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Estimation of the complexity of a digital image from the viewpoint of fixed point theory

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  • Han, Sang-Eon

Abstract

The present paper introduces and estimates the complexity of the fixed point property of a digital image (X, k) for any k−DC-self-map f of (X, k), where a k−DC-self-map f of (X, k) means a digitally k-continuous self-map of (X, k) with a digital version of the Banach contraction principle. To do this work, we need to study some properties of iterations of a k−DC-self-map f of (X, k) and to establish the notion of complexity of (X, k) denote by C♯(X, k) (see Definition 7 in the present paper). According to C♯(X, k), we can estimate complexity of the fixed point property of (X, k) for any k−DC-self-map f of (X, k). Based on this approach, the present paper investigates some relationships between the k-adjacency of (X, k) and C♯(X, k). Furthermore, we prove that C♯(X, k) is not a digital topological invariant. Besides, we develop the notions of uniform k-connectedness and strict k-connectivity to calculate C♯(X, k) for some digital images (X, k). In the paper each (X, k) is assumed to be a k-connected and non-empty set and 2 ≤ | X |≨∞, where | X | means the cardinal number of the given set X.

Suggested Citation

  • Han, Sang-Eon, 2019. "Estimation of the complexity of a digital image from the viewpoint of fixed point theory," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 236-248.
    • Handle: RePEc:eee:apmaco:v:347:y:2019:i:c:p:236-248
    DOI: 10.1016/j.amc.2018.10.067
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    Citations

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    Cited by:

    1. Sang-Eon Han, 2020. "Fixed Point Sets of Digital Curves and Digital Surfaces," Mathematics, MDPI, vol. 8(11), pages 1-25, October.
    2. Sang-Eon Han, 2020. "The Most Refined Axiom for a Digital Covering Space and Its Utilities," Mathematics, MDPI, vol. 8(11), pages 1-21, October.
    3. Sang-Eon Han & Saeid Jafari & Jeong Min Kang, 2019. "Topologies on Z n that Are Not Homeomorphic to the n -Dimensional Khalimsky Topological Space," Mathematics, MDPI, vol. 7(11), pages 1-12, November.
    4. Sang-Eon Han, 2021. "Discrete Group Actions on Digital Objects and Fixed Point Sets by Iso k (·)-Actions," Mathematics, MDPI, vol. 9(3), pages 1-25, February.
    5. Sang-Eon Han, 2020. "Digital k -Contractibility of an n -Times Iterated Connected Sum of Simple Closed k -Surfaces and Almost Fixed Point Property," Mathematics, MDPI, vol. 8(3), pages 1-23, March.
    6. Sang-Eon Han, 2019. "Fixed Point Theory for Digital k -Surfaces and Some Remarks on the Euler Characteristics of Digital Closed Surfaces," Mathematics, MDPI, vol. 7(12), pages 1-19, December.
    7. Sang-Eon Han, 2020. "Digital Topological Properties of an Alignment of Fixed Point Sets," Mathematics, MDPI, vol. 8(6), pages 1-18, June.
    8. Sang-Eon Han, 2019. "Remarks on the Preservation of the Almost Fixed Point Property Involving Several Types of Digitizations," Mathematics, MDPI, vol. 7(10), pages 1-14, October.
    9. Sang-Eon Han, 2020. "Fixed Point Sets of k -Continuous Self-Maps of m -Iterated Digital Wedges," Mathematics, MDPI, vol. 8(9), pages 1-26, September.

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