Integrating fuzzy and bipolar fuzzy metrics into digital topology for enhanced image analysis

Abstract

Digital topology provides a mathematical foundation for analysing discrete image structures. It is considered to be a very critical part of processing image data because it provides us with information about the properties of images. The approaches to digital topology have been studied to great detail by several researchers and each approach has been seen to have advantages and limitations. In general, the approaches that exist, which are the Graph theoretic and axiomatic methods, are both found to stmggle when it comes to handling the imprecision that is present in images and other digital structures. This is because such methods do not account for dual uncertainty. Consequently, this limits their effectiveness in real-world applications. While fuzzy and bipolar fuzzy set theory provide flexible tools for addressing these limitations, their integration into digital topology remains underdeveloped. This study addresses this gap by developing a mathematical framework that extends traditional digital topology into fuzzy and bipolar fuzzy logic for image analysis. The research formalises fuzzy and bipolar fuzzy metric spaces on the digital plane, and attempts to integrate these bipolar fuzzy metrics into digital topology. Topological properties, including connectedness, adjacency, and surroundness, are also extended into the fuzzy and bipolar fuzzy domain. This has been done in order to ensure mathematical consistency in representing digital images. Additionally, the study proposes new similarity, distance, and entropy measures to refine the important aspects of image analysis such as image segmentation. Through computational implementation, the findings contribute to a more realistic approach to image segmentation that is also mathematically sound. This research greatly improves the applicability of digital topology in image processing technique since it establishes a framework that handles uncertainty in digital images. The study’s outcomes will pave the way for deeper analysis of digital images and hence strengthen the intersection of topology, fuzzy mathematics, and artificial intelligence.