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Preliminaries

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Fuzzy Sets Methods in Image Processing and Understanding

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Abstract

This chapter introduces some concepts, notations, and terminology, which will be useful subsequently in this book. Section 2.1 summarizes the main sources of imprecision in the context of image processing and understanding. They concern both data and related knowledge. The various types of imprecision encountered constitute a compelling reason for exploiting fuzzy sets theory for image processing, analysis, and understanding. In order to make this book as self-contained as possible, Sect. 2.2 reviews selected definitions of the fuzzy sets theory. Only the important notions that are useful for the topics of the next chapters are given. The reader may refer to existing texts for further insight on fuzzy set theory (e.g., Dubois and Prade (Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York, 1980)). The main operators are then given in Sect. 2.3. The concept of linguistic variable is the subject of Sect. 2.4. A core notion in fuzzy set theory, this is used to capture both the conceptual and concrete levels. General methods to extend an operation applying on crisp sets to the corresponding operation applying on fuzzy sets are given in Sect. 2.5. Finally the main notations used in the book are summarized in Sect. 2.6.

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Notes

  1. 1.

    The interval [0,  1] is the most used. However, any other interval, or other set (typically a lattice) could be used. This also allows for extensions such as L-fuzzy sets [15], where L is a lattice defining the co-domain of the membership functions.

  2. 2.

    The convexity of a function f is defined as ∀(x, y), f(λx + (1 − λ)y) ≥ λf(x) + (1 − λ)f(y).

  3. 3.

    We draw the reader’s attention to the fact that since a possibility distribution and a membership functions have similar mathematical expression and given the links that exist between them, the same operators can apply to both of them. However, they have different meanings and origins, and this should not be under-estimated. We do not further consider possibilistic interpretations in this book.

  4. 4.

    Note that weaker forms of conjunctions can be used, not detailed here.

  5. 5.

    Note that it is not a dilation in the morphological sense.

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Authors and Affiliations

  1. Sorbonne Université, CNRS, LIP6, Paris, France

    Isabelle Bloch

  2. LTCI, Télécom Paris, Institut Polytechnique de Paris, Palaiseau, France

    Isabelle Bloch

  3. Department of Computer Science, University of Cincinnati, ML 0030, Cincinnati, OH, USA

    Anca Ralescu

Authors
  1. Isabelle Bloch
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  2. Anca Ralescu
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Bloch, I., Ralescu, A. (2023). Preliminaries. In: Fuzzy Sets Methods in Image Processing and Understanding. Springer, Cham. https://doi.org/10.1007/978-3-031-19425-2_2

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