Thesis (PDF) - Signal & Image Processing Lab

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4 List of Abbreviations and Symbols SE Structuring element RAG Region Adjacency Graph AS Alternating Sequences BTD Boundary Topographic Distance BTVT Boundary Topographic Variation Transform OCR Optical Character Recognition EWT Extrema Watershed Tree Z Set of integers E Euclidian space f, g, f(x), g(x) Functions from an Euclidian space to the reals τ Tree Transform t Tree T Tree Representation M Mapping function L Set of “labels” V Vertex of a tree E Edge connecting two vertices of a tree ⊆ Contained or equal ∪ Union ∩ Intersection ≤, �, ⊑, � Order relation in a complete semilattice ∧, �, ⊓, △ Infimum in a complete semilattice ∨, �, ⊔, ▽ Supremum in a complete semilattice δ Dilation in a complete semilattice ε Erosion in a complete semilattice γ Opening in a complete semilattice φ Closing in a complete semilattice Ψ Arbitrary operator ∆ Gray levels delta
Chapter 1 Introduction 1.1 Self-Duality Morphological operators are widely used for image analysis and processing. The clas- sical mathematical morphology theory was originally developed by Matheron and Serra [1], and later generalized to a complete-lattice framework by Serra [2]. A prac- tical description of mathematical morphology and its uses can be found in Soille’s book [3]. An extension of mathematical morphology from complete lattices to complete semilattices was developed by Keshet [4], and later further studied by Heijmans and Keshet in [5]. Most of existing morphological operators in complete lattices have the major dis- advantage of not being self-dual. A grayscale operator ψ is called self-dual when ψ(f) = −ψ(−f). Operators that are not self-dual typically do not treat bright and dark “objects” in a similar way, which is often an undesirable feature. For example, consider the problem of denoising the simple binary image in Fig. 1.1(a). A simple erosion operation (as a first step of an opening) does remove the bright components of the noise, but the dark components are actually dilated, as seen in Fig. 1.1(b). The standard approaches of opening-closing or closing-opening approxi- mately achieve self-duality, but are not actually self-dual; the filtering results depend on the order in which the opening and closing operators are applied. For instance, opening removes bright noise, but damages fine bright details of the image. Closing after opening can not restore the details damaged by the opening. In addition, lack of symmetry in the treatment of bright and dark components often causes damage to 5
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 20 and 21: 6 CHAPTER 1. INTRODUCTION the image
Page 22 and 23: 8 CHAPTER 1. INTRODUCTION disadvant
Page 24 and 25: 10 CHAPTER 1. INTRODUCTION 3. A gen
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Page 28 and 29: 14 CHAPTER 2. THEORETICAL BACKGROUN
Page 52 and 53: 38 CHAPTER 3. IMPLEMENTATION OF BTV
Page 58 and 59: Chapter 4 The “trench” problem
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54CHAPTER 4. THE “TRENCH” PROBL
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Chapter 5 Tree Semilattices In this
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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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