Topological interpretations of fuzzy subsets. A unified approach for fuzzy thresholding algorithms
Introduction
A new look at the classical definition of a fuzzy subset, introduced by Zadeh in his seminal work [59] (see also [9]), reveals topological and other structures that were hidden in the original definition.
Let us consider the standard definition (see [59], [9]) of a fuzzy subset1 X (of a set U) as a function μX: U → [0, 1], where U denotes a nonempty (crisp) set of reference, usually called universe. This approach is obviously equivalent to the definition of a fuzzy subset as a nested family {Uα: α ∈ [0, 1]} of crisp subsets of U, called the α-cuts. Here Uα = {t ∈ U: μX(t) ⩾ α}, for each α ∈ [0, 1].
But a nested family of subsets of a given crisp set U immediately induces a topology that is also nested. Thus, the classical definition of a fuzzy subset could be (re)-interpreted as a nested topology of a certain kind defined on a given crisp set. Other remarkable topological interpretations of fuzzy sets and related items have already appeared in the literature (see e.g. [1], [31], [39]).
Moreover, different kinds of fuzzy subsets would lead to different sorts of nested topologies. To put an easy example, observe that when X = U, so that X is crisp and μX(t) = 1 for every t ∈ U, we have that Uα = U for every α ∈ [0, 1]. It is then obvious that the topology induced by the subbasis {Uα = U}α∈[0,1] is the trivial one on the set U. Conversely, it straightforward to see that fuzzy subsets that are not crisp induce non-trivial topologies. Note also that when X = ∅, i.e. μX(t) = 0 for each t ∈ U, the induced topology is also the trivial one. But, for any other fuzzy set, the topology is not trivial (included the crisp sets X ⊂ U).
Which additional information is furnished by this new setting?
On the one hand, there are important classes of topologies (e.g., the lower preorderable ones) that are, by definition, nested. Also, a key fact related to nested topologies is that of defining different kinds of orderings (e.g. linear orders, total preorders) on a given nonempty crisp set. Having this in mind, we wonder if those orderings can directly be defined by means of suitable fuzzy subsets.
Moreover, on the other hand, this theoretical framework allows us to provide a general description for the most widely used fuzzy thresholding algorithms, such as those of Otsu [43], Huang and Wang [35] or the algorithm of the areas, which in some cases can be shown to be equivalent to Otsu’s algorithm [17]. Moreover, although we do not develop it here, these developments admit a straight extension to those fuzzy image thresholding algorithms based on the use of extensions of fuzzy sets, as it is the case for Tizhoosh’s algorithm [53].
The structure of the paper goes as follows:
After the Introduction (Section 1), the key notions of a fuzzy subset, a nested family and a nested topology are introduced in Section 2 of Preliminaries. In Section 3 we analyze the relationship between fuzzy subsets and nested topologies on a crisp set. In Section 4 we study, in terms of fuzzy subsets, particular classes of topologies, mainly related to the collection of total preorders that could be defined on a crisp set. In Section 5 we show an application of our theoretical developments to provide a unified framework for fuzzy image thresholding algorithms. Section 6 (Conclusion) closes the paper.
Section snippets
The classical concept of a fuzzy subset
Throughout the paper, a set in the standard sense is said to be crisp, in opposition to the term “fuzzy” introduced in Definition 2.1.
The standard definition of a fuzzy subset follows now. Definition 2.1 Let U be a nonempty set, usually called universe. A fuzzy subset X of U is defined by means of a map μX: U → [0, 1]. The map μX is said to be the membership function (or indicator) of X. The support of X is the crisp subset Supp(X) = {t ∈ U: μX(t) ≠ 0} ⊆ U, whereas the kernel of X is the crisp subset Ker(X) = {t ∈ U: μX(t) = 1} ⊆[59]
Nested families as regards [0, 1] vs. fuzzy subsets
As commented in Remark 2.4, a fuzzy subset generates a family of subsets of a crisp set, that is nested as regards [0, 1] in the sense of Definition 2.8.
What about the converse?
We wonder if any nested family as regards [0, 1] can be identified to a fuzzy subset, by agreeing with its collection of α-cuts. The answer is negative as Example 3.1 proves. Fortunately, we can characterize nested families as regards [0, 1] that correspond to the α-cuts of a fuzzy subset, as stated in Proposition 3.2. Example 3.1 Let
From general nested families to total preorders
In Proposition 2.19 we have seen that total preorders induce nested families. But the converse is also true: as a matter of fact, a nested family , defined on a set U, immediately endows U with a total preorder , as Proposition 4.2 shows. First we introduce a preparatory definition. Definition 4.1 Let U denote a nonempty crisp set. Given a nested family of subsets of U, the relation given by is said to be the natural preorder associated to the nested family .
Let us
A unified framework for fuzzy thresholding
A key factor for vision systems is the identification of regions (which represent objects) on an image. This operation, which is very simple for humans, is very difficult for machines. The division of an image into regions is known as segmentation; i.e., the segmentation of digital images is the process of dividing an image into disjoint parts, regions or classes so that each one of them has similar attributes or properties. Each of these classes represents an object of the image. There exist
Conclusion
The interpretation of several kinds of nested topologies, as well as total preorders on a nonempty crisp set U called universe, as suitable fuzzy subsets X of U carries a large variety of unexplored and maybe astonishing possibilities. Indeed, using classical well-known operators arising in fuzzy set theory (i.e., triangular norms and conorms, ordered weighted averaging aggregation operators [57], etc.) we may define operations directly on total preorders or nested topologies. The procedure is
Acknowledgement
This work is supported by the research projects MTM2009-12872-C02-02, TIN2010-15055 and MTM2012–37894-C02-02 (Spain).
Thanks are given to the editor in chief (Hamido Fujita Ph.D.), as well as to two anonymous referees, for their valuable suggestions and comments that have led to a substantial improvement of the manuscript.
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