Thesis (PDF) - Signal & Image Processing Lab

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20 CHAPTER 2. THEORETICAL BACKGROUND 1. x ≤ a0, ∀x ∈ X, 2. if there exists y, such that x ≤ y ≤ a0, for all x ∈ X, then y = a0. One defines the greatest minorant ∧X (also called infimum) of X, dually. A partially ordered set P is an inf-semilattice (resp. sup-semilattice) if every two- element subset {X1, X2} in P has an infimum X1 ∧ X2 (resp., a supremum X1 ∨ X2) in P . If P is both an inf and a sup-semilattice, then it is called a lattice. An inf- semilattice (resp., sup-semilattice) is complete, when every non-empty subset B ⊂ P has an infimum ∧B (resp. supremum ∨B). In this case, there exists in the semilattice a unique element 0, called zero element, (resp., U, called universe), such that, for any X ∈ P , 0 ∧ X = 0 (resp. U ∨ X = U). A complete lattice is a lattice, which is both a complete inf and complete sup-semilattice. 2.3.2 Morphology on Semilattices Some basic notions and tools of mathematical morphology on complete lattices were extended to semilattices in [21] by Keshet. In this section, we shortly review the extensions. Proofs of propositions will be omitted, but they are available in [21]. Without loss of generality, we restrict our discussion to inf-semilattices, since all definitions and results are valid in sup-semilattices as well, by duality. Erosions and openings are basic notions that are directly and naturally extendible from complete lattices to complete inf-semilattices. Erosion is defined in complete inf-semilattice S as any operator that distributes over the infimum, see (2.4). Preservation of the universe is not required, because it does not necessarily exist in S. Definition 1. A binary operator ε in an inf-semilattice S is an erosion iff, for all {Xi} ⊆ S: ε( � Xi) = � ε(Xi) (2.4) Proposition 1. Erosions in inf-semilattices are increasing 1 . i The extension of algebraic openings to semilattices is also straightforward: 1 A transform Ψ is increasing ⇔ ∀f ≤ g ⇒ Ψ(f) = Ψ(g). i
2.3. THEORETICAL BACKGROUND ON SEMILATTICES 21 Definition 2. (Algebraic Opening): A binary operator in an inf-semilattice S is an algebraic opening, iff it is idempotent 2 , increasing, and anti-extensive 3 . Compared to the above extensions, that of morphological opening, presented be- low, is a little less direct. This is because: i) In complete lattices, the morphological opening associated with an erosion is defined using its adjoint dilation, and ii) since there is no general definition of supremum in an inf-semilattice, one cannot generally define dilations there (however, limited versions of supremum and adjoint dilation will be defined in the sequel). Nevertheless, morphological opening can be completely extended to complete inf-semilattices, without help of dilation. Definition 3. (Morphological Opening): In a complete inf-semilattice S, the morphological opening γε associated with an erosion ε is defined, for any X ∈ S, by: γε(X) � � {Y ∈ S|ε(X) ≤ ε(Y )} (2.5) Proposition 2. Given any erosion in a complete inf-semilattice S, the associated morphological opening γε(X) of any element X ∈ S exists in S and is unique. Proposition 3. The morphological opening in an complete inf-semilattice is an algebraic opening, meaning it is idempotent, increasing, and anti-extensive. 2.3.3 Supremum and Dilations We now relate to basic morphological notions that are not directly extendible to inf-semilattices, namely, supremum and dilation. Although it is impossible to define the above operations generally in inf-semilattices, Keshet provides limited versions of them in [21]. Supremum Unless it is a complete lattice, a complete inf semilattice does not have a well-defined supremum for all its subsets. Nevertheless, a supremum does exist for some subsets of the semilattice. 2 A transform Ψ is idempotent ⇔ ΨΨ = Ψ. 3 A transform Ψ is anti-extensive ⇔ I ≥ Ψ, where I is an identity transform.
Page 1 and 2: SELF-DUAL MORPHOLOGICAL OPERATORS B
Page 3 and 4: The Research Thesis was done under
Page 5 and 6: iv CONTENTS 3 Implementation of BTV
Page 7 and 8: List of Tables 2.1 The BTV transfor
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Page 11 and 12: x LIST OF FIGURES 4.17 A Simba imag
Page 13 and 14: xii LIST OF FIGURES 6.20 A Lena ima
Page 15 and 16: Abstract This research addresses a
Page 18 and 19: 4 List of Abbreviations and Symbols
Page 20 and 21: 6 CHAPTER 1. INTRODUCTION the image
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Chapter 7 Conclusions and further r
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Appendix A 114
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Bibliography [1] J. Serra, Image An
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118 BIBLIOGRAPHY [21] Renato Keshet
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