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.

Table of 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 4 1.2. Morphological filtering 12 1.3. Residues 22 1.4. Distance transform, skeletons and granulometric curves 24 1.5. Hierarchies and the watershed transform 30 1.6. Some concluding thoughts 33 Chapter 2. Algebraic Foundations of Morphology 35 Christian RONSE, Jean SERRA 2.1. Introduction 35 2.2. Complete lattices 36 2.3. Examples of lattices 42 2.4. Closings and openings 51 2.5. Adjunctions 56 2.6. Connections and connective segmentation 64 2.7. 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 82 3.2. Watershed cuts: watershed on the edges of a graph 90 3.3. 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 111 4.2. General requirements 112 4.3. Convex ring and Minkowski functionals 113 4.4. Stereology and Minkowski functionals 119 4.5. Change in scale and stationarity 121 4.6. Individual objects and granulometries 122 4.7. Gray-level extension 128 4.8. As a conclusion 130 Chapter 5. Stochastic Methods 133 Christian LANTUÉJOUL 5.1. Introduction 133 5.2. Random transformation 134 5.3. Random image 138 Chapter 6. Fuzzy Sets and Mathematical Morphology 155 Isabelle BLOCH 6.1. Introduction 155 6.2. Background to fuzzy sets 156 6.3. Fuzzy dilations and erosions from duality principle 160 6.4. Fuzzy dilations and erosions from adjunction principle 165 6.5. Links between approaches 167 6.6. Application to the definition of spatial relations 170 6.7. Conclusion 176 PART III. FILTERING AND CONNECTIVITY 177 Chapter 7. Connected Operators based on Tree Pruning Strategies 179 Philippe SALEMBIER 7.1. Introduction 179 7.2. Connected operators 181 7.3. Tree representation and connected operator 182 7.4. Tree pruning 187 7.5. Conclusions 198 Chapter 8. Levelings 199 Jean SERRA, Corinne VACHIER, Fernand MEYER 8.1. Introduction 199 8.2. Set-theoretical leveling 200 8.3. Numerical levelings 209 8.4. Discrete levelings 214 8.5. Bibliographical comment 227 Chapter 9. Segmentation,Minimum Spanning Tree and Hierarchies 229 Fernand MEYER, Laurent NAJMAN 9.1. Introduction 229 9.2. Preamble: watersheds, floodings and plateaus 230 9.3. Hierarchies of segmentations 237 9.4. Computing contours saliency maps 252 9.5. Using hierarchies for segmentation 255 9.6. Lattice of hierarchies 258 PART IV. LINKS AND EXTENSIONS 263 Chapter 10. Distance, Granulometry and Skeleton 265 Michel COUPRIE, Hugues TALBOT 10.1. Skeletons 265 10.2. Skeletons in discrete spaces 269 10.3. Granulometric families and skeletons 270 10.4. Discrete distances 275 10.5. Bisector function 279 10.6. Homotopic transformations 280 10.7. Conclusion 289 Chapter 11. Color and Multivariate Images 291 Jesus ANGULO, Jocelyn CHANUSSOT 11.1. Introduction 291 11.2. Basic notions and notation 292 11.3. Morphological operators for color filtering 299 11.4. Mathematical morphology and color segmentation 312 11.5. Conclusion 320 Chapter 12. Algorithms for Mathematical Morphology 323 Thierry GÉRAUD, Hugues TALBOT, Marc VAN DROOGENBROECK 12.1. Introduction 323 12.2. Translation of definitions and algorithms 324 12.3. Taxonomy of algorithms 329 12.4. Geodesic reconstruction example 334 12.5. Historical perspectives and bibliography notes 344 12.6. Conclusions 352 PART V. APPLICATIONS 355 Chapter 13. Diatom Identification with Mathematical Morphology 357 Michael WILKINSON, Erik URBACH, Andre JALBA, Jos ROERDINK 13.1. Introduction 357 13.2. Morphological curvature scale space 358 13.3. Scale-space feature extraction 359 13.4. 2D size-shape pattern spectra 359 13.5. Datasets 364 13.6. Results 364 13.7. Conclusions 365 Chapter 14. Spatio-temporal Cardiac Segmentation 367 Jean COUSTY, Laurent NAJMAN, Michel COUPRIE 14.1.Which objects of interest? 368 14.2. How do we segment? 369 14.3. Results, conclusions and perspectives 372 Chapter 15. 3D Angiographic Image Segmentation 375 Benoît NAEGEL, Nicolas PASSAT, Christian RONSE 15.1. Context 375 15.2. Anatomical knowledge modeling 376 15.3. Hit-or-miss transform 378 15.4. Application: two vessel segmentation examples 378 15.5. Conclusion 383 Chapter 16. Compression 385 Beatriz MARCOTEGUI, Philippe SALEMBIER 16.1. Introduction 385 16.2. Morphological multiscale decomposition 385 16.3. Region-based decomposition 389 16.4. Conclusions 391 Chapter 17. Satellite Imagery and Digital ElevationModels 393 Pierre SOILLE 17.1. Introduction 393 17.2. On the specificity of satellite images 394 17.3. Mosaicing of satellite images 398 17.4. Applications to digital elevation models 400 17.5. Conclusion and perspectives 405 Chapter 18. Document Image Applications 407 Dan BLOOMBERG, Luc VINCENT 18.1. Introduction 407 18.2. Applications 409 Chapter 19. Analysis and Modeling of 3D Microstructures 421 Dominique JEULIN 19.1. Introduction 421 19.3. Models of random multiscale structures 431 19.4. Digital materials 440 19.5. Conclusion 444 Chapter 20. Random Spreads and Forest Fires 445 Jean SERRA 20.1. Introduction 445 20.2. Random spread 448 20.3. Forecast of the burnt zones 451 20.4. Discussion: estimating and choosing 453 20.5. Conclusions 454 Bibliography 457 List of Authors 499 Index 501