Thesis (PDF) - Signal & Image Processing Lab

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24 CHAPTER 2. THEORETICAL BACKGROUND Figure 2.4: Shape decomposition and associated Tree of shapes. (a) Original grayscale image f and (b)-(f) its shapes. The white shapes correspond to connected components of the collection of upper levels sets {θt(f)}, whereas the black shapes correspond to connected components of the collection of lower level sets {θt(f) c }.
2.4. SHAPE TREE SEMILATTICE BACKGROUND 25 Figure 2.5: Tree of shapes associated with Fig. 2.4. (a) Original tree; the numbers in parenthesis are the gray-levels of the shapes. (b) Modified tree; the number in parenthesis are the differences s(τ) between the levels of each shape τ and its parent p(τ). Notice that shape F was duplicated. original image, up to a local change of contrast. In order to perfectly reconstruct the original image, one can also store the original gray levels of the shapes [as shown in Fig. 2.5(a)], or modify the tree by replicating the shapes for all gray-levels they appear [as shown in Fig. 2.5(b)]. In the latter situation (which is the one adopted here), f(x) is equal to the sum of s(τ) for all shapes that contain τ(x). 2.4.2 Inf-Semilattice of Binary Sequences Keshet defines in [17] a partial ordering of images that takes into consideration the data of the tree of shapes. However, he does not use the tree itself in order to provide such ordering. He proposes a representation that keeps the datum of the tree readily available, but makes it simpler to compare images, and develops an inf-semilattice of binary sequences. Consider the set S formed by the null sequence and all finite binary sequences, with elements in {+1, −1}. Definition 7. Let s = {sn}, r = {rn} ∈ S. Sequence s is a prefix of sequence r, marked s � r, when ℓ(s) ≤ ℓ(r) and sn = rn, n = 1, . . . , ℓ(s), where ℓ(s) is the length
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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Page 60 and 61: 46CHAPTER 4. THE “TRENCH” PROBL
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76 CHAPTER 6. EXTREMA-WATERSHED TRE
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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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