Thesis (PDF) - Signal & Image Processing Lab

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Chapter 5 Tree Semilattices In this chapter, we propose a general framework for tree-based morphological image processing, which unifies the schemes presented in previous chapters, and enables the definition of new morphological operators that are based on tree representations. The heart of the proposed approach is a novel complete inf-semilattice of tree representations of images. The proposed image processing scheme is shown is figure 5.1. This chapter is organized as follows. First, Section 5.1 introduces the complete inf-semilattice of tree representations. Then, in Section 5.2, morphological image processing using the complete inf-semilattice of tree representations is developed. In Section 5.3, we address the problem of whether the complete inf-semilattice of tree representations induces a complete inf-semilattice structure on the set of images as well. Finally, the chapter is summarized in Section 5.4. Figure 5.1: Tree-based morphology. 62
5.1. THE COMPLETE INF-SEMILATTICE OF TREE REPRESENTATIONS 63 5.1 The Complete Inf-Semilattice of Tree Repre- sentations Let L be an arbitrary set of “labels”, and let t = (V, E) be a rooted tree, with root r, such that V ⊆ L. Therefore t is a tree of labels. Moreover, let M : E ↦→ V be an image of vertices, mapping each point in E to a vertex of t. As before, E is either an Euclidean space or a discrete rectangular grid within the image area. Definition 10. (Tree Representation) The structure T = (t, M) shall be called a tree representation. The set of all tree representations associated to the label set L and to the root r shall be denoted by T L r . Figure 5.2 depicts an example of a tree representation. Consider the following relation between tree representations: For all T1 = (t1, M1) and T2 = (t2, M2) in T L r , T1 ≤ T2 ⇐⇒ t1 ⊆ t2 and M1 �t2 M2, (5.1) where ⊆ is the usual graph inclusion, and �t2 is the partial ordering of vertices (see Figure 5.2: An example of image tree representation. An image with V1, V2, V3 and V4 zones is represented as a tree. Each pixel in this zone is mapped to a corresponding label.
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SELF-DUAL MORPHOLOGICAL OPERATORS B
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The Research Thesis was done under
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iv CONTENTS 3 Implementation of BTV
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List of Tables 2.1 The BTV transfor
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viii LIST OF FIGURES 2.9 An example
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x LIST OF FIGURES 4.17 A Simba imag
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xii LIST OF FIGURES 6.20 A Lena ima
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Abstract This research addresses a
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4 List of Abbreviations and Symbols
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6 CHAPTER 1. INTRODUCTION the image
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8 CHAPTER 1. INTRODUCTION disadvant
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10 CHAPTER 1. INTRODUCTION 3. A gen
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Page 58 and 59: Chapter 4 The “trench” problem
Page 60 and 61: 46CHAPTER 4. THE “TRENCH” PROBL
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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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