Generalisations of the Algebraic Theory Of

GENERALISATIONS OF THE ALGEBRAIC THEORY OF CONNECTIVE SEGMENTATION Seidon Alsaody1 Project Report for the Degree of Master of Science ... Supervisor: Prof. Em. Jean Serra, ESIEE Paris Examiner: Prof. Volodymyr Mazorchuk, Uppsala University ... Spring Semester 2010 1(Visiting) Universite´ Paris-Est, Laboratoire d’Informatique Gaspard-Monge, Equipe AS3I, ESIEE Paris (2, boulevard Blaise Pascal, Cite´ Descartes, BP 99, 93162 Noisy le Grand Cedex, France). Abstract We begin by defining the set-up and the framework of connective segmentation. Then we start from a theorem based on connective criteria established for the lattice of subsets of an arbitrary set and formulate and prove its analogue for more general lattices. This also requires generalising the underlying definitions. Secondly, we consider partial partitions and partial connections. We recall the definitions, and quote a result that gives a characterisation of (partial) connections. We link this approach to the above one by means of a commutative diagram; then, as a continuation of the work in the first part, we state generalised versions of the concepts and results that underlie this characterisation, thereafter giving the characterisation in the generalised setting as well. Finally we generalise the commutative diagram to the general setting in two manners. Contents 1 Introduction 2 1.1 Outline of the Paper . 2 1.2 Origin of the Problems . 3 1.3 Note on Terminology and Notation . 3 1.4 Fundamental Conventions . 3 1.5 Note on the References . 3 1.6 The Author’s Acknowledgements . 4 1.7 Basic Definitions . 4 1.7.1 Some Examples . 6 1.8 Fundamentals of Connective Segmentation . 7 2 The Characterisation Theorem of Connective Segmentation 9 2.1 Serra’s Theorem . 9 2.2 First Generalisation: Viscous Lattices . 9 2.2.1 Further Definitions . 9 2.2.2 Adapting the Set-Up . 12 2.2.3 Generalising the Theorem to Viscous Lattices . 13 2.3 Main Generalisation . 15 2.3.1 Adapting the Set-Up . 16 2.3.2 Step I: Atomistic Lattices . 17 2.3.3 Step II: (Generally) Non-Atomistic Lattices . 19 3 Partial Connections & Partial Partitions 21 3.1 Main Definitions . 22 3.2 Openings on Partial Partitions and Ronse’s Corollary . 23 3.3 Relating Ronse’s Approach to Serra’s . 24 4 Generalising Ronse’s Corollary 25 4.1 Basic Concepts . 26 4.2 Underlying Results . 27 4.3 The Corollary Generalised . 33 5 Relating Serra’s & Ronse’s Approaches in General Lattices 34 5.1 Maintaining the Non-Partial Setting . 35 5.2 Using Partiality . 36 6 Concluding Remarks 37 A Appendix: On Partitions, Partial Partitions and Partial Connections 39 B Appendix: On an Augmentation to the Theory 41 B.1 The Non-Partial Setting . 41 B.2 The Partial Setting . 43 1 1 Introduction 1.1 Outline of the Paper This paper lies within the field of complete lattices in the theory of partially ordered sets. We start by recalling some basic properties, and then consider the collection of all subsets of a given set, and recognise its proprties of a complete lattice. We then pass to the theory of mathematical morphology1. Two themes, namely the notion of connectivity and that of a partition, are central in this work, and their definitions are quoted. We also quote the definition of a criterion, and thus arrive by the end of Section 1 at the apparatus of connective segmentation. This theory has been introduced and studied thoroughly by Serra and, among others, Ronse. Having done this, we generalise in Section 2 a theorem of Serra that characterises segmentation in terms of so called connective criteria, in that we replace the subset lattice (to be understood as the lattice of all subsets of a given set) by an arbitrary complete lattice. The first step is the case of viscous lattices, which are images of the subset lattice under a certain class of maps. The importance of this case is that such lattices lack several properties that hold for the subset lattice. To this end, we generalise the set-up and prove an analogue of the original theorem, thus establishing that it is, in this sense, independent of these properties. We then take the final step of generalising the set-up, now to the case of arbitrary complete lattices, and state the theorem. As the reader will see, the theorem holds in its original sense in a large sub-class of such lattices, namely atomistic lattices (defined below); however, for the most general case, we prove a weaker version, as well as present an example showing that the original version of the theorem does not generalise as far. In Section 3, we present an approach to segmentation that uses a broader notion, namely so called partial connectivity instead of connectivity. This comprises the introduction of partial partitions; indeed, it has been shown that partial connections are characterised by a class of operators on partial partitions. An important aim of Section 3 is to demonstrate, through a commutative diagram, the link between this result (obtained by Ronse) and the theorem of Serra from Section 2. This section is entirely concerned with the case of the subset lattice. Motivated by the generalisation in Section 2 and the link of Section 3, section 4 is dedicated to generalising the set-up of partial connections and partial partitions to arbitrary complete lattices. Ronse’s result is generalised as well; it holds in full strength even in the most general case, due to the notion of partial partitions. Having done this, we return in Section 5 to the commutative diagram of Section 3, aim- ing at generalising it as well. The treatment of this section consists of two parts. In the first, we generalise the diagram as faithfully as possible, whereas in the second, we use the adaptation of Serra’s theorem into partial connections and partial partitions [4]; as we will have seen in Section 4, this adaptation generalises completely to arbitrary complete lattices. Thus we reformulate it in terms of connective criteria, and produce the commutative diagram in question. The first Appendix is dedicated to the proofs that the sets of partitions, partial partitions, and partial connections, as defined in the general setting, are all complete lattices. These 1See section 1.2 2 results underly various arguments in different sections of this work. The second appendix generalises an augmentation made in [4] to the theory of connective segmentation. 1.2 Origin of the Problems The notion of connective segmentation, introduced in Definitions 5–9 below, is a subfield of the field of mathematical morphology, which is, briefly, the study and analysis of images, using a lattice-theoretic framework. Image segmentation consists of subdividing, or segment- ing, an image into its different components, e.g. foreground and background. This is the practical application of the concepts studied below, and hopefully explains the terminology. The principle of mathematical morphology is to express the problems of image analysis in mathematical terms, using this theoretical set-up. It should be stressed that this is a paper in mathematics and hence is entirely concerned with the theoretical apparatus of mathematical morphology, in particular its algebraic parts. The applications into image analysis are outside the scope of this paper, and the interested reader is recommended to consult the appropriate literature listed in the bibliography. 1.3 Note on Terminology and Notation In the field under study, it sometimes occurs that different terminologies are used, and many notions have more than one name. The choices made in this work aim at maximising legibility and accuracy. Whenever possible, alternative terminology will be provided, in brackets, in the definitions. For more details on differences of terminology, Ronse [5] has made an extensive account. 1.4 Fundamental Conventions Throughout this paper, all sets considered, that are not explicitely stated to be subsets of other sets (i.e. all “ground sets”, usually denoted by E) are assumed to be non-empty, unless otherwise stated. Analogously, all complete lattices (to be defined below) that do not arise from a construction on other complete lattices (i.e. all “ground complete lattices”, usually denoted by L) are assumed to contain at least two different elements. 1.5 Note on the References At the formulation of Definitions 19, 24 and 26, the author was unaware of the existence of the articles [2] and [6]. Thus, those definitions were made independently of these sources, which explains the parantheses at each occurrence. The articles are however listed as references since they were, to the author’s knowledge, the first to introduce the notions of Definitions 24 and 26. 3 1.6 The Author’s Acknowledgements I am most grateful to Professor Emeritus Jean Serra for his invitation and efforts in order for this Master Thesis project to take place, as well as for being attentive to my research interests, and supervising and nourishing the work in progress with valuable remarks and ideas, and I am moreover much indebted for his warm welcoming and friendliness during the progress of the work. I would further like to express my deep gratitude to Doctor Hugues Talbot for arranging the financial support without which this work could not have been done, and for steadily and patiently being willing to help. I also wish to express my thankfulness to the entire team at ESIEE Paris among whose members I have spent a very enriching period of time. Moreover, I would like to thank Professor Christian Ronse for bringing the sources [2] and [6] to my attention. I am sincerely grateful to Professor Volodymyr Mazorchuk for undertaking the responsi- bility of examining this Master Thesis, and for offering me the possibility to give a presenta- tion at the Algebra and Geometry Seminar at Uppsala University.

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