Thesis (PDF) - Signal & Image Processing Lab

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6 CHAPTER 1. INTRODUCTION the image edges, as can be seen in Figs. 1.1(c) and 1.1(d). The most popular approach to approximately achieve self-duality in mathematical morphology is to use alternating dual operators. For instance, opening-closing and closing-opening filters, or the more general alternating sequential filters (ASF’s) [1, 3]. However, these are not really self-dual, and some of the difficulties related to the lack of self-duality still appear (as in the above example of Fig. 1.1). Development of self-dual morphological operators can be found, for instance, in the works of Serra, [1, chapter 8], Heijmans [6], and Mehnert & Jackway [7]. Al- though most of these operators are morphological filters (i.e., they are idempotent and increasing, see [1, 3]), the underlying approaches do not provide for adjunctions, or designing extensive or anti-extensive operators, see [1, 3]. As a consequence, the ability to design basic morphological operators (erosions, dilations, openings, and closings) is lost. Useful morphological tools, such as skeletons, granulometries and gradients [1, 3], are not available either. Another important branch of research is that of self-dual connected operators. Connected operators remove objects from an image without affecting the edges of the remaining objects. In [8], Heijmans provides a summary of the activity in this area, and characterizes in depth binary connected operators. A summary and char- acterization of grayscale connected operators can be found in the works of Serra and Salembier [9]. A subclass of connected operators are operators based on tree representations. Salembier and Garrido proposed a Binary Partition Tree for hierarchical segmentation in [10, 11]. A tree of shapes was proposed by Monasse and Guichard [12, 13] (see also [14, 15]). These representations are all self-dual and very powerful. However, the above methods do not define non-connected morphological operators, such as an erosion. 1.2 Background and motivation A new semilattice, based on the so-called Boundary-Topographic-Variation (BTV) Transform, was defined by Keshet in [16]. The importance of that new semilattice is that it allows to define self-dual non-connected morphological operators (like openings, erosions, etc.) without need of a reference image. In contrast, the other morphological approaches have one or more of the following
1.2. BACKGROUND AND MOTIVATION 7 (a) (b) (c) (d) Figure 1.1: (a) A noisy binary image, (b) Erosion by a 3 × 3 squared structuring element, (c) The result of opening-closing with the same s.e., and (d) The result of closing-opening with the same s.e.
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
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56CHAPTER 4. THE “TRENCH” PROBL
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Chapter 5 Tree Semilattices In this
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64 CHAPTER 5. TREE SEMILATTICES sec
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66 CHAPTER 5. TREE SEMILATTICES Fig
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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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