# Mathematical Morphology: From Theory to Applications

ISBN: 978-1-118-60078-8

Feb 2013, Wiley-ISTE

507 pages

Select type: O-Book

## 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.

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

2.6. 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

6.6. 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

20.3. 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