Mathematical Morphology

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1 Brochure More information from Mathematical Morphology Description: Mathematical Morphology allows for the analysis and processing of geometrical structures using techniques based on the fields of set theory, lattice theory, topology, and random functions. It is the basis of morphological image processing, and finds applications in fields including digital image processing (DSP), as well as areas for graphs, surface meshes, solids, and other spatial structures. This book presents an up to date treatment of mathematical morphology, based on the three pillars that made it an important field of theoretical work and practical application: a solid theoretical foundation, a large body of applications and an efficient implementation. The book is divided into five parts and includes 20 chapters. The five parts are structured as follows: - Part I sets out the fundamental aspects of the discipline, starting with a general introduction, followed by two more theory focused chapters, one addressing its mathematical structure and including an updated formalism, which is the result of several decades of work. - Part II extends this formalism to some non deterministic aspects of the theory, in particular detailing links with other disciplines such as stereology, geostatistics and fuzzy logic. - Part III addresses the theory of morphological filtering and segmentation, featuring modern connected approaches, from both theoretical and practical aspects. - Part IV features practical aspects of mathematical morphology, in particular how to deal with color and multivariate data, links to discrete geometry and topology, and some algorithmic aspects; without which applications would be impossible. - Part V showcases all the previously noted fields of work through a sample of interesting, representative and varied applications. Contents: Preface xv PART I. FOUNDATIONS 1 Chapter 1. Introduction to Mathematical Morphology 3 Laurent NAJMAN, Hugues TALBOT 1.1. First steps with mathematical morphology: dilations and erosions Morphological filtering Residues Distance transform, skeletons and granulometric curves Hierarchies and the watershed transform Some concluding thoughts 33 Chapter 2. Algebraic Foundations of Morphology 35 Christian RONSE, Jean SERRA 2.1. Introduction Complete lattices Examples of lattices Closings and openings Adjunctions 56

2 2.6. Connections and connective segmentation Morphological filtering and hierarchies 75 Chapter 3.Watersheds in Discrete Spaces 81 Gilles BERTRAND, Michel COUPRIE, Jean COUSTY, Laurent NAJMAN 3.1. Watersheds on the vertices of a graph Watershed cuts: watershed on the edges of a graph Watersheds in complexes 101 PART II. EVALUATING AND DECIDING 109 Chapter 4. An Introduction to Measurement Theory for Image Analysis 111 Hugues TALBOT, Jean SERRA, Laurent NAJMAN 4.1. Introduction General requirements Convex ring and Minkowski functionals Stereology and Minkowski functionals Change in scale and stationarity Individual objects and granulometries Gray level extension As a conclusion 130 Chapter 5. Stochastic Methods 133 Christian LANTUÉJOUL 5.1. Introduction Random transformation Random image 138 Chapter 6. Fuzzy Sets and Mathematical Morphology 155 Isabelle BLOCH 6.1. Introduction Background to fuzzy sets Fuzzy dilations and erosions from duality principle Fuzzy dilations and erosions from adjunction principle Links between approaches Application to the definition of spatial relations Conclusion 176 PART III. FILTERING AND CONNECTIVITY 177

3 Chapter 7. Connected Operators based on Tree Pruning Strategies 179 Philippe SALEMBIER 7.1. Introduction Connected operators Tree representation and connected operator Tree pruning Conclusions 198 Chapter 8. Levelings 199 Jean SERRA, Corinne VACHIER, Fernand MEYER 8.1. Introduction Set theoretical leveling Numerical levelings Discrete levelings Bibliographical comment 227 Chapter 9. Segmentation,Minimum Spanning Tree and Hierarchies 229 Fernand MEYER, Laurent NAJMAN 9.1. Introduction Preamble: watersheds, floodings and plateaus Hierarchies of segmentations Computing contours saliency maps Using hierarchies for segmentation Lattice of hierarchies 258 PART IV. LINKS AND EXTENSIONS 263 Chapter 10. Distance, Granulometry and Skeleton 265 Michel COUPRIE, Hugues TALBOT Skeletons Skeletons in discrete spaces Granulometric families and skeletons Discrete distances Bisector function Homotopic transformations Conclusion 289 Chapter 11. Color and Multivariate Images 291 Jesus ANGULO, Jocelyn CHANUSSOT

4 11.1. Introduction Basic notions and notation Morphological operators for color filtering Mathematical morphology and color segmentation Conclusion 320 Chapter 12. Algorithms for Mathematical Morphology 323 Thierry GÉRAUD, Hugues TALBOT, Marc VAN DROOGENBROECK Introduction Translation of definitions and algorithms Taxonomy of algorithms Geodesic reconstruction example Historical perspectives and bibliography notes Conclusions 352 PART V. APPLICATIONS 355 Chapter 13. Diatom Identification with Mathematical Morphology 357 Michael WILKINSON, Erik URBACH, Andre JALBA, Jos ROERDINK Introduction Morphological curvature scale space Scale space feature extraction D size shape pattern spectra Datasets Results Conclusions 365 Chapter 14. Spatio temporal Cardiac Segmentation 367 Jean COUSTY, Laurent NAJMAN, Michel COUPRIE 14.1.Which objects of interest? How do we segment? Results, conclusions and perspectives 372 Chapter 15. 3D Angiographic Image Segmentation 375 Benoît NAEGEL, Nicolas PASSAT, Christian RONSE Context Anatomical knowledge modeling Hit or miss transform Application: two vessel segmentation examples 378

5 15.5. Conclusion 383 Chapter 16. Compression 385 Beatriz MARCOTEGUI, Philippe SALEMBIER Introduction Morphological multiscale decomposition Region based decomposition Conclusions 391 Chapter 17. Satellite Imagery and Digital ElevationModels 393 Pierre SOILLE Introduction On the specificity of satellite images Mosaicing of satellite images Applications to digital elevation models Conclusion and perspectives 405 Chapter 18. Document Image Applications 407 Dan BLOOMBERG, Luc VINCENT Introduction Applications 409 Chapter 19. Analysis and Modeling of 3D Microstructures 421 Dominique JEULIN Introduction Models of random multiscale structures Digital materials Conclusion 444 Chapter 20. Random Spreads and Forest Fires 445 Jean SERRA Introduction Random spread Forecast of the burnt zones Discussion: estimating and choosing Conclusions 454 Bibliography 457 List of Authors 499 Index 501

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