Persistent homology of graph-like digital images
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Persistent homology of graph‑like digital images Ozgur Ege1 · Ismet Karaca1 Received: 24 March 2019 / Accepted: 15 February 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we introduce persistent homology for graph-like digital images as a new type of computation of digital homology groups. We calculate persistent homology groups of some graph-like digital images. Moreover, we prove some theorems related to a singleton point and 𝜅-connected graph-like digital images. It has been shown that identity and composition axioms are satisfied for digital persistent homology groups. Finally, we give some applications of the new theory to image processing. Keywords Digital topology · Digital image · Persistent homology Mathematics Subject Classification Primary 55N35 · Secondary 55M20 · 68R10
1 Introduction Homology is a significant topological invariant for topological spaces and detects the connected components, tunnels, voids of a topological space. Homology groups have been introduced by Poincare in 1895. They were not used in applied areas until the appearance of persistent homology [14]. Persistent homology is a way to describe topological features of a topological space X with filtration and is an extension of homology. In recent times, the theory of persistent homology [14, 34] has become a useful tool in different areas, such as topological data analysis [29], mathematics [18, 31] and shape analysis [23]. Zomorodian and Carlsson [34] study persistent homology and give an algorithm for computing persistent homology groups. Qaiser et al. [30] introduce a novel tumor segmentation approach for a histology whole-slide image by exploring the degree of connectivity among nuclei based on persistent homology profiles. In [12], a number of methods for thinking about data are given by using topological methods. Takiyama et al. [33] show that persistent homology index, a novel index for immunohistochemical labeling using persistent homology, can produce highly similar data to that generated by a pathologist * Ozgur Ege [email protected] Ismet Karaca [email protected] 1
Department of Mathematics, Ege University, 35100 Bornova, Izmir, Turkey
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using visual evaluation. For detailed knowledge related to persistent homology, see [16] and more recent papers include different applications, see [15, 17, 24]. Digital topology with a great number of applications continues to rise in many fields of science and engineering such as mathematics, image processing, information and computer science. The main goal in this area is to obtain new results on digital images in ℤn using different methods from algebraic topology. During the recent years, many papers have been revealed in digital topology. Digital versions of some concepts from algebraic topology have been studied in [4–8]. Arslan et al. [1] introduce the simplicial homology groups of n-dimensional digital images. Boxer et al.
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