A unified framework for knowledge measure with application: From fuzzy sets through interval-valued intuitionistic fuzzy sets
Introduction
Atanassov [1] suggested the notion of intuitionistic fuzzy sets (IFSs) as a natural extension of Zadeh’s [2] fuzzy sets (FSs). Atanassov and Gargov [3] further extended IFSs to the case of interval-valued IFSs (IVIFSs). For its great flexibility in coping with uncertainty, the theory of IFSs/IVIFSs has been widely explored in the last 10 years from different perspectives. Meanwhile, intuitionistic fuzzy entropy as an active research topic has always been receiving much attention from researchers. In this paper, we will not do further research on fuzzy entropy. Instead, we go in the opposite direction and focus on the modeling of intuitionistic fuzzy knowledge measure (IFKM), with which to help tackle some tough problems that are difficult to handle by using entropy alone.
Let us briefly review some aspects of fuzzy entropy first. As far as its basic structure is concerned, there are three types of entropy, i.e., the intuitionistic-type [4], the probabilistic-type [5], and the non-probabilistic-type [6]. The first one considers the modeling of the hesitancy aspect of uncertainty. The second one borrows the idea from the Shannon entropy. Both of them differ greatly from the last one in the basic structure of the axioms. Although having recently gotten more attention from researchers, the last one still suffers from some drawbacks, such as failing to distinguish between such IFSs that have equal degrees of membership and non-membership. In fact, it is derived directly from the context of FSs. It thus seems difficult to handle the aspect of the unknown well in measurement of entropy. The interested reader may consult Guo and Song [7] for more details about this topic.
In view of the inherent defects of intuitionistic fuzzy entropy as mentioned above, the pioneering exploration of the measurement of knowledge conveyed by an IFS was made by Szmidt et al. [8]. The notion of knowledge here is related to such information treated as useful in a particular context, characterized by a sort of regularity, certainty, and novelty. It is generally believed that a measure of knowledge can be viewed as a negation of entropy in a fuzzy system [9], [10], [11]. Guo [12] argued that, however, a measure of knowledge should not be simply regarded as a negation of entropy in the context of IFSs since there is no natural logic between these two kinds of measures with the introduction of hesitancy. So far there have been several attempts to deal with the IFKM from different perspectives. Some of them put emphasis on the information content conveyed by membership degrees plus non-membership degrees [10], [11], while the others gave more attention to the inherent fuzziness of an IFS/IVIFS [9], [12], [13]. A detailed overview can be found in [14]. We must indicate that all of the axioms and measuring models as mentioned above are entropy-dependent, and none of them has considered both the information content and the inherent fuzziness simultaneously. In the latest research findings, Guo and Xu [14] pointed out and justified that there are at least two facets of knowledge associated with an IFS, i.e. the information content and the information clarity. They then axiomatized the IFKM with the two facets and provided an entropy-independent axiomatic definition in the context of IFSs. On this basis, Guo and Xu [15] further developed a bi-parametric IFKM with which to reveal some significant aspects of psychological cognition hidden in the handling of IFSs.
This work aims to further recognize the nature of knowledge and establish a unified framework for knowledge measure (KM) from classical FSs through general IVIFSs. The normalized Hamming distance combined with the technique for order preference by similarity to ideal solution (TOPSIS), is used as the foundation for IFKM modeling. On this basis, new entropy-independent axioms of KM and resulting models are presented in the contexts of IVIFSs and FSs, respectively. This helps us to achieve the theoretical goal. The developed technique is then applied to image thresholding as a brand-new KM application, and new analytical models/algorithms are proposed to solve some crucial problems in the process. Theoretical achievements are strongly supported by experiment, and the outperformance of the developed technique with application is shown. The main application of IFKM presently is in decision making under uncertainty [9], [10], [11], [12], [13], [14], [15]. So far there have never been any attempts to apply this new type of measure to image processing or other fields.
The outline of the paper is as follows. Section 2 carefully reviews the literature on image thresholding. Section 3 briefly recalls some basic concepts and definitions. In Section 4, an innovative framework for KM is developed. The developed technique is then applied to image thresholding in Section 5. Section 6 experimentally shows the outperformance of our technique with application, followed by concluding remarks in Section 7.
Section snippets
Related work on image thresholding
Segmentation as a basic and important technique in image analysis, is a process of dividing an image into disjoint regions or classes for further recognition. There are many techniques in the literature to deal with it, such as gray-level thresholding [16], [17], [18], [19], [20], [21], [22], edge detection [23], region growing [24], fuzzy clustering [25], and others based on some particular theories [26]. Notable among these is the first one, which classifies the pixels into two categories:
Basic concepts
Zadeh [2] defined the notion of FSs as follows.
Definition 1 A FS in a universe can be expressed as where represents a membership degree of . Atanassov [1] further defined the notion of IFSs below.[2]
Definition 2 An IFS in a universe can be expressed as where and such that for , and they represent a membership degree and a non-membership degree of , respectively.[1]
An additional concept related to an IFS is
New non-linear IFKM
We develop first a new non-linear IFKM by using the normalized Hamming distance. The idea of the TOPSIS is introduced to achieve the goal. Consider a single element . Given the role of the two facets of knowledge as mentioned before, it is natural to determine the following two positive-ideal solutions, i.e. and , since both of them have the maximum values of the information content and the information clarity. The negative-ideal solution is
Application to image thresholding
We now use the above measures for image thresholding. Overview of this process is illustrated in Fig. 1.
Performance metrics and evaluation
The images used in our experiment are all from Weizmann reference library at the website: http://www.wisdom.weizmann.ac.il/ vision/Seg_Evaluation_DB, where over 200 original grayscale images are provided, along with ideal/ground truth (GT) ones. Eight representative models/algorithms are used for comparative analysis, including minimum entropy (MinEnt2013 [20], MinEnt2014 [21], MinEnt2018 [22]), fuzzy clustering (KIFECM2020 [25]), probabilistic neural network (PNNBRL2020 [26]), and the
Conclusion
In this paper, the nature of knowledge is further recognized, and new axiomatic definitions of IVIFKM and FKM are presented, respectively. The normalized Hamming distance combined with the TOPSIS is used as the foundation for KM modeling, thus leading to a unified framework for KM from FSs through IVIFSs. The developed technique is then used for image thresholding as a brand-new KM application, and a knowledge-driven thresholding methodology is proposed accordingly. We argue that the nature of
CRediT authorship contribution statement
Kaihong Guo: Conceptualization, Methodology, Validation, Formal analysis, Investigation, Writing - original draft, Supervision, Project administration, Funding acquisition. Hao Xu: Software, Resources, Data curation, Writing - review & editing, Visualization.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work is supported in part by the National Natural Science Foundation of China under Grant No. 71771110, and in part by the Humanities and Social Science Fund of Ministry of Education of China under Grant No. 16YJA630014. The authors would like to thank the Editor-in-Chief, Professor Mario Köppen, the Associate Editor, Professor Witold Pedrycz, and the anonymous reviewers for their constructive comments and suggestions, which have greatly improved the presentation of this research. The
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