Thesis (PDF) - Signal & Image Processing Lab

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18 CHAPTER 2. THEORETICAL BACKGROUND end up with a non natural description of the inclusion. Naturally, in the example of Fig. 2.3(a), one would have expected to have the two small squares included into the gray rectangle, and included into the white background. But the inclusion for these trees is mostly driven by the gray level rather than by the geometrical inclusion. Finally, we see that we need both trees if we want to have the two small squares represented, since each of them appears in one description, and not in the other. Now, instead of upper and lower sets, lets us introduce a term of shape. A shape is a connected component of upper or lower set without any holes in it. If a hole exists, it is filled. The shape corresponds to the connected component and its filled ”holes”. The sorting of shapes can then be made thanks to their geometrical inclusions. We can then create a tree structure as follows: each node corresponds to a shape; descendants are the shapes included into it, and the parent is the smallest shape that contains it (see Fig. 2.3(b) ). Each shape can specify either a gray level difference between it and its father or an absolute gray level. This will give us one single inclusion tree describing the image, in which a white object on black background is represented in the same manner as a black object on a white background. 2.3 Theoretical Background on semilattices Mathematical Morphology is a nonlinear image processing theory, which was based on complete lattices. In [21] Keshet has extended its scope to complete semilattices, which are more general. This section provides a brief overview of Mathematical Morphology on complete semilattices. 2.3.1 Complete Semilattices and Lattices A partially ordered set A is a set associated with a binary operator ≤, satisfying the following properties for any x, y, z ∈ A: reflexivity (x ≤ x), anti-symmetry (x ≤ y, y ≤ x ⇒ x = y), and transitivity (x ≤ y, y ≤ z ⇒ x ≤ z). In a partially ordered set A, the least majorant ∨X (also called supremum) of a subset X ⊆ A is defined as an element a0 ∈ A, such that:
2.3. THEORETICAL BACKGROUND ON SEMILATTICES 19 (a) (b) Figure 2.3: (a) The trees of connected components of upper and lower level sets of a simple image. (b) The resulting tree of shapes corresponding to the same image. Notice that D is a hole in F.
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
Page 9 and 10: viii LIST OF FIGURES 2.9 An example
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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