named the resultant graph as the neighborhood graph. But this representation contains two paradoxes, namely connectivity and boundary paradoxes. Kovalevsky [4] introduced the notion of abstract cellular complexes to study the structure of digital images and introduced axiomatic digital topology [8], which has no paradoxes. Moreover, he showed that every finite topological space with separation property is isomorphic to an abstract cellular complex. Further, Kovalevsky [5, 6] introduced the notions of a half plane, a digital line segment, etc., while using the notions of open sets, closed sets, closure, and interior. The concepts of semi-open set and semi-continuity were introduced by Levine [10]. The half-open intervals ða; b and ½a; bÞ are characterized as semi-open subsets of the real line. Though the collections of semi-open sets do not form a classical topology on R, it satisfies the condition of a basis in R, and hence, both half-open intervals of the form ½a; bÞ and ða; b generate two special topologies, namely lower limit topology and upper limit
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