Abstract
Since its genesis, fuzzy sets (FSs) theory (Zadeh, 1965) provided a flexible framework for handling the indeterminacy characterizing real-world systems, arising mainly from the imprecise and/or imperfect nature of information. Moreover, fuzzy logic set the foundations for dealing with reasoning under imprecision and offered the means for developing a context that reflects aspects of human decision-making. Images, on the other hand, are susceptible of bearing ambiguities, mostly associated with pixel values. This observation was early identified by Prewitt (1970), who stated that “a pictorial object is a fuzzy set which is specified by some membership function defined on all picture points”, thus acknowledging the fact that “some of its uncertainty is due to degradation, but some of it is inherent”. A decade later, Pal & King (1980) (1981) (1982) introduced a systematic approach to fuzzy image processing, by modelling image pixels using FSs expressing their corresponding degrees of brightness. A detailed study of fuzzy techniques for image processing and pattern recognition can be found in Bezdek et al and Chi et al (Bezdek, Keller, Krisnapuram, & Pal, 1999) (Chi, Yan, & Pham, 1996). However, FSs themselves suffer from the requirement of precisely assigning degrees of membership to the elements of a set. This constraint raises some of the flexibility of FSs theory to cope with data characterized by uncertainty. This observation led researchers to seek more efficient ways to express and model imprecision, thus giving birth to higher-order extensions of FSs theory. This article aims at outlining an alternative approach to digital image processing using the apparatus of Atanassov’s intuitionistic fuzzy sets (A-IFSs), a simple, yet efficient, generalization of FSs. We describe heuristic and analytic methods for analyzing/synthesizing images to/from their intuitionistic fuzzy components and discuss the particular properties of each stage of the process. Finally, we describe various applications of the intuitionistic fuzzy image processing (IFIP) framework from diverse imaging domains and provide the reader with open issues to be resolved and future lines of research to be followed.
TopBackground
However, and despite their vast impact to the design of algorithms and systems for real-world applications, FSs are not always able to directly model uncertainties associated with imprecise and/or imperfect information. This is due to the fact that their membership functions are themselves crisp. These limitations and drawbacks characterizing most ordinary fuzzy logic systems (FLSs) were identified and described by Mendel & Bob John (2002), who traced their sources back to the uncertainties that are present in FLSs and arise from various factors. The very meaning of words that are used in the antecedents and consequents of FLSs can be uncertain, since some words may often mean different things to different people. Moreover, extracting the knowledge from a group of experts who do not all agree, leads in consequents having a histogram of values associated with them. Additionally, data presented as inputs to an FLS, as well as data used for its tuning, are often noisy, thus bearing an amount of uncertainty. As a result, these uncertainties translate into additional uncertainties about FS membership functions. Finally, Atanassov et al. (Atanassov, Koshelev, Kreinovich, Rachamreddy & Yasemis, 1998) proved that there exists a fundamental justification for applying methods based on higher-order FSs to deal with everyday-life situations. Therefore, it comes as a natural consequence that such an extension should also be carried in the field of digital image processing.
Key Terms in this Chapter
Fuzzy Set: A generalization of the definition of the classical set. A fuzzy set is characterized by a membership function, which maps the members of the universe into the unit interval, thus assigning to elements of the universe degrees of belongingness with respect to a set.
Crisp Set: A set defined using a characteristic function that assigns a value of either 0 or 1 to each element of the universe, thereby discriminating between members and non-members of the crisp set under consideration. In the context of fuzzy sets theory, we often refer to crisp sets as “classical” or “ordinary” sets.
Fuzzification: The process of transforming crisp values into grades of membership corresponding to fuzzy sets expressing linguistic terms.
rNon-Membership Function: In the context of Atanassov’s intuitionistic fuzzy sets, it represents the degree to which an element of the universe does not belong to a set.
Intuitionistic Fuzzy Index: Also referred to as “hesitancy margin” or “indeterminacy index”. It represents the degree of indeterminacy regarding the assignment of an element of the universe to a particular set. It is calculated as the difference between unity and the sum of the corresponding membership and non-membership values.
Intuitionistic Fuzzy Set: An extension of the fuzzy set. It is defined using two characteristic functions, the membership and the non-membership that do not necessarily sum up to unity. They attribute to each individual of the universe corresponding degrees of belongingness and non-belongingness with respect to the set under consideration.
Image Processing: Image processing encompasses any form of information processing for which the input is an image and the output an image or a corresponding set of features.
Fuzzy Logic: Fuzzy logic is an extension of traditional Boolean logic. It is derived from fuzzy set theory and deals with concepts of partial truth and reasoning that is approximate rather than precise.
Defuzzification: The inverse process of fuzzification. It refers to the transformation of fuzzy sets into crisp numbers.
Membership Function: The membership function of a fuzzy set is a generalization of the characteristic function of crisp sets. In fuzzy logic, it represents the degree of truth as an extension of valuation.