Analysis and Visualization of Complex 3D Structures_JorgeMarquez

Analysis and Visualization of Complex 3D Structures:a discrete boundary-based approach

Analyse et visualisation des structures complexes en 3D :une approche par frontières discrètes.

par

Jorge Alberto Marquez Flores

Rapport de soutenance de thèse -PhD. Dissertation

Soutenu le 6 avril 1999 devante le jury composé de :Isabelle Magnin RapporteurChristine Graffigne RapporteurJean-Marie Rocchisani ExaminateurGilles Bertrand PrésidentFrancis Schmitt Directeur de thèse

Departement du Traitement du Signal et des ImagesEcole Nationale Superieure des Telecommunications

c� JORGE A. MARQUEZ PARIS, FRANCE, LE 6 AVRIL 1999

Abstract

Keywords: 3D-image processing, morphometry, discrete boundaries, voxels, structural complexity,Visible Human Project, amorphous deposition, scientific visualization.

The aim of this work was the segmentation, morphometry and visualization of three-dimensionalcomplex structures, in particular porous micro-structures and blood vessels with complex branches.We studied the different problems arising in the analysis of this kind of volumic data, fromcomplex-features definitions, as described in Chapter 1, todiscrete boundary representations (Chapter 2) andspecific applications (Chapters 3, 4 and 5).

As a first step, we identified and characterized morphological features of complex structuresrelated to specific problems, e.g.: coherency, tubular structures, ramifications, fractal dimension,interfaces, porosity, roughness and others. Then, we choose a discrete geometry approach for surfaceextraction, processing and visualization of volume data. In this approach, described in Chapter 2,external surfaces or boundaries of detected objects are represented by lists of facets (voxel faces).An important amount of 3D image processing consists in analyzing and modifying these lists andtheir associated data structures by examining neighboring facets and voxels, in order to performthe desired operation or measurement. Boundaries are then extracted by traversal of the directedgraph associated with the surface of each object. We then introduce several entities and tools foranalysis; in particular: facet, voxel and boundary neighborhoods, geodesic neighborhoods, geometricinterior traversal and boundary-based filters for mathematical morphology operations. Some of theseoperators (or subsets) are activated (or extracted) during graph traversal; we describe, for example,how the local facet configuration allows estimations of local normal and local surface area.

Concerning the application domains, we have worked on two 3D-image datasets, correspond-ing to different issues in Biomedicine and Physics. The generality and versatility of the boundaryrepresentation model allowed to use the same software tools for both subjects.

The first application, described in Chapter 3, was developed with a biomedical dataset from theVisible Human Project (VHP) of the National Library of Medicine, comprising both lungs. Anatom-ical cryo-sections of the VHP posed special problems of segmentation and visualization of internalstructures, hence, we developed an innovative technique for correcting regional radiometric inhomo-geneities. We also extracted the vascular pulmonary tree, as described in Chapter 4, and obtainedrealistic visualizations after different stages of segmentation and 3D image processing, using thesame boundary representation introduced earlier.

The second application (Chapter 5) comes from a collaboration with the Surface Physics labo-ratory of the Centre Nationale d’Etudes en Télecommunications (CNET), at Bagneux. The problemconsisted in measuring properties of porosity, roughness, connectivity and distribution of morpho-logical parameters extracted from computer simulations of amorphous photo-deposition of Nitride-Silicium. Our work included the 3D reconstruction of volume data, connected component analysisand morphometrical parameter extraction. Visualization and quantification of these data allowed re-searchers of the CNET to assess: the size and form of pores and deposition clusters, the porositydistribution in function of temperature and the inter-penetration quality of several atom species. 3Drenderings enable to notice critical transitions of the cluster morphology, clearly organized by colorlabels.

Many of the techniques we developed and their application took advantage of the tools of ourboundary-based approach as well as of several concepts and features of complex structures presentedin the first part.

Resume

Mots cle : Traitement d’image 3D, morphometrie, frontieres discretes, complexite structurale, Visible HumanProject, depots amorphes, visualisation scientifique.

L’objectif de ce travail est la segmentation, la morphométrie et la visualisation de structures com-plexes tridimensionnelles, en particulier des microstructures poreuses et des vaisseaux sanguins quipresentent des ramifications compliquées. Nous avons étudie ainsi les differents problèmes poséspar l’analyse de ce type de données volumiques, comprenant les définitions desattributs de com-plexite examinees dans le chapitre 1, lesrepresentations par frontieres discretes (chapitre 2) et desapplications spécifiques (chapitre 3, 4 et 5).

L’ etude de tels problèmes nous a amenés d’abord à identifier eta caractériser les attributs mor-phologiques des structures complexes, tels que : cohérence, structures tubulaires, ramifications, di-mension fractale, interfaces, porosité, rugosite et d’autres. Ensuite, nous avons choisi une approcheen geometrie discrete pour l’extraction des surfaces, le traitement et la visualisation des donnéesvolumiques. Dans cette approche (chapitre 2), les surfaces extérieures ou frontières des objets dé-tectes sont représentees par des listes de facettes. Une partie importante des traitements consiste àanalyser et modifier ces listes et les structures de données associées, en examinant les facettes et lesvoxels voisins pour effectuer les opérations ou mesures nécessaires. L’extraction de telles frontièresse fonde sur une méthode de parcours du graphe dirigé associé a la surface de chaque objet. Diversoutils sont alors introduits, tels que les voisinages des facettes et des frontières, les voisinages géo-desiques, le parcours de l’intérieur geometrique et des filtres de morphologie mathématique fondéssur la representation par frontière.

En ce qui concerne les domaines d’applications, nous avons travaillé sur deux ensembles de don-nees volumiques correspondant à deux problématiques différentes respectivement en biomédecine eten physique. La géneralite et la souplesse du modèle de représentation par frontières nous a permisd’utiliser les memes outils informatiques pour les deux problématiques.

La premiere application a éte developpee sur des données biomédicales correspondant aux pou-mons et provenant du Visible Human Project (VHP) de la National Library of Medicine. Les coupesanatomiques du VHP posent des problèmes particuliers pour la segmentation et la visualisationdes structures internes et nous avons ainsi développe une technique novatrice pour la correctiond’heterogeneites radiométriques regionales. Nous avons obtenu des visualisations réalistes de l’arbrevasculaire pulmonaire obtenu à la suite des différentes étapes de la segmentation et du traitement 3D,toujours en utilisant la représentation par frontières.

La deuxieme application a pu être abordée dans le cadre d’une collaboration avec le laboratoirede Physique des Surfaces du Centre Nationale d’Etudes en Télecommunications, à Bagneux. Leprobleme a consisté a mesurer les propriétes de porosité, rugosite, connexité et distribution des pa-rametres morphométriques extraits des résultats obtenus par simulation numérique d’un photo-dépotamorphe de Nitride-Silicium. Nous avons effectué des reconstructions 3D, l’identification des com-posantes connexes et l’extraction des paramètres morphométriques. Les visualisations et les mesuresobtenues ont permis aux chercheurs du CNET de connaˆıtre : la taille et la forme des pores, la tailleet la forme des amas du dépot, la distribution de la porosité par rapport à la temperature, ainsi quela qualite d’interpenetration de plusieurs espèces d’atomes. En outre, les rendus 3D permettront dereperer des transitions critiques de la morphologie des amas, identifiés clairement par des étiquettesen couleur.

Au total, plusieurs des techniques développees et les résultats présentes profitent des outils denotre approche par frontières, ainsi que des concepts et attributs des structures complexes étudies.

Acknowledgements - Remerciements - Agradecimientos.

This page mentions people at the roots, branches, and fruits of my PhD research. I include thosefriends or acquitances who, even without knowing, have contributed in no obvious manners to thiswork, ranging from academic interactions to personal relationships. My apologizes to those I mayhave forgotten.

First of all, I am in great debt withFrancis Schmitt, my thesis advisor andIsabelle Bloch, whoco-directed this work. They were always patient, critic and available for any discussion on all topics,and encouraged me at all times.Henry Ma ıtre was also a constant presence. Thanks to him, aswell as Pierre Duhamel for their hospitality in the former Image Department and the present TSIDepartment.

Thanks to the jury members for their criticism and advice:Isabelle Magnin, Christine Graf-figne, Gilles Bertrand and Jean-Marie Rocchisani.

Thanks

✦ To Sandrine Pata and Jean Flicstein (CNET-Bagneux) both of whom encouraged a fruitfulcollaboration and created the deposition simulation software described in Chapter 5.1. I alsoaknowledgeMarc Sigelle for continuous discussion on various subjects on mathematics, ran-dom fields, deposition and textures.

✦ For fruitful discussions, comments and insights since 1994, to: Michel Roux, Lars Aurdal,Yon Harderberg, Bert Verdonck, wonderfulFlorence Tupin, Hilmi Rifai, Raouf Benjemaa,Emanuel Trouve, Stephane Chauvin, Olivier Coulon, Sudha Kunduri, Alex Winter, MiguelMoctezuma, Ma.Elena Algorri, Geneviève Dardier, Sophie Paquerault, Anne Robert, HansBrettel, Jean-Pierre Veran, Mehdy Eyvazkhani, Aymeric Perchant, Wirawan, Yucel Yemez,Jean-Pierre Crettez, all from the ENST, and Seth Hutchinson (University of Illinois). Thanksalso to the wizards of the ”Tivoli” software team: Thierry Géraud, Yan Cointepa and DmitriPapanopoulus (plus some already mentioned). I spent a wonderful time at the Image Depart-ment!

✦ For their technical support, to: Alain Clainchard, Patrick Horain, Dominique Asselineau, ”lecentre de calcul de l’ENST” and the best secretary of the ENST: Patrica Friedrick.

✦ To Michel Ackerman and Richard Banvard (from The National Library of Medicine), whoprovided access to the Visible Human Project (VHP) dataset, remarks and advice for the specialwork on the VHP.

✦ For technical advice and discusion on discrete topology, voxels and visualization, to: Pe-tra Wiederhold (UAM-CINVESTAV), Klauss Voss (GMD-SCAI), Paolo Clementi (SiliconGraphics), Bernhard Pflesser (U. Hamburg) and Rafael Lacambra (LabVis-UNAM).

✦ From the Universidad Nacional Autónoma de Mexico, to: Andrés Buzo, Francisco Ugalde(signal and image processing professors), Cristina Piña (DEPFI/1982-1992),Francisco Cer-vantes (CI/1990-1993, ITAM, my former master thesis advisor in Mexico), and Jaime Pi-mentel (Centro de Instrumentos, UNAM -Cibertrol), all of them criticized, inspired or encour-aged some ideas that now constitute my line-work. In the same vein, thanks also to: Erick Petit,Laurent Cohen, J.L. Moretti and Jean-Marie Rochissani from the ”DEA en Génie Biologiqueet Medical” at Paris XIII.

✦ To friends and colleagues in Mexico, who have influenced, supported or inspired my workthese years in France in one way or another (approximated chronological order): Javier Sierra,Armando Solar and Manuel Estevez,Laura SanchezandDiana Servın (who also becamevery close friends), Carlos Yañez, José Saniger, Kent Brailovsky andGabriel Corkidi (whocontacted me with Etienne Hirsch in 1992 and signed as ’asesor nacional’), Mar´ıa Garza, Patri-cia Ostrovski and Ricardo Toledo -the five of us started our close encounters with applicationsof image processing, back in 1988, and coauthored many works, forming also a great team.

Grant formalities and other support are debt to: Comité de Becas del CIUNAM, Gerardo Ruiz(academic secretary), Claudio Firmani and Felipe Lara (directors of the CI), Maria Elena Hernándezand Patricia Vidal (DGAPA), Mauricette Teti, Stephane Bonenfant (ENST), and Patrick England(France Ambassy in Mexico).

Other friends who don’t appear above did contributed with other kind of support, and I sharedwith them 6+ years in Paris:Philippe AngladeandEtienne Hirsch (DEA advisors, from the Hôpi-tal de la Salpêtriere); Michel Gouéry, Michel Salsmann, Michaéle Schatt, ”Tara” and Cécile Theisen(from the Ecole de Beaux Arts), and finally, from the ENST or elsewhere: Asli and Emanuel, Miung,Claudia, Fabien, Ingeborg, Selim, Gérard, Cuahutemoc, Olga Odgers, Olga Sodolova, Céline, Car-men, Gabriel, Lorence, Florent, Muriel, Luc, Mariel, Marc, Mr. and Madame Tupin, Madame Dart,Benoit and Aline, Alicia, Plotr, Alfredo, Linda, Lety, Laura V., Pablo, and Göran.

This work is specially dedicated to my sisterCarmen and my best friends (alphabetically): Cé-cile, Chantal, Diana, Florence, Laura and Paty.

A la memoria de mi Madre,A mi Padre y mis hermanos,

Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1 Characterization of sets with complex structures inR� 21

1.1 Preliminar Notions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.1.1 Neighborhoods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.1.2 Boundaries and Interfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.1.3 Connected Sets and Objects . . .. . . . . . . . . . . . . . . . . . . . . . . 26

1.2 Intrinsic Properties of Solid Shapes. . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.2.1 The Approximating Ellipsoid, Isotropy and Deformed Shapes. . . . . . . . 28

1.2.2 Shape Factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1.2.3 Shape Decompositions and Local Descriptions. . . . . . . . . . . . . . . . 34

1.2.4 Convexity . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.2.5 Coherence .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

1.2.6 Scales of Representation and Characteristic Lengths .. . . . . . . . . . . . 42

1.3 Extrinsic Properties of Solid Shapes . . .. . . . . . . . . . . . . . . . . . . . . . . 45

1.3.1 Complex Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

1.3.2 Many Bodies and Clusters. . . . . . . . . . . . . . . . . . . . . . . . . . . 46

1.3.3 Complex Objects in Feature Space. . . . . . . . . . . . . . . . . . . . . . . 47

1.3.4 Complex Representations. . . . . . . . . . . . . . . . . . . . . . . . . . . 47

1.4 Fractal Dimensions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

1.4.1 Dimension Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

1.4.2 Morphological Features of Fractals. . . . . . . . . . . . . . . . . . . . . . 54

1.4.3 An Example: Particle Aggregates. . . . . . . . . . . . . . . . . . . . . . . 56

1.5 Notions and Measures of Complexity . .. . . . . . . . . . . . . . . . . . . . . . . 56

1.5.1 *Kolmogorov Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

1.5.2 Information Theory, Algorithmic Entropy and Scientific Observations . . . . 60

1.6 Shape Taxonomies .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

1.6.1 Tubular Structures (I). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

1.6.2 Ramified Thin Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

1.6.3 Surfaces, Boundaries and Interfaces. . . . . . . . . . . . . . . . . . . . . . 68

1.6.4 Description of Other Shape Features. . . . . . . . . . . . . . . . . . . . . . 69

1.7 Other Properties of Complex Information. . . . . . . . . . . . . . . . . . . . . . . 70

1.7.1 From Structure to Texture. . . . . . . . . . . . . . . . . . . . . . . . . . . 70

1.7.2 Noise, Blurring, Distorsion and Inhomogeneities . . .. . . . . . . . . . . . 71

1.8 Discussion and Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2 From Raw Data to Models inN�: A Boundary-Based Approach 77

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

2.2 Discrete Volume Representations and Boundaries. . . . . . . . . . . . . . . . . . . 80

2.2.1 Boundary Representations. . . . . . . . . . . . . . . . . . . . . . . . . . . 81

2.2.2 Relevance and Potential Uses of Discrete Boundary Representations. . . . . 82

2.3 A Facet-Based Model for Discrete Surface Representation . .. . . . . . . . . . . . 84

2.3.1 Discrete Elements: Scenes and Voxels. . . . . . . . . . . . . . . . . . . . . 84

2.3.2 Discrete Elements: Faces and Labels. . . . . . . . . . . . . . . . . . . . . 87

2.3.3 Discrete Elements: Facets and Boundaries. . . . . . . . . . . . . . . . . . . 89

2.3.4 Discrete Adjacency and Connectivity. . . . . . . . . . . . . . . . . . . . . 96

2.3.5 Voxel (Euclidean) Neighborhoods. . . . . . . . . . . . . . . . . . . . . . . 102

2.3.6 Facet (Geodesic) Neighborhoods .. . . . . . . . . . . . . . . . . . . . . . . 104

2.3.7 Voxel Geodesic Neighborhoods .. . . . . . . . . . . . . . . . . . . . . . . 106

2.3.8 Discrete Boundary Neighborhoods. . . . . . . . . . . . . . . . . . . . . . . 108

2.4 Surface Tracking . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

2.4.1 Surface Tracking Algorithm . . .. . . . . . . . . . . . . . . . . . . . . . . 114

2.4.2 Facet Traversal by Lateral Circuits. . . . . . . . . . . . . . . . . . . . . . . 117

2.4.3 Interior Traversal and Object Labeling. . . . . . . . . . . . . . . . . . . . . 119

2.4.4 Run-Length Interpretation of Interior Traversal. . . . . . . . . . . . . . . . 124

2.5 Boundary-Based Processing and Analysis of 3D Objects . . .. . . . . . . . . . . . 126

2.5.1 An Example of Boundary-Based Morphometry: Euclidean Area Estimation . 128

2.5.2 Boundary-Based and Volume Processing (Examples) .. . . . . . . . . . . . 129

2.6 Visualization of Complex Structures . . .. . . . . . . . . . . . . . . . . . . . . . . 135

2.6.1 Surface Rendering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

2.6.1.1 Hidden Surface Removal and Surface Shading. . . . . . . . . . . 135

2.6.1.2 Discrete Normal Estimations. . . . . . . . . . . . . . . . . . . . 135

2.6.2 Three-Dimensional Data Structures.. . . . . . . . . . . . . . . . . . . . . . 137

2.7 Summary and Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

3 Radiometric Homogenization of the Color Cryosection Images from the VHP Lungs for3-D Segmentation of Blood Vessels 143

3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

3.2 Problem Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

3.2.1 Gamma Correction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

3.2.2 Visualization of radiometric inhomogeneities. . . . . . . . . . . . . . . . . 146

3.3 Local Adaptive Homogenization. . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

3.3.1 Inhomogeneity Characterization by an Auto-regressive Model. . . . . . . . 150

3.3.2 Local Coherence Hypothesis . . .. . . . . . . . . . . . . . . . . . . . . . . 152

3.3.3 Correction Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

3.4 Discussion and Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

4 Extraction of Bronchial Blood-Vessels from Color Cryosections of the VHP 161

4.1 Anatomy of the Branching Structures in the Lungs. . . . . . . . . . . . . . . . . . 161

4.1.1 Tree Structure of the Pulmonary Artery. . . . . . . . . . . . . . . . . . . . 164

4.1.2 The Lungs of the Visible Human Male. . . . . . . . . . . . . . . . . . . . . 166

4.2 Blood-Vessel Extraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

4.2.1 Other Imaging Techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . 174

4.2.2 Detection Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

4.2.3 Fusion of Cross-Sectional Segmentation. . . . . . . . . . . . . . . . . . . . 176

4.2.4 Local Thresholding with Hysteresis. . . . . . . . . . . . . . . . . . . . . . 179

4.2.5 Region Growing Segmentation . .. . . . . . . . . . . . . . . . . . . . . . . 185

4.3 Conclusions and Perspectives.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

5 Analysis and Visualization of Computer Simulations of Amorphous Photo-Deposition 191

5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

5.2 Objectives of the Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

5.3 Monte-Carlo Simulation of Amorphous Deposition. . . . . . . . . . . . . . . . . . 194

5.3.1 Model and Simulation Parameters. . . . . . . . . . . . . . . . . . . . . . . 195

5.3.2 Morphological and Topological Dynamics. . . . . . . . . . . . . . . . . . . 196

5.4 Characterization of Deposition Morphology. . . . . . . . . . . . . . . . . . . . . . 197

5.4.1 Surface Features and Bulk Features. . . . . . . . . . . . . . . . . . . . . . 197

5.4.2 Complex Internal Features. . . . . . . . . . . . . . . . . . . . . . . . . . . 199

5.5 3D Methodology . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

5.6 Analysis methods and feature extraction .. . . . . . . . . . . . . . . . . . . . . . . 203

5.6.1 Other Possible Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

5.6.2 Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

5.7 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

5.7.1 Porosity, Roughness, Connectivity, and Anisotropy . .. . . . . . . . . . . . 213

5.7.2 Surface Density, Interfaces Surface. . . . . . . . . . . . . . . . . . . . . . 213

5.7.3 Interpenetration Quality and Segregation of a Three-Species Deposit. . . . . 215

5.7.4 Experimental Validation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

5.8 Conclusion and Perspectives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

Conclusions and Perspectives 225

APPENDICES 228

A Further Definitions of Object Subsets and Boundary-Related Subsets 229

B Tubular Extraction Methods and Branching Analysis 239

C Synthesis of 3-D Phantoms of Complex Porous Structures by Diffusion-Limited Aggre-gation with Relaxation 249

Resume long 252

Bibliography 259

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

Notation 11

Notation

The following list refers mainly to notation used in the theoretical foundations of discrete spaces,our boundary-based approach and the specific applications in the present work. Some literals mayhave a local meaning; for example,I is used to represent the tensor of inertia in Chapter 1, andd

refers to distances or fractal dimensions, depending on context.

Symbol Description First used in page #

X�� ”expression” ”X” is defined as”expression”. 24

a iff b, a �� b ”a if and only if b”. 25

∅ � �� φ The empty setfg, the Greek literals ”phi”, ”Phi”, anddiameter. 25

�a� b�� a� b� The closed and theopen intervals froma to b, with a� b � R. Ifa� b � N, a � b then�a� b� � fa� a� �� bg.

26

�a� b� An ordered pair (or vector) of entitiesa� b. 26

fangNn�M Finite sequence (set) of�N �M � �� elementsfaM � aM�� aN�� aNg, withN�M � N andM � N .

85

�a�� a�� an� An ordered set (n-tuple or vector) ofn entitiesfakgnk��. 29

� x � Expected value (average) ofx � X , withX � Rn. 27

int�x� Greatest integerint�x� � x, with x � R. 103

frac�x� Fractional part ofx � R, i.e., frac�x� �� x� int�x�. 103

card�A� Cardinality of setA (number of members). 27

A nB� A �B The set difference between setsA andB. 25

Ac� a The complement of setA; the 1-complement of booleana. 23

�� XOR Boolean logic operators: OR, AND and exclusive OR, wherea XOR b

�� a � b� � �a � b�.

24

AB� A B� A �B� A �B Mathematical-morphology operators (dilation, erosion, opening,closing) of setA by structuring elementB.

51

An B� n-Iterated dilations of setA by structuring elementB. 133

O�P�Q�U �V�W Sets inRn (objects, regions, scenes). Also, discrete sets inN� 23

V�W Scene domains inR�. 86

V� Cubic grid (scene domain inN�). 86

∂� Partial differential operator (e.g.,∂f�x�y�∂x ). 50

�V The boundary of the setV. If V � N�, �V is the discreteface-basedboundary of V.

24

�V Continuous point boundary of a voxelized setV. 92

V Thevoxel-based boundary (orsolid boundary) of V � N�. 93

KV The voxel-based boundary ofV, underK-connectivity. 93

12 Notation

�jV Thej th connected component of the (face-based) boundary ofV. 231

jKV Thej thK-connected component of the voxel-based boundary ofV,underK-connectivity.

112

V�j �� interior��jV� �jV Positive and negativesolid interior of �jV. 229

�V�j �� V� Filled interior of V�j , V (cavities are blocked out). 235

A b B BoundaryA is nested withinboundaryB; A � interior�B�. 231

Nfacets� Nbf �V� �� card��V� Number of boundary faces of�V. 92

Nvox � Nvb�V� �� card� V� Number of boundary voxels ofV. 93

N��V� �� card�f�jKVg� Number ofK-connected components of the face boundary�KV �SN�

j�� jKV� WhenV � N�, thenK � f� �� g.

231,101

NV�V�� Vol�V� �� card�V� Discrete volume (voxel cardinality) ofV � N�. 101

pq� pq�� pq�� Line segment between pointsp� q � R� and its two half-line ex-tensions excludingpq, with: pq��

�� pq�� n fp� qg (i.e.,

excludesp� q).

27

d�p� q�� kp� qk Euclidean distance between pointsp� q � R�. 24

d��p� q� Geodesic distance between pointsp� q � �V�V � R�. 26

dK Discrete distance underK-connectivity between grid pointsp� q �V� � N�, andK � � �� .

98

X�Y� Z� �x� y� z� Orthogonal axis (right-hand trihedron) inR� orN�; point coordinatesand vectors.

85

u�� u� Refers to oriented faces in positive and negative directions of axisu � fx� y� zg.

85

F�� fx�� x�� y�� y�� z�� z�g Partition of�V into sets of six face orientationsff kg�k� in the or-

thogonal gridN�. Each subset forms afacet wall.91

U�� e�� e��e��e��e��e�� Ordered set of unit vectors grouped by opposite pairs of orthonormal

orientations:�e� � �� e � � � �� e� � � � � ��.85

U� � f�ukg�k� Refers sequencially to each of the six orientation vectors assigned tooriented boundary facesF�.

85

v�x� y� z�� vm Voxels at points�x� y� z� andboundary voxel elements vm � V,with m � �� Nvb�.

86

f k Oriented face,f k � F�, with k � � � ��. 87

fn�� vm� f k� Discrete facet or boundary-face element fn � �V, with n �

�� Nbf �� k � � � ��, v � V. Index n depends onm� k and theshape of�V. Unit vector�uk � U� is assigned tof k.

89

v� v� -voxel in Vc (void or background label ”0”) and�-voxel in V (objector foreground label ”1”).

89

f � f�x� y� z� k� Short notation for a facet at boundary voxelv��x� y� z�, and orientedface �f k� �uk�� k � � � ��uk � U�. Isomorphic to a particularordered pair�v��v�.

91

opface�f� Opposite face operator. Given a boundary pair�v��v� andf �

�v�� f i�, orientation is reversed byopface�f�� v� f j�, with

j � i � ��i.

91

opface��V �� V � Opposite operator on an oriented facet boundary. Swaps all facet ori-

entationsu� � u�, andu� � u� (with u� � F�), in a discretefacet boundary�V� with a choice of facet orientation vectorsU�.

92

Notation 13

�V �� opface��V� Negative boundary, the reverse-oriented version of�V. 92

v�� v��x� y� z�� uk� main front voxel and main back voxel ofv��x� y� z� at a given facet

f�x� y� z� k�� k � � � ��.109

f �� f��x� y� z� � �uk� main-front facet and main-back facet of facetf�x� y� z� k�. 109

��V Theback-face boundary ofV. 110

��V The front-face boundary ofV. 110

�V Theback-voxel boundary (orendo-boundary) of V. 110�V The front-voxel boundary (orexo-boundary) of V. 110��V Thenext-front-voxel boundary ofV. 110

�� Any of the two facet-based boundaries,�� or ��. 110� Any of the four voxel-based boundaries,, �, � or ��. 110

facettovox��V� �� V Facet-to-voxel operator to buildV from �V. 110

voxtofacet� V� �� V Voxel-to-facet operator to build�V from V. 94

�O �

� O n �O Geometric interior of continuous setO. 25

�V �� V n V Discrete Geometric interior of voxelized setV. 93

interiorx��V� Discrete solid interior of a face-based boundary�V by x-runs ofvoxels.

96

G�V Fat-boundary neighborhoods of �V, defined by union-set combina-tions of voxel boundaries�V. For example, G�V �

� � V �V�is theimmediate neighborhood of �V.

112, 102

Bn �r� p� Euclidean sphere or spherical neighborhood in Rn of radius raroundp � Rn; also(closed) n-ball, or n-disk.

24

A nB If B is an euclidean sphereBn , thenA nB � A n B. 133

NK�p� Discrete neighborhood aroundp, includingp, under a specificK-connectivity.

102

EK � NK�p� Same as above, used asstructuring element for Mathematical Mor-phology operations, withK � � �� .

102

N ��p� Neighborhood aroundp, excludingp. 24

Nbf �p�N �� card� �V � N � Number of boundary faces in a neighborhoodN �p�. 208

N��r� p�� B��r� p� Geodesic neighborhood of radiusr around a pointp � �V� 104

Ngeo�� card�N��r� p�� Number of points ofN��r� p�. 104

�VA�B Interface between regionsVA andVB (for example, the face-basedinter-species boundary).

25

�Vjcondition Boundary �V restricted to ”condition”. 236

�Vjout The outmost boundary components of�V (all sub-boundaries with-out parents). Thefilled interior of V is defined as: �V� �

�interior��Vjout�.

235

�Vjz� Boundary patch corresponding to initial conditions (sustratum geom-etry) before deposition along axisZ.

203

14 Notation

�Vjzlast Boundary patch corresponding to the last deposited layer (axisZ). 206

�Vjz�� Projected boundary (top surface) as seen fromz � ��. 207

I�x� y�� f�x� y� Intensity 2D images; each one refers to a projected boundary�Vjz�� interpreted either as a rough surfacef�x� y� or an intensityimageI�x� y�.

207

V�z��y��x� Marginal index array representation of a sceneV (specific to soft-ware implementation).

137

umax� umin Maximum and minimum values of a setfuigNi�, with ui � R. 154

argmaxv

�f�v�� Argument value (or vector)v which maximizes a scalar functionf�v�.

argminv

�f�v�� Argument value (or vector)v which minimizes a scalar func-tion f�v�. In discrete sets, the indexiarg

�� argmax

ifuigNi�

is such that uiarg �umax � maxfuigNi� (similarly formin�� argmin�� ).

33

ROI, VOI Region Of Interest (usually in 2D) in a given image andVolume OfInterest in a 3D scene.

62

fIn� g Volume as a set ofn stacked color images aftergamma correction. 144

Hn�u� � Histo�In� Histogram (function) and histogram operator of a color imageIn (orROI) with u � fR�G�Bg, the intensity per channel.

147

mode�� Themode of a distribution curve in some defined domain. 41

XY� Y Z� ZX Orthogonal planes inR� orN�. 102

IXY Axial set: a scene or ROI organized (resampled) as a set of slices(imagesIn�x� y�) orthogonal to theZ-axis.

176

IYZ Coronal set: ... (as above)... orthogonal toX-axis. 176

IZX Saggital set: ... (as above)... orthogonal toY -axis. 176

Sunder�V�� Sover�V� Under-segmentation and over-segmentation output volume. 178

J�XY Conjunctive combination of segmentation results. 178

J�XY Disjunctive combination of segmentation results. 178

J��XY Associative combination of segmentation results. 178

�n� �t� �tk Normal, binormal and tangent vectors to a curve of a surface�V �R�. Usually,�t � �tk � �n).

68, 239

�k� � Characteristic feature lengths (e.g., peak or valley width and height,respectively) tangent (any direction in planeXY ) and perpendicular(axisZ) to a surfacez � f�x� y�.

45

hRMS RMS roughness of a 2.5D surfacez � f�x� y� defined as theRootMean Squared deviation from mean height� h �).

198

Notes about text notes, captions and figure symbols:

✦ The meaning of acronyms is first introduced in italics. For example:Mathematical Morphol-ogy (MM), and indicated at the external margin with a triangle, as reminder.” I

Notation 15

✦ Bibliographic references are listed in square bracket format: [LastNamesYY,...].

✦ Caption labels are set in parenthesis: (a). Outside the figure, they may be found in a� � �array to the right of the legend, representing subfigure components.

✦ The symbol “��” is used to tag an object in the figure, a reference vertex or point.

✦ Bolded arrows with a white tip represent special vectors in 3D. Some directions (orthogonalaxes, for example) appear represented by black tip arrows.

16 Notation

17

Preface

The first “complex” objects to be throughly analyzed by Science were fairly simple. Very high num-ber of components or features, dimensionality, and the length of descriptions demanded computertools and special mathematical treatment (e.g., many dimensions, fractals, measures of complexity,etc.), which have been developed only this century.

On the other hand, biological, natural and medical objects have been studied in rather qualitativefashion, with poor mathematical treatement until recent years in which non-linear phenomena andseveral advances permitted more formal approaches to study and synthesize irregular shapes.DigitalSpaces, Discrete Topology problems (also calledDigital Topology) and their applications are oneexample of the topics increasingly present in current literature, converging with computer and theo-retical advances, but also fostered by the increasing interest in formal approaches to difficult subjects.One explanation of the late appearence of such topics, besides technological achievements for the lasttwenty years, is their location at theboundary between Computer Engineering and Discrete Mathe-matics, the first being specially pragmatic and the second highly formal. If interdisciplinary activitiescould be modeled as any dynamic system, it might be seriously argued (and not just as metaphor) that“unmiscible domains tend to develop complex interfaces” even in scientific and engineering fields.

To know the context of the present work, some personal informations may help. My coming toFrance and my research subject have their background in the National Autonomous University ofMexico (UNAM), where I worked as an engineer in the Image Laboratory1 of theCentro de Instru-mentos of the UNAM2. By 1993 we were doing very simple signal- and image-processing applica-tions in Biomedicine, and worked close-together with biologists and physicians. Yet the problemswere interesting and results very fruitful. A particular project with theElectron Microscopy Labo-ratory of the School of Sciences of the UNAM motivated the design and development of tools ofanalysis of three-dimensional (3D) chromatine bodies in animal cell nuclei. Shapes were very com-plex, numeric measures on slices and contours did not show clear patterns and no commercial toolswere available for this analysis. During development of methods for discrete-surface extraction, Icame to France for a similar application, in the study of the Parkinson Desease, at the U-INSERM238, where the main concern was the geometric distorsionsof images and the lengthy microtome ses-sions. A grant from the UNAM allowed me to obtain aDEA in Genie Biologique et Medical and startthis thesis work in theDepartement Images of theEcole Nationale Superieure des Telecommunica-tions (ENST) by end 1994. (Details about people and laboratories appear in theAcknowledgements).

1http://www.cinstrum.unam.mx:8001/pub/imaimagenes.html2http://serpiente.dgsca.unam.mx/rectoria/htm/demo2.html

18

Problem Definition, Applications, (and what came of it:)Thesis Organization.

A personal aim for my DEA and PhD was to start and develop a line of research about complex-shaped objects from natural Sciences and Physics, and their analysis in 3D with image-processingmethods3. This is a too general goal for a single thesis: how to delimit the subject while studyingthe topics I was interested in? At present, this also had the inconvenient of limiting interdisciplinarywork and quantitative results, becoming more concerned with the state of the art, methodology,developpment, theory and visual results.

Since we did not have in mind a very specific problem, nor wanted to focus in a unique, singleapplication, we state here what the initial aims looked-like, knowing that the present work onlyachieved some of them, and even partially. We also summarize how the problems and their treatmentconstitute the chapters of this manuscript.

A first objective was to do an extensive bibliographic research, in order to identify and havepreliminar knowledge of how complex shapes are defined, their feature descriptors, key-conceptsof what is “shape complexity”, and “complex”, in general. From this survey, a number of specificcharacterization problems with 3D data was expected to arise. We illustrated some of the subjectswith our own past experiences, in the UNAM and during the DEA. Continuous and discrete repre-sentations and their morphological parameters were studied, with emphasis in the latter.

A broad idea was to usediscrete boundary representations (3D borders or frontiers) as the mainparadigm, since, excluding force fields and internal sources, most objects and phenomena (natural,or generated by man or computer) have important interactions at their boundaries and it is our first(or only) approach to them in several cases. Thus, one of the initial objectives was to present a largesample of shape features described and analyzed through discrete representations and processingto organize data and extract morphometric characteristics. Together with the first objective, thisproduced the material for theChapter 1 andChapter 2.

Chapter 2 also includes a description of the model of boundary representations in digital spaces(Z��, some aspects of algorithm implementation and analytic methods based on boundary traversal.

Having obtained access to theVisible Human Project (VHP) database, we had the opportunityto analyse complex objects, such as the lungs and the brain, from several imaging modalities, inparticular anatomical color photographs. Degraded radiometric information of high resolution posedenough problems to occupy a significant part of our time, andChapter 3 is devoted to the solutionof this problem.

The second part of the previous work on theVHP consisted in the extraction of the blood-vesseltree of the VHP lungs. This lead us to consider methods of segmentation of these structures, and topropose specifictubular detection methods.Chapter 4 presents our progress in these directions.

While developing the extraction methods of the blood vessels, a second application in SurfacePhysics attired our attention due to its multiple aspects in shape-complexity problems from a col-laboration with the laboratories of CNET-Bagneux. It is the three-dimensional structures producedby computer simulation of amorphous deposition. Such data and the aims of the CNET laboratoriesare representative and illustrative of the kind of problems we were trying to study, with the aim ofaccessingbulk androughness information about porous structures, in function of simulation param-eters. The collaboration also may give us later the opportunity to test and validate results againstexperimental data, which were not warranted to be produced during the thesis work. The problem,methods and preliminar results of this work are described inChapter 5.

3http://www.tsi.enst.fr/ ˜marquez/

19

Time limitations did not allow us to present here other results and related applications; but mostof them are roughly described and referenced, specially in Chapter 1.

A symbolic notation list is given in page 11. All chapters have a short abstract and an overview atthe end of each introduction describes the sections, goals and development of the exposition. Thereare sections in small font which can be skipped in a first lecture. A section of conclusions and someperspectives are also found at each Chapter.

Jorge Marquez, ENST, Paris, February 1999.

20

Chapter 1

Characterization of sets with complexstructures in R

�

Abstract

In this chapter we make a survey on complexity in continuous space (R�), from the point of view ofshapeand morphological featuresof physical, computer generated or digitized objects from differ-ent application domains. These features comprise simplified representations, form factors, convexity,coherence, decomposition into simpler elements, scale invariance properties (fractal dimensions),and other notions useful to describe and measure complex objects or sets, and complex relationshipsbetween them. An emphasis is given to concepts related to boundary representations, in view of theanalytical methods presented in Chapter 2, and the applications described in the other chapters.

1.1 Preliminar Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.1.1 Neighborhoods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.1.2 Boundaries and Interfaces. . . . . . . . . . . . . . . . . . . . . . . . . 24

1.1.3 Connected Sets and Objects. . . . . . . . . . . . . . . . . . . . . . . . 26

1.2 Intrinsic Properties of Solid Shapes . . . . . . . . . . . . . . . . . . . . . . . 28

1.2.1 The Approximating Ellipsoid, Isotropy and Deformed Shapes . .. . . . 28

1.2.2 Shape Factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1.2.3 Shape Decompositions and Local Descriptions . .. . . . . . . . . . . . 34

1.2.4 Convexity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.2.5 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

1.2.6 Scales of Representation and Characteristic Lengths. . . . . . . . . . . 42

1.3 Extrinsic Properties of Solid Shapes. . . . . . . . . . . . . . . . . . . . . . . 45

1.3.1 Complex Relations . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 45

1.3.2 Many Bodies and Clusters. . . . . . . . . . . . . . . . . . . . . . . . . 46

1.3.3 Complex Objects in Feature Space . .. . . . . . . . . . . . . . . . . . . 47

1.3.4 Complex Representations. . . . . . . . . . . . . . . . . . . . . . . . . 47

1.4 Fractal Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

1.4.1 Dimension Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . 48

22 Characterization of sets with complex structures inR�

1.4.2 Morphological Features of Fractals .. . . . . . . . . . . . . . . . . . . 54

1.4.3 An Example: Particle Aggregates . .. . . . . . . . . . . . . . . . . . . 56

1.5 Notions and Measures of Complexity . . .. . . . . . . . . . . . . . . . . . . 56

1.5.1 *Kolmogorov Complexity . . . . . . . . . . . . . . . . . . . . . . . . . 58

1.5.2 Information Theory, Algorithmic Entropy and Scientific Observations . . 60

1.6 Shape Taxonomies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

1.6.1 Tubular Structures (I). . . . . . . . . . . . . . . . . . . . . . . . . . . 64

1.6.2 Ramified Thin Structures. . . . . . . . . . . . . . . . . . . . . . . . . . 64

1.6.3 Surfaces, Boundaries and Interfaces .. . . . . . . . . . . . . . . . . . . 68

1.6.4 Description of Other Shape Features .. . . . . . . . . . . . . . . . . . . 69

1.7 Other Properties of Complex Information . . . . . . . . . . . . . . . . . . . . 70

1.7.1 From Structure to Texture. . . . . . . . . . . . . . . . . . . . . . . . . 70

1.7.2 Noise, Blurring, Distorsion and Inhomogeneities .. . . . . . . . . . . . 71

1.8 Discussion and Conclusions .. . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Introduction

We review in this chapter some concepts closely related to complex objects in three dimensions.Since the subject ofComplexity is too large for any single work, we have selected those conceptsdefining complex shapes that can be studied in a framework of biomedical and physics applications,with medium-level computer resources. At the same time, we study those morphological featuresrelated to the quantitative analysis and visualization approaches presented in Chapter 2, and the spe-cific applications of Part II. Our choice is based on what we believe is relevant and common to manyapplications, hence, theoretical details and related topics are to be examinated in the bibliography.Some of our own experiences are also cited.

Complexity, in the context of our work, will meanstatic complexity, as defined in Section 1.5,but also we restrict the subject to morphological or shape complexity, in object space. However, wereview in this chapter some other related meanings (non-convex shapes, ramified structures, fractalsand Kolmogorov Complexity, mainly), how they may be useful in shape description, and how toformulate some of them with thediscrete boundary-based approach we present in Chapter 2. Wefurther restrict ourselves to the class of rigid, non-deformable, non-fuzzy and constant-density ob-jects, considering mainly their geometrical shape and spatial configuration. The terms ”structure”or ”structural” are employed often. The intended meaning is not onlyintrincacy but alsoshapeorganization, that is, how an object is constructed, described or how its parts are arranged.

It is worth noticing that in many cases morphological descriptions follow general rules that relateshape and structure in the physical world tomorphogenic process. The latter includes, for example,water and wind erosion in geosciences, growth and homeostasis in Biology, or a set of user-specifiedprobabilities and parameters in the computer simulation of some model. The construction rulesmay be non-deterministic, i.e., subject to stochastic process and noise. They may howeverdeterminestatisticalproperties of the shape of a physical object, from branching structures (trees, blood-vessels)to porous media (particles, deposits), to shapes and configurations calledlaminar, granular, tubular,convoluted, etc.

Consider also the fact that other aspects of the shape remain unpredictable, such as the exactlocation of each pore, fold or protuberance, number and length of branches, etc. This fact maybe irrelevant, but a general characterization should provide estimations of averages and RMS (Root

1.1 Preliminar Notions 23

Mean Square), values of specific features. Physical pressure and temperature are typical examples ofmacroscopic and statistical properties of many-particle systems. In a similar way, the final morphol-ogy or configuration of many dynamic systems is usually the result of underlying non-deterministicprocess, impossible to fully track along several spatial scales, and requiring statistical models topredict macroscopic properties, as well as morphological features. Image processing examples ofmacroscopic features aretexture, roughness, entropy and fractal dimensions. The quest for under-standing most dynamic process and the validation of their mathematical models strongly relies ondescription and understanding of their associated morphology. It is in this spirit that this work wasconceived.

Sections in this chapter are organized as follows:

✦ In Section 1.1 we define some concepts that will be used often in the present report. Theseincludeboundaries, neighborhoods, andinterfaces. Appendix A, in page 229 further presentsinteriors, cavities, nested boundaries and other notions.

✦ In Section 1.2 and Section 1.3 we describe a set of features that have always posed somekind of problem to shape understanding, manipulation, and modeling, and make a generalcharacterization of what is commonly described as ”complex three-dimensional objects”. Wehave divided them intointrinsic properties andextrinsic properties of solid shapes.

✦ Scale invariance, auto-similarity, self-affinity, and several fractal dimension definitions arereviewed in Section 1.4. Morphological features of fractals are also discussed.

✦ In Section 1.5 we review some notions and measures of complexity and their relations withshape. These consist mainly of Kolmogorov Complexity, Algorithmic Complexity and En-tropy.

✦ In Section 1.6 Classifications of different shapes are analyzed in terms of local or global fea-tures characterizing a specific class of shapes. We suggest some ways to describe and quantifythose features, by analyzing relations between internal, external and boundary points of theobject.

✦ In Section 1.7.1 and Section 1.7.2 we mention some problems (subsampling, deformation) andparasitic information (noise) that contribute to ”the complexity” of an object studied through aset of data observations.

✦ In Section 1.8 we summarize all reviewed notions and high-level characteristics common tofeatures describing complex shapes.

1.1 Preliminar Notions

We introduce in this section some basic definitions specially applied to solid shapes. LetO be in gen-eral an object, a scene (set of objects) or a set of pointsp inR� (continuous) or inN� (discrete). Anysubset (region or points) ofO is often considered as the ”foreground” with respect to some predicateabout pointsp � O, or some operation onp (e.g., edge extraction). In this case, the complementOc

is called ”the background ofO”. Note that if there are other objects inOc, the background dependson definitions of what was called foreground. These definitions are predicates and operations onpto determine ifp � O. BothO andOc may be composed of one or more connected components(several objects, cavities, etc.). We introduce discrete connectivities in Chapter 2, Section 2.3.4,


and restrict here most definitions to the continuous case. Since we often use the concept of ”neigh-borhood”, ”boundary” and ”interior ofO”, definitions of these and related concepts are given hereand in Appendix A. As general references we cite [Weisstein98, Kiyosi96a, Kiyosi96b, Iyanaga68,Koenderink90, Coster89, Barnsley88, Herman98, Klette98]. Symbolic notation is summarized inpage 11.

1.1.1 Neighborhoods

DEFINITION 1.1 (Continuous Spherical Neighborhood inRn )Let p � Rn (with n � N) and let k � k denote the Euclidean distance in Rn. Let Bn�r� p� �

�fq � Rn j kp � qk � rg. We call Bn �r� p� the closedn-ball of p, closedn-disk, closed sphericalneighborhoodor closed Euclidean sphereof radius r.

An open n-ball or disk is defined byBn�r� p�� fq j kp � qk � rg. If nothing else is

indicated, ”ball”, ”disk” and ”sphere” will mean by default ”closed ball”, etc. We sometimes use alsoBr�p� to denote a discrete neighborhood ofp � N� and radiusr. In general, a neighborhoodN �p�or NK�p� aroundp may have a shape other than the spheric (e.g., cubic or rhombicuboctahedralif NK�p� � N� ). SubindexK is a parameter characterizing connectivity, scale, shape or otherproperties (see also below,geodesic neighborhood). We setr � � to indicate thatr � � is ”small”.It is also referred in mathematical literature as anepsilon-neighborhood. We notice that some authorscall ”neighborhood ofp” the setN ��p�

�� N �p��fpg. This is also a special case of anopen annular

neighborhood, defined as follows:

DEFINITION 1.2 (Open Annular Neighborhood inRn )Let p � Rn, with r�� r� � R. Let

Bn�r�� r�� p�

�� fq � Rn j r� � kp� qk � r�� r� � r�g

We call Bn�r� p� the openn-ring or openn-annular neighborhood of radii r�� r�.

Note that the special caser� � � corresponds to an openn-ball withoutp. We use discrete annu-lar neighborhoods as 2D sliding windows or roller memory buffers for local analysis in Section 3.3.3,and for 3D tubular detection in Section B. Tubular and boundary neighborhoods are defined and dis-cussed in Section B and Section 2.3.8.

1.1.2 Boundaries and Interfaces

DEFINITION 1.3 (Continuous Boundary inRn )Let O � R

n and let

�O �� fp � O j �� q�� q� � Bn �� p� such that �q� � O� �q� � Oc��g

where ”” denotes the logical ”and”. The set �O is called the boundary of O.1

Remark 1. Some authors setq� � p, which defines a boundary as a subset of the object.

Boundaries are also called ”frontiers”. The set�O of a single object may be composed of severalconnected components which make up the external and internal surfaces (cavities) ofO (see Def-inition A.6 below). Such properties are examinated later. The discrete version of�O poses some

1We use the symbol∂ for partial derivatives and�� and for various boundary definitions.


problems in relation with membership of a point (pixel, voxel) toO. Membership and neighboringrelations depend on connectivity definition inNn. We deal with the latter in Section 2.3.4, page 96.Figure 1.1a-d illustrates the definitions of continuous and discrete boundaries.

DEFINITION 1.4 (Open/Closed sets, Closure.)Let O a set and �O its boundary. Then

✦ O is a closed setiff2 �O � O.

✦ O is an open setiff �� p � �O�� p �� O.

✦ The set O �� O � O is called the closure ofO.

Let the set difference ”n” (or ”�”) between setsA andB be defined by:

A nB � A�B�� fa j �a � A� �a �� B�g�

DEFINITION 1.5 (Geometric Interior)

The set�O �

� O n �O is called the geometric interior of O.

Some authors use the definition of interior to denote geometric interior (orstrict interior). In thefollowing definitions we deal only with closed sets, that isO � O. Thus, any pointp � �O belongsalso toO. We also make some remarks on the discrete case but leave the equivalent definitions forChapter 2.

DEFINITION 1.6 (Interior and Exterior Points, Boundary Points)Given a setO � R

n and a point p � Rn:

p is an interior point of O iff p � O, and

p is an exterior point of O iff p � Oc.

p is a boundary point of O iff p � �O.

Thus, interior points ofO may belong to the boundary or to the geometric interior ofO. It may

happen inNn that a setO does not have geometric-interior points, i.e.,�O� ∅ . Some authors use

the notationint�O� andext�O� referring to the geometric interior or exterior ofO (the exterior isactually its complement).

The notion of boundary separatingforeground (O) from background (Oc) can be extended tothat of interface, as the set of points in common between two closed regionsOA�OB of differentnature (e.g., different labelsA�B, material or color). More precisely (see Figure 1.1e):

DEFINITION 1.7 (Continuous Interface inRn )Let OA�OB be closed sets such thatOA�B �OA OB �� ∅ , and define:

�OA�B�� fp � OA � OB j � � � � � q�� q� � Bn �� p� such that �q� � OA� �q� � OB��g

If �OA�B n �OA�B� � ∅ , then �OA�B is called the interface between regionsOA and OB (oralso, common boundary), and sometimes denoted ��OA�OB�.

The definition of adiscrete interface depends again on connectivity relations and boundary rep-resentations and is given in Chapter 2.

2”iff”�� ”if and only if”, we also use the symbol “�� ”


U

AO BO

U

B8O

A

OOU

4

U

O

O

p(a) (c)p

p(b) (d) (e)

ac

b

O

c

N(p )O

N(p )b

a

Figure 1.1:Boundary �O of an object O � R. (a) Point pa is a boundary point iff for any neighborhood

N �pa�, there are points p� q � N �pa� such that p � O and q � Oc. (b) An external point pb has aneighborhoodN �pb� � Oc, and (c) an internal pc point has a neighborhoodN �pc� � O. (d) In the discretecase (O � N), the boundary definition and neighborhoods NK �p� depend on connectivity relationships,(”K-connectivity”, with K � f�� g). (e) Interface �OA�B between two regions OA andOB .

DEFINITION 1.8 (Geodesic Neighborhood)Let p � �O. Then the set:

N��r� p�� fq j q � �O kp� qk � rg

is called a geodesic neighborhood ofp, with radius r.

Remarks. Strictly speaking, the Euclidean distancek � k should be replaced by ageodesic metric,with paths between boundary points strictly following geodesic lines of�O (which are minimum-length paths, using the geodesic metric), instead of taking the intersection with an-ball. For smallneighborhoods and very smooth surfaces we consider the approximation as acceptable3. The moregeneral definition uses then ageodesic distance dQ�� defined on a setQ:

Ngeo�r� p�� f q � Q j dQ�p� q� � r g�

In a boundary-based representation,Qmay in turn be the boundary set�O, and the geodesic distancebetween pointsp� q � �Q is denoted byd��p� q�.

1.1.3 Connected Sets and Objects

We introduce further definitions concerning boundary components, interior and their organization.In these definitions we deal only with closed sets inR� and their boundaries as closed surfaces. Weneed again some basic notions first:

NOTATION 1.1 (Closed and Open intervals)Let a� b � R, with a � b, then �a� b� denotes the closed intervalfx � R j a � x � bg, and�a� b� denotes the open interval fx � R j a � x � bg. If a� b � N, �a� b� is the set of integersfa� a �� a �� b� �� bg.

Notation�A�B� is reserved for ordered pairs of entitiesA�B (numbers, sets, vectors, etc.).

A connected set is classically defined as a set that can not be expressed as the union of disjoint,open sets. We give an alternative definition in terms of connected paths between points in a set.

3Such approximation is inadequate for example in describing neighborhoodson convolutedsurfaces, (e.g., cortex sulci).


DEFINITION 1.9 (Connected and Disconnected Set)An open set A � R� is called disconnectedif there are two open, non-empty sets U� V � R� such that:

U V � ∅ and U � V � A (1.1)

A set A � R� which is not necessarily open is also called disconnectedif there are two open

U� V � R� such that:

�U A� �� ∅ �V A� �� ∅ � (1.2)

�U A� �V A� � ∅ � (1.3)

�U A� � �V A� � A� (1.4)

Finnaly, if A is not disconnected, it is called connected.

It is generally easier to show that a set is disconnected than showing connectedness: if there is a pointx �� A, then that point can often be used to ’disconnect’A into two new open sets with the aboveproperties.

We will rather use aconnected path definition for discrete sets inN� in Chapter 2. Many non-connected sets may be expressed as the union ofN connected sets, which we name theconnectedcomponents of A, andcardA � card�A�

�� N is its cardinality.

DEFINITION 1.10 (Object)Let O � R� be a set of points. We callO an object, iff

N � N�� fOk � R� j Ok is a connected set k � �� Ng�

such that O �N�k��

Ok

Remarks. If we let Nk � � for each set Ok � k � �� N we verify that they are themselvesobjects, which we call connected componentsof O. Note that �O� �Ok are also objects under thisdefinitions.

DEFINITION 1.11 (Body, Solid object)

An object O is called a body, solid or solid object iff�O �� ∅ that is, if it has geometric interior.

We also refer allp � O as the foreground andp� � Oc as the background. Note that a single-component object may have several boundary components. We give bellow an definition of con-nected components as ”fragments” ofO in terms of its boundary components.

Finnally,� x � will denote the average or expected value of a population of samplesx � Rn,andpq will represent a line segment joining anyp� q � R�; i.e.,pq

�� fs j s � p t�q� p�� t �

R� �� t � ��g.These definitions and some presented in Appendix A (which may be skiped on a first lecture)

are extensively used in the following sections and other chapters. We propose in particular someequivalent concepts in discrete spaces, in Chapter 2. None of the traditional description methodsfrom Fourier analysis or from differential geometry was included, nor used, having in mind a “more”discrete approach, and now proceed to present the intrinsic properties and features of solid shapes.


1.2 Intrinsic Properties of Solid Shapes

Single objects may present several shape characteristics that make description and analysis lengthyor difficult. Traditional classifications and names for shapes such asovoid, flat, potato-like, spongy,trabecular, denticulated, fasciculated, etc., depict proportions, repetitive patterns, part arrangements,and salient characteristics by intuitive similarities with real-life objects and geometrical abstractions.A repetitivity study alone would deserve an exposition on frequency-domain techniques of analysis,together with concepts such asspatial correlation. Shape diversity and complexity has turned popularterminology imprecise and ambiguous, and formal, useful descriptions are still arising. We mentionin this section some of the most elementary features of global and local descriptions of isolatedobjects: principal axes (or diameters), decomposition into simpler parts, convexity properties, andways to proceed from coarse to fine descriptions with these features. We have called themintrinsicproperties, because they do not depend on external attributes, such as position and relationships ofthe objet with its context and other objects.

1.2.1 The Approximating Ellipsoid, Isotropy and Deformed Shapes

A simple but powerful global feature of an object refers to the axes of thetriaxial ellipsoid thatbest fits (or approximates) the object, whatever its shape. One of the roughest assumptions about abounded shape is that it can be always approximated by an ellipsoid. This extreme simplification ad-dresses the question of proportions between length, breadth and width (or height and depth). Fitnesscan be expressed as some optimal criterion, asLeast-Square differences, but a mechanical approachis already available from Physics4: When a physical solid has an homogeneous density of matter(for the sake of morphological description), there exists a virtual ellipsoid representing the object (itssolid contentsand its shape), denominated theellipsoid of inertia. It is suitable in Classical Mechan-ics to describe universal dynamic properties of all solids, no matter how intricate they are. Solidsor sets of solidsO are first reduced to a dimensionless point that corresponds to the center of massCM�O�)� �xCM� yCM� zCM� or geometric centroid (if density is uniform), and then represented by an”CM” Iellipsoid with angular momentum and other attributes which depend on shape and mass distribution.To simplify further, we assume uniform mass density equal to unity, to concentrate description onlyin morphology.

LetO be an object inR� and�O its boundary, as already defined. Without loss of generality, welocate the origin of the coordinate system at the centroid ofO, thus�xCM� yCM� zCM� � �� . Letu� v� w � fx� y� zg be three coordinate axes, in permutated order (right-handed system). We define:

uCM�� u �

R R RO u dx dydzR R RO dx dy dz

� (1.5)

�uv �

R R RO uv dx dy dzR R RO dx dy dz

� u �� v� (1.6)

�uu �

R R RO�v

� w��dx dy dzR R RO dx dy dz

� v� w � �fx� y� zg� fug�� (1.7)

The integrals�uv are theuv products of inertia (sometimes refered as ”moments about planew”),and integrals�uu are the moments of inertia about theu axis. Note the normalization with respectto the total mass ofO (volume element times mass density, which we assumed here to be 1). For

4In 2D, the best-fit ellipse is defined equivalently as the ellipse whose second-order moments equal those of the object[Jain89].

1.2 Intrinsic Properties of Solid Shapes 29

varying mass density�x� y� z�, the centroid refers to the geometric center (as when is constant),and the center of mass integral has a weighting term�x� y� z�, as well as integrals�uv and�uu.For a set ofN identical point masses located at positionsx i� i � �� N , the centroid is given byxCM � ��N

PNi�� xi. The matrix ortensor of inertia in Classical Mechanics is defined by:

I �

�� xx �xy �xz�yx �yy �yz�zx �zy �zz

�A (1.8)

SinceI is square and symmetric, it is possible to find a coordinate transformation to expressI in adiagonal form. LetA the matrix formed by the three column eigenvectors ofI ; then:

ID �A��IA �

�� a � �� b �� c

�A (1.9)

whereID is a diagonal matrix whose entries are the eigenvaluesf�a� �b� �cg of I, and the columnvectors ofA correspond to the three principal axes of the ellipsoid (also calledprincipal momentsof inertia). The equivalent concept in Statistics is thescatter matrix or covariance matrix, and itsdiagonalization is called thePrincipal Component Analysis (PCA) of the object, conceived as a J ”PCA(random) distribution of points inR�. The mass density is then substituted by the probability density.

To obtain in practice the product moments, the integrals�uv are numerically calculated just overfinite subsets ofO. A uniform, isotropic and orthogonal discretization of space provides a set ofN

samplespn � N�, with pn � O� n � �� N (see Figure 1.2 for the special case of the center

of mass, CM�O�). Samplespn are voxels with unitary volume and unitary density. Then�uv isreplaced by the discrete moments�uv , with u� v � fx� y� zg andx� y� z � N, and normalization isdone over theN samples. We do not demonstrate the latter in this document, but only give an intuitiveargument: given a partition ofO �

� �Nn��On, linearity permits to rearrange the integral terms of�uv , in order to calculate partial centers of mass for eachOn� n � �� N (see Figure 1.2b). Ina dense sampling, the error between these partial centers and the nearest sample from a set ofN

random samples is small and cancel itself on the average.

As a special case, the tensor of inertia of the discrete boundary�O may be an acceptable ap-proximation of the tensor of inertia ofO, (except for a proportional term). Figure 1.2e illustrates thisapproximation in 2D. In general, a ”filled” object and its boundary alone present differentphysicalmoments of inertia with respect to most axes. When morphological considerations are the only con-cerned, the boundary approximation allows to compare different shapes, using only their boundary-moment representation. To distinguishmoments and tensor of inertia calculated either overO or over�O, we add a superindex� to indicate ’boundary’:��uv andI� . We recall that under some conditions,volumetric integrals�uv may actually be precisely calculated from the boundary information, usingtheGauss Theorem as described in Section 2.5.1.

Isotropy There are simple measures obtained from the principal axes (principal components oreigenvalues) that describe eccentricity and global isotropy of a shape. While eccentricity is describedin two dimensions by one single parameter (usually the ratio� � a�b of semi-major-axis/semi-minor-axis),three-dimensional eccentricityrequires a three-component feature vector for a complete de-scription. Letf�a� �b� �cg denote the eigenvalues of the ellipsoid of inertia associated with an object,set or regionO. These are the principal axes, equal to twice the semiaxesa� b� c. Without loss ofgenerality, we assume� � �c � �b � �a. At least three independent isotropy indexesI�� I�� I� can


nOCM( ) = CM ( { CM( )} ), n=1,...,N

np{ }

On

O

O

{ }

On= { }

N

n=1U

(a)

*

O

. . .

*

*

*

***

*

(b)

**

* *

*

~~~~n

pN 8

OLim CM( { } ) CM( )

*

(c)

*

np ∋

O, n=1,...,N

(e)

O O

pn n=1,...,NO,

∋

(d)

*

CM( ) CM( )

{ (x , y )}n n

Figure 1.2:Center of mass (CM) approximation by finite samples (uniform mass distribution). (a) CM(O):the centers of mass of the elements of a partition (b) of a continuous object O � R can be averaged to obtainits CM(O). (c) The CM can be approximated by random samples of O, or (e) from its boundary �O. (d)Discrete samples in N are a particular case between (b) and (c).

be defined:

I� ��a

�b �c

I� � ��a �b�c

I� � �� a � �b�� a � �c�� b � �c��

��a ��b ��c�

These indexes are best descriptive when the following combinations of conditions apply:

✦ if �c � �a and�b � �a, thenI� measures an elongated, prolate uniaxial anisotropy. Equiv-alently, it measures 2D isotropy of mass distribution along the plane orthogonal to axis� a, asin the case of planar ecentricity�. The ratioI� is high if one direction predomines.

✦ if �c � �b and�c � �a, thenI� measures a ”flat”, discoidal or oblate uniaxial anisotropy.Equivalently, it measures 2D isotropy of mass distribution in the plane orthogonal to axis� c

(note differences and similitudes withI�). I� is high if two directions predomine at the sametime.


(c)(b)

(a)

Figure 1.3:Ganglion neuron 3D reconstruction from confocal microscopy imaging. Approximate contour ofthe global ellipsoid of inertia (a) (re-scaled); local ellipsoid of inertia for one dendrite (b). The dotted profile(c) represents the 3D convex hull(see Subsection 1.2.4), whose ellipsoid of inertia do not match with that ofthe neuron. 3D data is public domain.

✦ I� is therelative intervariance eccentricity; if it is high (case�a � �b � �c), isotropy isomnidirectional; if it is low, anisotropy occurs between axes directions taken two at a time.

In all cases, a perfect isotropic shape (not necessarily a sphere!) satisfiesI� � I� � I� � �.An application example is presented in Section 5.7.1, page 213 for measuring pore distribution ina deposit. The interest in isotropy measures is that they can be obtainedlocally, allowing to knowisotropy and orientation of shape features, and textural characteristics at several scales.

In a cross-sectional and bidimensional approach to 3D problems, we could have also defined avectorial eccentricity� �

� ��a�b� b�c� a�c�, with at leastb� c �� . The normalization factor��is introduced to havek � k � � for a sphere.

Finally we note that when 2D features depend on scalar properties on two axis (e.g., the moment�uv , or the eccentricity ratioa�b), it is often the case that in 3D the equivalent are vectors or tensorsto include diverse pairwise combinations of properties along coordinatesu� v� w � fx� y� zg. Secondorder combinations includeuv, u� andu� v� terms, for example. In approaches for studying a 3Dobject with 2D morphometry on cross-sections, the analysis may end up with too many parametersthat are only meaningful in the plane of definitionXY� Y Z, or ZX . Quantitative Stereology is aparticular way to do something similar: to obtain estimates of 3D measures from 2D samples, usingstatistics, integral and stochastic geometry.

Features that Result from Shape Deformation Several shape anisotropies and morphologicalfeatures in general can be understood in terms ofdeformed simpler features. When the shape charac-teristics can be described as deformed patterns, several deformation modes exist:torsion (rotationaldeformation),shear in one or more directions, inhomogeneousshrinking and expansion, bilinearwarp (that which maps parallelepids into trapezoids), and higher order warps. Most of these de-formations may be modeled or approximated by a series of linear transformationsT�� T�� Tnon each pointxi � O, as: y�i � T�T�Tnx

�i . This is an example of how certain shape features


can be represented in terms of other, simpler features, as the result of a geometrical transforma-tion. Biological descriptions and models of shape morphology and morphogenesis often proceed interms of adaptive change (deformations) of basic structures, or entire systems. Bookstein describesa quantitative approach to biological shape change in [Bookstein78] and Murray examines patternformation, development and morphogenetic rules in chapters XVII and XVIII of [Murray93]. Wedescribe morphological deformations in dopaminergic cell neurons due to pathological stress andneural death-compensation phenomena (”Apoptosis”) in [Anglade93, Anglade95, Anglade97].

If a 3D object is digitally represented as stacked sliceszk , rigid and non-rigid transformationsmay be applied in 2D, slice-by-slice, with parameters varying withzk , in order to model severaldeformation combinations. Many common reconstruction and shape restoration problems constitutethe opposite process: a set of deformed individual slices have to be aligned and/or unwarped torecover the original shape, which may in addition present intrinsic deformations. A severe problemis the lack of information about the set of transformationsTi, or whether their inverses exist and arewell-defined (insensitive to noise). We discuss further this subject in Subsection 1.7.2.

1.2.2 Shape Factors

Compacity A first measure of shape, quantifying some degree of complexity (”irregularity”), iscompactness. It describes how efficiently the shape is embedded in space. ”Roundness” measuresare discussed later.

Conventional compacity is defined by shape area/perimeter ratios in 2D, and by volume/surfaceratios in 3D. However, the inverse ratios are widely used under the name ofshape factors. LetP , Adenote the perimeter and area of a 2D shape (a closed contour)C; a normalized shape factor is thendefined by:

ff�D � P ��A� (1.10)

The smallest value,ff�D � �, corresponds to a circle, and higher values to more irregular (lesscompact) shapes. For this behaviour, some authors callcompacity, the inverse offf�D , and a non-compact object is defined as havingff�D � � .

In 3D, LetS, V be the (external) surface area and volume of an objectO. A normalized shapefactor is defined by:

ff�D � S��p��V � (1.11)

In this definition the surface and volume of any sphere constitute the lowest boundff�D � �,andff�D increases with irregularities (e.g., bumps, protuberances, crevices). However, there existobjects with the same shape factors and enormous differences in morphology. For fractal shapes,the surface, as well as the perimeter in 2D depend on the measuring scale and tend to infinity withincreasing resolution. The common practice is to introduce some smoothing, which fixes a boundingscale of measurement. Compacity and sphericity (circularity in 2D) are often treated as differentnotions, the latter being defined by morphological operators with a structuring element which sizeplays the role of a resolution bound [Serra82], pp. 336.

A second measure of shape complexity is given by the difference between an object and its ap-proximating ellipsoid. Measures of goodness of fit, dissimilarity or distance to any specific primitiveor other shapes can be obtained in several ways. For example, Bribiesca proposes in [Bribiesca96]to estimate the ”work done” while transforming one shape into another, after spatial registration andscale normalization (both shapes must have the same volume). Similarity measures between two


objects or with respect to a selected model of its shape provide thus a relative-complexity measure.For example, a relative form factorfI with respect to the ellipsoid of inertia is readily calculated: letS�O�, V �O�, S�I�, V �I� be the surface area and volume of the objectO and its ellipsoid of inertia,represented byI . We first normalize the ellipsoid principal semi-axesa� b� c to have the same volumeas the object; thusV �I� � ��abc � kV �O�, with k the normalization constant. LetS�I �� be thesurface of the normalized ellipsoid. The ratio

fI�� S�I��S�O� (1.12)

satisfies� �fI� �, and is close to 0 for irregular shapes. To distinguishfI from shape factorff�D ,we callfI theEllipsoid-of-inertia Form Factor.

Other methods to estimate goodness of fit are part of a spatial registration or matching process,and are based onmathematical-morphology (MM). These methods use various distance measures andJ ”MM”distance fields, or other ”potential” fields. Malandain et al. and Mangin, review and evaluate severaldistance transforms for 3D shape matching (city block, chess board, octagonal and Chamfer distance,which all give reasonable approximations of the Euclidean distance) [Borgefors84, Malandain93a,Mangin95].

Shape factors based on measurements over distance fields have been proposed. They are closelyrelated to goodness of fit, since shape factors are normalized with respect to a common primitive(e.g., circles and spheres). Danielsson proposed in 2D a ”G-factor” measuring the average distancebetween an interior picture point and the nearest boundary point [Danielsson78]. An average distanced can be calculated from the first-order moment (or center of mass) of the object, normalized to itsareaA:

d �

�Z ZOkp� q k dx dy

� A (1.13)

wherep � �x� y� � O and

q � argmin�q � ��O

kp� q � k

is the closest boundary point top (see Figure 1.4). TheG-factor is defined in such a way thatG � �for the circle, andG � � for other figures:

G �A

��d��(1.14)

In comparison with the common 2D shape factorff�D , theG-factor is insensitive to small intrutionor protrution variations, and describes better large features. Its robustness allows for quick estima-tions, using only a fraction of allp � O in a similar way to second moments approximations. Equa-tion 1.13 shows that the average distanced to �O can be obtained from integration of theinternaldistance map on allO, defined by:

d�O� ��

��p� kp� q k� j �p � O� �q � argmin

�q ��Okp� q �k �

��

The generalization toNn is straighforward, once discrete distances have been defined. Note that setsof all equidistant points at a given distance definelevel sets, such asisocontours in 2D anisosurfacesin 3D.


p

dA

dA = dx dy

∋O

q

∋

O

Figure 1.4:Computation of G-shape factor of an object O, by measuring the average of the distance of allpoints �p � O to the closest point in the boundary (�q � �O). Note the area element dA � dx dy.

The interest in our work of factorG or similar shape descriptors, is their definition in terms ofinterior and boundary points, which we are able to extract, traverse or sample in efficient ways, asdescribed in Chapter 2. To our knowledge, no application ofG has been reported in current literature.

Besides shape factors, there exist particular characteristic lengths (e.g., the caliper and Frênetdiameters). Some may be obtained from measures on the equivalent parallelepid enclosing the el-lipsoid of inertia, or from the convex hull or from other fitting primitives as the maximal includedellipsoid and the minimal ellipsoid containingO. We will come back later with the ellipsoid-of-inertia concept and the object’s principal axes, as we examine its potentialities in our work.

1.2.3 Shape Decompositions and Local Descriptions

A first step in refining the ellipsoid approximation is to break the object into two or more compo-nents, usually moreconvex than the original (convexity is examined in the next subsection). In turn,these may be approximated by elemental geometric shapes. This ”divide and conquer” or construc-tive approach can be further refined by recursion, as suggested by Cordella and Sanniti di Baja withCAD primitives [Baja84a], by Ranjan and Fournier, and Rourke and Badler who analyze unions ofspheres [RourkeDS79, RanjanUOS95], and by Phillips et al. and Chaudhuri et al., who use ellipsoids[Phillips86, Chaudhuri91]. Complexity of the description is defined by thestructure of a decomposi-tion into elliptic elements, its number and their spatial relations. This description can be based onlyon connectivity or proximity relationships (see Figure 1.5-right); or hierarchical, through ellipsoidapproximations at different scales (see Figure 1.5-left). In each case, the decomposition may becoded in aconnectivity tree structure or ahierarchical tree structure and further detailed by mul-tiresolution or scale-space representations of the object. Topology notions are naturally introduced,beginning with the simple counting of components and subcomponents, including holes and cavities,and how they relate with their neighboring elements, as well as their organization in nested sets (seein Appendix A the Definition A.2 and compare with thecontainment tree structure of Figure A.3 inpages 231 and 234).

In consequence, if the shape parameters of single primitives constitute thefirst order elementsto describe complex sets, then the relational elements such as trees, graphs, adjacency matrices andother discrete mathematical entities become thesecond-order elements that describe complex sets.The decomposition or partitioning may be subject to levels of detail or to the number of desiredprimitives to approximate a shape. In this case, the problem has similar characteristics to clusteranalysis when the number of classes is unknown. The problem has been studied the other way


a

d

c

hi

e f

b

gj

i

f

h

c d e j g

a b

Figure 1.5:Elliptic decomposition of a 2D shape (left) and connectivity tree representation(right). Labelsidentify each fitting ellipse and the corresponding clumps are separated by narrow necks. Concavities of theshape (or, equivalently, background convex components) can be represented by ”negative” ellipses (oneis indicated by the arrow at left). Note that other kind of decompositions are possible (e.g., by proximity,perceptual groupings, and hierarchical approximations). The criterion for considering ”connectivity” betweencomponents is the intersection of the fitting ellipses.

around: clusters of scattered points (”clouds” of points, usually in feature space) are grouped andrepresented as closed regions, whose shape is analyzed (see for example [Edelsbrunner83, Radke88,Preparata77, Diday70, Diday76]).

It is clear that complexity, understoodas ”high number of elements”, has also a Newton-mechanicscounterpart: systems of many bodies (N � �) are difficult to treat by deterministic analysis becauseit ends up with high-order and non-linear differential equations with no analytic solutions, and too-sensitive numerical solutions (i.e., not useful for precise predictions). Thus, image processing orquantification strategies to treat many-interacting component problems in a mechanistic-like frame-work may step into a similar many-body problem. Such frameworks may arise in deformable models,for example.

The decomposition of an object into simpler parts is itself an important topic closely relatedto segmentation, multiresolution and MM. If the object cannot be decomposed by discriminatingdifferent densities or colors, an hypothesis or assumption must be made on the grounds of shapecomposition. These assumptions may have a functional or physical basis, or consist of the degree ofconnectivity or narrowness between clumps, and the size of the separation ”necks” to be identified(the zones between labels in Figure 1.5-left). Literature repports how superposed discs generate anarbitrary shape in 2D, and how watershed segmentation or clump splitting through concavity analysisachieve separation into convex components [Russ90, YeoCLUMP94]. We have ourselves used thistechnique for astronomical image analysis [MarquezNGC91, Santiago92]. Held and Abe proposea decomposition scheme for binary shapes by maximal approximated convex subparts, which mayoverlap [Held92]. They also extend the idea of decomposition intomeaningful parts [Held94].

Similar techniques exist in 3D (see [Bloch90b, Mangin95] for watershed segmentation exam-ples). When some degree of non-convexity is allowed, we ourselves have used a partial watershed


3D segmentation by simple morphological iterated erosion (N -times, withN � � � �) and thenconditional dilation, as described by Márquez et al. and López et al. in [Marquez93b, Marquez94b,Lopez95, Marquez96a]. Further discussion and a 3D-implementation of this procedure is explainedin Section 2.5.2.

A complementary approach in shape composition analysis is the study of thelocal tensor ofinertia behavior. We intentionally change the denomination of ”ellipsoid of inertia” to (local) ”tensorof inertia” to stress that the corresponding matrixI and moments�uv are no more measured over thewhole objectO, but over “small”, “medium” or “large” neighborhoods of each sample point, withscale categories defined in terms of the size of the smallest and greatest features of interest, (or atleast resolvable). Each point in space then has a local tensor associated with it and its neighborhood.If O is embedded in a heterogeneous region, with other objects,I can be measured even in pointsexternal toO.

1.2.4 Convexity

If the ellipsoid of inertia or scatter matrix may well represent an object in some situations (ClassicalMechanics), it is far from describing further details. The next step to study local and global propertiesof a shape is examinating its global and local deviations from the approximating shapes, such as theellipsoid of inertia, tangential planes, or from another simplified object that approximates the objectunder some criterion.

Before entering into a brief discussion about local convexities, we recall the following definitionof ”convex” inR� generalizing the one given by Stoyan in [Stoyan94], pp. 353, forR

�:

DEFINITION 1.12 (Convex)A set O � R� is said to be convexwhen for each two points x�� x� �

O, the line segment x�x� lies in O, i.e.

f�x� �� x�g � O� with � � �� (1.15)

At the surface level, details at a pointp are constituted by variations in local surface curvatureK��p�, at direction , that may be accentuated, diminished or sign-inverted. Such variations are sim-ply known as ”convexities” or protruding regions, when curvature is positive for all (all directions)and ”concavities” when they are all negative.

Anisotropic curvatures occur whenK��p� changes sign in function of . In this case ”convexity”and ”concavity” are no more the adequate concepts. The largest and smallest curvatures�� areused to describe local curvature and always occur in mutually orthogonal directions. Their sign de-termine three types of local surface curvature (see Figure 1.6):

convex: �� concave: �� saddle: �� (or �� )

There is still a fourth type of local surface curvature called a ”monkey saddle” with principal curva-tures having an inflection point, i.e.,�� at some pointp. A typical example of the presenceof ”monkey saddles” are the networks of tubular structures (e.g., capillar vessels and mouths ofalveoli in the lungs [Weibel79b], pp.89). Examples of the application of these notions are given inSection B.

Differential Geometry offers a set of concepts for a proper analytical description in terms of localprincipal curvatures; the features that can be so described include discontinuities whenj��j or/andj��j are infinite or very large. The combinations of all possible cases (e.g.,�� ) are known


asridges, valleys, cusps, humps, dimples andcorners in surface patches [Koenderink90, Clarysse97].Differential Geometry becomes a ”heavy” tool for the simplest of cases, and cumbersome to use atdiscontinuities and very rough surfaces, but its analytic power is best exploited in models of physicalphenomena on surface boundaries.

(a)

(b)

(c)

Figure 1.6:Surface curvatures in function of local positive and/or negative radii of curvature. (a) Convex,(b) concave), (c) saddle (extracted from [Weibel79b]).

The size and distribution of concavities and convexities on a surface can be subject to harmonicanalysis. There are many physical phenomena that give rise to repetitive patterns, but also to har-monic deformations of an object, in accordance to natural oscillation frequencies for example, orelastic coefficients. Fourier-descriptors in 2D allow to characterize frequency distribution of har-monic components in a closed or open contour [Zahn72]. Its extension to 3D has not been easy,because the field of complex numbers extends rather to 4D quaternions.

The harmonic analysis also exemplifies another approach to shape description, treating a shapeas a deformated version of a reference norm or template. In Medicine and Biology, the basis foranatomical atlases is the construction of a representative specimen or prototype for identification ofmajor structures and interpretation of the differences, which may be intra-individual (the same subjectat different times or conditions), inter-individual, inter-race, or abnormal. This kind of referencemodels are built regardless of other differences, based in specific deformations. The latter maybe natural (from morphogenics, development, environment adaptation or evolution), pathological,congenital or just statistical. An ambitious goal is then to model and identify the nature of shapevariations themselves.

Convex Hulls and Alpha-shapes. Non-convexity is the most familiar feature of an object irreg-ularity and can be expressed in several ways. LetO be a solid with volumeV �O� and surfaceboundary�O (cf. Definition 1.3). The importance of concavities and regions with saddle points canbe characterized by the volume differences between the object and itsconvex hull, defined as follows[EfronP65, Edelsbrunner85, Santalo76, Chassery83a]:


DEFINITION 1.13 (Convex Hull.)Let O � R

� be an object or a set of points (either connected to form a single object, or with nostructure at all). Let Ci be any convex set such thatO � Ci, with i � �� . Then, the set

co�O� ��

�i��

Ci

is called the convex hull of O.

A particular case is to consider forCi the set of half-spaces5 containingO. Another way to stateDefinition 1.13 is that:for each non-convex set O there exists a smallest convex set, the convex hullco�O�, such thatO � co�O�.

A concept that generalizes convex hulls is that of an�-shape, or �-complex [Edelsbrunner94].It can be defined as a simplicial complex, that is, a polytope6, which is not necessarily convex norconnected, and can be derived from the weighted Delaunay triangulation of a point set. The param-eter� controls the level of detail. An�-shape can be defined for a single solid object or for a set ofunstructured points, or small objects.

DEFINITION 1.14 (Alpha-shape)Let O � R� an object. Consider the surfaces S��O� defined by a MM rolling-ball closing operationon O by a set of spheres R�, with � � �� a real parameter, and R� ranging from data resolutionto infinity. S��O� is called �-shape, with the convex hull the special case of R� � �. The surfaceS��O� � R� is defined by [Chassery83b]:

� � � �� p� q � S��O� � ��p �� q� � S��O� (1.16a)

In the context of shape decomposition into smaller shapes, this definition illustrates an ap-proach consisting in measuring concavities or other features in function of MM notions [Vincent90,PitaVene90a, Heijmans94b]. This is interesting in applications where MM operations can be imple-mented in efficient ways (parallelization or boundary-based).

We note the importance of a surface or boundary representation of the solids of interest. It iscommon to define�� , andR� as��. In this case, an�-shapeS��O� is defined asfollows [Edelsbrunner94]:

DEFINITION 1.15 (Alpha-shape bis)Let B�R� p� a closed ball of radius R � ��, centered at p � O, and Bc�R� p� its complement, then,

if � � �� S��O� ��Tp�O B�� p� such that O � B�� p�

if � � �� S��O� �� co�O�� (the convex hull)

if � � �� S��O� ��Tp�O Bc�� p� such that O �� B�� p�

A method to generate discrete approximations to convex-hulls and�-shapes from boundary rep-resentations is suggested in Chapter 2. Unfortunally, we were unable to test a program implementa-tion of this algorithm due to time limitations.

The same as with the fitting ellipsoid, shape features can be extracted from the comparison ofconvex hull or the alpha shape of an object and the object itself. This can be done in a global orin a local fashion. We present in the following paragraph just an example, together with a practicalestimation of the convex hull in 3D.

5Any plane divides the spaceR� in two half-spaces.6A polytope is defined as the intersection of a finite set of half-spaces.


Convexity-based Shape Factors In 3D, Phillips et al. proposed a simple approximation of theconvex hull of an objectO, called theconvex enclosure [Phillips86]. The object is projected intoorthogonal planes, and the 2D convex hull of the projections are used to construct, by boolean inter-section, a 3D convex shape, the convex enclosureeco�O� of the object (see Figure 1.7). They furtherpropose a shape factor that measures the ”convexity deficiency” ofO, defined by the ratio:

Cdef�O� ��

V � eco�O�� V �O�V �O�

Two important remarks: ramified thin objects and networks have a very highCdef�O�, as in theexample of Figure 1.3, whereV � eco�O�� V �O�. The second observation is that the constructionof eco�O� and the ratioCdef�O� strongly depend on orientation of a coordinate system, since they arebased on features from projections on the orthogonal planesX� Y andZ. A way to compensate forthis bias is to align the coordinate system with the principal axes.

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

(a) (b)

(c)

Figure 1.7:The convex enclosure. (a) A 3D particle and (b) its orthogonal plane projections. The convexenclosureis the intersection of the three orthogonal prisms defined by displacing the 2D convex hull (c) ofeach of the projections.

In the case ofglobal features, form or shape factors as those presented in Section 1.2.1, can bebetter analyzed in the case of convex shapes. The surface areaA and the volumeV of a bounded con-vex setSn in Euclidean n-spaceRn are connected by theisoperimetric inequality,An � nnUV n��

where U is the volume of then-sphere. If the setSn is itself then-sphere, then this becomes anequality.7

The relations between different-dimensional measures, such as compactness (sphericity or shapefactors) are a special case of the problems on isoperimetric inequalities, which link measures onboundaries and their contents in multiple-dimensions (thus, they are ratherisovolumetric). In re-cent decades this concept has been extended to include inequalities connecting geometric or physicalquantities that depend on size, shape and configuration of a set [Kiyosi96b], pp. 857. It even includesinequalities on eigenvalues of partial differential equations under given boundary conditions. For ex-ample, vibrating membranes possess characteristic modes satisfying different eigenvalue inequalities

7that shape factorsff�D � ff�D correspond to casesn � �� .


in function of the shape and area of the membrane, and its proportions (as with the length of vibratingstrings)[Polya51].

In a local approach, surface features can be approximated by polynomials, splines, superquadrics,or other surface patches. In these approaches a new problem is introduced: the compromise of localfitting versus continuity across patch boundaries. Complexity is then determined by the number ofcoefficients, the number of patches and their assembly under matching constraints. Sophisticated so-lutions have been proposed, as that by Keren et al. who construct a shape model with superquadricson an orthonormal wavelet basis, allowing the representation of surface singularities (e.g., cusps,umbilics, abrupt ridges and other differential discontinuities) [KerenDCO92].

It is clear that these problems and a local differential-geometry approach lead to the concepts ofcontinuity andcoherence; we treat only the latter in the following subsection, in the context of shapecomplexity.

1.2.5 Coherence

Differential geometry of curves and surfaces treats mostly smooth, continuous shapes in a ratherlocal, differential way. Surface discontinuities or singularities are topics of active research, requiringa special treatment. As the present work deals with discrete representations, we only mention inthis section the relevant issues concerningcoherent structures. Most of the other sections in thischapter deal with some lack of continuity and irregular boundaries that are not ”smooth” at someparticular scale of analysis (or at all scales, in the case of mathematical fractals). Koenderink notedfirst the importance of ”scale of analysis” of a shape through the concept of blurred derivatives[Koenderink90] and scale-space filtering [XinLimHong95]. In fact, most methods dealing with ill-posed problems involve smoothing and regularization techniques to avoid singularities. Many ofthese methods exploit a principle of coherence.

The termcoherence in general is defined as ”natural or due agreement of parts”. It denotes thepresence of repetitive patterns, and is the measure in which two or more things stick together, ormaintain a constant relationship or harmony. In optics, coherence refers to certain phase, orientationor wavelength relationships of electromagnetic waves, and is often considered for a specific wave-length range. Coherence can also name the similarities or simple relationships between two or morefeature distributions. Such relationships may consist in constant differences, for example. Thus, wecan deduce two general concepts, following Gröller and Purgathofer [Groller95]:

✦ Spatial Coherence refers to spatial homogeneity, consequence of constant or slow varyingrelationships in the spatial arrangement of data features, or of the objects themselves.

✦ Object Coherence refers to known local relationships between objects or parts of an object.

As a contribution of the present work we introduce two complementary principles that hold in com-mon situations:

✦ the Object-Coherence Principle, which stipulates that local neighborhoods are likely to beoccupied by the same objects, and

✦ theSpatial-Coherence Principle, which refers to local constancy of feature values.

These principles can be formulated as hypotheses to be tested for a particular problem. In terms ofpixel or voxel populations with a gray-value label, this local constancy results for example in localhistograms of intensity values remaining constant or slow-varying from one region to another, or


from one slice of an object to the next one. Letp�� p� be two voxels such thatkp� � p�k � �, with� � � ”small” (p�� p� are located in contiguous slices, for example); letN �p� denote a neighborhoodaround voxelp, andH�� a value/label histogram of a specified region; then we postule aLabelCoherence hypothesis forp�� p�:

� � � �� such thatkmodefH�N �p��g �modefH�N �p��gk � �(1.17)

wheremodef�g is an operator to extract the principal mode of the histogramH�� (the most frequentvalue of samplesfN �p��g�. After several experiences, we observed that this principle holds even ifthe object is very complex and even if undersampling or partial volume effects are significant.8 Wepostpone to Section 3.3.2 a further discussion and an application of theSpatial-Coherence Principle,using a particular color version of Equation 1.17.

Special cases of the object- and spatial-coherence properties have been widely used in computergraphics algorithms for incremental update operations, that is, to make calculations only when animportant change occurs. This cases include scan-line coherence [Foley92] and span coherence[Crocker84], but there are also depth, area, temporal, frame, projective, ray, cluster and volumecoherence [Groller95, Max90, Wilhelms91, Hubschman82, Horvath92].

A formal definition of coherence in signal processing and statistics summarizes some of theabove notions as follows:

DEFINITION 1.16 (Coherence)Let Xt � �X

��t � X

��t � � � � � X

�p�t � � C p � p � N� t � R be a complex-valued stationary process,

and let Sk�l�� be the cross Spectral Density Function (SDF) of X �k�t and X �l�

t at frequency �, with J ”SDF”k� l � f�� pg and k �� l. Then, the quantity:

�k�l�� jSk�l��j�

Sk�k��Sl�l��(1.18)

is called coherenceofXt at frequency �.

In this definition, coherence�k�l�� is a measure of the strength of association betweenX�k�t and

X�l�t at frequency�. Generally, we have� � �k�l�� . Coherence is not a measure of information

contents, but it is rather a measure of order. Complex signals or process far from thermodynamicalequilibrium may present different levels of organization in which coherent structures appear. Severalproperties of time series data and stochastic process are expressed in terms of the SDF, the cross SDF,correlation, autocorrelation and coherence, and are related to moving averages and autoregressivemodels (also known asAuto Regressive Moving Average (ARMA) models). It is worth noting that J ”ARMA”analysis of complex 3D objects in a discrete grid with coordinatesi� j� k can be done in terms ofsignal sequences if one coordinate, sayk is treated as ”time”, and the other coordinates are treatedas a sequence of image slicesi� j. In this way, the general definition of coherence is applicable.

We present in Chapter 3.3.2 a problem where thespatial coherence principle is employed asa working hypothesis for radiometric inhomogeneity correction of color images, when a complexstructure (the bronchial-vessel tree) is present. An ARMA model is also used for description of theproblem and the correction method.

8We see in Section 3.3.2 that this holds for� � � smaller than shape features whose variations are smaller thanH�N �p��.


Directional Coherence We introduce another definition of coherence, in terms of a ”similar-orientation” principle. Observing the normal vectors on a smooth surface, or the tangential vectorsalong a smooth tubular segment, small variations (or no variations at all) correspond to a coherentbehavior: the same average direction. Other feature vectors can be also coherent if they are ”similar”enough. This can be stated as a local or a global property as follows (see Figure 1.9):

DEFINITION 1.17 (Local Directional Coherence)LetO be an object, p � O, andN �R� p� a ball-neighborhood of radiusR. LetNN

�� card�N �R� p��

be the number of points q, with q � N �R� p�. Let Vp �� fvig� i � �� NN a set of vectors in R�

(e.g., normals, tangent or feature vectors), associated with p. Let

v �� vi �� NN

X�vi�N �R�p�

vi

be the average of fvig and let �v�� kvi � vk� � its standard deviation vector. Then, the set Vp

is locally coherent inN �R� p� iff � � � such that �v � �.

DEFINITION 1.18 (Global Directional Coherence)Let Vp a set of vectors associated with each p � O. The pair �O�Vp� is globally coherent iff

� p � O� Vp is locally coherent in N �R� p��

A convenient sizeR of neighborhood of analysis must be chosen. Note that, as with the notation”�-shape”, a more accurate nomenclature is ”R-local coherence”. TheDirectional Coherence defi-nitions allows a characterization of tubular structures (see Appendix B, Section B and Figure B.1).

Note that the average� v � may vary at eachp, but coherence exists if its standard deviation�v remains small. In the case of normal vectors, the lack of coherence is a measure of rough-ness. The RMS roughness is in fact the standard deviation from average height, a scalar feature(see Section 5.6). Figure 1.9 illustrates coherence of the normal vector as describing smooth andtextured, rough surfaces. In the case of surface tangent vectors, coherence measures for examplethe irregularities of a 3D filament with respect to a smooth version of it. Bifurcation points of afilamentary structure are characterized by at least ”two directions”, thus, two coherent measures ofthe local tangent vector. It also may happen that coherence exists only in some orientations, hence itis anisotropic coherence. 3D columnar structures and oriented textures are also examples in which acoherent direction characterizes distributed ”parallel” components.

Local Directional Coherence can be used as a distinctive characteristic of thin tubular structuresand was applied in this work for the design of a discrete tubular detector of bronchial blood-vessels(see Appendix B). Adifferential coherence definition could serve to characterize local variations.In this case, instead of small neighborhoodsN �R� p�, we must consider surface elementsdS on thetubular structure.

1.2.6 Scales of Representation and Characteristic Lengths

We conclude this section illustrating the notions of scales of representation (usually it is also thescale of analysis) and of characteristic lengths with an example. We first introduce the most relevantlevels of description used in literature.

Barret separated shape description into three general aspects that depend on scale [Barret80]. Heused the terms:


V

(a) (b)

N( )

��

< >

n

order

n

r

p

R, p

d

p

Figure 1.8:Some uses of the normal vector �n of a surface. (a) Local depth or thickness maps in the normaldirection at surface points p also characterize average branch diameters � � �r � in tubular structures. (b)The normal behavior in geodesic neighborhoodsN �R� p� of surface points p help to characterize roughnessfeatures (see Figure 1.9).

k

nk ∼σ( ) ∼ 0

k

σ( ) >> εn

nn

k

Figure 1.9:Coherence of the normal vectors of a surface. A small standard deviation of the normal in a localgeodesic neighborhoodN��R� p� (or surface element dS) defines a high local coherence, corresponding tosmooth surfaces (left), ��n�k � , with k indexing the points inside N��R� p�. A high standard deviation(��n�k � for some � ) in a local geodesic neighborhood (low normal coherence), characterizes tex-tured or rough surfaces. Note that coherence definitions depend on the scale of analysis, that is, the size ofthe neighborhood. To define differential coherencesmall geodesic neighborhoods N��R� p� become surfaceelements dS on the tubular structure.

✦ shape for large-scale variations (the general form of an object);

✦ roundness for the degree of smoothness in vertices and smaller variations (concavities, protu-berances); and

✦ surface texture for short-range fluctuations superimposed to the latter variations.

In terms of 2D Fourier coefficients, some authors describe “shape” retaining the first 5 Fouriercoefficients [Serra82], while what is called texture starts at some level that depends on application


and resolution limits. IffAk� k � �� g are the Fourier coefficients of a radial functionr�x� y�,Beddow et al. suggested the following shape measures of 2D contours [Beddow77]:

✦ global structure:Ln� �

n�Xk��

A�k

✦ roughness parameters:Rn�n� �

n�Xk�n�

A�k

wheren�� n�� n� � N are chosen in function of the shapes to analyze (typically, Beddow et al.repport the values ofn� � �� n� � �� n� � ��, as those coefficients bearing “enough”information, but no justification has been given).

(e)

(d)

(f)

(g)(j)

(h)

(i)

(a)

(b)

(c)

(k)

(l)

(m)

(n)

(o)

Figure 1.10:Complex shape characteristics in 2D slices as found in hystological preparations. (a), (b) and(f) show features described at different scales, with arrows indicating characteristic lengths , tangential andperpendicular to the ”average” contour. (k) and (o) are shape representations at different scales of analysis.See the text for a more detailed description.

Figure 1.10 shows an example of these and several other aspects of shape description that donot fit well in the scale-based description of Barret. This shape is representative of hystologicalpreparations. Circles in Figure 1.10 depict scale variation corresponding to: (a)-(b) surface textureor roughness details, (c) smooth areas (roundness features), and (f), (h), and (j) large shape features.Other features, as (g) have with one dimension belonging to subsampling scale (texture scale) andother dimensions to larger-scales; these features are difficult to detect and to integrate in a shapemultilevel description (notice aperceptual alignment of some (g) features with cavity in (j)). Thereare also features that suffer a topology change (connectivity) at different scales of analysis, as from(d) to (l), and from (g) to (m). We observe that pores (g) bear similarities to surface texture in (b), be-longing to a smaller scale of analysis, but they may be in 3D the components of thin tubular sections,or surface variations in other slices (as in (d)). At coarser resolutions as in (m), they disappear or co-alesce into single gaps, in function of proximity relations. A 3D object may appear in a cross-sectionas composed of several features (i), that are mistaken as part of the 2D shape at lower resolutions (k).

1.3 Extrinsic Properties of Solid Shapes 45

At larger scales of analysis some structures are identified as: (j) concavities, (f) protruding regions,and (h) cavities, tunnels or transversal sections of cavities appearing as holes. With lower resolutionsall these features may remain or gradually fuse into the global shape (o).

Characteristic lengths are sometimes analyzed in terms of an average shape contour astangentialor perpendicular to this average contour (see Figure 1.10b). The degree of contour smoothing isa relative notion. Both lengths are also called, respectively,characteristic width �k (if parallel ortangential) andcharacteristic height or depth � (if orthogonal). Note that�k in 3D gives rise to twoindependent lengths. These concepts for 3D irregular surfaces are introduced and used for roughnesscharacterization in the application described in Section 5.6. page 5d. In this case, instead of anaverage contour used as reference, the mean height of a multiple-valued functionzk � fk�x� y� isused to measure�k and�. We have detailed another application in 2D texture discrimination in[Corkidi98].

High level characteristics in Figure 1.10 such as the alignment of features (h), (g), (e), or thefisure in (g) and the cavity (j), areextrinsic properties of shapes, and are briefly described in thenext section. The detection and quantification of such characteristics belong to Pattern Recognitionanalysis, a subject not deeply studied in the present work.

Scale-Spaces. We have already broadly justified in another section the (vague) terms “small”,“medium” and “large” applied to scales of analysis (neighborhoods) as being relative to the size ofthe features of interest or available resolution. Multi-resolution is the concept introduced to deal withseveral scales of representation and analysis at the same time. It has evolved in the last years into avery rich and growing field in Computer Vision including concepts from human vision, differentialgeometry and dimensional analysis. The core of this theory is the study of the structure of the discon-tinuities in thescale-space, in such a way that analysis is done considering the structure features insuch scale-space and their differential properties. We cannot enter here into details, but just mentionthat relevant literature includes works by Koenderink, Lindberg, Haar Romeny, Alvarez and Morel,Florac, Kimia, and others [Haar90c, Lindeberg92b, Florack92, Alvarez93, Kimia93, Lindeberg94,Romeny94, Florack95b, Romeny96].

1.3 Extrinsic Properties of Solid Shapes

We mention in this section some features not belonging to a shapeper se, but which depend of itscontext or configuration in a larger space. This features can in turn be treated as if they constituteda physical object, having themselves ”intrinsic” properties similar to those already presented (coher-ence, convexity, isotropy, characteristic lengths, etc.). Most aspects of extrinsic properties belongto the fields of Pattern Recognition, Gestalt Perception and high-level descriptions. Since we havefocused our work on the intrinsic properties, this section is only a rough overview, but some pointsare discussed in the Section 5.8 of Perspectives.

1.3.1 Complex Relations

Timely problems in industry include those posed by highly engineered synthetic fibers (fatigue con-trol, optimal textile machining [Subhash94]), ultrathin carbon fiber threads, composite parts, or newmaterials which require understanding of microstructure, and even of complex woven and braidedstructures. Textured layers of fibers (like fiberglass, paper or ”knitwear” fabrics) pose related prob-lems [Groller95b]. In these researches, few 3D data are produced in similar ways as medical imaging


techniques, but computer simulations of process and material configurations do produce volumic dataand virtual objects that have to be analyzed and compared with available observations of real proto-types.

When applying ”divide and conquer” strategies to decompose an object into simpler components,shape complexity is translated into multiplication of elements, features and relations (e.g., intercon-nections, assembly). There may be interpretative, functional, or physical justifications of how theanalysis or decomposition has to proceed, identifying the object itself as an aggregate, a composite,or the result of some manufacturing process. In the latter case, prior blueprints, CAD models andassembling protocols may exist already (prototyping), describing how constituent parts go together.Additional complexity problems arise due to assembly protocols, in which not only a specific or-der is required, but several operations must be performed at the same time, or following intrincateschedules.

Reverse Engineering (RE) consists in obtaining or completing such blueprints and protocolsfrom an existing man-made object, and it is considered as a complex discipline, relying heavily onrestoration and the tackling of inverse problems. Even in the absence of documentation, a lot ofprior information exists for similar objects and functionalities. Either inverse or direct, it is clearthat engineering and modern technology require very complex descriptions of products and devices,and these descriptions are attached to the object itself, not only for its replication but for its use andmaintenance.

Virtual copies of museum pieces pose similar problems, but in this case the goal is a faithfuldocumentation. Available data from sensors (laser scanners, for example) may consist in unstruc-tured clouds of points, and optimized matching between multiple views has to be done [Benjemaa98,LiaoMedioni95]. In this case, several reconstruction, filtering and segmentation techniques are ap-plied, and complex morphological features pose several problems, such as occluded views, physicalaccess of the sensor, and camera calibration.

1.3.2 Many Bodies and Clusters

When considering a set or aggregate of several objects, the distribution of morphometric attributesmay present homogeneities and reflect an internal structure, a ”shape”. For example, the presenceof two populations of shapes (big and small, or round and irregular) may reveal two superposedmodes of a physical phenomenon, or two distinct phenomena giving shape to a global population.In a dynamical system, a state transition may be characterized by abrupt changes in function ofsome parameter. Section 5.7.3 presents an example of application in which the quantity and inter-connectivity of components reflect features of a physical process, in function of temperature.

To reduce complexity in a given description, there are at least two situations: a single complex ob-ject may be described as the result of the aggregation, assembly or interaction of many simpler andheterogeneous parts. The converse situation arises equally often: clouds of thousands of particlesmay be treated as one single (virtual) object, or an associated shape (their convex hull, a morpholog-ical closing, or other), or as a cluster of objects. In this approach intrinsic properties (e.g., topologyand compactness) are analyzed on the virtual shape.

Complex features may thus appear as perceptual structures of higher order (comprising manycomponents), and be characterized by proximity and neighboring relationships, distance distributionsand zones of influences. ”Shape analysis” is then translated into studying the spatial organization orarrangement of the composing elements. Spatial relationships are coded by adjacency graphs andmatrices, where adjacency is defined in terms of contact, distance or proximity measures. Theserepresentations also comprise the topological features of the set in the form of nodes or vertices,

1.3 Extrinsic Properties of Solid Shapes 47

branches, circuits, cycles, Euler number, connectivity, vertex degree, etc. Topology invariances areone of the most important properties, because two similar objects or patterns can be matched ifcorresponding topological features (bifurcation points, holes, centroids) are correctly paired, even ifone pattern is deformed. Hierarchical approaches allow to relax topological conditions, to accountfor small variations or noise in these features.

Mathematical morphology operators are often used to intentionally modify topology: a clusterof nearby objects is treated and measured as a single ”big shape”. As an example in biomedicalapplications, we have intensively used traditional closings and openings to identify mitosis for cell-proliferation analysis [MarquezMIT90, Garza92, Corkidi98], where a virtual shape, the mitosis, isdetected when a definite number of small characteristic components appear clustered together (dataconsists of gray-level images of� �� chromosome particles). If the cluster is too open or too closed,it is counted apart, and textural features are then considered to discriminate them from artifacts.

1.3.3 Complex Objects in Feature Space

From the above paragraphs, when comparing or characterizing several objects, either similar, distinctor composing together a single object (real or virtual), it is clear that the morphological features andattributes themselves give ”shape” to clusters, patterns and objects infeature spaces, and the numberof attributes determines its dimension. A familiar case is color, which has several vector represen-tations in different 3D color spaces (RGB, HSL, CIE,L*a*b, etc.). When a high number of objectsis present, histograms account for feature populations. Another remarkable example are dynamicalsystems, whose behavior is studied in parametric spaces, where stability or equilibrium structures(attractors) appear and present complex, fractal shapes. An important tool to reduce feature-spacedimension is PCA, as described in Section 1.2.1, where only the axes (equal to attributes) that havethe highest value (i.e., those that concentrate most energy) are retained for further analysis.

A particular phenomenon related to shape description is that of feature dependence on spatialposition. Let� be a morphometric parameter, say, mean diameter of a particle distribution. Insome situations� � f�x� y� z� depends on position in some local coordinate system, and there isa ”gradient” of sizes and, at constant spatial density of particles, mean mass density varies, even ifeach particle has constant density. Color and texture distribution on an object or a set of objects areother common examples of spatial varying attributes.

1.3.4 Complex Representations

When analyzing several representation models, it is not clear how the tradeoffs, approximations,and rough assumptions (e.g., linearity, symmetry, scale-invariance, first-order relationships) may in-troduce complex features attached to the model and not belonging to the object or structure beingrepresented. However, several examples are at hand in which such features are useful. In ScientificVisualization and Engineering the trianguled surfaces give rise to many models and approaches thatemphasize a particular aspect of a problem, like efficiency, realistic renderings, or physical proper-ties, and leave room for introducing a priori information. Their complexity allows to introduce, forexample, analytical formulations. Another reason for introducing some degree of additional com-plexity is the search of linear representations. For example, 3D rigid transforms and perspectiveprojections are best expressed by 4D homogeneous vectors and quaternions (a special case of 4x4matrices, or 2x2 matrices with entries inC ). Complexity is thus present in the ways an object isrepresented, and the study of minimal-length description is the subject of Section 1.5.1.


Even if fractal sets ideally possess infinite details, they may in turn constitute a simplification inshape description, since they have scale-invariance properties, easy to describe. If a fractal modelmay imply an infinite number of elements (thus, resulting in a ”very complex” representation fromthe point of view of ”number of details”), their interrelations are often described in a very com-pact way, by iterative process. The same as with Fourier and Wavelet analysis (global periodicitiesand spatial-localized periodicities), fractal models constitute a well-adapted representation for manyshape-descriptionproblems. To understand this and other aspects of irregularity features, we examinein the following section the subject of fractal dimensions in relation with shapes and morphologicalcharacterizations.

1.4 Fractal Dimensions

Literature on the subject of fractal sets and applicationshas often reported fractal dimension measure-ments that are meaningful, over reasonable scale ranges, for virtually all kinds of irregular objectsand data sets. They are strongly associated with intricacy and infinite details. Even if fractal dimen-sions (or multifractal dimensions) are not a universal morphological feature, its very absence wouldconstitute a singular feature of complex objects, at least as important as non-convexity, anisotropy,asymmetry, incoherence, and other ”negative” properties. As many fractal dimensions rely on op-erational definitions which end up with a graphical plot, even in extreme non-fractal cases, use-ful information is obtained in the form of a ”signature” (measure versus scale behavior), when alogarithmic-regression slope is not meaningful (the slope is the fractal dimension).

This section presents some excerpts from a fractal tutorial [Marquezftuto92], updated after fur-ther bibliographic examination. Abundant illustrationsand examples of all dimension definitions andfractal features can be found in the basic bibliography [MandelbrotFGN82, Barnsley88, Armin91,Armin94, Armin96, Takayasu90, Marquezftuto92], and we present only the main definitions andproperties concerning irregular structures and sets.

We make first a survey of dimension definitions of a setF � Rn, and then discuss properties offractals in terms of complex shapes.

1.4.1 Dimension Definitions

Exceptingd anddT , all the following dimensions take values inR, between the associated integerdimensionsd anddT . SometimesD alone is used to represent a generic fractional dimension, but isalso identified with the HausdorffDH or the capacity, or box-countingDC dimensions [Mattila84].In the followingL denotes a linear length size, andL-box a square or cubic box of sideL.

1 Remarks. Literature notation (Dsubscript, or dsubscript), is not homogeneous. We simply adopted lowcased for (apriori) integer dimension definitions, and capitalD for (potentially) fractional dimension definitions.

Euclidean dimensiond Also known asEmbedding Dimension d, (or D, when usingd for ”dis-tance”) refers to the dimension of the Euclidean spaceRn in whichF is embedded (thusd � n).The symbolDE is sometimes used. For a warped surface inR

�, for example, the topological dimen-sion (see below) isdT � � and the Euclidian dimension isd � �. ”Non-Euclidean fractals” is anambiguous term (by definition of fractal dimension, all true fractals ”are not Euclidean”), it shouldrefer to fractal sets embedded in hyperbolic geometry (or Lobachevsky-Bolyai-Gauss geometry), orelliptic (Riemannian) geometry, but it is sometimes misused in basic literature. All non-Euclidean

1.4 Fractal Dimensions 49

dimensionsD satisfyD � d. WhenD is very close tod, the irregularity and roughness features areaccentuated and the fractal set tend to fill the embedding space.

Topological dimensiondT This dimension is defined on local properties for all points in a setF � R

d, and corresponds to the intuitive notion of dimensiondT � � for points,dT � � for linesand smooth curves,dT � � for surfaces, etc., without regard of howF is embedded in a higherdimensional space (that is, it is invariant under homotopic transformations). Thus,dT varies infunction of the scale of analysis. A recursive definition can be given:

(a) dT � � if F is not connected (e.g., points),

(b) dT � k� k � � if any p � F has arbitrarily small neighborhoodsNr�p� whose boundary�Nr�p� has dimensiondT � k � �,

and the intuitive notion follows, because fork � �, all neighborhood of points in a smooth curve isjust an interval whose boundary consist of two points with dimensiond T � k � � � �. A closedcurve may be the boundary of a point on a surface, and a closed surface defines a solid, etc. Thetopological dimension is preserved when a homeomorphism deforms the object. Though there aresevere difficulties for the notion of dimension which would behave that way [Peitgen92].

Similarity dimension DS This dimension measures invariance ofF to scale transformations (di-lations and shrinkings). It is strictly applied only to sets with exact autosimilarity orself-similarity.However, a non-deterministic definition (stochastic self-similarity) exists, andself-affinity is a moregeneral terminology denoting statistical scale-invariance. Autosimilar fractals are built from scaled(maybe also deformed) copies of the whole, and this process is repeated at smaller scales. InR

d, apower law with exponentd�DS measures how characteristic feature lengths vary with scale.

Hausdorff dimensionDH Also known as Dimension of Hausdorff-Besicovich, it is defined bythe most efficient covering, that is terms of theHausdorff measure �d��: Let d� � � R, and letN�� f�d��d a set oftest functions f � R � R, such thatN�� is the number of discs, balls,boxes, or any covering, of diameter (size)� needed to coverF . Then, there exists a critical, uniquereal valued � DH , which we callHausdorff dimension of F , such that:

d � DH � N�� and d � DH � N��

�d�� is the measure obtained from the covering formed by all test functions, which can be interpretedas a value between length and area, area and volume, etc. A square has an infinite number of points��, an infinite ”length content”9 ��, it has a finite, non-zero area��, constant for any�, and a zerovolume��. The Hausdorff dimension is the most useful definition of fractal dimension, but it isdifficult to calculate. So, it is customary to estimateDH using the similarity dimensionDS , or thecapacity dimensionDC , defined in the next paragraph.

Mandelbrot gave three definitions of fractal. The first comes from the intuitive etimology of hiscoined term (fractus: fractured (irregular),fragere: fragment, fraction). The second introduces theconcept ofself-similarity: a fractal is a shape made of parts similar to the whole in some way. Thethird definition is the technical accepted one, usingDH anddT : A fractal is by definition a set forwhich dT � DH � d; that is, the Hausdorff Besicovitch dimension of the set strictly exceeds thetopological dimension [MandelbrotFGN82].

9Defined for example as the perimeter of a Peano curve filling the square, the covering being segments of size�.


Capacity DimensionDC

Also namedbox counting dimension [Voss86, Falconer90], defined by covering with identicalspheres, cubes or other test elements.DC is usually estimated by counting the numberNbox of cubicL-boxes, containing at least a point (or pixel, voxel, etc.) of the set. For example, ifLmax is the sizeof a square gray-level image, its capacity dimensionDC is defined by:

Nbox�L� �

�Lmax

L

�DC

WhereNbox is the number of cubes occupied by a pixel value in the 3D space defined by the imagecoordinates and the intensity axis.Nbox is calculated for different sizesL, and a linear regressionon the coordinate plane�log�Nbox�� log�L�� allows to estimateDC . As an example of application,Chuang et al. report [Chuang91] the measurement ofDC of the brain surface, as a measure of itsconvoluted shape. The drawback in their method is to measure discrete surface area as ”number ofvoxel faces”, introducing an important error which may explain the mismatch between their resultand other measurements [Majumdar88]. We introduce in Section 2.5.1 a corrected surface estimationfor anisotropic voxels.

Information dimension DI

Probability distributions of samples fromF allow to defineDI . It applies to stochastic distribu-tions of points and can be calculated by uniform sampling of space intoNcell cells of lengthL (or”L-boxes”). LetPi�L� the probability that one randomly chosen point is in theith cell. Then theinformationI�L� is defined by [Takayasu90, Taylor91]:

I�L� � �NcellXi��

Pi�L� � logPi�L�� withNcellXi��

Pi�L� � ��

and the information dimension (orentropic dimension) DI is defined ifI�L� follows a logarithmiclaw:

I�L� � I��DI logL

qth-order information dimension Dq This is an extension ofDI . DimensionsDC , DI and thecorrelation exponents result to be special cases ofDq. Several dimension features associated withfractals can be obtained from the probabilityP �m�L� of havingm points (pixels, voxels) inside aL-box. Mandelbrot [MandelbrotFGN82], Sarkar and Chaudury [Sarkar92] proposed to calculate themomentsMq�L� of the distributionP �m�L�, for allL:

M q�L� �

� PNm��m

qP �m�L�� if q ��

� if q � �

M�L�� M��L� is called themass dimension. The qth-order information dimensionsDq are

then defined from the expected value of the logarithmic derivative of the moments, with respectto L, which is a formal way to express linear regression, that is, the estimation of a fitted line byleast-squares, with slopeDq, to a cloud ofn point samples�logLi� logM

q�Li��, with i � �� n�:

Dq �

�� q

D∂ logMq�L�

∂ logL

E� if q ��

��q

�∂�PN

m�� logmP �m�L��∂ logL

�� if q � �

(1.19)


where� X � is the expected value of the random variableX , and corresponds in practice toselection of some (few) samples ofL, covering the range of scales where the fractal is defined. Theoperation �

∂ logM q�L�

∂ logL

�is the expected value of the slopeDq at a particular point�L�M q�L�� in log-log scale. To avoidconfusion with the boundary of a set,�O, we use symbol∂ to denote partial derivatives but alsoemploy rather the discrete notation �

� logM q�Li�

� logLi

�with i � �� n� to indicate linear least-squares fitting ofn samples.

Lyapunov dimensionDL

This measure is used to characterize the dimension of thechaotic attractors10. It is defined interms of the first two Lyapunov Characteristic Exponents (LCE)�� as:

DL��

The LCEs�� give the rate of exponential divergence from perturbed initial conditions of a sys-tem [Chirikov79]. LCEs are defined in terms of the average deviationU�t� from the unperturbedevolution of the system at timet:

�i�� lim

t�log kU�t�k i � f�� g�

We only mention here the relevance of dynamic systems in the study of complex structures asa two-way relation: non-linear dynamic process give rise to very complex physical structures (fromatoms to storms, and even organisms, viewed as stable (homeostatic) processes). At the same time,a geometrical description of thephase space (or state space) of a dynamic system involves complexshapes or point configurations, often fractal, which constitute the chaotic attractor.

Spectral dimension�D

It is related to random walk properties on fractal structures. ”Spectral” orfracton dimension isuseful to describe the kind of physical process confined within a fractal network or generating such astructure (aggregation process). As computer simulation of these systems involve Brownian motion(the random walk trajectories of interacting particles), understanding of such process is related toshape of the resulting structures. An example of this relation is given in the Appendix C, wherea Kardar-Parisi-Zheng (KPZ) equation predicts roughness in amorphous deposition. It is to notethat �D and other dimensions are different of each other because they contain different information(temporal behavior, probability transitions, harmonic composition, and many others, some not yetfully identified).

Minkowski or mathematical-morphology dimensionDM

A Mathematical-morphology dimension may be formulated in terms of Minkowski addition andsubtractionoperators, thus, we give an operational definition. LetX�ELi represent the mathematicalmorphology dilation of setX by a structural elementELi of characteristic sizeLi� i � �� Nscales�.Here,ELi plays the role of theL-box coverings. UsuallyELi are disks of variable radius�i �

10The attractor of a perfect elliptical orbit of an object around a planet, is a single point in phase space (velocityand position), while systems far from equilibrium possess fractal-shaped (or ”chaotic”) attractors, also calleddissipativestructures, because energy is dissipated from the system. In general attractors are the limit set of an iterated map


Li, with i � �� Nscales�. Each sampled point on the boundary of an object (�F ), is dilated (theequivalence of being covered with disks) to create a dilated boundary called theMinkowski sausagewith an area:

A�Li�� A��F � ELi�

The Minkowski dimensionDM , is calculated from the relation:

A�L� � L��DM �

by linear regression onlogA�L� vs. logL. Note thatDM is a particular case of the box countingdimensionDC , but with emphasize on MM operations. The wide use of discrete MM in ImageProcessing and in our discrete boundary approach makesDM an interesting shape feature.

An alternative definition was given by Serra in [Serra82], suggesting a linear regression onlogA��L� againstlogL. He uses�i as the resolution scales,i � �� Nscales� of a sequence ofimages, andNscalesdilations with a single structuring elementB, the unitary disk. Using his nota-tion, we replaceLi by �i, andELi by �iB. He then calls ”length of the boundary set” the followingquantity:

L��F� �i� ��

A��F � �iB�

��i

which allows further interpretations. In the continuous case, for example, a self-similar fractal con-tour has infinite length (perimeter), at the limit of�i � �.

Table 1.1 summarizes the dimensions above discussed and includes some other definitions foundin the literature. Several other dimension definitions are possible but it is clear that many of themare not conceptually different, and some of them are just different ways to estimate a few importantfractal dimensions:DH � DC� DS � Dq or to measure very particular features, or scale and lengths indifferent spaces or different metrics. Just to cite an example, thechemical distance introduces thegraph or chemical dimension, meaningful in the study of percolation clusters.


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1.4.2 Morphological Features of Fractals

In addition to formal definitions, several types of fractals have been identified, in function of howthey are embedded in space and to what features refer their fractal dimension. To mention threeexamples:

Surface or boundary fractals: Dense objects where only the surface or boundary is scale invariant[Russ94]. Its fractal dimension characterizes ”roughness” of the surface. A classical examplein 3D is the brain cortex and mountain reliefs.

Mass fractals: Fractals in which both the internal structure (mass distribution) and surface are frac-tal (Pfeifer, 1989; Gouyet, 1991; Russ, 1994; Zahid and Ganczarczyk, 1994). Its fractal di-mension characterizes the space-filling ability. Natural examples are the blood-vessel systemof the lungs, botanical trees, and lettuces. An artifical example is found in some communica-tion networks.

Pore fractals: Dense objects in which hole or pore structure and its distribution are fractal (Gouyet,1991; Russ, 1994) or the pores and surface have similar scaling properties (Pfeifer, 1989; Zahidand Ganczarczyk, 1994). Some natural sponges constitute an example.

We finally give in this section a summary of qualitative salient morphological features of fractals.

A. Roughness.Fractals are too irregular to be locally described with conventional Euclidean geom-etry.

B. Fine structure. Details appear at arbitrary small scales; i.e., the Fourier spectrum of a (mathe-matical) fractal signal has infinite suport.

C. Fractional associated measures.Not only the spectrum of a fractal signal is infinite (details atall scales), but the autocorrelation (power spectrum) function of a fractal time seriesf � R� R

has a particular curve shape: if linear, it may have fractional slope. Similar features can beobserved for 2D or 3D curves and surfaces. In order to describe signal or system properties offractal process by integral transforms and differential equations, the need arises to introducefractional orders of integration and differentiation. For example, afractional Brownian motion(fBm) can be described as a fractional ”integral” of white noise. fBm has been successfully”fBm” Iused to describe natural reliefs, for example.

D. Scaling. Fractals often exhibit ”scaling”, self-similar or autoaffine properties, that is, features areunchanged by scale transformations, or they can be described as being copies of the sameset at a different scale. This scale-invariance may be exact (periodic fractals), approximative(deterministic fractals) or statistical (random fractals). Conversely, and for a fractal object orset of points, its dimension is a scale-invariant feature, describing the ratio of the number offeatures at one scale to the number of features at another (larger) scale. A deterministic andreal fractal exhibit a constant ratio, independent of the scale.

E. Recursive construction. Autosimilar fractals admit a recursive description and construction, orcan be defined by component organization rules using conventional geometries, becoming thelimit set of iterative processfk�� g�fk�. Even non-deterministic fractals may admit anapproximative model through stochastic constructions. An example is theIterated FunctionSystems (IFS), introduced by Barnsley [Barnsley88], and often used in fractal compression.”IFS” IThese systems specify affine transformations and sets of probabilities to make ofany shape ordata set theattractor set of an iterative process that approximates the shape.


F. Non-integer dimension. The ”fractal dimension” (as defined operationally or theoretically esti-mated) is a non-integer number, larger than the topological dimension of the object. There aresets that exhibit some properties of fractals (self-similarity, infinite detail), while having integerdimensions (the Peano curve, the Hilbert curve and other space-filling sets, for example).

G. Space-embedding properties.Fractal objects may be considered ”to lay” between points andsegments, segments and planes, planes and solids, but they may tend to fill or occupy space inmany other ways: aggregated clusters and porous media, branching structures, complex sheetsand manifolds. In function of the fractal, its dimension measures roughness features (high orinfinite length/surface or surface/volume ratios) and itsspace-filling capabilities.

H. Measure of contents.Their content measure depends on the working dimension: it may be ”in-finite” in the greatest lower integer dimension next to their fractal dimension, or ”zero” in theleast upper dimension, and finite at the fractal dimension of the set. A fractal contour has”infinite” perimeter, even if its area is bounded. Discrete contours samples (polygonizations)yield perimeter estimations which depend on the sampling resolution, and increase logarith-mically with the negative logarithm of the resolution ”yard-stick” (in other words,in a log-logscale, perimeter is proportional to resolution). The implication is that there exist content mea-sures between cardinality (number of points) and length, or between length and surface area,between surface and volume, etc.

For the objectives of our work, we will keep in mind propertiesG andH , because they relatefractal properties with boundary morphology, and warn us against dependence of boundary contentson resolution, when the boundary is fractal. It is interesting to note that early works in QuantitativeMicroscopy noted this problem, advising biologists and material scientists to well indicate the res-olution of their morphometric measurements such as the form factor, which depends on perimeter[Russ90].

It is interesting to note that fractals and other complex objects are not necessarily described by”complex” or even long descriptions. Simple formulas for� allow to obtain billions of digits of itsinfinite decimal representation. Short descriptions are often the case of popular fractal sets, whereprocedural descriptions allows a short piece of computer code to recreate with any desired precisionotherwise infinitely-describable sets. The classic example is the Mandelbrot setM, mathematicallydefined by:

M �� fc � C j lim

n�jfn�z�j �� z � C g�

wherefn�z� denotes the iterative process:

fn��z� � f�fn�z�� for any n� and

f��z� � z� c� with c� z � Cfn�z� is also known as thequadratic map, often written as:

zn�� z�n c� with zn� c � C � z� � ��

A finite region ofC is usually explored and zoomed-in, showing unending details. These formula-tions, and the corresponding computer program to visualizeM are not only finite but very short. Inthe other hand, non-mathematical descriptions (graphical and verbal) are unable to exhaust the infi-nite details ofM. The boundary�M is extremely complex at any resolution, with curled, ”hairly-like” features, and the ”main theme”, the overal shapeM of Figure 1.11a, reappears frequently atdifferent scales (see for example, at the center of the left inset, Figure 1.11).


In Section 1.5 we describe how complexity measures are based on the minimal length of a de-scription. In this sense,M should be a ”primitive”, as one could intuitively expect from a simple2nd-order mapping.

1.4.3 An Example: Particle Aggregates

One of the richest models of irregular fractal structures is theDiffusion Limited Aggregation (DLA)process [Witten83]. This process considers random walk realizations of particles on a diffusive”DLA” I

medium, under a set of (physical) constraints (e.g., force fields, interactions, viscosity and stickness).It is simple to describe and has enabled successful Monte Carlo simulation of many physical and bi-ological process, such as electric discharges, dendritic growth, bacterial colonies, deposition processas those described in Chapter 5.1. Mathematical models exist to predict some associated macroscopicquantities: size, fractal dimension and roughness of an aggregate in function of time process.

Figure 1.12 shows a typical simulation of discrete, two-dimensional DLA in a square lattice.A massm of about 15000 random-walking particles fall into the central cluster (initially a singlefoxed point) forming dentritic branches that restrict movement of incomers. In a ”solid” circularaggregate, mass and radius at timet (in cycles equal to particle life) follow a squared lawmt � r�t ,for all t. In a DLA cluster, radius increases faster, due to unfilled gaps, following for longt rangesa fractional-power lawmt � rDC

t , whereDC turns out to be the fractal capacity dimension of theaggregate. In this exampleDC was calculated by linear regression ofN sampleslogmk versuslog rk� k � �� N .

1.5 Notions and Measures of Complexity

There are more than 30 measures of complexity proposed in the literature [Gell-Mann95, Gunther94],but they may be broadly classified into two categories:Static or Dynamic Complexity [Turney90,Turney90b]. The first addresses the question of how an object, set or system is composed and whatfeatures best describe it. These questions do not depend of the process by which structural informa-tion is encoded. The second category tackles the question of how much dynamical or computationaleffort is needed to describe the information content of an object, set or state of a system. While thefirst measures may influence the second (the shape of an object or the configuration of a system deter-mines its description), both complexities are not equivalent. An object or system may be structurallysimple (i.e. have a low static complexity) but it may have a complex dynamical behavior. As we arealso using the term ”dynamic system”, to avoid confusion we refer toStatic or Dynamic Definitionof Complexity.

High cardinality is perhaps the most important property of what is called ”complex”: high num-ber of elements, inter-connections or features, number of dimensions, size (data volume or high-resolution representations, in order to preserve fine details), high density of details (related to highfrequencies), and other information in the form of structured relationships andlengthy descriptions.Simple shapes or configurations may be related to dynamical complex problems (combinatorial com-plexity): the problem may become intractable because of the huge number of combinations of basiselements, or bancs of filters for specific data extraction.

Literature on ”Complexity” is highly devoted tocomputational complexity theory. Besides math-ematical interestper se, one reason is that complex objects or phenomena are best studied throughsystematic descriptions (i.e., computerized), and experimental observations that produce digital data.Since their study proceeds also through isomorphisms with data structures, computer algorithms pro-

1.5 Notions and Measures of Complexity 57

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vide testable methods of analysis and description, and give quantifiable means to synthesize and/oranalyze such data.

Computational Complexity (CC) is related to the time or memory storage requirements for com-”CC” Iputing some function, and Algorithmic Information Content (AIC) is the most widely used measureof CC. Measure AIC is defined as the length of a computer program required to generate the data.For instance, consider various time series of 1’s, 0’s. The sequence�� has minimum AIC, and arandom sequence has maximum AIC. However, this definition of complexity is unsatisfactory. Thisis because a page of randomly generated characters would have higher AIC than a classic-literaturetext. It is rather desirable to discover the regularities in the time series data, and be able to compressthese regularities in the form of models. Effective complexity is the AIC of the description of the setof regularities and their associated probabilities (for a detailed description, see [Gell-Mann95]). Inpractice, this is computable only as an upper bound.

1.5.1 *Kolmogorov Complexity

*These two Subsections can be skipped or just browsed in a first reading.

Kolmogorov Complexity (KC) is a modern notion of randomness dealing with the quantity of”KC” Iinformation in individual finite objects; that is,pointwise randomness rather than statistical random-ness [LiKolmo93]. Such notion is not possible to define by applying Shannon’s information theory


because, unlike Kolmogorov measure, information theory is only concerned with the average in-formation of a random source. The KC measure is known variously as ‘algorithmic information’,‘algorithmic entropy’, ‘Kolmogorov-Chaitin complexity’, ‘descriptional complexity’, ‘shortest pro-gram length’, ‘algorithmic randomness’, and others.

We saw in Section 1.2 a common example: geometric primitives are used to approximate shapes,and data analysis follows often a pattern recognition approach where raw data are organized into ”fa-miliar” shapes: signals, time-series, images, cluster of points in feature spaces, and so for. Template-matching is a formal approach in Computer Vision to generate reference data, and evaluate thematching of a given sample with the template. In more general terms, data from diverse phenom-ena can often be expressed as a decomposition into unities from a dictionary or basis of primitiveelements, functions or procedures with known properties (templates, Fourier cosinusoids, wavelets,affine transformations, banks of filters, orthogonal functions, and spline-meshes are common ex-amples). We suggest that primitives have not to be ”simple” in the classical sense, and ”complexprimitives” can be used as far as their description and realization is compact.

In this analytic approach, once a descriptive basis is selected, comparisons and measures of sim-plicity or complexity can be sometimes stated in terms of the decomposition spectrum of the objectin a given basis of primitives. For example, IFS fractals are described by a set of geometric trans-forms (a ”basis”) and a set of probabilities in a Markov-1 sequence, that play the role of (stochastic)coefficients [Barnsley88]. Fractional Brownian motion is a related example, using Fourier filtering[MandelbrotFGN82, Mandelbrot68]. A third example is found in botanic trees and blood vessels,which can be roughly described as being composed of tubular elements of different radii and lengths,whose relationships are distinctive of the species [West97]. When the basis of primitives is very large(starting at the hundreds, to say a number, but it may be infinite), its specific distribution can alsotake the form of a space of possible states, and then a probabilistic description allows to formulatethe decomposition approach of a set of data in terms of information theory and statistical mechan-ics, with the Shannon entropy (H � �Pi pi log pi) in mathematics, or equivalently in physics, theBoltzmann-Gibbs entropy (S � k logW ), with pi, discrete probabilities,W the number of possiblestates with the same energy as the state of the system, andk the Boltzmann constant. In physical sys-tems, entropy quantifies the amount of randomness or disorder, and the unavailability of a system’senergy to do work, while the more abstract probabilistic definition is rather linked with informationcontents, and efficient codification. As data compression depends on randomness, it results again tobe a measure of the latter: an absolutely random sequence cannot be compressed (reduced).

Randomness or regularity of the basic features chosen to represent an object can be studied andsimulated, and the complexity measure of the computer program to reproduce such an object (or aninstance of its class, given a probabilistic description) can be used to characterize the complexity ofthe object itself. The problem of using such description is thatif the random law is completely known,the program may be very simple and the object not “complex”, under this approach. Theknowledgeof a random law becomes also closer to a deterministic description; thus the available information,and how is it coded, is also an element to measure complexity. There are still other definitions ofcomplexity, randomness and disorder, linked to dynamical systems and thermodynamics, such as theKolmogorov-Sinai entropy.

With all these notions in mind, we can state the concept of Kolmogorov Complexity (KC), firstlyin words: KC deals with measures based on thelength of minimal description of a set of data.Given a finite stringx of characters (the postscript file of this thesis, for example), its KC measureis the length of the shortest program in a Turing machine code, in binary bits (or say, in C langage),which prints stringx without any other input. It is assumed that any pattern can be somehow coded(described) into a sequential stringx of symbols. In the KC sense, an objectO is more “complex”


than an objectQ, if the minimal descriptionx of O is longer than the minimal descriptiony of Q.This intuitive idea can be precisely formulated (see below).

A last remark is that difference of KC with Shannon’s entropyH , as a measure of disorder, canbe stated as follows:H is a quantity for measuring the randomness of random variables, while KCis a measure of randomness of individual objects based on logic arguments instead of probability[Kolmo68, Chaitin77]. We now give a more rigorous definition, following Kolmogorov’s one andour interpretation:

DEFINITION 1.19 (Kolmogorov-Chaitin complexity) Let x � X� y � Y , with X� Y sets of con-structive objects (as strings of primitives, ordinal numbers, ...), and let A � Y �f�� Ng � X

and B � Y � f�� Ng � X be partial recursive functions, and N � NWe define:

KA�xjy� ��

minn�N�log� njA�y� n� � x�� if � n � N� A�y� n� � x

� otherwise (i.e., no ”n” exists)(1.20)

The function A is said to be asymptotically optimal iff

�� functionB� � constant C� �� x � X� y � Y � j KA�xjy� � KB�xjy� C(1.21)

For an asymptotically optimal A, which is known to exist, KA�xjy� is simply denoted K�xjy�, andis called the Kolmogorov-Chaitin complexity of x, given y.

The interpretation (or at least one intuitive interpretation) of this definition is similar to that al-ready given asminimal description length (of the strings of primitives). ConstantC is the worst-case“offset” between two sets of descriptionsA�B, given a certain knowledge or information (repre-sented byy� Y ). A�B are a mapping between interpretations (e.g., descriptions in a human lan-guage) of objectsx and ordinal numbers in order to enumerate them, then compare two indexedstringsA�y� n�� A�y�m�� n �� m, and choose the shortest corresponding to the described objectx.Note that constantC ensures that in a great number of descriptions, two setsA, B give (if A isasymptotically optimal) similar complexity measuresKA�xjy�� KB�xjy�.11

1.5.2 Information Theory, Algorithmic Entropy and Scientific Observations

It is worth to note that The Shannon Information Theory doesn’t allow to directly define the entropyof a single state in a physical system (for example), among a number of them (the ”ensemble”, forwhich H has a clear meaning). This difficulty is solved by the notion ofalgorithmic entropy, inaddition tophysical entropy, and for completeness, we just give its definition, using KC [Li92]:

DEFINITION 1.20 (Algorithmic entropy) The algorithmic entropy of a microscopic state of a sys-tem is the Kolmogorov complexity of that state.

The relation of scientific observations and test of hypotheses with probability theory is natural:to make a compact description of a set of observations or hypotheses, an observer has an infinite

11The enumeration of strings, propositions, models and theorems (i.e., an isomorphism withN) and the application ofmathematics on such numerical encoding, comes from thediagonal method introduced by Cantor to show that rationalsare infinitely countable. Kurt Gödel used a similar strategy to build his theorem about Incompleteness or Inconsistenceof formal systems. It is thus not surprising that other works on Kolmogorov’s literature deal withmetamathematics,interpretation, model theory, and metalanguages [Li92, Li95, Gunther94].


binary sequenceX as the outcome of an infinite sequence of possible experiments on some aspectof Nature (decidability is assumed). Description ofX is done through its underlying regularities,and a ”true random” or ”absolutly random” sequence is its own minimal description, and cannot becompressed [LiKolmo93], pp 46. An investigator tries to formulate a theory governingX , which isconsistent with past experiments (observations), and candidate theories or hypotheses are identifiedwith computer programs that compute binary sequences starting with an observed segment ofX (theN first outcomes). Randomness modeling and simulation continues to be a critical step in hypothesistesting and prediction [Bosch94], so again, computer programs and Turing-machines enter directlyinto descriptions of complex data and structured hypotheses about them (dynamic definition of com-plexity). The search for regularities is an effort towards simplification, understanding and extractionof relevant features.

Thus, complex sets of data, even if obtained from static spatial objects, relate with physicalsystems through their (minimal length) descriptions. The question of how to code descriptive infor-mation is addressed by concepts such as entropy and KC, which are associated with computation andinformation theory. The association between objects and physical process is particularly verified forbiological objects, which result from growth process and shape formation (morphogenesis), throughlong interaction with their environment. This interaction follows ”iterative construction rules”, asthose to generate fractal shapes, and they may appear at different scales. Literature on complexdynamical systems has been enriched with further examples, the most popular being the Julia sets,mathematical objects generated by iterative mappings (see Figure 1.11). When a data set is the resultof a natural stochastic process, it becomes natural for a researcher to simulate and try to recreate itsstructure and features, through computer models of its originating process.

In this very sense, many features of a complex object are statistically predictable from a completedescription of the process, and it suffices for most practical purposes the study of the generatingprocess, rather than the detailed morphology of the outcomes.

In the Shannon’s classical Information Theory, a quantity of information is assigned to an en-semble of possible messages. When all messages are equally probable, this quantity is the numberof bits needed to count all possibilities. To introduce the notion of Kolmogorov Complexity, let’stake the example of number description with four large numbers:�� , ”one followed by fifteensevens”,�� and ”the surface of Jupiter, inm�”. These four numbers are of similarorder of magnitude but their description is of different length and spelled in words the first and thethird have a different complexity, which comes from some kind of regularity. Note that the seconddescription does not provide exactly the same information. Such regularity can be made relative toa system of reference given by a numerical basisor a language. Taking for example ”the surface ofJupiter” value as the unity, and the other numbers as the surface value of other masses, all numericexpressions change, and the first can be as long as the third.

In the approach of Shannon, themeaning of a message is ignored, as we are only concerned withthe communication between sender and receiver.Algorithmic Complexity was then introduced byKolmogorov as a measure for the information content of individual objects (messages) [LiKolmo93].Other important measures of complexity closely related to KC and to shape complexity are:Com-putational Complexity (CC), Logical Depth, Instance Complexity and Long distance correlation[SerraR88, LiKolmo93].

KC and related measures have shred light into several vague notions that have been used in infor-mal fashions, and made clear their relationshipswith randomness and order. For example, complexitymeasures have aided to formulate precise definitions of ”simplicity”, relating this concept with theempirical principle of economy (or ”Occam’s razor”) used to choose, from several hypotheses, the”simplest”. It has been demonstrated [Bosch94] that ”simple” not necessarily corresponds just to


”minimal-length descriptions”. Another practical notion is that of ”complicated”, which qualifies thefact of using a long description, when there exists a much shorter.

Relevance of KolmogorovComplexity in Shape Analysis It has been argued by Edmonds [Edmonds95]that KC is a weak definition of complexityin Biology, because it describes the difficulty of compres-sion of a representation, as limited by storage and algorithmic difficulties in computational biologyand morphogenetic description methods. He argues that this difficulty has little direct connectionwith the practical aspects of a functioning organism, and that is presumably used because it is a well-understood formalism. Literature presents few quantitative applications of KC, since it is difficultto extract minimal-length descriptions. KC is also a relative measure which depends on availableknowledge and of sets of symbols (in fact of languages and grammars that code the measured de-scription), all which have prompted rigorous formulations and several levels of abstraction, difficultto conciliate with natural-Science and real-world applications.

Besides better ways to measure KC, and to be a useful parameter in shape analysis (e.g., tex-ture discrimination), KC and other complexity measures demand an effort to design minimal-lengthdescriptions, thus, optimal computer algorithms. In practice the concern of speed and memory costtends to produce additional length, since data analysis and modeling are inherent to description. Thisoverhead appears in the form of theoretical considerations to justify ad-hoc algorithms. To exploitKC theory results, it also demands higher level descriptions, abstraction and codification into existingmathematical structures.

In a first approach, we propose to measure the length of the discrete-boundary representationof a scene. A way to do it is directly linked with a standard compression method, therun-lengthcoding of binary objects in 3D. In some applications, octrees are more efficient, but it suffies totranslate an object by one voxel to have a very different octree representation. We describe three-dimensional run-length coding in Section 2.4.4 and how we relate this compression method withboundary representations, enabling to obtain a complexity measure based on a description length.

Fractal Sets in Computational Complexity (CC) When considering the CC size of an algorithm,an estimation of the order of the number of operationsO�L�, is made, withL the number of elements.In 3D,L may be the number of voxels of aRegion Of Interest (ROI) , or the resolutionL � NX �”ROI” I

NY � NZ . For simplicity, assumeN�� NX � NY � NZ , thusL � N �, and an algorithm

which depends linearly on volume size is said to be of CC equal toO�N ��. Independence of the”complexity of the scene” is claimed as an advantage in the performance of many rendering methods,for example. When dealing with boundaries, perimeters and surfaces, performance estimations maystrongly depend on the shape and the analysis resolution, in a similar way as the shape factorff�D ,defined by Equation 1.11, in Section 1.2.2. In this case, the fractal dimension of the shape (or moreprecisely, the Hausdorff measure of contents) gives a better estimation of the number of operations.In such a case, combinatorial analysis of a 3D algorithm for processing and information extractionshould use for exampleL � ND as a measure of the contents of elements to process, withD thefractal dimension of the dataset, anddT � D � d. In the case of a surface boundary, say a mountainrelief, typical values aredT � �� D � �� d � �. An algorithm whose length depends linearly onthe extend of this (discrete) fractal surface has a CC ofO�N�� operations.

1.6 Shape Taxonomies 63

1.6 Shape Taxonomies

Reductionist approaches for describing complex shapes have been around for centuries. A pletora ofad-hoc terms and adjectives exist. As a sample we mention:tubular, laminar, t-shape, denticulated,rosette-shape, knotted, tangled, intertwined, ruffled, granular, vermiculated, trabecular, spongyform,etc. It is more precise and useful to translate distinctive features into mathematical descriptions thattake advantage of more abstract characteristics, such as principal-axes, surface/volume ratios andspecific pattern relationships. Thus, the adjectivesovoidal (egg-shaped), prolate, oblate, spheroidal,lenticular and discal are ”coded” by the eccentricities, i.e., the relative proportions of the three prin-cipal axesa� b� c� of an object, when its shape is well approximated by an ellipsoid. It is ”spherical”or ”spheroidal” iff a � b � c; it is ”ovoidal” (prolate-ellipsoidal, cigar-shaped, etc.) iffa � b

anda � c (an axis length predominates), and ”lenticular” (oblate-ellipsoidal, pill-shaped, etc.) iffa � b� b � c (one axis is significantly smaller than the other two).

Similar formulations can be made by examinating local and global properties, the shapes ofconcavities, the local and global behavior of the normal vector, and the shape of the very distributionof different components and features. When a graph representation is available, shape description istranslated into description of graph characteristics.

An interesting approach, using MM operators, is to characterize a shape in function of its “re-sponse” to filtering with specific structuring elementsEk, with k an index indicating either size ortopological characteristics (e.g., connectivity). This is the basis formorphologic granulometry, inwhich the filter-response function, (an histogram of connected components after morphological fil-tering with different structuring elements), produces profiles that may be very distinctive of shapecharacteristics to the point of being a “signature” of that object (see [Coster89] pp. 119–160 and[DougPelz92]).

Another approach is the use of morphological “probes”, in thestereological sense. Quantitativestereology provides a set of tools and methods for estimation of 3D features by measurements onsets of 2D or 1D samples. Several geometrical probes (called in general ”test systems” [Weibel79b],pp.11,116–130) are possible: sets of random points, lines, semi-circles, sweeping tangent planes,structuring elements, intersecting planes, spheres and particles (also called point or line processs,etc.). The distribution properties of their (random) intersections with the object (statistics of chordlengths and angles, for example) provide several means to statistically characterize many of its shapefeatures, and formulas exist to accurately estimate some morphometric parameters. The study oftest systems and stereological morphometry are also topics from Stochastic or Integral Geometryand Geometric Probability [Santalo76, StoyanSG87, Stoyan94]. Applications related to the presentsubject include complexity estimation of 3D scenes for ray-tracing optimization [Cazals97], analysisof randomness in geometric features [Pennec96], and our own works on textural feature extractionby intensity-profile sampling [CorkidiAIP98, Corkidi98, Corkidi98b]. Since the present thesis dealsrather with boundary representations, we did not use stereological tools and we restrict this sectionto a boundary-based analysis.

The next subsections do not pretend to be a systematic coding of terminology into mathematicaldescriptions. We only chose some representative concepts and some ways they can be described oridentified in a discrete representation of a structure, in view of our approach. We start by formulatinga way to explore an object, given some pre-segmented subset (the whole scene, a region of interest,or the object or set of objects themselves), and then we propose methods to study typical structureand shape notions.

LetP �� fpk � N � �Og the set of all discrete mappingspk (sequences of point samples) on

the surface boundary of a connected objectO � N�, with a parameterk � N. If pointspk� pk�� are


connected, then at “instant”k � t, pt is a point walking the surface boundary of a connected objectO, and in general it may constitute a 3D traversal of any subset of the boundary: a patch (or a setof disjoint patches), a circuit (idem), a random walk, or a discontinuous jump or sub-sampling ofO,when no sequential connectivity is defined. Such subsets, also calledsubmanifolds whenO � N

n,have topological dimensionDT � n� � (see Section 1.4). In the case of surface patches (DT � �),it is natural to introduce two parametersi� j and mappingssi�j � N� � �O, but for the sake ofsimplicity, we examine here a linear set of pointspk � P � k � �� N , on�O (a path). Besides,there is no unique, analytic parametrization ofO, but we show in Section 2.4.2, how to find a numberof them, using a Hamiltonian traversal of the directed graph associated toO.

Let pk� ql � P � k� l � N, which constitute two different sets of point samples of�O. We givenow some examples of how features and statistical properties ofpk describe topological and morpho-logical characteristics ofO. More complex combinations and features could be found with severaltest sets likeP and more parameters.

1.6.1 Tubular Structures (I)

When considering boundary points, tubular structures or generalized cylinders, such as blood vessels,are characterized by the profusion of (small, circular) closed loops around the object. Other loopsmay be much longer and irregular, thus their mean diameter satisfies� � maxmf�mg, with formfactor ff�D � �. From this intuitive approach, we see that there exist ’circuits’fpkg, such thatpk � pk�m for all k and somem � N� m � N . Suchm (the length of a path) can be identifiedwith a generalized local perimeter in discrete point units12. ’Thin’ filamentary structures may bedefined bym � �N , with � � ��, for example. Let a setC� � O, be constituted byM circuitsfpk�hg� h � �� M . If m� � m� � � � � � mM , thenC� is a conic structure (it maynot necessarily be a cone, because we have just chosensome circuit samplespk�h ). Principal axescharacteristics of sample setsfpkg allow to determine if tubular features are more or less round, orflatenned (eccentricity).

Tortuous tubular structures may be then defined as those structures where a sequence of contourperimetersmh form an irregular shaped profile (sometimes called ”generalized cylinder”), whenmh

is plotted againsth. Tortuosity parameters exist for continuous contour representations of bloodvessels [BaskinAA96], and may be characterized through star-polygon contours, as described in[Verdonck96], pp. 65.

Further discussion on tubular structures with a regional-based approach is found in Appendix B.page 240.

1.6.2 Ramified Thin Structures

Fine ramifications, in the form of ”spongy” networks arestatistically characterized by high inter-connectivities, and very high surface/volume ratios. Taking a sequence of pointspk in the boundaryof such an object, its geodesic and spatial neighborhoods (as defined in Section 1.1) should showa high density of pointsp �k (surface density by volume unit). We study and apply our methods ofanalysis to a concrete example in Section 5.7.1.

Several approaches are possible for medium and large branching structures, where a more ac-curated representation is needed. When considering a set of random planesPi, i � �� N in-tersecting one or several objectsT , branching structures are globally characterized by an elliptical

12until now we have not definedhow pk lies on�O. A possibility is to placepk along geodesic paths.


cross section in some orientations, and splitting into two or more of these sections, when followingthe axis of the branches. IfPi is a sequence of planes along an axis, then a continuous trackingof each tubular component can be done, except where branches are parallel to some planePi (seeFigure 1.13-right).

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In a local characterization, ramifications are highly non-convex. Ideally, there are at least onesaddle point in the boundary of the bifurcations, since a junction presents at least one zone whereprincipal curvatures change of sign (see Figure 1.14b,d). Saddle points also exist in tubular bent sec-tions (see pointsp� p in Figure 1.14c). ConsideringN internal pointspj � T , with j � �� Nthere is a systematic way to univocally identify bifurcations, by studying ball-neighborhoodsB�r� p�and their intersections with the object itselfO. LetN k

�� card��B�r� p� �T � be the number of

connected objects (usually contours) of the boundary intersections at radiusrk� k � �� M . Ifthere is somek, such that�N k

� � �� AND �Nk�� , then the pointpj is at a bifurcation region

(see Figure 1.14a). Since small bumps at the surface may be taken as ”branches”, other selectioncriteria must be added, as examining larger neighborhoods. It may be possible to haveN k

� � � atmultiple branching points.

Optimal detection of individual branching points, and extraction of a skeletal graph have beendifficult subjects in 3D. Approaches using theMedial Axis Transform (MAT) give often poor re-sults since they are too sensitive to boundary noise [Ogniewicz94, Schmitt94, OgniewiczTR95,KresMala94]. More robust representations useCentral Axis detection for pulmonary tree extrac-tion by region growing plus heuristic localization of bifurcations [Niblack90, Wood95, Pisupati95],and scale-spacecores (or medialness representations) with ridge detection [McAuliffe96]. None ofthese approaches is suitable for boundary scanning or probing which is more adapted to analysis ofcompact structures, where branching is not a dominant feature. A successful strategy for bifurca-


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Figure 1.14:Branching points are characterized in several ways. (a) Their ball-neighborhoods ideally havea defined number of intersection with the tubular components. The number of connected-component intere-sections define the branching degree equal to card(intersections)-2, with 0 indicating no branching, and -1indicating a terminal point. (b,d) Saddle zones also locate branching points.

tion detection is to execute detection from inside (tubular tracking), and when more than one path ispossible, then spawn new tracking detectors for each path.

Lindenmayer Formal Grammars

From the point of view of modeling ramified objects, iterative, constructive schemas have beensuccessful in accurately reproducing botanical trees and many other natural structures representedby graphs with complex topology (besides trees, the brain sulci [Mangin95], and certain cellular net-works [CorkidiMarquez93, CorkidiMarquez94] and lattices [Prusinkiewicz89], for example). Themost popular construction models are theLindenmayer grammars L-systems or string rewriting sys-tems, that introduce a formal language�G�W� P �, with a set of symbolsG (constructive blocks), astarting string or axiomW (a pattern to follow, an initial configuration), and a set of rewriting orproduction rulesP [Prusinkiewicz89]. The rules indicate how the symbols are arranged on differentiterations, and control the generation process. Since some decisions (e.g., branching, tubular diam-eter and which primitive to use) can be randomized following empirical or theoretical distributions,non-deterministic, natural diverse objects can be modeled, and the production rules may be setup tosimulate physical events (e.g., growth, adaptative change, offspring). Oustanding examples of accu-rate plant models for several botanic specimens are illustrated in [Prusinkiewicz90, Kaandorp94].

We illustrate L-systems with a 2D ”tree” (Figure 1.15) generated by the following set of pseudo-instructions at iterationn:

function Plot Branch( n, state)

set diameter=f�n�

plot(segment, diameter)

advance y�n�

if rnd()�thresh1(n)


?

θn

n-1

(d)

(a) (b) (c)

Figure 1.15:Branching structure generation with a 2D Lindenmayer-system. (a) Starting string or ”axiom”;(b) symbol to use; (c) production rules to arrange symbols in varying orientations and sizes (only location andangle is represented) (c); grammatical construction result after n � �� steps (d).

rotate ��n� degrees

if thresh1�rnd()�thresh2(n)

rotate degrees;

Plot Branch( n-1, state);

if thresh2�rnd()�thresh3(n)

Plot Branch(n-1, state)

else

plot(last segment, diameter);

turn� degrees;

if(n�0)

Plot Branch( n-1, state)

return (state)

end

A main program defines a branch segment to be ploted (the ”axiom”) and calls the recursivefunctionPlot Branch() with an initial valuen. The presentstate is passed, modified and returnedto control recursion. If no bifurcations are decided to take place, this function plots full segmentsfollowing a single curled branch (see Figure 1.15a,b). Otherwise, given a bifurcation event, two newbranches are started (the old one is reset to 0, at smaller scale).��n� controls ”curling”, in functionof n, and is the branching angle plus a random bias. Parametersthresh1, thresh2(n), thresh3(n)are empirical adjusted, the last two being decreasing functions ofn, thus probability of branchingdecreases until a full branch is completed and another is continued.


In Section 2.4.4 we further describe tubular thin structures as a “dual” of flat manifolds, char-acterized by high rate of normal vector changes per volume unity, as opposed to interfaces andconvoluted surfaces. A particular case of branching structures in the bronchial tree of the humanlungs is presented in Chapter 4.

With respect to shapes in natural sciences, a whole discipline calledmorphogenetics studiesgrowth phenomena and process that give rise to forms in nature. These are considered as adaptive,evolutive process, since form or structure changes appear in response to environment exchanges.Thus dynamical systems play again a central role in complex form understanding and modeling.Cellular automata [Wolfram84b], genetic or evolutionary computation [FogelEC94] and other recenttopics also exploit biological-inspired paradigms, and have served as models of pattern complexity.

1.6.3 Surfaces, Boundaries and Interfaces

Laminar structures . Plaques or laminar manifolds may be characterized by existence of short pathsin one predominant direction (the local average normaln��, with� a standard deviation vector thatmay account for roughness features). We again postpone further discussion to Section 2.4.4.

Convoluted surfaces. As laminar structures, these present short paths, distributed in several di-rections. A discrete line traversing the scene may intersect several times the same object; thus acharacterization may be given in terms of instersections. Letflkg be a set of random lines (the”probes”),k � �� M , withM to be determined for meaningful statistics. Let� the mean thick-ness of the intersected object subsets with all probes, and� the mean free path between intersections(empty space). Then, the number of total intersections and the ratio�� provide quantitative in-formation about convoluted surfaces, and their distribution in space. For example, the brain cortexsurface should possess at many places a high number of intersections with a high intersection ratio(the sulci being slender and more or less curved). A lettuce would have a high number of intersec-tions everywhere, with a constant small intersection ratio (the leaves being very thin). Intersectionswith probing planes, spheres, etc. would provide other kind of informations, which may statisticallycharacterize many other complex features. An example of convoluted surface is presented below, inthe paragraph on complex interfaces (Figure 1.16).

When convoluted surfaces are rather smooth, and constitute height fluctuations of a plane surface,expressed asz � z�f�x� y�, with f�x� y� a uni-valued function they are calledroughness [Wong86,Briers93]. The simplest measure of this feature is the root mean square value of fluctuations aroundthe mean height. Most analysis has been traditionally done by spectral techniques, but other textural-based approaches are feasible. An example is presented in Section 5.7.1, where we show how towork with real convoluted surfaces, wheref�x� y� is multi-valued, giving rise to surface ’meanders’or ’overhang’.

Interfaces. Boundaries between objects identified as different, under some criterion form them-selves virtual surfaces or submanifolds shared by two or more bodies. Thus, interfaces are second-order structures, arising from neighboring and connectivity relationships.

Interfacial sheets ormembranes may be described in a similar fashion as laminar manifoldsand convoluted structures, but we need to re-define segmentation and filtering methods in order tointerpret one side of the interface as ”background”, and the other side as ”foreground” (definitionsmay be switched during processing). Discrete representation is also extended, it may still consist ofa list of voxels, but also a list of 18-connected facets (voxel faces interconnected at least at one edgewith another facet).K-connectivity is defined in Chapter 2.


HD ( a) < D ( b) =>

(a) (b)

Flux << Flux a b

| a| << | b|H

Figure 1.16:Given two physical media (or bodies), their common boundary defines the exchange rate ofinformation or material flow. (a) A smooth boundary�a with low Hausdorff dimensionDH �a� (or any otherfractal dimension) characterizes a low surface-area interface and a low flux of information, energy or materialF luxa between both media (small arrows). (b) A very rough, convoluted or fractal boundary�b, with ahigher Hausdorff dimensionDH�b� has also a much larger surface-areaj�bj, and in consequence, the flux ofinformation, energy or mass (denoted byF luxb, across the boundary is much higher (large arrows). Boundaryshapes may take several different morphologies to fill up the embedding space in efficient ways, followingminimal energy configurations, for example.

A particular example of complex interfaces is illustrated in Section 5.7.2. We only mention thatspace-filling interfaces are very important in multi-component mixture understanding, interpenetra-tion, and seggregation and their proper mastering, i.e., a proper morphological quantification andmeasurement ofmixture quality determine valuable mechanical or other physical properties in ap-plications from material science, solid state physics and chemical compounds. As present in naturalphenomena, their interpretation provides clues to material, energy or information exchanges acrossthem, as illustrated in Figure 1.16, and in a real application, with one example from Cell Biology inFigure 2.21, page 134.

1.6.4 Description of Other Shape Features

Convex structures. The conventional definition of Paragraph 1.2.4 can be directly applied, usingonly point samples at the boundary ofO. Let pk� ql � �O and

L�� frtjrt � pk t�pk � ql�� t � �� g�

thenO is ”convex” iff �pk� ql � �O, L � O, that is,�t rt � O. Degrees of non-convexity may be

established by measuring the percentage of pointsrt ��O, the geometric interior.


Surface Texture. Let a population of ’coupled’ pairsfpk� qlg� k� l � N, with a coupling rule,and (eventually) value labels�k� �l assigned to eachpk� ql. The label is not necessarily a gray-leveldensity, but any local measure. Then we consider the probabilitiesof co-occurrence of�k� �l. Surfaceirregularity characteristics can then be extracted in a similar way to co-occurrence textural parametersin planar images. Usually,pk� ql constitute the endpoints of chords, and distribution of chord lengthsunder certain random process is a common 2nd-order analysis in the case of 2D shapes. In a 3Dcontext, texture refers to variations in many scales, examined statistically. For example, a distributionprofile of the random variablejjpk � qkjj may have two different populations in convoluted andbranching objects: the larger population denotes separated ”sheets”, and the shorter, neighboringpoints. A large average� jjpk � qk jj � and a small second peak in the profile denote both a veryintrincate, filamentary structure (a sponge). Shape interpretations of ’contrast’, ’2nd moment’, etc,may also be investigated. See Section 5.7.1 for a particular example. A related approach is that ofrun-lengths, where thepk� ql in 2D shapes of grey-level images are the intersection of sets of randomor horizontal test lines [GallowayRL75, EllisDRL81, Chu90]. Run-length analysis is adapted tothe structural approach to texture, but we did not considered this for the present work. We presenthowever other considerations relating texture and complex shapes in the following section.

1.7 Other Properties of Complex Information

Until now, we have limited our attention to solid-shape characteristics for modelling and morpho-metrical purposes. We have set aside multiresolution and scale-space characteristics, topologicalproblems and Gestalt or Pattern Recognition problems, which also describe complex information,and we shall tackle outside the present work. Other kind of characteristics are inherent to complexdata, such as noise, sampling limits, partial surface or volume effects, artifacts and distorsion. Sincethese features affect segmentation and shape measurements, we mention in the following paragraphsaspects of two of these problems, in relation to shape information.

1.7.1 From Structure to Texture

Real-world objects are characterized by a number of features or patterns difficult to describe, even ifthe human visual system is able to recognize and distinguish different visual patterns that appear ina close examination of real objects. These features or patterns are called ’textures’ and ’roughness’.

We make in this Section only some observations. The first concern the transition from what wemay call ”deterministic structure” to what we call ”texture”. As we actually employ digital computersand finite-resolution measuring devices for data analysis and representation, all measurements andresults are discrete samples of what we try to analyze (plus convolution byPoint Spread Functions(PSF) of other resolution uncertainties in the system). In the case of frequency-domain analysis, the”PSF”INyquist-limit is attained when data resolution lies over the smallest details of the object. If the objectis very intrincate at certain scales, it may then appear as ”noise”. When some structure is present,a texture appears (for example, when even weak spatial frequencies produce anti-aliasing artifacts).When trying to model the normal surface of a fractal object to be rendered in 3D, there may be noway to obtain an average distribution of smooth normals, because the averaging is done on a non-differentiable surface. A solution would consist in modeling light interactions with physical fractals.An example of light interaction with fine structures is given by butterfly wings, where in addition topigments, the microstructure of wing scales refracts and diffracts light, producing iridescent, metallic,or pearly effects; in most cases, characteristic feature widths correspond to visible wave-lengths.

1.7 Other Properties of Complex Information 71

As texture may be present at different scales of observation, it is pertinent to ask for their scale-invariance properties, such as fractal dimensions, and we have already showed some examples.

Lacunarity. Fractal dimensionDq is related to first-order statistics of mass distribution, in func-tion of scale. Second-order statistics provide information about spatial relationships, and discrepancybetween real and expected measures. Several definitions have been devised for quantifying ”gaps”(or lacunas), and irregularity fluctuations in textures. Lacunarity��L� at scale lengthL is a tex-tural feature that quantifies deviation from translational invariance by describing the distribution of”gaps” within a set at multiple scales. The more lacunar a set, the more heterogeneous the spatialarrangement of gaps [MandelbrotFGN82].

A way to estimate��L� is by using the momentsM q of the qth-order information dimensionDq, defined in Paragraph 1.4.1, withq � � [Peitgen88]:

��L� �

�M��L�

�� hM�L�i�hM�L�i�

At fixed scaleL, ��L� measures the width of distributionP �m�L�. It is possible for differentfractal surfaces to have the same dimensionDH , and present a different lacunarity��L�. Kelleret al. report a measure of lacunarity better adapted for small texture patches, and more suited forsegmentation [KellerChenCrownover89]:

C�L� �M�L��N�L�

M�L� N�L�

In this equation,M�L� is the average mass density within aL-box, andN�L� is proportional13 to thenumber ofL-boxes needed to cover the fractal set. From fractal dimension definitions, several lacu-narities can be proposed. Note that the notion of ”gap” was defined in terms of foreground (”filling”components of the texture) and background (no foreground). Nothing prevents from considering arelative lacunarity, in the same vein of interfaces with respect to boundary between two regions (twoforeground classesC�� C�, implying background gaps plusC�-gaps orC�-gaps).

1.7.2 Noise, Blurring, Distorsion and Inhomogeneities

Data produced by measure devices contain many kinds of information that has to be extracted andwhich has also suffered some transformation (distorsion) or degradation, in the form of added noise.The concept of noise includes unmeaningful information from the source data, extraneous signalsdue to isolation problems, and information from uncontrolled sources. Besides introducing distor-sion and imprecision, the less this information is identified, the more it is useless and consideredas noise that has to be filtered. Precision limits play also an important role, specially if observa-tions are indirect (low or noobservability), and the intermediate device or process, modeled as acommunication channel, is non-linear.

As noise is part of the data, and it is often influenced by data features, it can be as complex asthe data itself. Noise may even be unidentified information, thus containing non-random structure,unsuitable for stochastical models of noise. The general model consists of an input signalx, and they observed output from the measurement device, satisfying :

y � � � x ��

13A dimensional constant was omitted for clarity.


where� is additive (random) noise,� is the convolution operation with�, the Point-Spread-Functionof the system, which blurs the input. More complex distorsions may exist, including multiplicativenoise (also calledstructured noise, e.g., speckle), and non-linear intensity or geometric deformationsof the input. Most of restoration processing consists in obtaining the best estimator ofx from yand some knowledge of� and�. The simplest approximations include Gaussian (white) noise andGaussian convolution.

For other kind of data degradation, such as radiometric inhomogeneities (background illumina-tion variations, physical degradation or other), an a priori knowledge of datax itself, or some reason-able assumptions besides those already mentioned allows to filter or reduce noise. Such assumptionsinclude continuity criteria, such a spatial coherence of data, a specific probability distribution, acharacteristic spectrum (in the shape description, or the density of the data), or other characteristicfeature. We present in Chapter 3 an example where the characteristic feature is the coherence of 3Dstructures, along the cross-section axis.

In terms of complexity, the relevance of noise understanding relies in discriminating noise fromrandomness. Given the available model about the shape of an object, some kinds of noise mayintroduce unpredictable variations not included in the model. When properly incorporated into themodel (e.g., a Gaussian noise�� ), it may explain the random fluctuations of some aspect ofthe data. Statistical, topological and linear-transform invariances, robust feature parameters, erroranalysis and knowledge of the noise sources are to be incorporated within complexity measures, asin any experimental routine.

Several generalizations of the above problems have been proposed in terms of physical measure-ment models and estimation theory (see for example [Serra82] chapter 1 and [Mintz1994]). Follow-ing an experimental approach, given a ’theoretical’ feature,X , any measurement device or algorithmhas access only to a certain set ofNobs observations (estimations, approximations) ofX :

Yk � �k�X�� k � �� Nobs��

where�k is a sequence of transformations induced by the observation or measure process. Ideally,�j � �k , for all j� k, and consists of affine transformations for geometric features, convolutionplus noise for spectral attributes, etc. In general�k are spatial and time-varying transformations,comprising several types of distorsion, noise, etc. The first concern is to select robust featuresX withhigh invariance under�k, and then measureYk , either as a substituting feature, or as an estimationof X . The latter approach is routinely used in mathematical morphology, where the transformationof isotropic closing regularizes the contour of certain shapes. The perimeterP of a digitized shapeis an example of a non-robust measureX , in which the sampling resolution is included in��,under the form of a scale transformation, or a sampling function (decimation, round-off), whichprovides different measuresYk � �k�P � � P . These measures appear for example in the form ofan estimator

�P �� Yk� �� k � �� Nobs��

sensitive to shape irregularities, the scale of observation, noise and indeterminacies in the process ofcontour detection and machine precision.

We have ourselves experienced two situations where several transformations altered data in-put. The first is described in [Anglade93, Marquez94b], where ultra-microscopic slices from cellnuclei were digitized from electron microscope negatives. The goal was to seekpathological-induced modifications in cell-nuclei morphology. These were mixed with important incertitudes inslice thickness, anisotropic mechanical and heat-induced deformations, radiometric inhomogeneitiesand offset due to contrast-product and irregular thickness, electron-beam control and blur, dust

1.8 Discussion and Conclusions 73

and chromatine artifacts, histochemical blur, photographic developpment incertitudes and artifacts,rigid mis-registration, camera acquisition calibration, and errors from bilinear-unwarping approx-imation of geometrical non-linear distorsions. Under these circumstances, only differential (rela-tive) analysis could be done, with poor statistical validation. The second situation was far moretractable, comprising color radiometric inhomogeneities and is presented in Chapter 3 and reportedin [Marquez96b, Marquez98].

Thus, even for ”simple” objects, there may exist complex noise, complex measurement proto-cols and incertitudes (the PSF of a sophisticated observation device), and complex methods of datafiltering and analysis.

1.8 Discussion and Conclusions

Being such a large and rather recent domain, we cannot give a detailed list of the various problems,approaches and concepts of Scientific Visualization and processing of complex objects and data. Wehave given an overall review of some key aspects, in view of a quantitative analysis. Many of thesecharacteristics can be handled in a medium computer environment, available in average researchlaboratories for applications that approach the image processing standards: morphometry, objectmanipulation and data visualization.

The following conclusions can be drawn:

✦ The notion of ”complexity” embodies and relates with the following properties:

– Cardinality: high number of elements or components.

– High number of topological features: connections, holes, genus, knots. A particularfeature isspace embedding and configuration with respect to other objects..

– Bifurcations (branching), either physical or abstract (graphs, behavior, response).

– Higher order properties: high number of inter-relations, interactions or interconnections,

– Non-linear descriptions.

– Long descriptions. The lengths of the shortest descriptions are important measures, andthe very structure of the description has its own complex features.

– Partial regularities, lack of them, or its presence implies lengthy descriptions.

– Partial scale-invariances, (local) non-convexities and concavities, hierarchies (groupingand clustering at various levels), repetitive or non-repetitive patterns, textures, fractaldimensions, degree of order and randomness, etc.

– These characteristics may be present in several ways: information (energy) present at sev-eral or all frequencies, locallly or randomly. Basis representations need several or infinitenumber of coefficients; high-order statistics are needed to describe complex objects, etc.

Whether it is length, information content or cardinality, it is clear that complex objects andprocesses give rise to large models. When considering evenseveral descriptive levels, morethan one single model may be required. Additional features of data include their ”natural”format, i.e. scalar, vectorial, tensorial, multidimensional, etc., and how they are embedded inspace.


✦ Some understanding of complex objects is attained by descriptions involving physical process(morphogenesis), hierarchization ordering, the amount of randomness, disorder and higher-degree descriptions (interactions, and the effect of several elements taken as a unit). Noise,subsampling and other phenomena/artifacts from data acquisition adds for what is ”complex”.

In a second-order analysis, the irregularities of the interconnections, relationships or interac-tions between the components of an object or system, may also be subject to a similar analysis,and turn out to be non-linear (not superposable). In fact, in dynamical systems, the objects ofinterest lie in multi-dimensional phase-spaces (chaotic attractors).

✦ Many of these notions involveboundaries. They may be the external surface of an object(frontier between background and foreground) or transition surfaces (interfaces) separatingtwo media or two components with different attributes. A particular problem (and still anothercomplex feature) is posed by fuzzy, undefined and ill-defined boundaries.

✦ An arbitrary object can be approximated by simple primitives (e.g., spheres, harmonic or poly-nomial surfaces), or even by complex known sets (”canonic” fractals, e.g., Iterated FunctionSystems, and iterated polynomials as the Mandelbrot’s quadratic map). As complexity hasbeen also defined at a state of organization between complete, deterministic order and com-plete disorder (noise) identified with ”chaos”, it may turn out that real complex objects areeven more complex that fractals, in the sense of the Kolmogorov Complexity (KC). This is thecase of self-similar fractals in which a very simple construction rule entirely describes all scalefeatures. Thus, ”infinite detail” is not necessarily a complexity attribute. The knowledge of anunderlying law or pattern (random laws) seems also to reduce complexity (description length)in function such knowledge. This simplification is in fact a desired goal in Science.

✦ The utility of complexity measures, such as KC, resides in re-orienting modeling strategiesin the identification and choice of minimal representations, algorithms, languages, etc. If apattern is too irregular for Fourier analysis description (spatial periodicities turn out to betoo weak), it may be more natural or economical to use iterative process such as deterministicfractals, to describe it. If the pattern is poorly scale-invariant, other paradigms must be devised.

✦ To be useful as descriptive tools, complexity measures need to be invariant (stable or robust)under common transformations and noise. A possible problem is that they tend to quantifythemselves some kind of invariance and repetitive patterns in the observed data, and not theirabsence. Still, another practical problem is the difference meaning and use of the terminologyin different disciplines, and the goals in studying complex sets (representational, interpretative,model-oriented, formal, application-oriented, philosophical, aesthetical, etc).

✦ In a final analysis, must features and concepts around complex or simple shapes and theirdescription turn to be rather a kind of invariance, continuity, linearity, regularity or any othersimplification more or less present. Real ”complex” attributes begin thus with substantial lackof some or many of these features.

We can say that many authors have tried several approaches to organize complex information,but some lack of uniformity and standard definitions can be seen. There are for example ’shapecomplexity’ and ’algorithmic complexity’ as two notions sharing some features –the latter is usedto code descriptions of the first. Application contexts strongly determine the language and selectionof morphometrical methods and concepts, but the number of shape features and possibilities makedifficult the choice and interpretation.

1.8 Discussion and Conclusions 75

As a summary, we list in Table 1.2 only the main features of a common data application problem,according to a difficult qualification of ”low” and ”high”.


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Chapter 2

From Raw Data to Models inN�: ABoundary-Based Approach

Abstract

In this chapter we discuss discrete boundary representations and present a general technique forprocessing, analyzing and visualizing three-dimensional objects with arbitrary shape and topology,taking into account several considerations of the role of boundaries in many research fields, fromdata analysis to modeling in N�. The technique consists in building first a boundary representationof objects detected by an implementation of the surface tracking algorithm of Artzy et al. [Artzy81].This algorithm provides the set of voxel facets for the boundary of each object. As their adjacencyrules are available during surface tracking, we also obtain efficient methods for boundary and vol-ume traversal, taking into account local information. Hence, boundary representation constitutes thebasis of some of the 3D operations that we perform. These operations comprise three groups: (1) 3Dimage processing and object manipulation, (2) measurements (morphometry) and (3) visualization.Our approach has allowed us to pre-process complex objects in order to reconstruct their 3D shape,to extract morphological parameters, and to clearly visualize their structure and spatial imbrication.Applications in the Biomedical and Physics domains are presented in Chapters 3,4 and 5.

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

2.2 Discrete Volume Representations and Boundaries . . .. . . . . . . . . . . . 80

2.2.1 Boundary Representations. . . . . . . . . . . . . . . . . . . . . . . . . 81

2.2.2 Relevance and Potential Uses of Discrete Boundary Representations . . . 82

2.3 A Facet-Based Model for Discrete Surface Representation. . . . . . . . . . . 84

2.3.1 Discrete Elements: Scenes and Voxels. . . . . . . . . . . . . . . . . . . 84

2.3.2 Discrete Elements: Faces and Labels. . . . . . . . . . . . . . . . . . . 87

2.3.3 Discrete Elements: Facets and Boundaries. . . . . . . . . . . . . . . . . 89

2.3.4 Discrete Adjacency and Connectivity. . . . . . . . . . . . . . . . . . . 96

2.3.5 Voxel (Euclidean) Neighborhoods . .. . . . . . . . . . . . . . . . . . . 102

2.3.6 Facet (Geodesic) Neighborhoods . . .. . . . . . . . . . . . . . . . . . . 104

2.3.7 Voxel Geodesic Neighborhoods . . .. . . . . . . . . . . . . . . . . . . 106

78 From Raw Data to Models inN�: A Boundary-Based Approach

2.3.8 Discrete Boundary Neighborhoods . .. . . . . . . . . . . . . . . . . . . 108

2.4 Surface Tracking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

2.4.1 Surface Tracking Algorithm. . . . . . . . . . . . . . . . . . . . . . . . 114

2.4.2 Facet Traversal by Lateral Circuits . .. . . . . . . . . . . . . . . . . . . 117

2.4.3 Interior Traversal and Object Labeling. . . . . . . . . . . . . . . . . . . 119

2.4.4 Run-Length Interpretation of Interior Traversal . .. . . . . . . . . . . . 124

2.5 Boundary-Based Processing and Analysis of 3D Objects. . . . . . . . . . . . 126

2.5.1 An Example of Boundary-Based Morphometry: Euclidean Area Estimation 128

2.5.2 Boundary-Based and Volume Processing (Examples). . . . . . . . . . . 129

2.6 Visualization of Complex Structures . . . . . . . . . . . . . . . . . . . . . . . 135

2.6.1 Surface Rendering . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 135

2.6.1.1 Hidden Surface Removal and Surface Shading. . . . . . . . . 135

2.6.1.2 Discrete Normal Estimations. . . . . . . . . . . . . . . . . . 135

2.6.2 Three-Dimensional Data Structures. .. . . . . . . . . . . . . . . . . . . 137

2.7 Summary and Conclusions .. . . . . . . . . . . . . . . . . . . . . . . . . . . 137

2.1 Introduction

It has been widely recognized that quantitative analysis and computer processing of 3D imagesstill lacks the advancement and understanding already acquired for 1D and 2D signal processing[Pommert93, Wood92, Kaufman94b]. Simple visualization has enabled some basic understandingof intricate 3D data in similar ways as humans do when they visually examine real objects. Liter-ature is rich in 3D realistic reconstructions with color, transparence, texture, shadings and specularreflections, but with limited analysis. More scientific –i.e.,objective, quantitative– approaches arestill recent and seldom.

These problems are not only due to increasing data volume, or even higher complexity (eithercomplexity of the data or of the task to perform), but also to the ways data are represented and ma-nipulated, taking into account well-defined but punctual objectives. Also, digital computers haveimposed their characteristics and limitations to data analysis, and this has been extended to represen-tation, modeling and processing, specially in three-dimensional data.Computer Aided Design andManufacturing(CAD, CAM e.g. solid modeling), and engineering have prompted the development”CAD, CAM” Iof computer graphics technologies for visualization, modeling and design of geometric and artificialobjects, while Medical Imaging (Tomography, Magnetic Resonance Imaging, Ultrasound Imaging,etc.), Quantitative Microscopy or Remote Sensing have stimulated the use and developement of tech-nologies and theoretical advances for Image Processing, Pattern Recognition, volume visualizationand true quantitative analysis.

A CAD approach typically consists of efficient imaging, through fast and simplified repre-sentations, such as vector rastering, contour extraction and plotting, triangular meshes, polyno-mial patches, texture-mapping, depth maps and other image buffers, and the corresponding pipelineand parallel architectures (geometric engines, transputers, etc.). Effective graphic interfaces al-low humans to grasp visual features of 3D data, and make semi-automated space manipulationsor computer-assisted segmentation and component isolation, while reliable automation of these pro-cesses is still in development.

2.1 Introduction 79

A discipline calledAlgorithmic Geometry (or Computational Geometry) has produced efficientmethods for dealing with most of the visualization and modeling problems, specially suited to math-ematical objects [LeePreparata84, Preparata85]. On the other hand,full quantitative analysis andprocessing have been difficult or impossible to incorporate with these approaches, or has shown tobe incompatible with required simplifications, as used for visual realism (e.g., triangulated meshesand Bezier surface), but difficult to process, in comparison to discrete volume-element representa-tions which extend the picture-element paradigm.

By “full quantitative analysis and processing”, we mean mainly: Mathematical Morphology(MM), local and global feature extraction and morphometric measurements, geometry-driven orJ ”MM”morphometry-driven filtering, enhancing or MM-filtering, roughness and fractal analysis, multi-resolution image processing, and other techniques. We saw in Chapter 1 that particular featuresare desirable in the study of complex data: measurements should be robust and present some kindof invariance (geometrical, topological); they also must permit experimental validation. Even if justMM is considered, it turns to be a very rich paradigm for analysis of 3D scenes represented as dis-crete 3D volumes: granulometry, distance fields, medial-axis-transforms, etc., which in turn openaccess to various techniques of complex analysis (in the sense of ‘sophisticated’). However, MM ispoorly or inefficiently suited to arbitrary triangular or simplicial meshes and difficult to apply on theanalytical representations used in CAD (e.g., splines, superquadrics, wavelets).

In certain cases, the model characteristics (smooth, analytic surfaces and blended ridges) over-ride the nature of the original data, hiding or distorting valuable details, or introducing undesirablefeatures. The surface normal and other differential entities are particularly sensitive to small per-turbations. Furthermore, after processing and filtering data to extract other pieces of information,modeling artifacts contaminate the results, requiring ad-hoc and costly correction and validation[Zelkowitz97, Machiraju96]. A similar argument could be applied to uniform discretization, in whichthe staircase artifacts superposes with original data, but very effective techniques exist to clean it up,using classical Fourier interpolation and decimation or other well-tested methods.

On the other hand, physical-based models and visualization (as for example mechanical andfluid deformations, fluid dynamics, and even radiosity algorithms), rely on continuous or hybridapproaches. As such, they have been tackled by analytic and CAD methods, specially those us-ing finite elements and boundary elements [ThompsonToga96, Gallagher89, Sethian96, Ebert94,Troutman93]. All these trade-offs have prompted proliferation of methods to pass from continu-ous to discrete representations and viceversa, in order to suit theoretical advantages to all modernapplications, scientific and industrial.

Discrete images, sets and operators inN� andZ� have been developed in the past in accordancewith data structures and computer-science concepts such as voxels, grids, random access locations,meshes, and their models with discrete mathematics: graphs, n-dimensional incidence structures,lattices and discrete functions [Voss93, Ritter96]. Besides understanding and representational ad-vancements, these tools have permitted to alleviate the lack of quantification and processing in theCAD-based approaches.

In Chapter 1 we have presented and given an overall characterization of geometric and non-geometric structures from low to high degrees of complexity, as found in the biomedical domainsand physical sciences phenomena. To study such objects and their associated properties, we havechosen adiscrete-geometry approach for surface extraction, volume processing and visualization.In this chapter we present some theoretical foundations, tools and methods using this approach torepresent and analyze data in scientific applications where fast and realistic visualization is not themain concern for a depth understanding of their nature.

The sections in this chapter are organized as follows:


✦ In Section 2.2 we discuss the state of the art of discrete volume representation problems andapplications, and their relationships with boundary representations.

✦ In Section 2.3 we present in detail all definitions of elements that constitutea discrete-boundaryrepresentation inN�. They include voxel faces, various boundary sets, connectivity1 and neigh-boring relations.

✦ In Section 2.4 we briefly present the boundary-extraction algorithm of Artzy et al. [Artzy81],using the above definitions and the modifications and enhancements we have incorporated.

✦ In Section 2.5 we discuss how diverse measurements are obtained from boundary representa-tions and some of our particular contributions.

✦ In Section 2.6 we present some aspects of visualization with boundary representations.

2.2 Discrete Volume Representations and Boundaries

The most successful paradigm of representation, processing and visualization of discrete objects intwo dimensions is the orthogonal grid of picture elements. Hexagonal grids have also become at-tractive in many applications, because they don’t pose neighborhood-connectivity paradoxes, whendefining foreground and background in the same way. We will examine this issue in 3D in Subsec-tion 2.3.4.

The direct equivalent of the pixel-based representation of 2D images is the volume-element, orvoxel-based representation, orcuberille model, introduced by Herman and Udupa [Herman83]. Inthis representation an orthogonal grid of cuboids or parallelepipeds is used as discretization grid, andmathematically treated as a 3D matrix array of integer coordinatesx� y� z � N� (or indices�i� j� k�)[Herman83, Chen85]. We introduce our version of this model in Subsections 2.3.1 and 2.3.2.

Most formal and well-tested algorithms and representations in Image Science and Discrete Topol-ogy use the orthogonal grid representations for digital picture (pixels) and volume data (voxels). Asimple reason is their direct isomorphism withNn. Fourier analysis has provided solid groundsfor conversions from or toRn, quantization, filter design, interpolation, sampling phenomena, etc.Equivalent advances for non-uniform tessellations ofRn and arbitrary triangulations are either toospecific, non-robust or often unavailable. These and other requirements strongly favor the discrete-boundary representations that are compatible with orthogonal grids where volume elements are par-allelepipeds. There are works on grids other than cubic in 3D, with similar capabilities (e.g., gener-alizations of the hexagonal grid in 2D [Ponce88, Herman98]).

Data volumes themselves are mainly produced as discrete samples inNn, or are converted so,from polar coordinates or interpolations. In many research fields, developments of measurement de-vices have enabled the obtention of huge matrices of 3D data, with increasing structural complexityand resolution (starting at�� 1 byte, for example), in the form of scalar, vector ortensor quantities, temporal sequences, multiple channels, and higher dimensionality. These featurespermit to digitize or scan information in great detail, and represent physical processes and abstractfeatures at all scales. Large scale simulations also produce increasingly complex data. Thus, 3Dvoxel-based graphics has a recent rapid growth, specially in Medical Imaging and Non-linear Dy-namic Systems. Theory and techniques based on the cuberille model, as applied to such data analysis,are also called “volumetric imaging”, “volumetric graphics”, or “voxel imagery” [Kaufman94a].

1Or connectedness.

2.2 Discrete Volume Representations and Boundaries 81

Exceptions to systematic coordinate sampling are laser scanners, and mechanical sensors thatproduce very high precision clouds of unstructured coordinates, but in many applications there is aneed to pass through voxel grid representations.

Large amounts of complex data with the mentioned problems are very difficult to analyze, requir-ing often a pre-processing stage, such as filtering, weak-connectivity component separation, or otheroperations in a selective and local fashion [Kaufman94c, Rossignac94, Udupa94b, Lopez93]. Tostudy such objects and their associated problems, we chose a discrete geometry approach for surfaceextraction, volume processing and visualization.

Voxels and octree representations have been extensively reviewed in the literature (see for exam-ple [Jackins80, Amans85]), and we will rather focus on boundary definitions and regional represen-tations, in order to introduce ourboundary-based paradigm.

2.2.1 Boundary Representations

Boundary-based representations (also known in Computer Graphics asb-reps ) were originated J ”b-rep”in the CAD-CAM engineering community, to simplify 3D object representation and manipulationthrough polygonal facets, patches or primitive boundary elements. Simplification of b-reps is widelydue to the systematic analytic formulation of models, intersections and algebraic operations betweenlines, planes, polyhedra, or analytical surfaces (e.g. conics, splines, Bezier curves). Ray-tracing andradiosity techniques for realistic visualization have successfully exploited this geometric approach toobject representation, even for fairly complex objects.

The same cannot be said about quantitative analysis, specially when signal and image process-ing techniques have to be employed (e.g., edge-detection operators, mathematical morphology andscale-space filtering). The geometric primitives introduce irregular biases to measures and requirecomplicated calculations. Rapid degradation may appear in some b-reps, which have then to be re-built and verified after processing (to preserve topology, for example). In real imaging applications,boundaries are not always well defined, textures and noise cannot be separated and the CAD-CAMapproaches poorly represent objects with intricate structures.

More primitive elements have been used to simplify the mathematical treatment without the lossof a faithful b-rep: triangle meshes and their dual meshes are routinely employed to represent objectsurfaces, and constitute a good example of a systematic solution for modeling and visualizationproblems of complex objects with numerous features (color, texture, components, contact areas, etc.)[Boissonnat96, Boissonnat93, Gueziec95]. Nevertheless, many triangulation methods start with amesh approximation based on discrete boundaries [Algorri95].

In their 2D continuous version, b-reps for 3D reconstruction from serial slices are based on pla-nar contours, (continuous) polygons [Boissonnat88], or Delaunay triangulations [Boissonnat84]. Indiscrete space, contours have been represented by chain codes in discrete grids [Rosenfeld82], orcoded into topologically consistent representations, such as the cellular complexes that comprise in-dividual vertices, edges and their inter-connections. Thek-chains generalize cellular complexes andhave associated states and an algebraic-topological structure suitable for dynamic systems modeling[Voss93, Kovalevsky89, Pailloncy97]. Together with other formulations, several b-reps have con-stitued the basis for important fields of research where data representation is no more the main prob-lem, but strong links arise with other concerns. This is the case with level-set methods [Sethian96]and the scale-space primal sketch [Lindeberg92b], where boundaries are organized or interpretedin higher dimensions. Dynamic boundary representations also appear in the form of solutions orisosurfaces of differential equations in image processing, and have produced advancements from de-


formable and active contours [Kass88] to curve evolution and the fundamental equations of ImageProcessing as an “interpolation” between the heat equation and classical morphological operators[Alvarez93, Brockett94].

2.2.2 Relevance and Potential Uses of Discrete Boundary Representations

There are at least six important reasons that have motivated our boundary-based approach, and willreappear frequently along this manuscript, often as arguments for tool design. As our main concernis the capability to deal with complex structure and data, we review below how a boundary-basedapproach can be useful.

Data reduction Visualization, shape understanding and interaction withn–dimensional objects isfrequently better achieved throughclever �n � k�-dimensional simplifications; in particularboundaries of the objectO, or through boundaries of subsets ofO. The adjective “clever” isnot in general true. An example are sequential slices, or even 1D-profilograms, that do providesome access to data, but poor understanding of three dimensional structure and inter-relations.When carefully applied, the simplification approach has provided successful results in 2D im-age processing, where contour extraction and analysis is a common task in many applications.In gray-level image analysis, the surface-intensity interpretation (with density seen as heightlevel), and the subsequent decomposition in level sets (isodensity curves), are also examplesof this dimension-reduction approach. It is worth noting that also texture analysis, for ex-ample, has been represented and treated as roughness of a “2.5-D” surface orrange images[Rosenfeld70], and frequency-domain analysis can be visually plotted with the amplitude as athird dimension. Finally, higher dimensions are only accessible to human perception throughsets of 3D representations (“level sets”), which imply intensive use of isodensity surfaces.

Fractal boundaries When complex objects are associated with fractals, a particular remark can bemade on the relevance of b-reps: the definition of topology dimensiondT of an objectOrelies on local neighborhoods and relations betweenO, as embedded in an Euclidean space ofdimensiond, and its boundary�O, which has topological dimensiondT � d� �. The fractalfeatures ofO are described by dimension quantitiesDx whose values satistfydT � Dx � d,as noted in Section 1.4.

Boundary-based measurements of fractals. From a more general point of view, the notions ofcomplexity and fractal dimensions of irregular sets are highly related to then� k dimensionalcomponents (orsubmanifolds) of an objectO embedded in an-dimensional space. Thus, the“material content” of fractal boundaries inn�k dimensions (cardinality, length, area, volume,etc.) has the property of being “infinite” at somen�m dimension (a boundary of an�m�subset ofO) and “zero” atn�m�. However, it possesses a finite, non-zero fractional valueatn� �, withm� � � � � m a fraction that codes the (n�m �)-dimensional space-fillingproperties of fractalO. In this sense, a complexity measure can be obtained to consider theirregularity and roughnessof boundaries only.

Interface phenomena Physical phenomena present particular characteristics precisely atinterfacesand boundaries of objects or sets. This observation directly links structure complexity andphysical interactions. A related remark is that space-filling structuresare interfaces (Subsec-tion 1.6.3). We shall see and illustrate some examples in Chapter 5.1 (Amorphous Photo-deposition) and 6 (Other Applications: Chromatin and Cellular Nuclei Reconstruction inTransmision Microscopy). In many physical phenomena, for a given constant volume, the sur-face extent of an interface augments with increasing rates of exchange of information, or flow

2.2 Discrete Volume Representations and Boundaries 83

of material or energy (e.g. heat dissipation) through convoluted boundaries. Fractal interfacestend to develop at active transition regions, andn-dimensional space traversal in a dynamicsystem (the blood-vessel networks that have to visit every cell in the organism, for example) isoptimized through�n�k� dimensional space-filling structures [Wood95, Kalda93]. Moreover,most differential equations of physics require the knowledge of defined boundary conditionsfor their proper solution or analysis, and no numerical method of solution is workable withoutsuch specification, thus boundary-traversal methods, manipulation and feature extraction (orspecification) on discrete boundaries is required in any modeling task.

Boundary-based filtering A set of mathematical relations enable to extract volume characteris-tics and do morphological operations using b-reps. These include mainly the discrete GaussTheorem (Section 2.5.1), and the equivalence bewteen morphological-filtering and boundary-labeling (Section 2.5). The generality of these relations encourages to further explore waysto process and extract information of an object or a Region or Volume Of Interest (ROIor VOI) from its discrete boundaries, and combine it with local, global and internal data(individual voxel information). Modern theory of image processing has explored relation-ships between curve-evolution equations, mathematical morphology and scale-space analysis[Alvarez93, Lindeberg93a]. The resulting mathematical frameworks bear similarities withboundary-based morphological operations. Besides, it has become natural to extend the curve-evolution framework on 2D-images to multidimensional manifolds, we only cite some exam-ples: continuous fronts and level sets [Sethian96], shape description based on incrementalcontour deformations [Kimia93], sheet models [Whitten93], estimation of principal curvaturesin cortical surfaces [Joshi95], and manifold mapping of the cortex [Szekely92].

Level-set representationsIn the framework of level-curve evolution discussed by Cohignac et al.in [Cohignac92], a 2D image can be either identified with the set of its level curves and thentheir evolution is studied, or the image itself can beevolved, using the evolution formulation ofthe fundamental equation of image processing, which goes outside the scope of our work, andwe only mention the fact that a similar approach may be devised in three or more dimensions,giving rise toboundary evolution in n-dimensional images.

A problem related to level sets comes from the fact that volumic data can be rearranged or ac-cessed in a number of ways, such as stacked cross sections (stereotomies), non-planar internalsurfaces, or scanline samples. These reorganizations allow different modes of volume access,and different processing and filtering methods: separable operations, dimension-reductionstrategies, data-compression and other specific tasks. Extending this idea, and having in mindthe interpretation of certain 2D-images as sets of level contours, it becomes clear that volumedata can be represented as sets oflevel boundaries, which may code some kind of homogene-ity (isolevel or isodensity surfaces, where “density” may comprise more general features, e.g.,texture measurements).

Layer models Recent works use a similar approach to level sets in the form of discrete fronts or ac-tive contours [Malladi95, Davatzikos95] and of coupled-surfaces propagation [Zeng98] usedto extract volumetric layers of the cortical gray matter of the brain. This volume representationis also named “a peeled-onion” model, and allows for some volume processing and under-standing through sets of discrete “piled” or “peeled” boundaries or layers. A characteristicdrawback is that topology is not well preserved through consecutive layer sets, and homotopicoperations are not always possible, but restrictive principles can be applied, as in the level curveevolution framework. Homotopy problems can be treated by proper tracking of connected andunconnected components. An image-processing example of a level-set representation is the


distance field isosurfaces (orisodistance surfaces in this example) of an objectO:

Li�� fp � O j minq��O�kp� qk� � zig� � i � �� N�

with zi a constant such that� � z� � z� � � � � � zN . Level setsLi corresponds to a constantdistancezi to the nearest boundary points inside an object. This level-set decomposition isused for example in watershed segmentation of weakly-connected binary objects. In this case,each distance unit correspond to an order indexi of the set of isosurfaces or level setsL i

that constitute the discrete distance field. In a chamfer distance fieldd�p� � N� p � N�,two discrete surfacesSd� Sd�� d�p� which are isodistance subsets separated by one-voxelunit distance, may share several voxels in function of shape concavities; this property allowsboundary-based characterization and detection of a number of shape features. Another recentexample is 3D thinning by successive nested boundaries2, generated by simple-point removal(homotopic thinning) [Marion-Poty94]:

��O b ��O b � � �b �NO

It is by now evident thatdiscrete boundary representation and manipulation constitute a corner-stone of numerous techniques of analysis and modeling in all kinds of disciplines. There is still alot of work to do for properly relating, integrating and exploiting diverse approaches or frameworkswhere b-reps are a timely notion: active (or deformable) models, finite-element and finite-boundarymodels, fronts and diffusion equations, discrete differential geometry, integral geometry on geodesicsets (the boundaries of Euclidean sets), geometry-driven processing, and many other active researchtopics. Together with the measurable features of complex shapes reviewed in Chapter 1, we have bynow a broad panorama of analytical methods and potential applications in Scientific Visualizationbased on boundary information. We introduce in the following section the elements of a model forvolume and boundary representations.

2.3 A Facet-Based Model for Discrete Surface Representation

The aim of this model is to represent inN� a set of solids or regions using their discrete boundaries.These consist of all the voxels (or voxel faces) that separate objects from background, permitting vol-ume processing operations and morphometrical calculations. In later sections of this chapter, we useseveral definitions which are here presented in a progressive fashion and commented within the con-text of our work. To this end, in the three following subsections (2.3.1 to 2.3.3) we list the elementsof discrete representations: scenes, voxels, faces, labels, facets and boundaries. In Subsection 2.3.4,we introduce the adjacency and connectivity relations needed between these elements, and then, inSubsections 2.3.5 to 2.3.8, the neighborhood relationships that allow to characterize local proper-ties in voxels (Euclidean neighborhoods), facets (geodesic neighborhoods, along the boundary), andboundaries themselves.

2.3.1 Discrete Elements: Scenes and Voxels

We review in these paragraphs coordinate systems, scene domains, and voxels. In this and latersubsections some definitions are elementary and can be found in basic literature, such as [Kong89,

2Recall Definition A.2, and Notation A.1 at page 231:A b B means “A nested withinB”.

2.3 A Facet-Based Model for Discrete Surface Representation 85

Rosenfeld91, Voss93, Udupa93, Udupa94b, Udupa94c, Klette98, Herman98], but we adapt and for-mulate them to build our model and make more evident their computer implementation. Small-fontremarks may be skipped in a first lecture.

In the following definitions we use real and integer coordinates: points�x� y� z� � R�, andindices�i� j� k� � N�. When�x� y� z� are integers, or form a 3D array with fixed distances betweenpoints, we say that they form adiscrete grid or cubic lattice.

DEFINITION 2.1 (Unit vectors)The ordered set of unit vectors

U� � ��e��e��e��

is called orthonormal basis or unit system of basis coordinates. The ordered set of axis-coordinates is denoted by �X� Y� Z� and corresponds to basisU � forming a right-handed systemor trihedron . When any two coordinate axes (or basis vectors) are exchanged or one axis di-rection is reversed, we obtain a left-handed system. The ordered set of negative unit vectorsU� � ��e��e��e�� is also a left-handed basis.

U�

�ei

DEFINITION 2.2 (Grid Orientations)The ordered set U�

�� e�� e��e�� e��e�� e�� comprises three orthogonal directions

in the space and six orientations, grouped by opposite pairs. We also define: U �� U�.

We call U� and U vector sets of grid orientations. For convenience, we use also set notationU� � f�ukgk��, U� � f�eig�i��, U� � f��eig�i��, with unit vectors �uk� ��ei, in coordinateorder as defined for each U�� U � U�� U�.

�uk

SetsU�� U are used in the next subsection to associate an orientation and a normal to face elementsof voxels. Other sets and properties will use these sets of vectors, mainly: coordinate offsets, crosssections, surface tracking and volume-traversal order.

DEFINITION 2.3 (Physical or World Coordinates)Let Lx� Ly� Lz � R, scale lengths. The set f�x� y� z� � �iLx� jLy� kLz� j �i� j� k� � N

� g iscalled system of physical coordinatesor world grid coordinates.

1 Remarks. Non-linear coordinate transformations are usually defined with a mappingW � V � R�, with

V � N�. Then the physical coordinates system is the setf�x� y� z� � W �i� j� k�� i� j� k� � N� g.


DEFINITION 2.4 (Scene Domain or Support)Let Nx� Ny� Nz � N, and let �� Nx � ��, �� Ny � ��, �� Nz � �� R be closed intervals. A 3Dscene domainV � R� (or support) is a 3D array or box of dimensionsNx �Ny �Nz:

V�� Nx� �� Ny � �� Nz � ��

with coordinate indices i� j� k � N, running from � to Nx � �, � to Ny � � and � to Nz � �,respectively. We also refer to p � V as the point coordinates p � �x� y� z�, where x� y� z may bereals or integers. In the latter case, we say that p is a grid point of V.

V

Remark 1. A n-dimensional image is usually a function from some scene domain to another set (codomain orrange). We will describe laterlabel attributes of a voxel to characterize information of a scene.

Remark 2. It is often convenient to consider a discrete subset ofV (or defineV inN� ) forming a regular grid ofpoints:

V�� f�xi� yj � zk� �R

� j �xi� yj� zk� � �iLx� jLy� kLz�� i� j� k� � N� g

with Lx� Ly� Lz � R. If Lx � Ly � Lz, thenV� is called acubic grid, otherwise it is arectangular grid ororthogonal grid. Real matrices with entriesfvijkg is the standard representation of 3D scalar images.

V�

”VOI, ROI” I

Remark 3. A Volume or Region Of Interest (VOI or ROI) of V is often defined as a“box” subset ofV as:

VROI�� Nx�� Nx�� Ny��Ny�� Nz��Nz� , such thatVROI � V. We consider a ROI in general to have any

shape, independent of its contents.

DEFINITION 2.5 (Voxel)Let V � R� be a scene domain and p � �x� y� z� � V a grid point. The set

v � v�x� y� z�� x� x �� y� y �� z� z ��

is called a volume elementor voxelat point �x� y� z�, which is then called reference vertexof v.

v

Remark 1. Many other grids are possible; they are in general calledspace tessellations. In many configurationsx� y� z �� N. For example the points�x�� y�� z�� x� y� z� � �� define the voxel centers in abody-centered-cubicgrid. Non-integer coordinates arise also with center-of-mass calculations, sub-voxel precision orpartial volume effect.

Remark 2. WhenLx � Ly � Lz, the voxels in world coordinates are calledisotropic voxels. Note that�Lx� Ly� Lz� are the physical lengths assigned to the three orthogonal edge dimensions of a voxel, depending ondata interpretation.

Remark 3. A voxel v is a spatial neighborhood of pointp � �x� y� z�, in the body-centered grid, but it isoften treated in literature asp itself. The expressions “kvk ”, “ v � �� ”, etc. refer to coordinate and vectoroperations, withk k the Euclidean norm.

Remark 4. In orthogonal grid coordinatesv is a cube (and the scene is called acuberille environment, afterChen et al. in [Chen85]) withVol�v� �, in grid units. In world coordinates it is a parallelepiped orprismaticvoxel with 12 side edges, 6 rectangular faces and volumeVol�v� � L x Ly Lz. Herman defines a voxel ingeneral as aVoronoi neighborhood, in other discrete grids (face-centered cubic and body-centered cubic, mainly),determined by neighborhood configuration. In such cases, voxel shapes are rhombododecahedra or other poly-hedra, presenting different advantages and properties [Ponce88, Herman98]. We limited our work to rectangulargrids and prismatic voxels. Since we have defined a voxels as closed 3D-intervals, including all 8 vertices and all6 faces; contiguous voxel intersections (shared faces) are not empty: they overlap (see below).


2.3.2 Discrete Elements: Faces and Labels

It is convenient to fix a systematic enumeration of voxel faces, in order to assign them several proper-ties as orientation and traversal circuits for b-reps and surface tracking. Our choice was the following:

DEFINITION 2.6 (Voxel Faces)Let v�x�� y�� z�� V be a voxel. The following subsets of v:

f � �� fx� y� z j x � x�� y � �y�� y� �� z � �z�� z� ��g�

f � �� fx� y� z j x � x� �� y � �y�� y� �� z � �z�� z� ��g�

f � �� fx� y� z j y � y�� x � �x�� x� �� z � �z�� z� ��g�

f � �� fx� y� z j y � y� �� x � �x�� x� �� z � �z�� z� ��g�

f �� fx� y� z j z � z�� x � �x�� x� �� y � �y�� y� ��g�

f �� fx� y� z j z � z� �� x � �x�� x� �� y � �y�� y� ��g�

consitute the 6 facesof v, and denote it by ff kgk�� . f k

Remark 1. Some authors call the voxel faces “3D pixels” [Perroton93], but some confusion arises when con-sidering voxels as 2D pixels in serial slices. We further introduce orientation properties, thus we retain the term“voxel face”. Notice that the words “a single voxel” implicitly definev as the foreground against the backgroundvc (see Definition 1.3).

Remark 2. The order numbering of facesff kg�k�� will be associated with the set of grid orientationsU� (SeeFigure 2.1).

Remark 3. Consider two voxels such thatkva � vbk � � (thus, “adjacent”, as defined later). Letff kag�k��

be the faces ofva andff kbg�k�� the faces ofvb. If, for exampleva� vb are voxels at�x�y� z� and�x � �� y� z�

respectively, thenf �a is the same set of points thatf �b , andva�vb share the same face (See Figures 2.1d-e). Wemake however a distinction offace orientations in order to introduce voxel faces at object boundaries asfacets.

Remark 4. Similar enumerations were done for other voxel elements, such as vertices, edges, diagonals anddiagonal planes, when needed in our implementation. Their definitions are straightforward.

It is important to notice that theface centersof a voxel at�x�� y�� z�� are located at points�x�� y�� z��uk �� .

DEFINITION 2.7 (Oriented Faces)Let ff kgk�� be the faces of a voxel v�x�� y�� z��. Let the grid orientationsU� � f�ukgk�� definethe six face orientations. Then, the set of pairs f�f k� �uk�g� is called the oriented faces of voxelv, and each �uk is thus a unit normal vector of the oriented face f k.

Remark 1. Let fvkg�k�� be the voxels at�x� y� z�� x�� y�� z�� uk, with �uk � U�. The set of pairs

f� v � vk � �uk�g also define the oriented faces ofv.

Oriented faces point outwards of voxelv. For two consecutive voxelsva�x� y� z� andvb�x �� y� z�, the oriented face�f �a� �u�� of voxelva points to the opposite direction than the oriented face�f �b � �u�� of voxelvb (see Figure 2.1). We found convenient to work in certain cases with a comple-mentary set ofinwards-oriented faces, associatingff kgk�� with U , instead ofU�. The choice ofone set or another define orientation on a discrete surface composed of voxel faces (described in next


(x,y,z)(x,y,z)

Y

Z

X

or

��

��

1

4

3 f

(b)

1

5

f

foreground

00 ^

2f

(a)

0

f 0

(f)(d)

(e) (g)

background

(c)f

u

f

f

f

f

Figure 2.1:Voxels and faces. (a) A voxel indicated at point p � �x� y� z�, by a “��”, chosen as the referencevertex. Faces f �� f � and f � are indicated though not visible at the opposite sides of f � f and f �.(b) Right-handed trihedron of axis coordinates �X�Y� Z� used to locate points, voxels and to number theirfaces. (c) Other grid and other voxel definitions: face-centered cubic (fcc) and body-centered cubic (bcc), at�x� y� z� � �� . The set of points of face f of voxel (d) is “the same” of the face f � of voxel (e),but an orientation and a unique choice are defined when foreground and background labels are assigned tovoxels (g) and (f) respectively.

subsection). We define theface-based boundaryof a single voxelv as:

�v�� ff kgk��

with its orientation�� given by the face orientation, either outwards (“”, choosingU � ) or inwards(“�”, choosingU ).

In order to refer to information in a scene support and identify specific subsets (objects, regionsor points), we define the following attributes of voxels and scenes.

DEFINITION 2.8 (Voxel Labels)Let fVngNLn�� V be a set of subsets of a scene domain. We can assign a label to each subset.

Let A � f�n � NgNLn�� and L �V � A, such that

�� n � �� NL�� v � Vn� � � �L�v� � �n��

We say that v has a scalar label�n, or that v is a “�n-voxel”.A labeled sceneor simply sceneis the pair �V�L �

�� V� fL�v� j v � Vg�.�n-voxel I

Remark 1. WhenV is partitioned intoNL disjoint subsets, i.e.,V � �NLn��Vn , with Vi � Vj � ∅ fori �� j, with labelsA � ��NL � � � N, and mappingL as above, we callV a scalar field, 3D scalar imageorgray-level volume, and voxelv�x� y� z� has assigned a gray-level valueL�v� � L�x� y� z� � ��NL � �.


Remark 2. Vector labels of dimensionD arise when�n � ND. A common example withD � are color labels

denoted by�L�v�� R�G�B�� describing red, blue, green and transparency intensities for eachv � V, a color

scene. Real-world data are obtained often as discrete samples of continuous domains, and producen-dimensionalscalar, vector or even tensor images.

Remark 3. The expression “v�x�y� z� � �” suggests that voxelv at �x�y� z� has assigned a label�. But v isa defined region of points inR�, not a function, and it would be better to useL�v� � � to refer to the label or“contents” of voxelv at point coordinates�x� y� z�.

Remark 4. Six voxel-face labels assigned to the facesff ka�g�k�� of a voxelva can be rather defined in terms of

a vector label�L of dimension 6 assigned tova, with scalar componentsf L�f ka� � �ka g�k��.

Remark 5. We callV� � �V�L� a set of�-voxels or�-set. Since it may happen thatcard�V�� , V� canbe composed of various connected components. Once definedK-connectivity, we will haveK-connected objects,with connectivity depending on adjacent voxels sharing the same label (see Section 2.3.4, page 96).

NOTATION 2.1 (Binary Labels: �-voxel,�-voxel)Let V � V, and let B � f�� g be a set of binary labels, and the mappingLB �V� B such that:

LB�v� � � �� v � Vc� then v is called �-voxel�

LB�v� � � �� v � V � then v is called �-voxel�

LB is called a binary function or labeling, V is the foreground, and V c is the background.

We also call a pair �V�LB � binary scene, with LB as above.

NOTATION 2.2 Let L�v� � f�� g a binary labeling. We denote its binary-label complement as

L�v� �� L�v�. Moreover, if v is a �-voxel at �x� y� z�� f�� g, we define the �-voxel v at

(x,y,z) as the voxel such that L�v� � L�v� � �� L�v�.

Remark 1. vc is a set complement (“all voxels exceptingv”), while v represents a logic 1-complement (the

same voxel with the opposite label ofv), defined only for voxels with binary labels.

Selection and classification of pointsq � V (or �V�L �) proceed by stating propositions aboutq andits attributes, such asL�q� (binary itself, gray levels or other information), and examining if thesepropositions are true or false (predicates of the proposition) [Pavlidis74]. A single proposition aboutq partitions the scene domain into two sets. This process is thescene (binary) segmentation, whichgeneralizes Definition 2.1. A�-object is anyW � �V�LB �, composed of�-voxels.

In summary, a binarization process partitions a scene into�-objects composing the foreground,and�-objects constituting the background. The first include all kind of particles, bodies, etc., andthe second comprise holes, pores, cavities, etc. Note that the starting sceneV may have other infor-mation and binary segmentation came out from a proposition about points inV and a segmentationprocedure to evaluate or estimate the predicate for all the scene.

2.3.3 Discrete Elements: Facets and Boundaries

From the different definitions given so far, we may now introduce the notion of discrete boundaryelements, which are the voxels and voxel-faces that separate foreground from background (undersome labeling process) in a scene. We state this notion in the form of an existence theorem in what


we shall see is the “�-connectivity”, but may be extended to other connectivities (to be introducedlater):

PROPERTY 2.1 (Boundary Voxels, Boundary Faces (Facets))LetV a binary segmented scene. Let V �V such that V is a set of �-voxels andV c its complementof �-voxels. Let V �� ∅ , Vc �� ∅ ,then,

� v� � V � v� � Vc� k � �� such that v� � v� � �uk� �uk � U��

thus, kv� � v�k � � in unitary grid coordinates.and,

voxel faces f i � v�� f j � v�� with i� j � �� such that j � i ��i�In fact, i � k (face and unit vector indices). We call

✦ the ordered pair f�� v�� f i�, a facet of V (or boundary oriented face),

✦ the ordered pair f�� v�� f j�, a facet of Vc,

✦ the �-voxels v�, boundary voxelsof V , and

✦ the �-voxels v�, boundary voxelsof Vc.

f�

Remarks. The rule “j � i� ��i ” is only a way to code face indices into pairs of opposed orientations:

f�f i� f j�g�i�� j�i��i � f�f �� f �� f �� f �� f �� f �� f �� f �� f �� f �� f �� f ��g

Indicesi� j could be related following any look-up table. See for examplef �� f � in Figure 2.1d-g.

By construction, boundary oriented faces (hereinafter simply denotedfacets) satisfy the followingproperties issue from the existence property. The proofs are trivial.

COLLORARY 2.1 (Facets as Pairs of Voxels)A facet, without regard to orientation (�), can alsobe defined as f j j

�� v� v��; an orientation �� or �� can be assigned from the expression

sign��v� � v�� uk�, with k � �� such that �v� � v�� uk� �� .

When orientation is assigned, the facetf�� v�� f i�� i � �� , can be also uniquely associ-

ated with the ordered pair of�- and�-voxels (see example of Figure 2.1f,g, in page 88):

f� �� v�� v��

conversely, theoppositefacetf� � �v�� f j�, with j � i��i can be associated with the orderedpair:

f� �� v�� v��

Thus, if we knowf , we knowfv�� v�g (see below thefacet-to-boundary-voxel operator ). A choicef� is calledoutwards-oriented. We have now the elements for the practical implementation of facetrepresentation:

COLLORARY 2.2 (Facet Representation)A facet f � �v� fk� of voxel v�x� y� z�, at oriented facef k � k � �� is uniquely specified by the vector �x� y� z� k�, or more explicitly by the pair ��x� y� z��uk�,with �uk � U�, the signed unit vector which is the normal to the oriented faces f k.


Remarks. In terms of discrete boundaries, a facet is most of the times a “boundary element” orbel, or also asurfel for “surface element”, in the multidimensional generalization introduced by Udupa in [Udupa94c]. Since aset of facet elements alone may constitute ribbons or open surfaces (not boundaries, which are closed), we preservethe term “facet”. We also use voxels as boundary elements (see below, Definition 2.12, page 93). While there aresome analogies with 2D pixels and pixel borders, 3D rectangular facets possess two relevant attributes: orientationand an associatedset oflateral traversal circuits, which are assigned in function of facet orientation, and introducedlater.

Rosenfeld et al. and Udupa used in [Rosenfeld91, Udupa90] an alternative notation to de-scribe oriented boundary faces. It recalls the setU� of Definition 2.2, but refers to the orderedset�f �� f � “combined” with the coordinates of the boundary voxelv�x� y� z� of a given facet.We change slightly some details for clarity; in essence, they defined

F�� x�� x�� y�� y�� z�� z�� (2.1)

as a partition of any set of facets into six subsets, in function of orientation. They further identifieddirectly a boundary face with a pair�v�� v��, with one ofx� y, or z in F� as follows: F�

Coordinate notation with F� Facets �v�� f k�, or f � �x� y� z� k�

�x�� y� z� �x� y� z� ��

�x�� y� z� �x� y� z� ��

�x� y�� z� �x� y� z� ��

�x� y�� z� �x� y� z� ��

�x� y� z�� x� y� z� ��

�x� y� z�� x� y� z� ��

With this classification, facetf�x� y� z� �� is a “y�-facet”, for example. Notation using “F�” isnot always clear (neither the one used by Udupa, and Rosenfeld et al., with “�x�x� � � � ”), but weexplain it for compatibility with literature where surface tracking implementations are proposed anduseF�. This partitioning is also used by Udupa to describe boundary opposed facets, for run lengthcoding. We useF� mainly with this connotation. Notice thatx� �� x�, etc., the sign indicatingonly opposed face directions on the axisX . Thus, we avoid the use of “�x�x, etc.” in this notation.The symbolsf and subindexedfn, n � N� are reserved to indicate any facet of the boundary withoutreference to its orientation (however comprised in its description by�x� y� z� k� ).

DEFINITION 2.9 (Opposite-Face Operator)Let v� � V , v� � Vc a �-voxel and an adjacent �-voxel, such that f�

�� v�� f i� and f�

��

�v�� f j�, with j � i ��i. We define a mapping opface�� between facets, such that f� �opface�f �. We call it the opposite face operator.

opface��

Remarks. Operatoropface�� swaps the face-orientation interpretation of facef i (thus, the sign of the facetnormal), and the face index off i from i to i� ��i, and the associated pair�v��v�� is substituted by its mirror�v��v��. This operator allows to interpret a foreground boundary as the negative of a background boundary, andconversely. In discrete convexobjects, they always enclose runs of voxels ofV, but in non-vonvexobjects, theymay enclose voxels of the foreground (concavities, holes, tunnels or cavities).

At this point we can define the boundary of an object in a scene (volume representation) in atleast three ways:

✦ Continuous point boundary ��. The (continuous) set of points defining the boundary.


✦ Discrete Face-Based Boundary �. The (discrete) set of facets (boundary-oriented voxel faces).

✦ Discrete Voxel-Based Boundary . The (discrete) set of boundary voxels (or grid points).

In the following paragraphs we provide a precise definition of each b-rep, their relationships, andhow to pass from one to another. Applications, advantages and limitations are mentioned on the way.

DEFINITION 2.10 (Continuous Point Boundary)Let V be a binary scene and V � V a �-object. Then, the set 3

��V �� f�x� y� z� � V j v� � Vc� v� � V � such that �x� y� z� � �v� v�� g

is called continuous point boundary of V .

�V

If V � V� (the cubic grid),��V consists of all points at intersections between�- and�-voxels. Thereare three ways in which the condition “�v� v�� ∅ ” arises in the cubic grid. Voxelsv�� v� (theobject foreground and its background) share either:

✦ a face (i.e.,kv� � v�k � � ), or

✦ only an edge (i.e.,kv� � v�k �p� ), or

✦ only a vertex (i.e.,kv� � v�k �p� ).

This distinction introduces different “K” connectivities in discrete representations (K � �� inthe above cases, withstrict equalities). We examine more closelyadjacency andconnectivity inSubsection 2.3.4. For the moment we consider only face intersections, since the surface area of theset�v� v�� is not zero whenkv� � v�k � �.

Remarks. Even if the voxels correspond to grid points,��V is a continuous boundary, since �x� y� z� �� N�,such that�x� y� z� � �v� � v��. A discrete boundary is then specified not by the totality of its real points but byenumerating either the facets or the boundary voxels. Notice that most definitions given in Section 1.1, pages 23-28 and Appendix A apply to��V, and some can be translated directly for discrete boundaries. Facet orientation(using eitherU� orU� ) provide the means to introduce oriented boundaries and allow a formal definition of solidinterior for discrete objects.

DEFINITION 2.11 (Discrete Face-Based Boundary)Let V a binary scene, V � V be a �-object. The set

�V �� f f � �v�� f k� j v� � V � f k � v�� f k � ��V g

is called face-based boundaryor facet boundaryof V .

Setting Nfacets�� card��V�, we also write � V � ffngNfacets

n�� .If facet orientation is outwards (choice U �), then �V � �V� is a positive orientedboundary.

� V

Remark 1. �V is a finite set ofNfacets subsetsfk, that is, of oriented voxel faces contained in the continuousboundary��V. Face-based boundaries are illustrated in the Figure 2.2a.

Remark 2. When the operatoropface�� is applied to allf � �V, the boundary orientation is inverted allowing

to definenegative boundariesalso as:� V � �� opface�� V� �, and conversely (see Definition A.4, page 232).

This definitions allow to characterize and manipulatediscrete cavities.

3For simplicity, we used symbol� to denote continuous and generic boundaries in Chapter 1.


DEFINITION 2.12 (Discrete Voxel-Based Boundary)Let V a binary scene, V � V be a �-object. The set

V �� fv � V j f k � v� k � �� such that �v� f k� � �V g

is called voxel-based boundaryor solid boundary of V .Setting Nvox

�� card� V�, we also write V � fvmgNvox

m��.

V

Remark 1. Using the solid boundaryV, thegeometric interior of Definition 1.5 (page 25) in the discrete case

is the set�

V�� V n V . There may be nonempty objects for which

�

V� ∅ , or it consists of several connectedcomponents, but a discrete version of theJordan property about closed, simple boundaries [Udupa94c, Herman96])allows to consider these cases. No discrete geometric interior is defined with face-based boundaries.

�V

Remark 2. The 2D equivalent of V are classic contours formed by pixels. The 2D equivalent of�V areedge-based boundaries, which are mentionned in some contour extraction methods, active contours, Markov-fieldsegmentation and cell complexes. The present work should be easily adapted to 2D applications.

Remark 3. No orientation was attributed toV in our implementation, but boundary voxels could be orientableif their facets remain adjacent (a non-simple point is a counterexample). If orientable, an average normal could beassigned. We conjecture that orientability is related to satifying or not the Jordan property.

Since there is one and only one voxel atp � �x� y� z�, the boundary V is isomorphic to a listof point coordinates�x� y� z�. An algorithm to extract V , without passing by�V , is presented in[Voss93]. Voxel-based boundary representation is illustrated in Figures 2.2b,d and the three bound-aries�� of a sphereVr are compared in Figure 2.3. Comparison of� and is also illustratedin Figures 2.2c,d by removing aside a (thin) wall of facets and a (thick) wall of voxels in each case.Conversion between the two discrete b-reps is possible, and we introduce two operators to do it.

We first observe that a unique boundary�-voxel is obtained directly fromf � �x� y� z� k� � �V �as the voxelv at�x� y� z�. To obtain the�-voxelv� “at the other side”, we recall that face-indexk isthe index�uk � U�; thusv� � v� �uk .

DEFINITION 2.13 (Facet to Boundary-Voxel Operator)Given a facet f�x� y� z� k� we define the facet to boundary-voxel operatoras

�v�� v�� facettovox�f�x� y� z� k�� v�x� y� z� �uk� v�x� y� z� �

facettovox��

Remark 1. As defined, this operator is consistent with the definition off� as the ordered pair�v�� v��.

Remark 2. Since several facets may share a single boundary voxel, it follows thatcard��V� � card� V�. Weconjecture that the case “�” corresponds to an infinite wall of voxels. In our implementation of thefacettovox��operator, redundancy of boundary voxelsv � is eliminated during conversions by labeling in a scene thesev �’s. If avoxel has not been visited, it is un-labeled and we add it to the list of boundary voxels. This process is independantof the voxel order chosen.

For the inverse operation, given a�-voxel v� � V , there is at least one�-voxel in contact withit, determining one facetf . To obtain all facetsfn � v�, withn � �, we identify allfv�g � Vc suchthatkv� � v�k � �, by neighbor examination (up to 6 cases). In other words, the correspondingindicesk of each facef k are thosek � �� such thatv� �uk is a�-voxel.


Z

X

Y

}={VV ={ }

ffacet

voxels

= ( p , )

face voxelf

p p

facets

(b)

(c)

(a)

(d)

vf 5

5

mvfa

fa

f n

fb

c cbf f f

Figure 2.2:Two boundary representations (b-reps) of object V. (a) A point p � �x� y� z� and an oriented facef k, k � � � ��, for example, f �; the boundary element is a facet f . (b) The voxel at p is a different, “3D” kind

of boundary element. (c) Face-based boundary�V, made by facets ffngNfacets

� ; the (thin)“wall” of facetswith face f � has been removed to show how �V is composed of facets. (d) Voxel-based boundary V, madeby voxels fvmgNvox

� ; the (thick) “wall” of voxels corresponding to facets with face f �, has been removed toshow how V is composed of voxels (becoming a “double wall” of facets). Notice the corresponding taggedfacets fa� fb� fc in both representations, with facet fa invisible in (d). Facets fb� fc share the same voxel.

DEFINITION 2.14 (Boundary-Voxel to Facet Operator)Given v� � V and all fv�gn� � Vc, with n � �, such that kv� � v�k � �, the faces f k ��v� v�� k � �� determine the set of m facets ffngn�mn�� v�� f k�� k � �� m � �.Such a procedure is called boundary-voxel to facet operator:

voxtofacet�v�x� y� z�� f fn�x� y� z� k�� some k � �� gn�mn��voxtofacet��

Remark . Facet indexn of fn�x� y� z� k� enumerates all facets of a boundary voxel (at least one, at most six). Itis not related to indexk indicating face orientation (f k).

PROPERTY 2.2 (Boundary Conversions)The following equivalences hold, when applying the conversion operators boundary-voxel byboundary-voxel or facet by facet, with redundancy eliminated:

�V � voxtofacet� V� (2.2a)

V � facettovox��V� (2.2b)


∋

V

V

f(x,y,z,k)

∋ V

V

(a)p=(x,y,z)

∋v(x,y,z)

(b)

(c)

Figure 2.3:Boundary representations of an object V and frontier elements for each b-rep: (a) points p ��x� y� z� (symbol “��”) for continuous boundaries �V; (b) facets (romboids, triangles, and oriented facesf�x� y� z� k� ) for face-based boundaries �V; (c) voxels v�x� y� z� for voxel-based boundaries V.

Remarks. If V is considered as isolated, it may haveN int cavities�

Vr� r � ��Nint . Thus, it has inner, negativeboundaries “�rV�”, and the last relations are unions of all components.

Solid Interior of a Face-Based Boundary Let �V composed by one single component; or take theoutmost boundary�Vjout (see Definition A.9, page 235) such that all boundary components ofV arecontained inside�Ojout . For simplicity we consider the first case (no cavities). Then, the discretesolid interior ofV is the setV itself, as defined operationnally (“traversed”) from its continuousboundary��V (see Notation A.2), when atraversal method exists. InN�, the discrete solid interior is

expressed in terms of its solid boundary and its geometric interior:interior�V� ��

�V � V . We give

now a way to expressV in terms of the face-based b-rep ofV ; that is,how to traverse V using facetinformation. When cavities are present (thus�V is multiply-connected) we then consider�Vjout forthe solid interior ofV , filling out all its cavities.

A traversal method to define all solid interior points ofV , given its boundary�V , is now de-scribed.

Consider runs of voxels between opposite facetsf � �x� y� z� k� of a boundary�V , sharing twocoordinates (e.g., y,z, thus colinear in axisX). We consider only runs starting at a boundary voxelof a given facet, running in the opposite direction of its normal, and ending at the opposite facet;thus the runs lie insideV . These runs may consist of at least one voxel. We could equally take otherpartitions of�V , as explained in page 91, usingF� (Equation 2.1), e.g., all the pairs “�y�� y��” and“�z�� z��”. Sorting of facets into corresponding opposites is done easily during surface tracking, asexplained later, since allx� y� z� k are available. These runs constitute all the interior of�V (hence,of V), including V . We have then:


DEFINITION 2.15 (Discrete Solid Interior of�V)Let �V be composed by a single connected component. Let also

F�� f�xa� y� z� �� xb� y� z� �� V j xa � xbg (2.3)

be a subpartition of �V into opposite facets f�x a� y� z� �� and f�xb� y� z� ��; then the set:

interiorx��V� �� f v�x� y� z� � V j �xa � x � xb� � �xa� y� z� �� xb� y� z� �� F�� g

is called the x-solid interior of �V (or solid interior by x-runs ��xa� y� z�� xb� y� z��).

interiorx

Remark 1. Solid interiors byy- or z-runs of voxels are similarily defined. In the “x�”-notation, opposite facetsin F�� are pairs of type�x�� x�� F�

� . Note that the direction of runs (two, fromx� to x�) are defined tomatch the normal�uk � U� of the ending facet.

Remark 2. When the boundary ofV is constituted by several connected components, i.e.,�V �SN�k �kV, then

each component interior defines a ”fragment” or ”cavity” ofV.

PROPERTY 2.3 (Volume Traversal)The x-, y-, and z-solid interiors of �V and V are equivalent:

interiorx��V� � interiory��V� � interiorz��V� � V

Hereinafter, we drop the subindex, and use by defaultx-runs from facets with orientationf � (i.e.,x�) to facets withf � (i.e., x�). Demonstration of this property is trivial and very similar to theProperty2.1, using either the ordered set of face orientationsff kg�, or the partitionF�, of facetcoordinates, which are equivalent. Cavities have not been considered here, but if�V is composedof several parts (nested boundaries), internal facets can be found to match by pairs with indicesj � k ��k. The way to find such nested boundaries is treated in Section 2.4. Since all voxelsare visited, we have thus defined avolume traversalmethod.

To simplify discussion, we have employed till now a strong voxel-adjacency condition:bound-ary voxels share a face, or equivalently a boundary pair�v�� v�� satisfies: “kv� � v�k � �”, in gridunits. In particular, this condition was required in several definitions about boundary voxels as setsof points (i.e.,v� v� is a facet). It is a special case ofK-adjacency, withK � �, as presented inthe following paragraphs. Several concepts need to consider the type of connectivity and adjacency.

2.3.4 Discrete Adjacency and Connectivity

A set of voxels with the same label (representing objects or background) may form one or moreconnected components. In discrete topology we must define what connectivity means, as in the2D case, because the number of components may change in function of the initial choice of fore-ground and background connectivities. This is equivalent to the mathematical assertion that cardi-nality of a set depends on its topology, and topology changes in function of connectivity relationships[Kovalevsky89, Voss93]. Thus a consistent choice must be made and preserved during analysis. A


+

x

y

- xxfacets

j=1x -

x N

0

x -

u

-( x , y, z, 0) j{ }V =

" wall"

(c)(a)

(b)

1uV

V

Figure 2.4: Facet walls. (a) Facets of �V are classified by their orientation �uk, forming opposed pairs�x�� x��. Object V is fully contained between any of its three sets of opposed facet walls. (b) Traversaldirection from x� to x�. (c) A ”facet wall” is the subset of all facets fj � fj�x� y� z� k�� k � � � � � � � inpartitionF; for example, �Vjx� .

convenient solution is to introduceK-adjacencyandK-connectivity , withK a parameter depend-ing on the geometric properties and the ways that volume elements are organized in a discrete space.The termconnectivity is broadly applied to regions, subsets of objects (for example a discrete path,or a discrete submanifold), or sets of voxels, andadjacencydenotes connectivity in the immedi-ate neighborhood of a voxel, that is, at the minimum discrete distance (grid resolution). Discretedistances are in turn defined for eachK, in function of space tessellation.

Since we adopted the cubic grid for our model, connectivity in 3D is an extension of the 2D case(the square grid): pixels, contours and regions with 4- or 8-connectivity become voxels, boundariesand regions withK-connectivity. Adjacency between voxels depends on neighborhood relations.They may either:

✦ share one face and then possess up to� neighbors (6-adjacency), or

✦ share only an edge, and contact up to 6+12=�� neighbors (18-adjacency), or

✦ share only a vertex and contact in this way up to 6+12+8=�� neighbors (26-adjacency).

Thus,K � f�� g in the cubic grid intersections. With these intuitive notions, we proceedwith the following definitions. Facet and voxel adjacency and connectivity relations are illustrated inFigure 2.5.

DEFINITION 2.16 (K-Adjacency of Voxels)Let va� vb � V be two voxels. We say that

✦ va� vb are �-adjacent iff kva � vbk � �,

✦ va� vb are ��-adjacent iff kva � vbk �p�,

✦ va� vb are ��-adjacent iff kva � vbk �p�.

Remark 1. The latter definitions are equivalent to those given above, since Euclidean distances between faces,edges and vertices satisfy the indicated inequalities. InN

n aK-adjacency is defined bykv a � vbk� � n, withKthe number of induced neighbors in the orthogonal grid and�n K �n � � [Malandain93].


Remark 2. Other connectivities are possible; for example, Gordon et al. defined a “Tu” connectivity where thevoxels are adjacent if they share an edge parallel to any two fixed coordinate axes orthogonal to axisu � fX�Y� Zg[Gordon89]. The justification for such anisotropic connectivity is that some faster implementations avoid a naturalredundancy found in the isotropicK-connectivity (see [Gordon89b, Perroton93]. Several problems arise when thesame object is rotated because the anisotropic version of the surface tracker is not orientation-invariant. While thisdrawback may be irrelevant in some applications (mainly visualization, using strictly cubic voxels), it is criticalfor homotopic shrinking, normal estimation, slice-interpolation and other processing and morphometry operations,specially when dealing with fine structures. We have thus restricted our implementation to the original choice ofisotropic connectivity used by Artzy et al. in [Artzy81].

Remark 3. In some applications, the above definitions are often defined as strict equalities. In such a case,Figures 2.6c, e, and f illustrate minimumK-connected paths between two points, withK � �� , respectively(see also below, Definition 2.19). Cellular complexes generalize these connectivity possibilities, counting edgesand vertexes as candidates for boundary elements, besides voxels and their faces.

Remark 4. Voxel adjacency can be also defined from grid coordinate differences of�� betweenK-adjacentvoxels; for example they differ in one or two coordinates forK � ��, and in exactly one coordinate forK � �.

Adjacency of the facets is defined in terms of their boundary voxels. The number of adjacentsmay vary in terms of voxel configuration.

DEFINITION 2.17 (K-Adjacency of Facets)Let va� vb � V be two K-adjacent boundary voxels. Let f�� f � �va � vb� (at least one pervoxel, at most six per voxel). We say that f�� f are K-adjacent if

✦ for K � �, f� and f share the same �-voxel (va � vb) and one edge (their faceorientations are orthogonal and the associated �-voxels are 6-adjacent),

✦ for K � �, f� and f share an edge (they are edge-adjacent, and they belong to 6-adjacentvoxels),

✦ for K � ��, f� and f share an edge, and belong to 18-adjacent voxels (they share thesame 0-voxel),

✦ for K � ��, f� and f share a vertex (they are vertex-adjacent, and each one belongs to26-adjacent voxels.

Since facets are boundary elements, facet adjacency is also called geodesic- or surface-connectivity.

Remark 1. Adjacency refers to neighboring voxels or facets sharing at least a point. Note that voxel and facetadjacency is asymmetrical, but nottransitive relation. K-connectivity is the condition of pairwiseK-adjacencyin a path bewteen voxels or facets, and turns out to be a symmetrical and transitive relation (thus, it is areflexiverelation [Klette98]).

DEFINITION 2.18 (Connected Facets)Let fa� fb � �V be two facets. We say that fa� fb are K-connectediff

n � N� ffkgnk�� V � such that (2.4)

�f� � fa� �fn � fb� �� k � �� n� �� fk is K-adjacent to fk��

where K � f�� g. The sequence ffkgnk�� which is the shortest among all such paths, iscalled a geodesic K-connected pathbetween fa and fb, and n is the length of the path.


Remark 1. True geodesics imply a notion ofshortest length, which is better defined in terms of a metric definedon ��V. In discrete spaces, the metric depends on the choice ofK-adjacency andK-connectivity, since differentpaths arise between surface elements.

DEFINITION 2.19 (Connected Voxels)Let va� vb be two voxels of object V . We say that va� vb are K-connectediff

m � N� fvkgmk�� V � such that (2.5)

�v� � va� �vm � vb� �� k � �� m� �� vk is K-adjacent to vk��

where K � f�� g. The sequence fvkgmk�� is called a K-connected path(or K-path) be-tween va and vb.

Remark 1. It may happen that for some otherj� k � �� n facetsfj andfk are alsoK-connected (idem forvjandvk). Multiple paths are useful to define ”non-simple” paths.

(b) (c) (d)(a)

0-adjacent facets26-adjacent facets

6-adjacent facets

18-adjacent facets

Figure 2.5:Adjacency relations between facets and between voxels: (a) single voxel, (b) voxel adjacencyby face (6-adjacent voxels), (c) by edge (18-adjacent voxels), (d) by vertices (26-adjacent voxels). Facetadjacencies are also indicated. A -voxel (background) is shown in (c).

Remark 2. Cardinality of a setV � V depends on connectivity and adjacency relationships. For example, inFigure 2.5, the number ofK-connected objects depends onK:

(a) and (b): 1 object for allK � �;(c) 2 objects ifK � �,(c) 1 object ifK � ��,(d) 1 object ifK � ��,(d) 2 objects ifK � ��.

Analogously, Figures 2.6c,e,f have one object forK � ��, three in (c) forK � ��, and three in (e) forK � �.

In summary, an object connectivity is defined by local adjacency of its elements. Several combina-tions of background connectivity (K�) and foreground connectivity (K�) are possible, but to avoidconnectivity paradoxes4, compatible choices are proposed:

4e.g., a 18-connected ’wall’ do not separate a 18-connected background.


(c)(a)

(f)

(b)

(e)(d)

Figure 2.6:Connectivity relations between voxels tagged with a “��”: (a) both, 18- and 26-connected, (b)6-, 18- and 26-connected, (c) 26-connected only, (d-e) both, 18- and and 26-connected, (f) 6-, 18- and 26-connected at the same time. Tagged voxels are K-connected if there is a path of pairwise K-adjacent voxels

foregroundK�-connectivity background K�-connectivity

18 66 18

26 66 26

18 2626 18

The combination “�� ”, even if consistent, is not very useful, due to loss of information and be-cause it does not allow to define surfaces satisfying the Jordan property (simple and closed, partition-ing the space into interior and exterior). Combinations such as�� or �� are not useful,too, causing holes and tunnels where separability should be expected. We denote the choice of aspecific ordered pair as�K�� K��-connectivity . If the opposite is also used, then we denote both asfK�� K�g-connectivities (idem for adjacency, in the immediate neighborhood).

Toriwaki et al. also introduced a conditioned�� -connectivity [Klette98]. Serra, Morgenthalerproposed choices of�K�� K�� f�� g, which are compatible with mathematical the-orems on discrete Jordan paths in the orthogonal grid [Serra82, Morgenthaler81], while Kong andRosenfeld further applied an algebraic topological approach to homotopic transformations inN

� andproposed to usef�� g-connectivities [Kong89]. ForZn, Malandain provided a formal characteri-zation for a choice of discrete connectivities in voxel surfaces (or solid boundary components)KVand cell complexes [Malandain93].

Face-based boundary representation is better suited to edge-sharing adjacency, with��- or �-connectivity for voxels, which is also a choice free of topological paradoxes. Since cavities are back-ground connected components, we prefered to treat them as similar as possible to foreground objects.We thus excluded vertex sharing as a connecting property and worked with choices�K�� K�� f�� g for boundary extraction and representation using mainly the pair�� . Thin,


filamentary structures (blood vessels) and medial axis transforms (skeletons) require a differenttreatment (thin fragments), where connectivity should be restored and topology preserved duringthinning. In that case a choice�K�� K�� enhances particle connectivity under “strict”segmentation. We will come back to these issues in Section 2.4.4 and in Chapter 4.

Another restriction arises ifK� � �, since facets can have different adjacents if 18-connectivityis choosen for�V . This is illustrated in Figure 2.6a, where two points are not 6-connected, butthey are 18-connected. When facet connectivity is analyzed, Rosenfeld and Kong distinguish 18-connectivity in Figure 2.6a from the “stronger” 18-connectivity of Figure 2.6b, where a 6-connectedpath of voxels exists between the tagged points [Rosenfeld91]. These voxels have at least one facetin the 18-connected boundary.

In the rest of this chapter, unless otherwise stated, we use�K�� K�� connectivities forfacets and voxels, and recall it explicitly (K) where connectivity for other entities (neighborhoods)are also used, or a particular property is only valid for that connectivity. Face-based boundaries�Vare always defined as��-connected (voxels sharing only a vertex are not connected).

We are now able to count all connected components of a scene.

DEFINITION 2.20 (A Scene as a Set of Objects)Let fVigNobj

i�� be a set of K-connected 1-objects inV. If��V �

Nobj�i��

Vi�A � Vi Vj � ∅ � � i� j � �� Nobj�� i �� j�

�then we say that scene V is described by its Nobj disjoint, K-connected components Vi, withi � �� Nobj.

Remark 1. In this definition the properties that scene components share are: ”being disjoint, 1-objects”, and”K-connectivity”. Other kind of decompositions and descriptions of a scene are possible, in function of the specifi-cation of the setsVi, which depend on re-labeling operations on a binary scene, that is, the attributes of two voxelsin order to beK-connected. Since different labels arise from different data processing (filtering, dilation, selection,sort-by-size,...), a sequence of scene decompositions can be obtained, in which the connected components change,and re-group into new sets.

We introduce in the Section 2.4 the means to extract the connected componentsVi of a (binary)sceneV, detecting their boundaries�Vi, or Vi by using a facet-based or a voxel-based representa-tion.

To quantify morphometric properties by different b-reps, we also need to distinguish the volumeand surface measures that arise from each b-rep asset cardinalities, and then find its measure inworld coordinates (usually one or more scaling factors).

DEFINITION 2.21 (Voxel Cardinality (Discrete Volume))Let V be a K-connected object in a scene V. Then, V � fvkgNVk�� is a set of NV K-connected

voxels, that is, the voxel cardinality, or discrete volumeof V is the integer: card�V� �� NV .

Remark 1. Recall thatNvox denotes the number of boundary voxels, i.e., the cardinality ofV, whileN is the

number of boundary connected components (cavities and nested boundaries) ofV, i.e., the cardinality of the setf jVg� j � �� N . Similarily for the number of boundary componentsN� .


DEFINITION 2.22 (Facet Cardinality (Cardinality Surface Area))Let V as above, withK � ��. Then, �V � ffkgNfacets

k�� is a set ofNfacets ��-connected facets, that

is, the facet cardinality, or cardinality surface areaof V is the integer: card��V� �� Nfacets .

Remarks. The discrete surface areaSurf ��V� in a cubic grid and scaled to world coordinates, is:Surf ��V��

Nfacets � L�, with L � Rbeing the side of the voxel faces (grid step). For the physical volume:Vol�V��

NV � L�. For anisotropic voxels, see Subsection 2.5.1. A solid-surface area is defined asSurf vox�V�� Nvox

in grid units. Since�N�� Nfacets �Nvox depends on the shape of an object, it could be useful to quantify in a

cubic grid the compactness of an object. We rather used discrete shape factors, as presented in Subsection 1.2.2.

Other measurements and analysis require the concept of discrete neihborhoods, which are intro-duced in the next subsections for voxels, facets and boundaries.

2.3.5 Voxel (Euclidean) Neighborhoods

We saw in Chapter 1 that local information is needed to define or measure specific shape features ofany object. Several problems in 3D require to traverse, analyze or modify not only pointsp � V , orp � �V , but also itsneighborhood. In discrete space the smallest neighborhoods are theK-adjacentvoxels (or facets), and a neighborhood can be restricted to points inV , �V , V , or other sets. Theycan also be generalized in several other ways: in theXY -plane, along the surface, along the localnormal, or other direction. Discrete neighborhoods are related to important analysis methods, suchas Mathematical Morphology (MM) in classical 2D- and 3D-image processing.

One of the principles of boundary-based processing, is the making of decisions about operationsor parameter adjustement at the instant of visiting each facet of a list (usually a list per object). Thesedecisions can also been made at a higher level, i.e., per whole objects, since the traversal of a list ofselected objects comprises the traversal of the list of their facets�V � ffn� n � �� Nfacetsg.While scanning the boundary or the interior of an object, analysis of neighborhood information (e.g.configuration, mean orientation, labels), for each facet or voxel, allows:

✦ to perform filtering,

✦ to modify a voxel label,

✦ to tag specific facets,

✦ to alliviate discretization problems in visualization (e.g., a local average allows to estimatesurface normals).

✦ to create special lists, or

✦ to accumulate a count or sum for a measurement, in function of facet or/and neighborhoodcharacteristics, which may determine the weight of the contribution.

Eventually, a whole object is modified, re-labeled or selected as result of a boundary-based anal-ysis. All such possibilities and boundary-conditioning of 3D processing or analysis can be calledgenericallyboundary-driven, filtering or analysis, belonging to the more general concept ofshape-, morphology-, feature- or geometry-driven filtering, analysis, etc. Related examples are given in[TerHaarRomeny94]. We present in these subsections Euclidean and geodesic neighborhoods forvolume and boundary elements and boundaries themselves.


From the adjacency relation between voxels we obtain sets of natural nearest and near neigh-bors that form the basis for MM operations as erosion (�) and dilation (�), as well as for localmeasurements.

DEFINITION 2.23 (First-Order VoxelK-Neighborhoods)Let v � V be a voxel. If v� � V , such that v� is K-adjacent to v, K � f�� g, then, the set

NK�v�� fvj j �vj is K-adjacent to v�� j � �� K g

is called the immediateK-neighborhood, or first-order K-neighborhoodof v.

In some applications the central voxel is not included in the neighborhood definition, and denotebyN �

K�v� the setNK�v� � fvg. We useEK � NK�v� to denote neighborhoodsNK�v� used as EK

structuring elements for MM filtering operations. Figure 2.7 shows the possible neighborhoods in2D and 3D forK � �� andK � �� in 3D.

Figure 2.7:K-adjacent voxel neighborhoods, with (a) K � �� in 2D, and (b) K � � �� in 3D (thecentral voxel is included in EK , but not in N �

K�v� � EK � fvg. These neighborhoods constitute alsostructuring elementsfor Mathematical Morphology operations.

Remark 1. A voxel neighborhood of a�-voxel may or may not contain�-voxels. We usually are interested inneighborhood voxels sharing a particular label. However, see in Section 5.7.3 a four-label example in the study ofatomic species interpenetration in amorphous-deposition simulations.

Remark 2. In MM operationsE�� is acubic structuring element,E� is anoctahedralstructuring element andE� is acube-octahedronstructuring element. These sets are used in literature for approximations of Euclideandistance transformations, and region growing for constructing Voronoi regions and zones of influence [Klette98].

Remark 3. In the terminology of Crystallography, solid and surface Physics, voxelsmay correspond tosites,the orthogonal gridV� to a lattice, all v � N�

� �va� are named thenearest neighborsof v a, while all v ��N �

� �va� � N �� va�� are thenext-nearest neighborsandv � �N �

��va��N�� va��N

�� va�� are thenext-

to-next nearest neighbors.

Higher-order neighborhoods of a voxel comprise discrete ballsBr�M�v� of radiusr and someparameterM which is related to the shape of the ball, since discretization allows for different ap-proximations. SometimesM relates to structural elementsEK to dilate a voxel into a “sphere”,alternating different values ofK. Let q � �x� y� z� � R�. In the discrete approximation of an Eu-clidean sphere, voxelsv are computed at integer samples�int�x�� int�y�� int�z�� to obtainBr�q� as


an approximation of the continuous ballB r�q�. Then the difference (under some metrics) betweenBr�q� and samples5 of the continuous boundary��B r�q� is minimized, but depends on positionsq,

which are clipped or rounded to discrete grid points.

In our applicationswe used the following discrete approximation of an Euclidean sphere of radiusr � R in a cubic gridV�:

Br�q� � fp � V� j round � kp� qk � � rgwith

round �x� �

�int�x�� if �x� int�x��

int�x� �� if �x� int�x��

Sets of discrete-ball neighborhoodswere implemented as precalculated offsets from a given centerq.6

These balls compriseBr�q�� B�r�q�, with r � f�� Nrg (for r � �: B��q�� q , B��q� �

� ∅ ).The maximum radiusNr depends on applications; we used up toNr � �� (see Chapter 3). We alsoadded means to read or write onBr�q� for update operations insliding windows (or rolling ballbuffers), as described in Section 3.3.3, page 156, and for the design of a discretetubular-structuredetector, as described in Section B, page 242. Other applications (e.g. local measurements andfiltering) are described later.

Facet Euclidean Neighborhoods of a fp � �V , with the associated voxelv atp � �x� y� z� couldbe defined as sets:

EK�fp� � ffq � ��V NK�p��g� (2.6)

with K � f�� g. In general, in a r-ball region centered atfp (recall facet centers are shifted to��uk �� ):

Er�fp� � ffq � ��V Br�fp��g (2.7)

Since boundary faces are placed on a surface, it makes more sense to define their neighborhoodalong�V , in the following Subsection. See later comments onsurface density, in paragraph 2.3.7,page 108.

2.3.6 Facet (Geodesic) Neighborhoods

Besides Euclidean neighborhoods, which do not depend on the shape of an object, we now con-sidergeodesicneighborhoods, which are defined along (or tangential to) the surface boundary of aconnected set.7�8

There are three ways to generate geodesic neighborhoods:

✦ By defining a discrete geodesic metricdgeo�p � q� and then to estimate surface curvatures toestablish a local reference system. Then:

N��r� p�� fq � �V j dgeo�p� q� � r g

5A mapping such asz �pr� � �x� � y�� givesz �Rfrom grid samplesx�y � N.

6recall that voxel centers are atq � �� , which adds a constant offset to allp � Br�q � �� 7Geodesics in general can also be defined with respect to any set.8Strictly speaking, surface “geodesics” are only defined in continuous surfaces with a local reference system based on

principal curvatures, a metric measure, and a minimal-path condition under this metric.


✦ As in Definition 1.8, by using the approximation (similarly forV):

N��r� p� � Br�p� �V �✦ By using a MM approach withgeodesic K-adjacent boundary elements (facets or boundary

voxels) and dilations (iterated adjacents of first-order adjacents).

The first approach is very accurate but difficult to implement and many applications require smallvalues ofr (up tor � � in geodesic neighborhoods). The other two are very similar and well adaptedfor boundary-based operations, thus we made an implementation using facets in a very simplifiedway, as explained in the remaining paragraphs (the voxel version was not tested, but it is presentedin the next subsection).

Since a facet has four edges and a face-based boundary�V was defined as a closed surface,any f � �V has always one facet adjacent per edge. They may belong to the same or to anotherboundary voxelsv � V . Connectivity relations allow several other neighbors. Discretization limitsorientations for facesff kg� to only three orthogonal directions (i.e., the discrete normals�uk � U�).Some local averaging is required to approximate local Euclidean normal vectors of�V , and otherproperties of discrete surfaces (e.g., curvature). The following property restricts what facet neighborsare possible and allows to organize the neighborhood, in function of facet orientation�u k:

DEFINITION 2.24 (First-Order Neighbors of Facets (or Geodesic�-adjacents))Let f � �v� k� � �V , then its four ��-adjacent facets ffmg�� satisfy :

� m � �� vm � V � i � �� such that:

fm � �vm� i� �i �� m ��m��

ffmg�� are called geodesic�-adjacent neighborsof f .

Remark 1. Some of thevm may refer to the same voxel.

Remark 2. Strictly speaking, “geodesic�-adjacent” (or Euclidean��-adjacent) facets may or may not exist(consider a single voxel, for example). There is a way to define them in terms of geodesic -adjacents of the facetsffmg�� themselves (excepting the centralf ). It may then happen that some of the geodesic�-adjacents are identicalto somefm (see Figure 2.8c). To have a way to visualize this and other local configurations of facets, we introducea “cut-and-fold paper” representation of boundary patches, illustrated in the same figure.

Remark 3. For each facet edge ( ), there are� possible facet neighbors (imagine three positions of a door,m � ��m � �� ), thus, a total of�� possible facet neighbors. Notice that the conditioni �� m � ��m

warranties that in voxel�- and��-adjacency, any facet has only one��-adjacent neighbor when considering theexample in Figure 2.5c (see also remarks of Property 2.1, page 90).

Orientability along�V is also needed, in order to introduce the traversal of�V . We also completethe full definition of geodesic neighborhoods after definition oflateral directed circuits along�V , inSubsection 2.4.2, which allow to select pairs of facet adjacents as “incoming” and “outgoing” facetsto traverse�V . We introduce however a preliminar definition, in terms of a partition of the fourgeodesic neighbor facets into:

✦ 2-adjacents: A choice oftwo �-adjacent facetsffleft� frightg at orthogonal edges off (theymay be at the same voxel; see Figure 2.8a,d).

✦ co-adjacents:The remaining two facets,ffright-co� fleft-cog.


(c)

(d)

(b)(a) right adj

right co-adj

left adj left co-adjF

F

F

order of faceneighborhoods:

1th

3th

2nd

F

> 3th

0

2

1

0 1

2

3

4

5

6

Figure 2.8:Geodesic facet neighborhoods in a “cut-and-fold paper” representation. (a) First-order neigh-bors, 2-adjacent and co-adjacent facets, relative to a starting voxel-face F . Numbers indicate the explorationorder for the adjacency subset. (b) Facets 3 to 6 are second-order neighbors (only 2-adjacents and secondadjacents are numbered). (c) Third-order facets are included (“semi-adjacent”) in a higher-order neigh-borhood. A local-facet configuration example in 3D is shown (d), where color indicates face-neighborhoodorder. Dashed facets in (c) are not necessarily “the same” facet in 3D, as shown in the example (d). See alsoFigure 2.9

The terms “left” and “right” are just to distinguish locally each facet9. There are up to four possiblepartitions of the four neighbors to obtain 2-adjacents. The choice is latter done by introducing alocal reference system trihedron with normalsU� and lateral circuits with anin-edgeandout-edgedirection.

Remark 1. GeodesicK-connected paths and geodesicK-connectivity could be also defined with -adjacencyand 2-adjacents, in order to define and manipulate connected subsets (geodesic patches) along a boundary�V.

Thus, hereinafter the first-order neighbors are either 2-adjacent facets, co-adjacent facets, or all 4-adjacent (see Figure 2.8).

2.3.7 Voxel Geodesic Neighborhoods

For completeness, we include in this subsection some definitions of geodesic neighborhoods onV .We did not use them in the present work, since facet neighborhoods were employed as described inthe last subsection, and other local calculations used a MM approximation, by erosions and dilations

9This notation used by various authors is moderately misleading when considering local displacements: a sequence ofchoices alternating ‘left’ and ‘right’ facets on a plane turns out to be a straight path (not a ‘zig-zag’). In the other hand, asequence of ‘lefts’ tends to be (correctly) a closed circuit


(voxel Euclidean neighborhoods) following voxels of the facets of a discrete boundary�V , or itsequivalent V , using the face-to-voxel operator.

The next definitions are a special case of those introduced by Bertrand in [Bertrand94b] as dis-crete geodesic neighborhoods related to simple point determination (those points whose removaldoes not change object discrete topology), and to find topological numbers, which characterize fur-ther topological properties of a cubic gridV� � N�. Bloch et al. made a fast implementation whichstill uses a volume approach to obtain a boundary representation [Bloch90, Bloch90b]. Recall thatneighborhoodsN �

K�v� andN �K�v�� v � EK define also a structuring elements for MM operations

atv. In the following we are interested in voxels on the solid boundaryV , as a subset ofV .

DEFINITION 2.25 (n-Order Voxel-Based Geodesic Neighborhood)Let V � V�, and v � V�. The n-order geodesic neighborhoodof v inside V is definedrecursively as: �

N �n�v�V� �

� N �n �v� V �

N kn �v�V� �

�Sw�Nk��

n �v�V� fN �n�w� N �

��v� V g�In the context of the other definitions,n � f�� g. Bloch as well as Bertrand express thisconstruction as a geodesic propagation insideN �

��v� V (we are restricting their definition toV).It becomes more intuitive when considering only the subsetV ; then the propagation is along thissurface [Bloch90]. In general, the new setN k

n �v�V�may be expressed as a conditional MM dilation

(d) (e)

n

(c)

(b)(a)

(f)

2

3

4

5

6

1

2

1

4

3

adj

co-adj

6

7

8

10

11

5

12

9

Figure 2.9:Access to geodesic neighboring facets is achieved by adjacency (a) and co-adjacency (d) rela-tionships, which assign a search order in first and second-order neigborhoods (b,e). An assignement is doneby LUT tables, following specific circuits. An estimation of the local normal vector is available from weightedaveraging of the normals of the neighboring facets (f). See also Figure 2.8, and Figure 2.12.


(�) ofN k��n �v�V�, inside (restricted or conditioned to)N �

��v� V , as:

N kn �v�V� �

��N k��n �v�V� � N �

n�w�� N �

��v� V � �whereN �

n �w� is the structuring element (excluding the centerw).

The suitable geodesic neighborhoods for different connectivity pairs allow to minimize the num-ber of possible candidates to simple points and compute several topological features. Since neihbor-hoods so constructed are represented by graphs, they also allow to use standard algorithms for search-ing connected components in graphs [Bertrand94b] (such as our boundary-traversal methods). Forthe connectivity pairs�� K�� K� and�� , with K � f�� g, several require-ments indicate thatN �

� �v�V��N ��v�V��N �

��v�V� andN �� v�V� are the optimal or compatible

choices of geodesic (voxel) neighborhoods, for each connectivity pair.

Geodesic distance fields may be obtained by geodesic propagation, as described by Bloch [Bloch90].Generalized geodesic distances on gray scale images are described in [Soille94].

Other Neighborhoods of Boundary Voxels. As with facet “Euclidean” neighborhoods (Para-graph 2.3.5, page 104), boundary voxelsv�p� � V , p � �x� y� z� have Euclidean neighborhoodsdefined as sets:

EK �p� � fv�q� � � V NK�p��g� (2.8)

whereK � f�� g. In general we can user-balls and define:

E r �p� � fv�q� � � V Br�p��g� (2.9)

which constitute small ROIs wherediscrete surface density of V at v�p� can be studied to extractirregularity measures or3D-roughness features in a neighborhoodE r �p�. We describe an applicationof discrete surface densities in Section 5.6.

2.3.8 Discrete Boundary Neighborhoods

Geodesic neighbors of facets or boundary voxels were defined along the surface; i.e. intangentialdirections. We consider now neighbor elements in thenormal directions of�V , that is, “inwards”and “outwards” ofV . Since no restriction is applied in the tangential directions, the whole boundaryis considered.

Once the face-based�V is extracted, further information of the shape ofV can be obtained fromboundary characteristics. Boundary�V , as a set of facetsffjgNfacets

j�� , can be specified even withoutknowingV itself, for example as a transformation on the boundary of another object. In order toperform some analysis and processing operations on�V �V , or a small neighborhood of them, wefound convenient to define the following elements in the first-order neighborhood of a facet, in orderto consider internal and external layers of facets and voxels adjacent to�V . Figure 2.10 illustratesome of these elements.


DEFINITION 2.26 (First- and Second-Order Neighbors of Boundary Voxels and Facets)Let U � f�ukgk�� be the set of unit vectors (Definition 2.2). Let f � f�x� y� z� k� � �V be a

facet, with �v�� v��, the corresponding boundary pair. Consider the neighborhood N �� v�� of

�-adjacent voxels to v�. Let us partitionN �� v�� into: N �

� �v�� Vc andN �� v�� V , which are

external and internal to V . Consider alsoN �� v�� Vc. Then10:

f�� f��x� y� z� �uk� k� is the front-facet11of f�x� y� z� k�

f�� f��x� y� z�� uk� k� is the back-facetof f

f�� f��x� y� z�� uk� k� is the next back-facetof f

v�� v��x� y� z� � v��x� y� z� �uk� is the main front-voxel of v� at f

v�� v��x� y� z�� uk� is the main back-voxelof v� at f

fv�j gj�j��

�� N �

� �v�� Vc are the front-voxels of v�

fv�j gj�j��

�� N �

� �v�� V are the back-voxelsof v�

fv��j gj�

j�� N �

� �v�� Vc are the next front-voxelsof v� at f .

Remark 1. The voxelsfv�j g� � � j � � in front of all boundary faces ofv� are also defined as the set:fv��x� y� z� � �uj�, with j �� kg.

Remark 2. No front-voxel and no back-voxel is a boundary voxel (v�j �� V� j �).

Remark 3. Consider facets ofv�. Then, the next back-facetf�� of v� atf is also definedby:f��x� y� z� k� �opface�f��x�y� z��uk� i�, with i � k��k; i.e., these facets have opposite orientations (Definition 2.9). Withthis alternative definition, only the first-order voxel neighborv

� is required ((see Figure 2.10b)).

Remark 4. By a similar argument, the back-facet ofv� at f�x� y� z� k�, that is,f� has the opposite orientationthatf��x�y� z�� i� � v�, with i � k � ��k. Thus:f� � opface

�f��x� y� z�� k � ��k�

�.

Only the last set constitutes a second-order neighborhood. In practice, all these elements are obtainedfrom offset coordinates�x� y� z� � �x�� y�� z��, with �� x�� y�� z� � � and look-up tables,usually in function of thek index of each facet. The implicit restriction toK � � comes fromthe available information from a face-based boundary�V , from which the front facet and the backfacet are directly accessible, while other facets require to know the local voxel configuration. Similararguments apply for the front, back andK-adjacent voxels, and an application constitutinga personalconstribution is afacet-driven implementation ofregion growing segmentation, using neighboringvoxels and facets, as described in Figure 4.23, Chapter 4. We now proceed to define several boundarysets obtained from the above neighboring elements:

10Recall that given only a boundary voxelv�, several facets may exist, as well as adjacent�-voxelsv �.11We used notationf�� f� to denote the choice of face orientation��uk. Heref� indicates the oriented face positioned

at offsets��uk. Recall that face centers are located at�x� y� z� � ��uk � �� .


f

f

(x,y,z)

f

f

Y

Z

X

5^

5u +v

0+v-

+v4u

0

v-

4

u

0

0

fv

v1

0

background (label )

foreground (label )0

1

(a)

v+

^

--

u

+

v

v

1

1(b)

(c)

(d)

Figure 2.10:Neighbors of boundary voxels and facets (example). (a) Voxel v� � V at �x� y� z�, boundarypair �v� v�� and unit vector �u � U�. (b) Facet f � f�x� y� z� � � �V, its front-facet f�, its back-facetf�, and its next back-facetf�� (the four facets share the same orientation �u�. (c) The main front-voxelv��x� y� z��u� of v� at f and the main back-voxelv��x� y� z��u� of v� at f . (d) Three possible front-voxels v� �v

�� v

�� of v� (selection depends on the boundary neighborhoodN �

� �v�� Vc). In this example, ifv�� v

�� V, then back-voxels v�� v

�� do not exist inside V.

DEFINITION 2.27 (Discrete Boundary Neighborhoods)Let V � N� be a discrete object formed by a set of (��-connected) voxels fvi�x� y� z�gNVi�� and

�V its ��-connected face-based boundary ffjgNfacetsj�� , with NV � Nfacets � N the number of

voxels and facets, respectively. We define the following union of neighboring elements, for allfj�x� y� z� k�� V , assuming a ��-connected foreground, and �-connected background:

�V �� ffjgNfacets

j�� The face-based boundary.

��V �� ff�j g

Nfacets +j�� The front-face boundary or face-exoboundary.

��V �� ff�j g

Nfacets -j�� The back-face boundaryor face-endoboundary.

V �� fv��jgNvox

j�� The voxel-based boundaryor solid boundary.

�V �� fv�j gNvox +

j�� The front-voxel boundary or front solid boundary .

�V �� fv�j gNvox -

j�� The back-voxel boundaryor back solid boundary.

��V �� fv��

j gNvox ++j�� The next-front-voxel boundary.

��V �� V � �V � ��V � The next-front-face boundary.

where Nfacets � Nfacets +� Nfacets -� Nfacets ++� Nvox � Nvox +� Nvox -� Nvox ++ are the number ofneighbor-boundary elements of each set (facets or voxels), and depend on the shape and connec-tivity of the original object V . We further denote by � �V and �V all the above face-based andvoxel-based boundaries, and in particular we denote the front- or back- boundaries by � � .


Remark 1. As defined, these neighborhood sets have not necessarily nesting relations with�V or V (childor parent), and in general do not preserve topology (e.g., number of connected components). Verification of theJordan property allows to deal with such cases.

Remark 2. Givenf � �V as a pair�v��v��, Udupa and Herman called [Udupa92, Herman98]:

✦ SurfaceS of V �Zn, the set of all pairs�v��v�� isomorphic inN� to �V,

✦ Reverse-oriented surface�S, the set of all pairs�v��v�� isomorphic inN� to �V� � opface��V �,

✦ Immediate interior II�S� all boundary voxelsfv�g equal to V inN�,

✦ Immediate exterior IE�S� all boundary voxelsfv�g equal to �V inN�.

✦ Immediate neighborhood IN�S�� II�S� � IE�S� equal to V � �V inN�.

Their generalization toZn includes two connectivities�� satisfyingdiscrete Jordan-surface properties whichwe do not consider here. Multidimensional surfaces are specified as��-boundaries. We adopted a more specificnomenclature, since we deal with various other boundaries, interiors and cavities.

Remark 3. The notation for the back-face boundary��V is not to be confounded withnegative boundary� V�

whose facets are reverse-oriented.

f

f

f

f

��

��

��

��

��

��

v

+

v1

VVc

1v

=

=

v

+

-

-

--

+ -

++ + -

(a)

(b)

(c)

V

Figure 2.11:Discrete boundary neighborhoods of an object V (2D view). (a) facet-based boundaries separat-ing neighboring voxel layers around the standard surface boundary �V; the faces in front of each face of �Vconstitute the front-face boundary ��V; the faces back to each face of �V constitute the back-face bound-ary ��V. (b) voxel-based boundaries: V corresponds to voxels with faces in �V; front-voxel boundary�V corresponds to ��V and back-voxel boundary �V corresponds to ��V; an additionalvoxel layer can

be obtained from ��V, the next-front-voxel boundary ��V, without knowing the next-front-face bound-ary ��V. (c) Corresponding neighbors of boundary voxels and facets (3D view). All coordinates and facesare directly obtained from �V, using look-up tables for a given K-connectivity (K � �, in the 2D figure, andK � �� in our implementation).

It is clear that we could also use MM operators directly for some definitions, since examiningN �K�v� � EK at eachv � V defines an MM operation onV , excluding the voxel boundary. From


facet and voxel boundary equivalences we have for example:

��V � ��V � E�� (2.10)

��V � ��V � E�� (2.11)

The difference in our approach is the method to obtain��V � �V directly from�V itself, thus we canreverse the definition and effectuate voxel MM operations by adding�V to V (“wrapping” a layerof voxels), or subtractingV from V (“peeling” a layer of voxels).

A number of relationships and properties arise from the above definitions; for example:

✦ Definition of facetsf as pairs of boundaries has a global equivalent, but orientation informationis lost: �V � � V �V� � f orientation off g. This information can be expressed as abinary decision of assigning indexj or k in the following set: f f�v� f j ork � � � V �V�� j � k ��k� k � �� g.

✦ Except for voxel labels, in�� -connectivity we have ��V � �Vc, which may consist of

several connected components ifV has cavities. It is also the nearest global neighborhood ofan object and we explore it to detect contact with other objects, to extract interface boundaries,label specific voxels and make morphological reconstruction.

✦ Chamfer distance fields (internal “�” or external “”) may be obtained from proper sequencesof sets �

KiWj , whereWj

�� interior� �

KjV� , and the internal chamfer field from sequences

�Ki�V n Kj

�, withKi� Kj � f�� g being proper sequences of connectivities that ensurethe best chamfer approximation to Euclidean distance.

✦ For a 6-connected objectW (Definition 2.28):

�W � E�� W n ��W “peeling ��W”

�W � E�� W � ��W “wrapping �

��W”.

Several other relationships and operations may be derived, such asmorphological reconstruc-tion for which we provide an example in Section 2.5.2, page 130.

✦ Fat-boundary neighborhoods are trivially defined; for example:

G

V �� V � V��

G

II V �� V � �V�� or

G

III V �� V � V � �V��

where G

�V equals theimmediate neighborhood IN��V� of Udupa and Herman [Udupa92],the set of all boundary voxelsf v�� v�g. These fat-boundary neighborhoods may not haveuseful properties from the point of view of discrete geometry (they may present holes and cav-ities), but provide a dense sampling of space around�V to measure local, statistical properties.Also, their comparative volume and surface area may give estimations of local roughness.

✦ Irregularity shape properties at scale resolution can be studied globally from analyzing “resid-ual sets”, such as�V V .

In order to better explain different ways to exploit these boundary-related sets, we present in thefollowing section the means to extract and traverse face-based boundaries.

2.4 Surface Tracking 113

2.4 Surface Tracking

We present in this section the stage ofboundary extraction or surface tracking, in order to detectconnected componentsV in a sceneV, and our implementation of the method proposed by Artzy etal. in [Artzy81].

Implicit Surface Extraction

The first surface-rendering methods for arbitrarily complex objects attracted much attention, as theydramatically enhanced 3D visualization of any data volume, but without further representation oranalysis. A popular method is the one proposed by Lorensen and Cline, themarching cubes, whichis based on iso-surface thresholding and reconstruction, giving as output a list of triangles at thevertices of a cubic mesh [Lorensen87]. In several ways, themarching cubes surface reconstructionis like 2D thresholding with binarization, without contour extraction. It is thus animplicit methodfor extraction and representation of object boundaries. Bloomenthal recently developed an advancedversion of the marching cubes, called a “polygonizer”; but it is still focused to surface and volumerendering [Bloomenthal94].

The main criticism tomarching cubes and similar algorithms for surface extraction is that it doesnot provide directly an explicit representation of each particular object in a scene, and processing andanalysis require further transformation of data into structured meshes. The output of the marchingcubes is a raw list of triangles (or codes to triangle configurations), which does not isolate each indi-vidual connected component; surfaces must be tracked from the list of triangles all over the volumein order to separate and process individual objects. Another solution for single object manipulationis to segment and label the scene beforehand, either by a human or by computer methods to detectconnected-components. The output is a volume where voxels bear information about the objects orclasses they are believed to belong to. Once a scene is labeled for object processing, the whole vol-ume has to be either traversed several times (once per label, with marching cubes), or bear lengthycodes per voxel (e.g., labels plus pointers to mesh configurations and surface normals) and case-by-case examination has to be employed. It turns out to be efficient for preliminar visualizations (asan interactive 3D-thresholding), or when one single object is easily localized, but information forprocessing and morphometric operations has to be obtained apart.

Explicit Surface Extraction

There are methods toexplicitly extract a 3D b-rep of each connected component in a discrete scene.These methods are able to perform surface detection and tracking at the same time, and may workon gray-level scenes, without previous segmentation (however, literature mainly reports binary im-plementations of such extraction algorithms). In this way, the b-rep of the scene is accessed andprocessed at a higher level, object by object. If morphometric characteristics are available (e.g. ob-ject shape), selection criteria allow to process only relevant features or apply for examplegeometrydriven techniques for filtering [Romeny94].

The most representative of the explicit surface extraction methods is thesurface tracking algo-rithm from Artzy et al., based ondirected-graph models of the surface of an object [Artzy81]. Wereview it together with our implementation in the next subsection. Bryant and Bryant report a simi-lar method for n-dimensions, but their description is very poor [Bryant90]. Shu and Krueger reporta faster implementation, which turns to be much more complex and oriented to CAD-like objectswith large planar surfaces [Shu91]. Other authors have proposed alternative discrete b-reps with


similar characteristics which are suitable for boundary-based processing, MM, and morphometrics.We just mentioncellular complexes [Elter92], which is well adapted for dealing with local-topologyproblems in modeling and homotopic processing, since the data structure specification of a cellularcomplex includes the incidence and adjacency relations among its elements [Rosi96]. This additionalinformation introduces the burden of managing complex and lengthy data structures for even simplediscrete sets, since not only faces and voxels are used as independent components of boundary ele-ments, but also edges an vertices. The approach is however well adapted for physical models wherethe cellular complex encodes model attributes that strongly depend on topology.

Since surface tracking methods proceed in a sequential progression to traverse a graph, a one-dimensional list is obtained for the 3D-surface of the object; aK-connected path of facets (or voxels,depending of the surface tracking method) which traverses the whole boundary. No preferentialorder exists and there are several possibilities obtaining a list, depending on the starting point, andthe particular set of surface tracking directions12. Once found, the list provides the means to traversethe surface for purposes other than its initial detection, such as:

✦ local filtering (smoothing), either of the original gray-level information, or other local infor-mation, such as normal vector calculations,

✦ collecting neighborhood information,

✦ labeling, either the surface boundary or the whole solid, and

✦ visualization with a Z-buffer or other object-rendering techniques for pipe-line processing.

In previous sections we introduced our facet-based model for b-reps; we now explain our implemen-tation of the surface tracking method of Artzy et al. in the following paragraphs.

2.4.1 Surface Tracking Algorithm

Effective methods for detection and surface tracking of�V were first developed by the it Medi-cal Image Processing Group (MIPG) of the Pennsylvania University [Artzy81, Herman83b], andlater justified and enhanced by discrete-topology advances [Herman83, Kong92, Udupa94]. Severalvariations and faster implementations have been reported in the bibliography [Gordon89, Udupa90,UdupaOdhner91, Shu91, Perroton93, Perroton95].

In a facet-based model the extraction of�V relies on a depth-first or breadth-first explorationof the graph model of the surface. The facet-based approach allowed Artzy et al. a significantimprovement, since their method retains only the nodes (facets) that are reached at most twice. Thetraversal properties of the graph representation are described in the next section.

We first describe the steps of the surface tracking algorithm, as implemented for 18-connectivity.Candidates of adjacent (or co-adjacent) facets are examinated in a specific order similar to contourtracking. It is illustrated in Figure 2.12 for the current detected facet in�V .

All facets to be explored are added to a queueQ. Those which have been already put inQ butwhich could be reachedonce more by another path are put into a second listP , since each facet hastwo incoming edges and two outgoing edges. If a new facet is encountered, it is searched inP , if itis not there, it is added to the queueQ and output to the list of boundary facets (the “else” in step (3)of the following table).

12We estimate per face-based boundary no more thanN facets � �� Nfacets different paths, having 8 possibles circuitsper facet, and 6 face orientations


The header of the algorithm, summarizes the relevant data structures, where the volume arraydesign forV is described in Section 2.6.2, page 2.6.2:

INPUT: Binary sceneV � fvoxels�z��y��x�g; optionally, a facetf� � �V , or a starting point neara boundary component. Other parameters:�K�� K��-connectivities, and thresholding method.

OUTPUT: List �V of facetsf whose voxels are 18-connected (allf is 18-connected withf�, eithergiven or found).�V is the face-based boundary�V .

KEY STRUCTURES:� QueueQ (circular FIFO), “left” and “right” faces adjacent tof�.� List P of marked faces (nodes of a bintree), which are the left and right adjacents to anew facef . In the following, arrows� �� indicate queueing (unqueueing), or link insertion(unlinking) of an element in a queue or a linked list.� Look-Up-Tables with coordinate offsets of neighborhood voxels to search, for each facef k � F�� with somek � �� . Adjacents are detected from pre-calculated LUT’s indicatingfacet candidates.� Voxel connectivity, face-adjacency and special cases (such as the “Tu”-connectivity of Gor-don et al. [Gordon89]) are handled in the LUT design and specific local tests.

SURFACE TRACKING (face-based boundary extraction).

1. Find initial facetf� � �v�� f �� V� f � � F�. It is eitherfound by user (interactive selection in cross-sections), orfound by automatic scanning if a ROI (thresholding, labels, local segmentation).

2. f� � Q, f� � �V , f� � P , f� � P (notice two copies).(Collisions are avoided with a hashing table and a linked list)

3. while(Q �� ∅ )f � Q (unqueue onef from Q)

find fleft�� left adjacent tof�withfleft � �V (offset LUTs).

if( fleft � P )fleft � P (unlink fromP )

elsefleft � Q, fleft � �V , fleft � P

find fright�� right adjacent tof�withfright � �V (offset LUTs).

if( fright � P )fright � P (unlink fromP )

elsefright � Q, fright � �V , fright � P

4. endwhile: �V is a list of the external boundary ofV , a 18-connected body.

Remark 1. At the input, the access mode to�V, which is being extracted, is random access, since a list is notavailable and the whole 3D scene must be present in memory. At the output, the boundary is organized in a linearlist �V of facets, and can be accessed in sequential access (boundary traversal).

Two modes of surface-tracking operation were implemented, with several options.


(f)

(i)

(g)

(h)

F

(a)

F

3

12

56

4

(b)

F

?

F

F

F

(d)(e)

F

?

??

(c)

F

/*** Adjacency : search the RIGHT-adjacent facet to F ***/

else if facet=5 V [3] := (h)

/*** Adjacency : search the LEFT-adjacent facet to F ***/

(f) If facet=4 V [3] := (g)

unqueue F from Q

queue V [2]

else if facet=2 V [2] := (d)

else V [3] := (i)

/*** facet = 3 ***/

/*** facet = 6 ***/queue V [3] in Q

(a) V [1] = F

else V [2] := (e)

(b) If facet=1 V [2] := (c)

Figure 2.12:Facet tracking order. Three possible candidates of the right-adjacent facet are examinated peredge in (a,b,c,d,e). The shaded facet (not visible in case (c)) is the next facet, if the queried voxel (shadedin gray) belongs to the object. The left-adjacent facet is searched (f,g,h,i) if no more “right” facets remainunvisited. The output list “�V �k�”, with k � �� N facets ” constitutes the extracted boundary. The pseudo-code illustrates one-single cycle (two adjacent facets). Attending candidates are queued in a list Q.

✦ In the first mode, a starting coordinate (a facetf� on an object) can be given (found by theuser), or the user may let the algorithm scan the whole scene (or a ROI of it) to extract allobjects satisfying selection criteria (size, gray-level value range, or having a specified label, ifassigned beforehand). The objective of this operation mode is to allow full automation of theextraction process while performing a fast segmentation (such as thresholding).

✦ In the second operation mode, a scene previously labeled (segmented) can be scanned to extractall boundaries of iso-label voxels, and the threshold value of segmentation is replaced by thenext object-labelLi found during scene scanning while instead of testing condition� � (or� � ) for boundary detection, the condition� � Li is tested, fori � �� Nobj�. If severalconnected components are present per each label, a third sub-mode (not yet implemented)would organize connected objects in function of their labels. The turn-around solution is toscan a scene with a given starting label as threshold (first mode), save the list, and repeat until


all labeled sets are extracted. This scanning mode allows to use the output of any segmentationoperator.

With either tracking mode, a set of 18-connected objects (connected components of a segmentedscene) can be described by a list of boundaries. The details are explained in Subsection 2.4.3. Weconsider in the remaining a single objectV .

The extracted boundary�V (a list of facetsfi) may in turn be organized in different ways. Whenonly the space coordinates of a facet�x� y� z� k� are retained and redundacy checked out, we obtaina��-connected voxel-based boundaryV , as the list of�-voxels in contact with the background (6-connected withV c). Another organization is possible when sorting facets by orientation typek, that,is by using the face partitionF� � fx�� x�� y�� y�� z�� z�g to represent�V by six lists of pointcoordinates corresponding also to oriented facesf�f k� �uk�g� (see Definition 2.7).

Any operation employingboundary traversal visits voxels or neighborhoods corresponding toall fi � �V , or a subset, such as the subpartitionF�� (Equation 2.3). Facet visualization is theperspective or orthogonal transform of the coordinates of eachfi, using either Z-buffer or back-to-front rendering. We examine these topics in later sections.

2.4.2 Facet Traversal by Lateral Circuits

Until now we have not explained how “left” and “right” adjacent and co-adjacent facets are assigned,and how is constructed the surface graph model. To establish a mathematical model of discretesurfaces in the boundary tracking algorithm, a discrete orientation is assigned to facet traversal,besides unit normalsU�, to oriented facesf k, in function ofk � �� . A full trihedron is formedand facets have a back and a front (U�), but also their edges arein-edges (incident) andout-edges(adjacents). Figure 2.13 illustrates this trihedron per voxel face, including normals, the referenceindex k � �� , and the alternative notation with the setF� (see Equation 2.1), which helps toindicate which axis direction is concerned, but does not describe the nature of these sets of points asvoxel faces.

Recall that Figure 2.9 in page 107 showed the access to adjacents through such circuits (whitearrows are interchangeable by black arrows in the opposed sens). Each facet has two orthogonaltraversal circuits assigned from one edge to the opposite and an adjacent edge, in function of itsorientationU� and a choice.

Boundaries themselves possess an overalllateral orientation,13 analogous to orientation of con-tours in 2D which have only two possible traversal directions: hourly (–), and trigonometric (+). In3D, we have 6 contour orientations in planesXY� Y Z� ZX .

Thus, each facet has 4 edge-neighbors; traversing in one sense or the opposite gives 8 possibilitiesfor assigning the “next” and “previous” facets. Since there are three possible adjacent faces perborder, we have 12 possible faces (next and previous) in all (see Figure 2.12b).

We talk of lateral circuits, and directed circuits relative to the coordinate axesX� Y� Z, to avoidconfusion with orientation of surface normal vectors, which remain orthogonal to circuit directions.It would be logic to specify “geodesic” circuits, but literature avoids the term to avoid cumbersomenotations. Boundary traversals are constrained to follow these circuits. With a predefined circuitdirection, in relation to XYZ, a consistent traversal order is introduced.

13not to be confounded with the discretenormal orientation of the facets, given by the setU �


-

-

+

-

+

+

x

f 3

f 1

f 5

z x

y

z

y

Figure 2.13:Face orientations and lateral directed circuits. Pairs of directed circuits are assigned to eachface f k (note that f � f � f � are at the opposite sides). Long axes represent the 6 face normals, and the tworemaining directions, tangent to each face, determine traversal circuits across the edges, to search the nexttwo adjacent faces. Recall that the ordered set of faces (as a vector) �f � � � � � f �� and assigned orientationsU� correspond to face orientations �x�� x�� y�� y�� z�� z��, in the alternative notation of Rosenfeld et al.Outgoing edges of facet with face f � are bolded.

In Section 2.3.6, we introduced geodesic facet adjacents for closed objects. Lateral traversalcircuits consist in specifying two adjacent facets (left and right)14 asoutcomingor successorfacetswhen the circuit crosses an edge from the facet under consideration to the adjacent. The other twoare co-adjacent (left and right) which constituteincoming or predecessorfacet. Here, “right” and“left” do not have anything to do with “left- or right-handed coordinate system”, being only a localchoice of direction.

Note that there are (6 face orientations� 4 circuit combinations =) 24 possible sets of directed-circuit choices, which can be organized into 2 complementary sets corresponding to left-hand orright-hand coordinatesX� Y� Z. We call the 12 possible choices of the left-hand setorbits, andits complement,co-orbits. If consistently chosen, orbits provide positive measures of volume forforeground objects (and negative for the internal boundary of background cavities), and co-orbitschange the sign. Otherwise, the difference between orbits is the permutation of surface directionsfollowed during surface tracking of traversal. As with the coordinate system (left-handed or right-handed), the co-orbit set corresponds to a specular inversion of the orbit set.

Surface tracking and most processings are all implemented in a “successor” direction, followingadjacent facets and one choice of directed circuits (one orbit out of 12 possibilities). A complemen-tary alternative is to use the co-adjacent facets, that is, those found following the “previous” directionof facet traversal (the co-orbit set of directed circuits). In our implementation adjacency rules andpossible decisions for each possible configuration of successors/predecessors, were thoroughly pre-calculated and coded into LUT’s in function of the 6 possible face orientations. This allowed tomake available the two directed-circuit sets (orbit and its co-orbit), but a single set has to be chosen

14If the coordinate system is itself exchanged from axes permutationsXY Z to XZY , left and right are inter-chanched.


for most operations. We remind that the net difference in traversing a boundary by adjacence or byco-adjacence, results in a change of sign in signed-surface or volume measurements, in a similar wayto 2D signed area and perimeter extraction of contours, in hourly or trigonometric senses. A dualityexists when the scene is 1-complemented and the tracked object is background, (a cavity, instead offoreground), orientation sign is inverted from�� to � for � � x� y� z (setF�); that is, from set oforientationsU� to setU , and facet traversal behaves as if using co-adjacency. To calculate signedsurface and other morphometrics, each term contribution is weighted by the facet sign.

Directed circuits for facet traversal allow to use directed graphs (digraphs) to represent and tra-verse discrete surfaces. Then, boundary extraction becomes a full-digraph traversal problem. Thefacets are then represented by the digraph nodes, and arcs are constituted by directed-adjacence rela-tions between pairs of facets [Artzy81, Voss93]. One or moreHamiltonian circuits‘, that is, circuitspassing through every node, may exist, but there is no warranty than nodes are visited only once,which is only desirable for optimization purposes. There exist conventional techniques to traversesuch graphs, as described in [Knuth68], and most implementations are equivalent to traversal of abinary tree spanning the digraph. Left branches correspond to left adjacents and left co-adjacents,and similarly for right branches (Figure 2.14).

The surface of object of Figure 2.14a is tracked by the algorithm, starting at facet number 1.The 2D-digraph represents connectivity and adjacency of all facets and it is traversed, building a tree(Figure 2.14d). Different trees are generated if the starting facet is other than facet 1, or traversalcircuits are inverted and lefts and right adjacents and co-adjacents are exchanged. In a planar surfacethe tracking follows a stright path, but it turns at corners. Artzy described the traversal as “cloningflies over sugar cubes” [Herman98], which means that a cellular automata algorithm duplicates it-self at each facet (branching in the bintree spanning the digraph of the surface), searching for newfacets. Such duplication succeeds more often in irregular surface locations than in planar of concavelocations. This remark could be useful in future applications for feature extraction.

In digraph representation, four oriented arcs link each node (facet) to two facet-successors (out-puts) and two facet-predecessors (inputs). This imposes four possibilitiesof traversal-order combina-tions, and hence, exhaustive ways to traverse the digraph given: (1) a starting facet, and (2) a choiceof oriented circuits, to determine which edge is an in-edge (precessor facet) or an out-edge (successorfacet). A full traversal cannot be completed by just following the left successors of a given face orjust the right successors. The correct choice of which successor follows –left or right–, is done bythe circuit definition, which depends on the next facet orientation itself (for example, if it is anx�

facet, the right adjacent is chosen but if it is ax� facet, the left is chosen).

Theory and justification of the surface-tracking algorithm has taken time to be satisfactory andclear. Details and further discussion can be found in the bibliography.

2.4.3 Interior Traversal and Object Labeling

When several objectsVk, k � �� NV are present in a sceneV, their boundaries are extractedone after another by a simple algorithm described in the next subsection (the setfVkgNVk�� is alsocalled theconnected components of V ). Since segmentation consists in labeling the interior of eachdetected�Vk, we explain in this section how such operation is performed and in general, how objectinteriors are traversed. There are two labeling modes:Geometric and Solid Interior Labeling andFull Interior Labeling.


hamiltonian path of the digraphbinary spanning tree of a

associated digraph to boundaryV := { facets 1, 2, ..., 14 }

"left" adjacent facet"right" adjacent facet

Discret object

9

(b)

V

12

14

8

107

5

(a)

21

34

6

11 13

9(c)

34

5

1

2

6

7

13

1211

10

814

2

9

4

3

3

4

5

6

7

8

10

12

14

13

11

X

X

X

1

1

(d)

Figure 2.14:Discrete surface tracking as a directed-graph traversal. Nodes of the digraph (directed graph)represent facets and links represent adjacency relations. (a) Discrete object V, (b) traversal circuits to findadjacent facets, (c) digraph associated to boundary �V , and (d) a binary tree that spans ONE Hamiltonianpath of the digraph (there are 2 only-left and 2 only-right paths in (c)). Other trees are possible.

x

y

z

x

yz

10

9

127

14

852

16

11

3

13

4

Figure 2.15:Directed lateral circuits. Sequence of visited (detected) facets ffjg��j�� following the LUT ofleft and right adjacents reproduce paths of the digraph and the object in Figure 2.14a,c. Facet orientationand object shape determine the paths, which are independent of starting facet fj� (j � � in the figure). AHamiltonian path passes through all facets (not necessarily once).

Geometric and Solid Interior Labeling The geometric interior�V and the solid interiorinterior��V�

that constitute all the objectV can be both traversed from the extracted face-based boundary�V . In


a first approach, labeling ignores present cavities and fragments (it “fills” them out). This opera-tion is enough for most applications and we detail it after explaining a “conditioned” version, whichaccounts for nested boundaries.

Full Interior Labeling The boundary of an objectV with several nested boundaries (cavities andfragments inside those cavities) is in fact its outmost boundary�Vjout (see Definition A.9, page 235),and has to be traversed in a different fashion to take into account background components insideV(cavities). We do not avoid the traversal itself in these regions, but we either:

✦ disable labeling, or other volume operations during traversal of cavities and fragments, or

✦ the voxels of these regions are labeled differently, or

✦ their boundaries (i.e., nested boundaries) are extracted and incorporated into the ”full” bound-ary�V (different conected components of the boundary), and cavities become true backgroundcomponents.

The nested boundaries (cavities, fragments, etc.) can be also extracted later, in a second scan of thesceneV, making a list of boundary connected components�jV , with j � � corresponding to thefirst detected, outmost boundary and then running iteratively for all nested boundaries, up to either amaximum depth or a maximum number (we denotemax�j� � N��V�. To this end, scene traversal issubstituted by interior traversal, as done in the labeling process (see below), but for surface trackinginside an object.

A full boundary�V is then specified by the list of all nested boundaries. The geometric interiorof this full boundary coincides with its solide interior, since all cavities and fragments are takeninto account. When traversing such full boundary, only voxels inV are visited. When traversingthe interior of the outmost boundary, all voxels inV are also visited, but also those voxels inV c

corresponding to cavities.

Let �V be partitioned into 6 sets inF�, facetsf�v� f i�, with i � �� constant in each set (facets“x�”, etc.). Each partition, such as�Vjx�, is called afacet wall (see Figure 2.4). Their voxels maycompose one or several connected components in non-convex objects or with nested cavities.�Vhas six wall sets. The interior traversal method starts from all voxels in a “x�-wall”, reading (orlabeling) the scene, and advancing towardsX until the scene limits are attained (open or incom-plete boundaries), or a special label is recognized. The special label can be previously marked; weoverlay acontention x�-wall in the scene with aStop label to each voxel, we can then examinefor foreground information, and cavities.

I NTERIOR TRAVERSAL (usually, for label operations).

Build a “stop” wall (partition�Vjx�):(1) for all v � facettovox��Vjx��

L�v� � Stop label(2) for all f � �Vjx�(3) WhileL�v� �� Stop label

walk from voxel atf�v� f �� in the�u� direction(optionally,L�v� �� final label, or do other operations onv).

(4) f�v� f �� is such thatL�v� � Stop label(optionally,L�v� �� final label, or do other operation).


Labeling (if done) can be further conditioned. If facets were previously sorted into opposed,colinear pairs�f�v� f i�� f�v� f j��, with j � i ��i, traversal is even simpler.

Labeling of Connected Components Udupa proposed in [Udupa90] an algorithm consisting inapplying boolean operations (1-complement and XOR15, mainly) to mask detected objects whoseholes have not been detected. The mask serves to render each detected component “invisible” to thesurface tracking operation, while holes are treated as foreground. For large volumes it imposes anoverwhelming load, if parallel methods are not employed.

In our implementation of this feature we added some capabilities for dealing with many ob-jects at the same time (the present limit is 32K objects, which have to share a maximum of 250different labels). Udupa reports a recursive approach [Udupa90], in which boolean operations aresequentially applied to the whole volume buffer to extract and label each connected component. LetNops the number of boolean and labeling operations as described by Udupa (typically,Nops � �:scan-detection, label, XOR, label, XOR). These operations are iterated to treat nested bodies insidecavities and conversely, exchanging background by foreground, until all nested levels are attained.Real-case applications require usually one single iteration [Udupa90].

For a scene of sizeN� voxels forming up toNobj objects to detect and label, even if they areone-voxel large, the method proposed by Udupa executes aroundNobj�N��Nopstotal operations.We label directly the geometric interior of each detected componend by a3D-fill operation (or “3D-flood”), using the boundary representation itself. Discrete fractal surfaces, with dimension� � D ��, say D=2.5, have a surface composed of� O�N�� facets, for each objectk � �� Nobj�. Weexecute a total ofMops � � operations: scan-detection and label, thusMops � ��Nobj. All thisgives a total average ofNobj � N�� Nops. Even for a very complex, such as space-fillingfractals (a 3D checkerboard of one-voxel size), the performance remains 2.5 times superior.

Our labeling algorithm consists of the following instructions:

INPUT: A sceneV (or ROI). Initial labelL� to detect. On a binary scene, usually ‘255’ representsthe foreground (�-objects), and ’0’ the background (�-objects). On a gray-level 3D image, itmay be a (local) threshold. On a labeled scene, it is defined by the first label found differentfrom background, and latter defined by each subsequent label found. Other inputs: numberof objects to detectNobj, minimum size ‘min surf’ as number of facets, and optionally, anoutput label (usually varying to indicate a different object). Other options: fill cavities anderase fragments, or detect nested boundaries.

OUTPUT: Lis‘t of boundaries consisting of lists of facets with the corresponding label, the cardinalsurface area (number of facets per each object), and a labeled sceneV. Optionally, volumemeasures can be extracted since volume traversal allows to count visited voxels inside eachobject.

CONNECTED COMPONENTS (detection and label of connected components by boundaries).

1. SetL � L�, the current detection label (usually 255 in binary scenes).Setk � �, allocate and initialize list of surfaces�Vk.InitializeLout, the output label, usually a sequence�� Nobj, with Lout �� L.

2. � v � V (or ROI),if ( f� atv, with labelL)

15Givena� b booleans,a XOR b�� a � b� � �a � b�; e.g., “one or the other, but not both”.


�Vk = SURFACE TRACKING(f� , L)if( card��Vk� � min surf) L �� background.LABEL OBJECT( �Vk� Lout )incrementLout; incrementk� Nobj � k.if( card��Vk� � min surf) discard�Vk

3. continue to scan allV (the labeled object is now ’invisible’ to detection).4. End.Nobj is the total number of connected componentsVk found inV

The ”LABEL OBJECT” operation consists of volume traversal ofV using the extracted boundary�Vk, and labeling interior voxels with a target labelLout.

Other Capabilities. In our version of the original algorithm of Artzy et al. we have incorporatedthe following enhancements, to take full advantage of boundary representations:

(a) Connected component labeling during scene or ROI scanning.This operation has already beendescribed.

(b) Concurrent operations. A number of processings, analysis and morphometrics is doneon-the-fly , while boundaries are being extracted, or just traversed when they are already known.Processing can be a simple re-labeling , or includeneighborhood operations such as MM di-lation and erosion. Because of the enhancement described in (a), morphometric operations areperformed concurrently with the labeling operation (“on-the fly”), allowing to include manyobject-feature extraction capabilities.

(c) Multi-label operations All operations were implemented to deal with different labels, (up to250 in the present implementation). This possibility allows to treat as “invisible” objects witha given label, or take a particular decision if one voxel or facet has a specific label. To thisend, “foreground” and “background” are relative, and defined in function of a specific goal,such as common boundary extraction between two regions, which may produce open surfaces(boundary patches). This allows to work also with several non-binary labels. We present anexample in Section 5.6, page 203 (see also Figure 5.9, page 204).

(d) Selectivity Most operations can be executed or not, by selecting specific criteria according tothe desired processing or measurement. Current criteria include size (volume-value thresholdsor windows), surface, form factor, center of mass, sphericity (or “spread”) or isotropy (seeTable 2.1).

(e) Non-solid objects.Some provisions have been taken to allow for manipulation of non-solid ob-jects. These are constituted by sets of special facets that may not have all four adjacents andare not defined by a pairf0-voxel� 1-voxelg. For example: a single-facet cloud, a ribbon offacetsx�, or an open subset of a boundary (a surface patch). Most morphological operationsonly make sense with closed-surface boundaries, (solid filling, for example), but in some casesparticular manipulations with surface patches are useful. A specific example is the label andvisualization of the contact interface between two objects.

Most of these operations use the boundary information available at each processing stage. If anoperation modifies the scene in such a way that the boundary-information update is not available forone or more objects, then a new boundary is completely re-extracted, given a starting point in theinterior or at the boundary of the target objects. If no starting point is known, the scene can alwaysbe re-scanned entirely in few minutes (1-5) for volume resolutions of the order of��.


2.4.4 Run-Length Interpretation of Interior Traversal

Definition 2.15 ofsolid interior and Property 2.3 about volume traversal defined runs of voxelsbetween colinear facets with opposed (antiparallel) orientation16. There are however two differenceswith real run-lengths: first, we indicate the initial and final voxels plus their oriented face. Thisallows to know the direction of the codification. The second difference is that, from the point of viewof data compression, our full representation is redundant, because run-length coding can be donein only one direction, while we keep three sets of run-lengths (from sets of wallsx� to wallsx�,and the same for wallsy�� y�� z�� z�. It would not be difficult to add a routine to eliminate thisredundancy and to recover, if needed, the orthogonal sets of run-lengths.

X

Y

(x ,y,z, )f -

(x ,y,z, )f + 1

0

. . .

. . .. . .

(f)

(f)

(e)

(d)

. . .

. . .

(d)

. . .(a)

(d)

. . .

(b). . .

. . .

x k- x +

k

. . .

(c)

Figure 2.16:A run-length representation of a 3D object V. (a) profile of �V; (b) a slice perpendicular toaxis Z; (c) pairs of Nx facets facets �x�k � x

�k �, from �V encode x-runs of voxelsthat fill the solid interior of

V; the kth x-run is the closed interval �x�k � x�k �. Shaded voxels indicate first and last voxels of each x-run.

(d) Cavities and concavities are taken into account by multiple x-runs for given values y� z (a set of bars); (e)a single x-run is compressed into two border voxels (two facets x�k�� x

�k�

); (f) a full specification of orientedboundary-face pairs � f�x�� y� z� �� f�x�� y� z� �� delimits starting and ending voxels in x-runs. Storageand volume traversal are possible by specifying the set f�x�k � x�k �gNx facets��

k� .

16A step towards justification of our ad-hoc condition of Definition A.2, page 229


Using the facet partitionF�, a facet pair with opposed oriented faces�f i� f j�, with j � i��i,for example, withi � �� j � �, the set of opposite facets of the formf�xa� y� z� �� f�xb� y� z� ��,with xa � xb, gives all information needed to identify anx-run of voxels (a ’bar’) between co-ordinates�x�� y� z�, and �x�� y� z�, where we use the notation of Udupa and partitionF�, as inDefinition 2.15 (see Figure 2.16).

Since the whole volume is represented, and pairs of sorted facets that differ only inx�� x� (i.e.,forming anx-run, and representing theinterior��V�), the information supplied by these facets isenough to calculate the volume of the object. These and other morphometrical parameters will bepresented in Section 2.5.


2.5 Boundary-Based Processing and Analysis of 3D Objects

Applying theGauss Theorem17, volume-integral calculations such as the volume itself, the center ofmass and higher-order moments can be obtained from surface-integral calculations [Udupa90b]. Letf a vector function,n the normal vector of a surface elementds. The setV is any spatial region in3D (the object interior, for example), and�V is its continuous boundary. Then, the continuous GaussThorem states that:

ZZZVrf�x� y� z� dxdydz �

ZZ�V

f�s� �n ds (2.12)

Volume integral surface integral

This theorem expresses a relationship between a volume integral of a region in the space, and asurface integral, measured along the boundary of that region. It is the 3D equivalent of theGreenTheorem in 2D, which allows to extract the surface area of a regionR from information of thecontourC � �R. In a discrete form, normaln corresponds to the three axes of orientation, plus sign,justifying the importance of data partition acoording toF�. Equation 2.12 gives rise to the followingformulae to calculate the volume and the first central moment (Center of Mass or “centroid”) for eachobject (see Figure 2.17):

Vol �

Nx facetsXn��

�x�n � x�n

� � LyLz (2.13a)

XC ��

�Nx

Nx facetsXn��

�x�n � x�n

�(2.13b)

YC ��

�Ny

Ny facetsXn��

�y�n � y�n

�(2.13c)

ZC ��

�Nz

Nz facetsXn��

�z�n � z�n

�(2.13d)

whereNx facets�� card��V�jx�� Ny facets

�� card��V�jy�� Nz facets

�� card��V�jz��, i.e., the

total of facets of typesx�� x�, etc. We deduce the equation 1.13a from the Gauss Theorem, but itshould be obvious from Figure 2.17, as well as the equations for the centroidXC and the run-lengthinterpretation of the facet partition.

Several methods we use come from a general interpretation of the Gauss theorem:analysis andmanipulation of a 3D object can be accomplished by analysis and manipulation of its boundary. Inthe cases where such correspondence is not possible, we complement ourboundary-based paradigmwith a volume traversal method, which takes advantage itself of boundary-representations, sincefacets constitute starting and ending points of voxel runs.

At each stepj of a series of (pre-)processings over a sceneVj , (withV� the starting scene), the

lists of objects and their boundaries can be updated, allowing to extract morphometrical informationuseful for processing, before final quantification, and allowing to select the objects to be furtheranalyzed. Surface rendering visualization at each stepVj is also feasible.

17also known asOstrodgradskiı’s Theorem, or Divergence Theorem

2.5 Boundary-Based Processing and Analysis of 3D Objects 127

Object

Volume element

Slice axisface

faceBoundary

Figure 2.17:Volume calculation of a discrete object V using the voxel face in �V . Volume elements are builtfrom face couples in opposite orientations.

Once all objects identified in the 3D scene are properly represented, the processing, quantitativeanalysis and visualization can be effectuated through their boundaries. We have already seen howto take advantage of the discrete Gauss Theorem to calculate the volume, centroids and moments ofinertia of the object. Processing operations can be thus simplified and made selectively on a singleobject, or on a list of them, rather than performing the operation in the whole volume. This selectivityallows to introduce a priori information concerning the kind of objects to be analyzed, or even onsubsets of their boundaries (e.g., concavities). Binary mathematical morphology inN

� is an exampleof boundary-based processing:


DEFINITION 2.28 (Morphological Boundary)Let W � V � N

� a discrete object in a 3D scene. Let �W be a full boundary (all cavities andfragment sub-boundaries of W must be included). Then, the morphological erosion � of W by astructuring 6-connected element E� can be defined in terms of the 18-connected voxel boundary( W) of the object because (see for example [Jain89, page 388], for the 2D case):

��W �Wn�W �E�� (2.14)

thus, ��W is also called a morphological boundary. Given ��W , the converse relation holds:

�W �E�� Wn ��W (2.15)

The erosion of W by E� is the ��-connected geometric interior�W. We call peeling this erosion-

by-boundary operation.The dual operation, that is, “wrapping the front-voxel boundary” comes from definition of �Win terms of a dilation by N� � E�:

��W � �W � E��nW (2.16)

Then, a dilation is done by wrapping �W to W:

�W �E�� W � ��W (2.17)

Recall that boundary neighborspm are defined and found in terms of�W (a facet boundary; seeDefinitions 2.26 and 2.27, pages 109-110. Thus, discrete morphological operations of an object inthe cubic grid can be effectuated by boundary traversal operations.

2.5.1 An Example of Boundary-Based Morphometry: Euclidean Area Estimation

Several morphological measures can be obtained or estimated from boundary information. We il-lustrate what can be done introducing a correction of non-Euclidean measures, based on geodesicneighborhoods (adjacent ornext-neighbor face on the boundary), which are analyzed on a boundarytraversal.

A discrete estimation of Euclidean surface area, for cubic voxels with discrete stepsLx � Ly �

Lz�� , is obtained from (see Figure 2.18) [Russ90]:

S � �� number of othogonal-adjacent voxels

p�� number of 2D-diagonal adjacent voxels

p�� number of 3D-diagonal-adjacent voxels�

(2.18a)


(a) (b) (c)

Figure 2.18:Euclidean approximations of area for staircase configurations of boundary voxels. (a) Orthogo-nal voxels (no correction), (b) two-dimensionaldiagonaladjacence, (c) three-dimensional diagonaladjacence.Light shadowed facets indicate the corresponding facet configuration.

Another approximation was proposed and tested by Smeulders and Beckers for 3D line lengths,which is directly applied to 3D surface area [Smeulders89]:

S � �� number of othogonal-adjacent voxels

�� number of 2D-diagonal adjacent voxels

�� number of 3D-diagonal-adjacent voxels

(2.19a)

Prismatic voxels of arbitrary physical dimensionsLx� Ly andLz demand more complex corrections.2D-diagonal and 3D-diagonal planes are formed from orthogonal faces in their six possible configu-rations (see Figure 2.19). Facet-adjacency information is obtained from the boundary representationand provides 2 adjacent and 2 co-adjacent next neighbors for each facet. A 3-bit code of their relativeorientation and a lookup table of correction factors allow us to device an Euclidean approximation oflocal surface area. Higher-order corrections with second-next neighbors are feasible, but codes andconfiguration combinations become much more complex. If higher precision is intended, it seemsmore appropriate to build mesh models of the discrete surface (marching cubes, adaptive triangula-tions or fitting of polynomial surface patches).

Sabcg � LyLz (2.20)

Scdeg � LxLy (2.21)

Sagef � LzLx (2.22)

Sacdf � ��Lx

qL�y L�

z (2.23)

Sabde � ��LypL�z L�

x (2.24)

Sbcef � ��Lz

qL�x L�

y (2.25)

Sace � ��qL�xL

�y L�

yL�z L�

zL�x (2.26)

2.5.2 Boundary-Based and Volume Processing (Examples)

These are some examples of 3D processing tools that make use of the principles already mentioned.


Figure 2.19:Prismatic voxel of sidesLx� Ly andLz and its 2D and 3D diagonals. Surface diagonal elements(equations 2.20) help to obtain Euclidean approximations of area for staircase configurations of boundaryvoxels.

✦ Post-labeling of segmented objects. This secondary labeling , after segmentation, can be em-ployed to mask the object on the original (gray-level) scene for densitometric measurements

(as a free-shape ROI). The geometric interior�V can be traversed, thus labeled or simply exam-

inated, guiding the interior traversal from boundary traversals of the facet subsetsfx�n g and

fx�n g which form the extreme points of an internal “run” of voxels. An entire object is filteredout from the scene by being re-labeled with the same value of the background by using a LUTand then by excluding its from the boundary traversal list.

✦ Separation of weakly-connected components. This tool exploits the boundary representationto create discrete watershed surfaces, in a similar fashion to 2D clump splitting, as explainedin Section 1.2.3, and detailed in [Bloch90b, Mangin95, Marquez93b, Marquez94b, Lopez95,Marquez96a].

Using the morphological equivalence of boundary trimming and discrete erosion by struc-tural elementsEK � K � �� , one or more erosions are firstly made on the object todecompose by simply relabeling to background the boundary voxels, and re-extracting thenew boundaries. These erosions are non-homotopic, giving rise to several fragments which arelabeled as different objects. Then, a number of conditional dilations are applied, preservinglabels locally (homotopic dilation) and extending the separating interfaces to the whole object.Conditional dilations are applied until all original object is filled with the new labels. Morecompact components are obtained with this method. The total numberN� of required erosionsis function of that of the minimum neck-size of the connections to break. In the biological ap-plication described in [MarquezCON91a, Lopez95, Marquez96a], serial images of cell nucleiform touching chromatine bodies with different shapes. In this example, we setN� � �, withE�, corresponding to the estimated size of the connection neck [Lopez95]. In general, thenumber of reconstruction dilationsN� depends on thin structures present and is determinedby convergence (no more new voxels are added). In the same biological application it wasN� � ��. The reason thatN� � N� and sometimesN� � N� is that a single erosion can


disappear a thin run of several voxels (a filament for example), and each conditioned dilationwill restore only one voxel at a time.

✦ Label manipulations. Two objects can be associated to be always treated together by beingassigned the same label. The opposite manipulation would consist in breaking a single objectinto two or more volume components that are weak connected. Such separation can be doneby watershed segmentation, marking each component with a different label (see Figure 2.20).This operation creates discrete interfaces constituted by the facets shared by each labeled com-ponent. In general, a voxel label can serve as special marker, be transparent to some particularanalysis, serve as “contention wall” for propagation processes or have other meanings. Ob-ject labeling is done by traversing the object’s volume by series of voxels comprised betweenfacetsx� and x � ; this replaces interior-fill (or “flood”) algorithms which do not exploitboundary information and require specification of an internal point to work.

a bc d

Figure 2.20:Separation of weakly-connected components. (a) A subset of cross sections ofchromatine clumps in animal cell nuclei; the gray-marked regions are seen as background (inblack) by the watershed segmentation. (b) Output subset; color labels indicate each of the 26weakly-connected objects that were detected (operations are done in 3D space). (c) View (POV)of reconstruction. (d) Another POV, selecting-out the biggest 3 components. Data from theMicroscopy Laboratory, Facultad de Ciencias, Mexico University.


✦ A more precise measure estimation of surface area. Facet representation imposes an importanterror in area calculations due to 2D and 3D staircases. A proper correction in function ofadjacency relations is possible, as explained in Section 2.5.1.

✦ Estimation of the Euclidean normal vector for each facet. This calculation is described insection Section 2.5.1.


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Figure 2.21: (Top) Complex interfaces example from a biological application: analysis of invagina-tions and endoplasmic reticulum interfaces with the nucleus of a dopaminergic neuron as described in[Anglade93, Marquez94b, Anglade95, Anglade97]. (Bottom) Use of voxel and facet boundary neighbor-hoods ��V � and ��V � to extract, analyze and display internal boundary components (nucleolus, periph-eral chromatin, nuclear wall, endoplasmic reticulum and invaginations). These examples are from displays atlow-resolution, with dithering on only 7 colors.

2.6 Visualization of Complex Structures 135

2.6 Visualization of Complex Structures

Visual interpretation of voxelized objects can also be done using discrete boundaries. A conic-perspective transformation is used, and a diffuse illumination model is applied for each of the discretenormal vectors. In the simplest approach, three discrete normal directions give three gray-levelsprecalculated in function of light direction. Back-to-front rendering was used to solve the hidden-surface removal problem in our first implementations. Z-buffer techniques are much more efficient,and have been recently available as public library functions [Karinthi95], or hardware-implementedin graphics workstations. At present we have not incorporated such techniques, but our boundary-based approach is compatible with Z-buffer rendering.

Common color displays have a limited number of colors (256 minus the colors allocated by theX-Window system). In order to have color shade gradations and be able to display ahigh number ofdistinct color labels, a dithering technique was employed to synthesize interpolated colors. Giventwo color shadesC�� C�, this technique allows to interpolate15 shades(plus two original extremevaluesC�� C�) denoted byf�ckg�� , betweenC�� C�, with �c� � C� and�c�� C�. In this manner, thesystem allocates only 7 original shadesfCkg�� for each color label, without noticeable Mach bandsor textural effects [Ulichney87a]. Edge sharpeness is also respected, since dither masks are appliedper pixel, preserving resolution at edges (see Figure 2.22).

Figure 2.22:A 4x4 dither matrix set of 17 patterns, used to interpolate between black and white. The size ofthe patterns has been exagerated (scaled by 3).

2.6.1 Surface Rendering

2.6.1.1 Hidden Surface Removal and Surface Shading

In visualization of complex structures, simple perspective projection does not provide an effectivedepth illusion. In our boundary-based approach, hidden surface removal is performed by simpleback-to-front displaying of single facets. This technique is compatible with Z-buffer techniques andspecial graphics architectures, which avoids the need of depth-sorting of facets. Depth or distanceshading cueing is done by letting the back-to-front rendering yield the coordinates of the surfacevoxels along the viewing axis instead of the encoded surface normal vector. A blending factor allowsto combine both shading techniques (depth shade and the normal orientation shading). Several light-sources are possible, because discrete normals are precalculated in function of facet orientation.

2.6.1.2 Discrete Normal Estimations

In a more general approach, discrete normals are assigned per voxel or per facet in function of localneighborhood averaging. The nearest facet neighbors (18-adjacent facets) allow for� possible nor-mal orientations, because at a given facet, three neighboring facets are possible per edge, and eachfacet has four edges. Second- and third-order neighbors determine a higher number of discrete nor-mals. Our implementation and data structures have taken into account this higher-order information.Figure 2.25 shows 3D rendering of a CT skull (resolution�� ), with normal estimationby first-order facet neighbors (geodesic 4-adjacents, Definition 2.24). Discretization artifacts are only


Figure 2.23:Smooth-gradations rendering using dither interpolation of 7 colors, at real display resolution.There are 120 interpolated shades. Note that dithering texture is almost of the same order than the printerhalftones.

Figure 2.24:Same as Figure 2.23, zoomed detail (X4). Edge borders are scarcely distorted and texturesremain minimal.

visible in theZ axis, where resolution is the smallest (146). Our facet-based approach for renderingpresents some advantages in comparison with several existing rendering methods; for example:

✦ Rendering is discrete-oriented; individual voxel details and inter-connections can be madevisible. At the same time, a smooth appearance is available. In continuous-orientedapproaches(splines, traingle meshes and Phong smoothing) voxel information is lost or distorted.

✦ Only lists of boundary elements are stored and traversed.

✦ Traversal and display may be very selective: it can be organized by ROIs, by objects of interest,by topological features such as cavities, by sub-boundaries, or specific sets of facets. Thisselectivity allows also for labelling specific facets.

2.7 Summary and Conclusions 137

✦ Discrete-normal estimations may be dependent of other local informations (shape, curvature,and neighborhood characteristics).

✦ Boundary traversal for visual rendering can be combined with object processing and morpho-metric feature extraction.

2.6.2 Three-Dimensional Data Structures.

We found illustrative to give some few examples of implementation details. In particular, whendealing with huge volumes (�� bytes for one database), we find useful the followingstrategy that we could not found properly reported in the literature.

During boundary extraction and volume processing (e.g. region growing, thresholding, back-ground noise remotion and volume MM operations), a volume dataset, such as coordinates withlabels, a computer-simulation output, voxelized surfaces, or a set of image slices, is read into a 3Ddata structure. Several data structures have been reported, such as:

Spatial array of voxels (or ”spatial occupancy enumeration” [Foley92]). A straightNx�Ny �Nz

array of numbers (or vectors). This is a conveniently indexed 3D structure, but very expensivein terms of storage, since a single continuous block of memory is allocated (for example acubic volume of�� 16Mb). We used these arrays in some small ROI operations, andvoxel neighborhoods, where radius is limited to no more than 40 voxels.

Octrees. This 3D-structures are difficult for implementing traversal and neighborhood operations.We did not use them at all.

Run-length encoding. This compact representation is implicitly used in our b-rep, as already de-scribed. During volume operations, an explicit run-length compression does not allow randomaccess which would require lengthy decoding and encoding to update linked lists (see how-ever the permforming methods reported in []). Run-lengths are implictly produced duringthe boundary extraction (”encoding”), even without pairing opposite facets; data also behavesas run-lengths during the boundary-based operations (”decoding”) and when boundaries arere-extracted, if needed (equivalent to re-encoding).

Marginal indexing. This is a hierarchical pointer-based sparse representation, which allows all ran-dom access operations as spatial arrays of voxels. It is illustrated in Figure 2.26, and describedfor 2D images in [Foley92]. We made a 3D-implementation consisting of:

✦ a pointer to a volume ”V ” (a scene),

✦ which consists of an array of pointers to slices (images ”V �z�” in theZ-axis);

✦ these slices are also constituted as a ”wall” of pointers ”V �z��y�” to

✦ rows of voxelsV �z��y�� V �z��y��x max�.

2.7 Summary and Conclusions

We described in this chapter a discrete boundary-based representation (“b-rep”) of objects inN�, and

provided definitions and tools for several manipulations and analysis.


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A key concept of digital spaces isK-connectivity, which partitions a sceneV into K-objects(K-connected components), whose frontiers or boundaries can be represented by points, facets orvoxels. We used in particular the face-based boundary�V as the basis of several manipulations andanalysis of discrete sets. Of particular importance is the partition of�V into sets with facets of thesame orientation (6 setsF�). The voxel-based boundaryV allows to define and make possiblemorphological characteristics (e.g., geometric interior, neighborhoods). Passage from one b-rep tothe other is done by examination of the immediate neighborhood (from boundary voxels to facets)and redundancy elimination (from facets to boundary voxels).

Several concepts can be grouped together; we havetraversal operations in a scene, the interior ofan object, a facet, or a boundary. Facets have a trihedron of discrete orientations assigned, (normalsand lateral circuits) and are partitioned in 6 orientation sets (F�). Neighborhoodsappear in Euclideanand Geodesic modes (along a surface), but also in the surface normal and tangential directions. Theyare also defined for voxels, facets and boundaries.

We started early implementations of the boundary extraction method of Artzy et al. for biomedi-cal applications (electron microscopy, [Marquez94b] in PC computers, and this lead us to review thetheory of digital spaces and discrete representations, as well as the manipulation of several objectsat the same time. We have dealed with other problems related to complex objects and boundaries.


Some of these are presented in this manuscript, in Chapters 3-5. At present the system was portedto UNIX, working under a Motif user interface. We have tested most algorithms in resolutions up to832x640x220 voxels.

A substantial part of the work in this chapter has been already done by other groups, usingsometimes different notation, or claiming generalizations ton dimensions, or faster algorithms (seefor example [Shu91] for a comparison table). They usually are reported to work for one singleobject which has to be indicated or isolated by the user. We describe synoptically our work and ourcontributions:

✦ We stated the b-rep model in terms of boundary elements: oriented-faces and voxels, and devel-opped a consistent description of objects in discrete space and several characteristics towardscomplex shapes, as presented in Chapter 1. Some of these definitions are not always clear inthe literature, such as partitions of face-orientations, and lateral circuits, and the difference intreatment between voxel faces and boundary elements (facets or voxels).

✦ We have built and optimized several Look-Up Tables (LUT) to look for facet adjacents and co-adjacents, and to be able in the future to work with different sets of traversal circuits (orbits),as a mean to sample in different ways a given surface.

✦ Our volume-traversal method using a subset of�V (a containing wall of facetsx� and a start-ing wall of facetsx�, not necessarily paired) is very simple and original, and we avoid thecomputation of logical operations on the whole scene (a method still used since its introduc-tion in [Udupa90]).

✦ Besides a paper of volume calculation of Udupa ([Udupa81]), and the original suggestionof Rosenfeld et al.[Rosenfeld91], we have not found actual applications of morphometricalparameter extraction from boundary information. We implemented some MM operations, andneihborhood analysis on�V , and applied them in Chapter 4 for local segmentation, regiongrowing and tubular detection, and in Chapter 5.1 for interface analysis.

✦ In particular, we proposed a better Euclidean estimation of boundary surface area, using thenearest facet neighborhood configuration. This information also provided a normal vectorestimation that allows smoother visualization of voxel-based volumes, and the means to probethe object, from its boundary, in the normal direction.

In the other hand, we did not study, tested, finished or incorporated the following topics andfeatures in the present work:

✦ The discrete Jordan property of surfaces.

✦ Topological considerations (homotopic operations, simple point removal, Medial Axis Trans-form, which is only briefly described in Chapter 4).

✦ Full validation and comparison of different measures and Euclidean approximations. Some ofthese were done, but not presented in this report.

✦ Nested-boundary trees, adjacency graphs, hierarchical decompositions and other descriptivestructures for several objects (having common boundaries and neighboring relations).

✦ Extension to the discrete case of all definitions in the preliminaries Section 1.1 and Ap-pendix A.


The latter list opens a very vast field of perspectives; but these will be summarized in Section 5.8”Conclusions and Perspectives”.


Chapter 3

Radiometric Homogenization of theColor Cryosection Images from the VHPLungs for 3-D Segmentation of BloodVessels

Abstract

This chapter deals with the problem of radiometric inhomogeneities found on the physical color im-ages of the anatomical cryosections from the Visible Human Project (VHP) male body. Our goal is toextract very thin structures, like the blood vessel tree from the lungs. Current segmentation methodsapplied to VHP color images are disturbed by discontinuous, inter-slice radiometric variations; wethus devised an adaptive correction that is propagated along a series of parallel slices, taking ad-vantage of structural coherence between consecutive slices [Marquez96b, Marquez98]. No blurringis introduced, and fine details and texture are respected. Results of 3-D segmentation of fine bloodvessels on the corrected volume are presented.

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

3.2 Problem Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

3.2.1 Gamma Correction . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 145

3.2.2 Visualization of radiometric inhomogeneities . . .. . . . . . . . . . . . 146

3.3 Local Adaptive Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . 150

3.3.1 Inhomogeneity Characterization by an Auto-regressive Model . .. . . . 150

3.3.2 Local Coherence Hypothesis. . . . . . . . . . . . . . . . . . . . . . . . 152

3.3.3 Correction Algorithm .. . . . . . . . . . . . . . . . . . . . . . . . . . . 154

3.4 Discussion and Conclusion .. . . . . . . . . . . . . . . . . . . . . . . . . . . 160

144 Radiometric Homogenization of Color Images from the VHP Lungs

3.1 Introduction

Many laboratories have extensive experience of the three-dimensional reconstruction of structuresfrom data collected from transmitted, emitted or reflected radiation (X-ray computed and emissiontomography, nuclear magnetic resonance imaging, ultrasound, confocal and electron microscopeimages, etc.). This is not always the case with traditional, physical, cross-section reconstruction,which is lengthy and requires block fixation and destruction of the specimen. Even if physicians arefamiliar with anatomic slices of the human body, until very recently there were no large color datasets to reconstruct whole organs for anatomy atlases and models.

Cryosection techniques have been used in the Visible Human Project (VHP) [Ackerman95,”VHP” ISpitzer96] of the National Library of Medicine to make a high resolution volume of anatomicalinformation available to scientists. 1871 axial-plane slices spaced at 1mm have been sampled at aresolution of 0.33 mm, with�� pixels of 24 bits. Three sets of images corresponding to thethree channels RGB have been recorded. The three channels are spatially well registered and a signalof one byte per pixel and per channel is available. The complete data files occupy 15 Gigabytes.Computer tomography and magnetic resonance imaging sets are also available.

This chapter deals with the problem of radiometric inhomogeneities that we have found on phys-ical color images from the VHP male body. These images correspond to anatomical cryosections1 ofthe pulmonary region. Interest for the detection, display and compensation of such inhomogeneitiesin the VHP is timely, since automatic procedures for feature extraction and elaboration of referenceatlases require volume data to be homogeneous and isotropic. Hence, any noise or artifact compo-nents must be properly identified or estimated.

We consider in the following our data subset or Region Of Interest (ROI) which is the croppedsub-volume which encloses both lungs. This ROI consists of 160 cross sections of�� pixelsof 24 bits, corresponding to slice numbers 1320-1479. In the following sections we describe theproblem of specific inhomogeneities encountered in the ROI, and present several methods whichwe have developed to visualize and characterize them. Then we detail the correction algorithm forwhich we use the fact that consecutive images are very similar in structural features. We finallypresent results of the correction and the automatic extraction of blood vessels by 3-D segmentationtechniques.

3.2 Problem Description

In order to obtain three-dimensional segmentation of the bronchial blood vessels or any other struc-tures in volume data from cross sections, the images must be spatially registered, but they must alsoberadiometrically registered or homogenized, i.e., color intensity distorsions must be eliminated orproperly compensated through the ROI. The VHP color images are correctly registered in rotationand translation, except for some images in which small translation offsets have to be corrected. Toensure radiometric homogenization, each color image was digitized by the VHP team including agray-scale test card for color camera calibration, to allow forgamma correction [Poyton96], andhomogenization of a stack of color images (see Fig. 3.1). The test card was used to homogenizeintensities in each imageIn of the ROI,n � �� , and to obtain a setfIn� g in which in-

1Hystological slice or polished tissue previously congelated.

3.2 Problem Description 145

tensities are normalized with respect to an average gray scale in each channel RGB. This correctionconsists of:

(i) a measurement of the 20 RGB gray-scale values2 (which present a non-linear response varyingbetween channels and between images),

(ii) a curve fitting of the average gray scale (a spline),

(iii) a gamma correction by using a specific 256-entry look-up table, for eachIn, and each channelu � fR�G�Bg (see Fig. 3.1c).

In each image and for each gray-scale bari � � � � �� of the corresponding test card, the extremevalues (over a window of�� pixels) were not taken into account in averaging operations as theywere considered as outliers due to acquisition artifacts or noise.

3.2.1 Gamma Correction

We observed that, even after a separate gamma correction of each image by using the accompanyinggray scale, there remained some radiometric discontinuities between slices which affect the three-dimensional segmentation of small blood vessels and bronchi (see Figs. 3.2 and 3.3). The originof these inhomogeneities is unknown, but they seem to be unrelated to acquisition artifacts (cameracalibration or variations in lighting, causing an uncontroled reflectance factor). Spitzer et al. notedin [Spitzer96]: “previous experience with willed cadavers who have died by court-ordered lethalinjection had revealed that such remains may undergo massive deterioration within 24 hours ofdeath”. Even if measures were taken to delay this deterioration and the body was frozen two daysafter death, some tissue degradation could have taken place during the 20 days of cutting sessions.Hence, radiometric inhomogeneities could be evidence of an irregular oxydation of the sliced surfaceof the frozen body between two image acquisitions. Furthermore, these acquisitions could havebeen unevenly spaced in time, or taken under different temperature conditions; but these are onlyhypotheses. Lung tissue differs from other parenchyma, since it has a porous structure. Thus, a highsensitivity to air exposure is expected. We have found no other regions of the VHP male body, exceptthe lungs, where a similar radiometric inhomogeneity problem is noticeable.

In a sequence of slices, a radiometric homogenization could be sought by ahistogram registrationwhere the main features, in particular the principal mode, are more or less aligned for small, mediumand large regions in each image. Since a shift of the main peak corresponds to an average color offset,radiometric inhomogeneities can easily be visualized by coronal or sagittal cross-section imagesresampled from the slices in the ROI (see Fig. 3.2). Inter-slice inhomogeneities then appear in suchcross-sections as horizontal artifacts.

Let fIng be the original color data subset andfI n� g the subset obtained by gamma correction,using the gray scale provided for each image. Figure 3.3-left shows a coronal slice sub-image fromfIng; each of its rows corresponds to one of the successive axial-plane images. Figure 3.3-rightshows the corresponding resampled slice fromfIn� g.

Even if some global homogenization has taken place, we observe that the most important re-maining horizontal discontinuities can not be interpreted as just a global color offset between twosuccessive acquisitions. This is further demonstrated by exhibiting inhomogeneities, before and aftercorrection, in the different visualization modes which we will now describe.

2A “gray” level is not perfectly colorless for a camera.


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Figure 3.1:Global gamma correction. (a) A gray scale (in color) is used for a global radiometrichomogenization of the image set fI ng. (b) Each gray scale image provides three RGB intensitycurves, with i the gray bar index from 0 to 19 and �u the mean value of their measured RGB values(�u � fR�G�Bg). (c) A 256-entry look-up table is built to gamma-correct each channel of each slicewith the linear scale defined by the gray bar indices.

3.2.2 Visualization of radiometric inhomogeneities

The remaining inhomogeneities as well as coherent features along the slices may be further studiedwith three modalities of visualization that we proposed consisting in images of

✦ stacked color histograms of the slice set (see Fig. 3.4),

✦ co-occurrence color images (see Fig. 3.5), and


��

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✦ checkerboard displays of adjacent slices (see Fig. 3.6).

These techniques were applied to exhibit and understand the radiometric inhomogeneitiespresentin our set of gamma-corrected color images. They allowed us to design a correction method. At first,before we realized the local character of inhomogeneities, we aimed at aglobal homogenization ofcolor intensities and developed two approaches which gave rise to correction strategies that did notwork. However, we consider it illustrative to describe them briefly.

For each color imageIn� , we obtain three histogramsHn�u�� u � fR�G�Bg (see Fig. 3.4a),which may be viewed in radiometric space as a row of a 2-D color image (see Fig. 3.4b). Thehorizontal axis corresponds to the intensity levelsu for each color channel RGB, the vertical axis


Figure 3.3:Residual inhomogeneities. A coronal sub-image of the original volume set (left), and aftergamma correction (right). Remaining inhomogeneities appear in the right as horizontal artifactspointed by arrows. Only 40 slices are displayed to show the inhomogeneities more clearly.

corresponds to the slice numbern, and the pixel color at abscissau represents the three frequencies(maximum value normalized to 255) of the three intensity valuesu in slicen. Thus each horizontalrow represents the color histogram of a cross-section image. Global inhomogeneities are then easilyvisualized as color shifts between rows. Alignment of extrema was attempted for a global homoge-nization by histogram manipulation, with acrest-line tracking technique (see Fig. 3.4c). Importantlocal extrema of RGB-color histograms were detected and linked to form continuous ridges, and his-togram stretching operations were then applied to each image. Unfortunately, this operation did notremove some local discontinuous variations.

Co-occurrence color images were also useful in exhibiting RGB-intensity disparities betweentwo consecutive imagesIn� � I

n�� (see Fig. 3.5). Both axes correspond to traditional histogram ordi-

nates, that is, intensity valuesun� un�� f�� g. In conventional histograms, the height of thefunction at locationu indicates the frequency of pixels with that intensity valueu. In co-occurrencecolor histograms, RGB intensities are plotted directly as a colored pixel, instead of coding them asheight profiles. Thus, the pixel at location�un� un�� indicates the frequency of coordinate pairs�x� y� in imagesIn� andIn��

� , having valuesun andun�� respectively, withun � fR�G�Bg. Whenboth images are exactly the same, the co-occurrence color plot consists of a diagonal colored line,whose RGB intensities correspond to the conventional color histogram of the image. When bothimages are highly uncorrelated, no diagonal appears, and the plot consists of scattered points. Im-ages from a series of thin slices should present co-occurrence images with predominant diagonalfeatures. Radiometric inhomogeneities appear as an irregular scattering of point clouds near the di-agonal, which is also distorted. Biases from the geometric diagonal reflect the presence of significantpopulations of pixels with systematic shifts in RGB values. A second global correction method couldthen be devised on diagonal properties, similar to crest-line alignment in stacked histograms, but itwould not allow us a correct homogenization which has to be performed locally in our case.

Checkerboard displays of adjacent slices allow us to exhibit local disparities more clearly beforeand after a correction method is applied. A checkerboard pattern is used to interlace two consecutiveimages, which are displayed alternatively in the ”black” and ”white” squares (see Fig. 3.6). Thepresence of remaining inhomogeneities notably enhances the perception of the checkerboard pattern


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��

��

n n+1H(u , u )

γ nI

γI n+1

u n+1

nu

0

255

255

Figure 3.5: Co-occurrence color image. Pairwise comparisons of consecutive gamma-correctedimages In� and In��

� allow us to visualize radiometric inhomogeneities as an irregular scatteringaround the diagonal. The RGB values of the co-occurrence image pixel situated at coordinates�un� un�� represent the frequency of the pixels with the same spatial coordinate �x� y� in images I n�and In��

� and having the intensity values un and un��, with u � fR�G�Bg.

(see Fig. 3.6-left). This pattern becomes unnoticeable when we apply the local adaptive homoge-nization method described in Section 3.3.1 (see Fig. 3.6-right).

3.3 Local Adaptive Homogenization

3.3.1 Inhomogeneity Characterization by an Auto-regressive Model

Co-occurrence images took us naturally to consider pairwise differential images and to study thepeak features of their local, stacked color histograms. To correct the discontinuities observed, wehave adopted the following first-order autoregressive model:

In� �x� y� � In�� x� y� �n �x� y� � (3.1)

whereIn� �x� y� is the gamma-corrected RGB-intensity vector of pixelp � �x� y� in slice numbern, and�n �x� y� is an offset vector (see Fig. 3.7a) that includes at least three kinds of radiometricdiscontinuities that may arise between two axial-plane slices:

✦ The first kind of discontinuity is the real texture and 3-D structure featureinnovations betweenslices (see Fig. 3.7b). This is the information that must be preserved.

✦ The second kind are smooth gradients due to lighting changes between image acquisitions,gradual specimen alterations during slicing and inhomogeneous blood distribution at the timeof freezing (see Fig. 3.7c).

✦ The third kind corresponds to the residual inhomogeneities which we need to remove. It con-sists of noise and some artifacts due to regional variations caused by an unknown physicalprocess which has altered the color of the tissue before digitization (see Fig. 3.7d). In some

3.3 Local Adaptive Homogenization 151

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imaging modalities such as MRI [Zijden95], this component of�n �x� y� can be further sep-arated into inter- and intra-slice intensity variations, but some a priori knowledge would berequired for an explicit formulation.

(b)(a)

(d)(c)

. ... .

Figure 3.7:An ideal, simulated decomposition of inter-slice radiometric inhomogeneities. (a) Dif-ferential image �In� � In� � In�� k�� (see footnote 1 in page 152); (b) 3-D structure innovationsand texture (to be extracted); (c) smooth 2-D and 3-D gradients (lighting mismatches and speci-men spatial and temporal variations); (d) inhomogeneity artifacts and noise due to discontinuousalterations. Unfortunately, in practice, such a decomposition can only be approximated.

3.3.2 Local Coherence Hypothesis

In order to correct the third kind of inhomogeneities, we have taken advantage ofstructural coher-ence (see Paragraph 1.2.5, page 40). First, a differential image�In� �x� y� is obtained by subtractionbetween two consecutive color cross sections1. Only innovations corresponding to anatomical dif-ferences between successive images, give rise to significant positive or negative deviations from theground level (0,0,0). They appear as contrasted thin features at the boundaries of 3-D structuressuch as organs, bones or vessels (see Figs. 3.7a and 3.7b). Large colored regions in the differen-tial images should then reveal radiometric discontinuities. As we are interested in segmentation offine structures such as the blood vessels, large unexpected innovations which suddenly appear mustbe removed. In the lungs such inhomogeneities will color differential images unevenly, as can beobserved in Figs. 3.7a and 3.9a.

1 In order to display and distinguish negative and positive deviations from (0,0,0), a constant offsetk�� is added toeach channel; but in the discussion we refer to the ”ground level” as (0,0,0).


np

np

n

H (u) := Histo( N )

H (u)

pn

N

p

p

n

n

and

∋

u(p)

Rp

where N = { q : |p-q| < R, q I }

∋ δ nγ

u = 0

δ nγI -

n-1γII :=

nγ

u { R, G, B }

Figure 3.8: Local coherence hypothesis: when consecutive color slices are radiometrically reg-istered, their differential images�In� have local histogram modes aligned around �R�G�B� �� , for all pixels p � �In� .

For radiometric homogenization, ourcoherence hypothesis consists in having the modes of localhistograms of�In� �x� y� centered at�R�G�B� � �� , for all pixelsp � �In� (see Fig. 3.8). Evenif very complex fine structures are present, two consecutive thin cross sections are similar enoughto expect that,locally, �In� �x� y� will have a histogram centered at (0,0,0). The local histogram atpointp � �In� is measured over a small circular regionN n

p in the differential image,N np � fq j

jp � qj � R� q � �In� g. The radiusR of this neighborhood is chosen large enough to guaranteemeaningful statistics (e.g., the histogram mode should be easily detectable). Furthermore, secondarypeaks present in this histogram correspond to innovations which represent the end or beginning ofa different tissue region between two slices (a slanted vessel, or the transition between two organs,


for example). Any shift of the central mode from (0,0,0) implies a radiometric discontinuity ofthe third kind at that pixel. In our implementation the region radiusR was chosen larger than thelocal area occupied by an innovation such as small blood vessels that we want to extract and whichhave a diameter of 1.0 to 6.0 mm; thus we setR � �� pixels. The coherence hypothesis can alsobe stated as follows: if no inhomogeneities are present, the area of innovations (thin, high-contrast(black) features in Fig. 3.7b) is negligeable compared to the area of all inter-slice common features(white background in Fig. 3.7b). Innovations are also highly contrasted, that is, their peaks in thedifferential histogram are far from (0,0,0), which facilitates further automatic detection of the centralmode (see Fig. 3.8). This property also applies to large discontinuities at the top or the bottom ofwell-defined objects (vertebrae, for instance). Thus, if the mode is too far from (0,0,0), no correctionis done and the inhomogeneity is treated as a large, natural discontinuity. The decision threshold forcorrection was fixed at (30,20,20), because no radiometric inhomogeneity higher than (26,12,8) wasdetected in the worst case. This was found around slicen � �� of the setf�I n� g when we checkedparenchyma transitions with other organs.

3.3.3 Correction Algorithm

Thus, we perform a local adaptive correction, for each color channel, which is propagated along theimagesIn� . Figure 3.9 shows difference color images, plus a global offset in order to display negativevalues also. The dominant gray value�R�G�B� � �k�� k�� k��) corresponds to ground level(0,0,0), and darker or lighter pixels represent innovations or inhomogeneities. A local histogram isobtained for each channel and for each pixelp�x� y� of the difference color image�In� ; the pixelneighborhoodN n

p of radiusR � �� consists of 1250 pixels. The main peak is detected and its shiftfrom zero is calculated for each pixelp:

erru�p�� argmax

u�Hn

p �u�� u � fR�G�Bg�

This shift represents the local mismatch between imagesIn�� andIn� at pixelp. A partial correctionof amounta � erru, with � � a � �, is made on the corresponding pixelp of the second imageIn� .OnceIn� is corrected, the third imageIn��

� is subtracted fromIn� to obtain�In�� , and the process is

repeated, until all images of the ROI have been processed. Figure 3.10 summarizes the algorithm.

A second pass is performed backwards to ensure a minimal deviation from original undistortedvalues while rejecting inhomogeneities which are not innovations. Thefading memory constanta � � that we have introduced weights the error correction. Such a constant comes from a generalautoregressive model of the form:

In� � a� a�In�� a�I

n�� a�I

n�� (3.2)

from which we have only retained the first-order term. Constanta plays the role ofa�, and thecombination ofa� together with the error of the first-order approximation is assimilated to the term�n�x� y� in Equation 3.1. We have found that a partial correction of 60% (a � ��) in one directionand another of 60% of the residual difference in the backwards direction (making a total of 84% ofthe original difference), perform better than a full 100% correction (a � ��) in one single pass, asthis may produce some color artifacts. The reason is that the starting sliceI n�� for error propagationmay be too inhomogeneous in relation to its neighbors, and a full correction may thus introduceunnatural color shifts, even if the final set is radiometrically homogeneous. Besides the memoryfading constant, a proper choice ofIn�� can be devised using homogeneity testing techniques [Wu93].The goal is to start at an image with low distorsion, and a simple criterion is to choose the slice with


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local histogram

size of innovationsR >

n = n

δ n

n

pnI

p

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p

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0

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a0< <1, u {R,G,B}

u

u(p) u(p) - a err (p)u

γγ

γ

A

u

γn

nerr (p) = argmax( H (u) )p

n

n -1

n n+ 1

Figure 3.10:Adaptive correction algorithm. Starting at slice n � n � selected as the slice whose nextdifference image is closest to ”zero” (Equation 3.3), shifts in the mode of local histograms H n

p �u�on differential images are compensated on image number n, for every pixel p. Neighborhood sizeR for Hn

p �u� is chosen greater than innovations. Corrections are propagated forwards and thenbackwards along the ROI.

minimal differences between its neighbors. In our case, we select the slice numbern� for which thedifferential image�In� � In� � In�� has the smallest contrast contents. A simple way to ensure thiscondition is selecting an imageI n�� for which the area of the differential histogram is minimal:

n�� argmin

n

�� Xu�fR�G�Bg

ZIn�

Histo��In� �

�A (3.3)

Finally, to optimize the local histogram calculations, arolling buffer (or sliding window) tech-nique has been implemented (see Fig. 3.11). It consists in updating the neighborhood informationby adding to the local histogram only the next entering front of pixels, subtracting the leaving pixels,and updating the central pixel information2. Updates of the buffer are done as long as windowN n

p

2The central pixelp itself is excluded fromN np in order to enhance statistical robustness when radiusR is diminished.

This improves the local radiometric homogeneity around small details such as in the extreme case of one-pixel width bloodvessels.


is slid on each row of the input image; the neighborhood is fully read only once per row to initializethe buffer. Local operations other than histogram measurements can also be performed.

Fig. 3.12-left is equivalent to Fig. 3.3-right. Figure 3.12-right shows the same coronal sub-imageas in Fig. 3.12-left, but after our local adaptive homogenization. Previous horizontal artifacts havealmost disappeared. We call the new color data subsetfInHOMOGg. Other coronal and sagittal imagesresampled fromfInHOMOGg show similar performances of the radiometric correction (see Fig. 3.13).

N = { q : |p-q| < R } =>

- {old list }p’ pn n

+ {new list }

Histogram update:

pn

H H

. . .

p = (x, y) 1p’ = (x+ , y)

precalculated list of offsets {q}

Figure 3.11:Rolling buffer technique for fast local histogram calculation. Only the boundary pixels(new or old) of the sliding neighborhoodN n

p are considered for the update of the histogram.

Let fInGRAYg be the set of gray-level images, taking the luminance fromfInHOMOGg. Using local (or”adaptive”) thresholding [Castleman90] onfInGRAYg, we have obtained a binary volumefInBINg, wherefine blood vessels can be isolated for three-dimensional reconstruction. A volume rendering viewof fInBINg is shown in Fig. 3.14-left, using a back-to-front approach with depth cueing [Frieder85,Tiede93]. An improper choice of the neighborhood radiusR for the analysis (too small or too large)or of other parameters as the constanta may affect homogenization results. But these results remainvery robust for a wide interval of the parameter values, as observed when setting the neighborhoodradius from 15 to 40 pixels, and the constanta from 0.6 to 0.85. In order to evaluate the performanceof the 3-D segmentation of the volume setfInHOMOGg with different homogenization parameters, weuse the inverse of an axialMaximum Intensity Projection (MIP) image [Udupa93, Napel92], as J ”MIP”shown in Fig. 3.14-right. MIP color images allow us to visualize the projection of the blood vesselstraces present in the volume and to verify their presence after the 3-D segmentation.

Simple local thresholding onfInGRAYg was useful for evaluation of the homogenization resultswith the help of coarse volume renderings, as shown in Fig. 3.14-left. Besides a more realisticvisualization, we aim at obtaining morphometric analysis and explicit representations of anatomicstructures. Since lung parenchyma and blood-vessel images present texture and partial-area varia-tions, local thresholding onfI nGRAYg is inadequate for fine blood-vessel extraction, and does not take


Figure 3.12:Local adaptive homogenization result: (left) coronal resampled image of the gamma-corrected set f�In� g; (right) the same cross section taken from fI nHOMOGg, i.e., after local adaptivehomogenization.

Figure 3.13:Local adaptive homogenization result: (left) another coronal resampled image of thegamma-corrected set; (right) the same cross section after local adaptive homogenization.

full advantage of 3-D homogenization and color information. Thus, a more robust method was usedfor thin-structure segmentation.

Using the gamma-corrected and radiometrically homogenized ROI we have obtained realisticreconstructions of the bronchial tree after different stages of segmentation and 3-D image processing(see Fig. 3.15), using a3-D region growing method, and a 3-D surface tracking with boundary rep-resentations [Pavlidis90, Artzy81, Udupa82, Thurf95]. These methods are described in Chapter 4.Correction of the inhomogeneities proved to be useful for the automated 3-D segmentation by regiongrowing because this method relies on local statistics of RGB values (mean and variance). With


Figure 3.14:Volume rendering: (left) back-to-front display with depth cueing of blood vessels tracesdetected in the radiometrically homogenized sub-volume; (right) axial minimum intensity projectionimage of slices 1420-1429.

this approach, the red channel information was enough to discriminate most bronchial blood vesselsfrom lung tissue. Gray-level luminance was also tested, with no evident improvement. Blue andgreen channels carry less contrast information, and linear combinations with the red channel tend todiminish the discriminating power of the latter, at least for blood vessels in the lungs of the VHP.Further work must be done to confirm these observations and to explore segmentation approachesusingPrincipal Component Analysis (PCA) of color distribution in parenchyma, and using colorspaces other than RGB.

Figure 3.15:3-D Visualization. Two views of bronchial blood-vessel tree extracted from the VHPdata set after radiometric homogenization.


3.4 Discussion and Conclusion

We have proposed and tested a non-linear, local adaptive algorithm for radiometric homogenizationof cryosection image series from the VHP dataset. The high resolution of the thin slices allowedus to exploit local coherence, in order to distinguish changes in 3-D structures from radiometricartifacts. This was done by: (i) selection of an imageIn� as reference, (ii) calculation of the colordifferential images�In from two consecutive slicesn � � andn, (iii) calculation, for each pixel, ofa local histogram of�In, (iv) examination of the RGB shift from zero of the principal mode of thehistogram, and (iv) correction of the error and setting the resulting image as a new reference in orderto propagate the correction from slice to slice.

An important advantage of the adaptive homogenization method is that no blurring is introducedin the resulting setfI nHOMOGg of corrected images. Since no low-pass filtering is done, fine 3-D detailsand textures are preserved. Pixel modification consists of local RGB offsets which depend only on thecoherence mismatch between regions of two adjacent images. The use of a fading-memory constantallows us to preserve smooth gradients across the volume set (inter-slice variations), while the localcharacter of our correction impedes the filtering of smooth variations inside an axial image (intra-slice variations). For these reasons, local thresholding or region growing techniques were neededto segment fine structures such as the blood vessels, as shown in Fig. 3.15. However, classicalsegmentation problems such as non-additive noise, sub-sampling, partial volume, etc., remain to besolved.

Remaining inhomogeneities still visible in Fig. 3.12b could be entirely smoothed out if smallerneighborhoodsof lung tissue were analyzed or a higher-order autoregressive model were used (Equa-tion 3.2)v. This type of very local analysis is difficult in the proximity either of important innova-tions or of parenchyma other than the lung tissue, because local histogram analysis works better withmedium sized regions where the radius is much larger than blood vessel innovations. A possiblesolution would be to perform first a coarse segmentation of large innovations, such as those fromthe heart and the stomach, in order to mask them out from differential images. In this way, differ-ential histograms could be freed from large innovation components, improving mode detection andmeasurement of its local shift from zero.

Our segmentation methods used only the red channel, as explained in Chapter 4. The homo-geneization method allows the correction of each color channel, which would permit a data-fusionapproach. Another solution is to work in an adapted color space in which blood-vessel characteriza-tion uses PCA.

There are many image processing and pattern recognition techniques designed for motion imagesequences. It is common practice to apply such techniques to 3D volume analysis or vice versa.The homogenization method presented in this paper may be applied to correct or detect certain in-homogeneities in time sequences; for example, in old motion film restoration where local distorsionsand artifacts appear often in individual frames or short sequences. This example is just a glimpse ofpotential applications of the techniques developped forVHP-imaging to other fields.

Chapter 4

Extraction of Bronchial Blood-Vesselsfrom Color Cryosections of the VHP

Abstract

We describe in this chapter the extraction of the bronchial blood vessels in the lungs of the male sub-ject from the Visible Human Project (VHP), as an example of complex 3D data. We review first someanatomical facts of the lungs, useful in any computer analysis. Then, we examine the characteristicsof the VHP database concerning the lungs, the general approaches for vessel segmentation found inthe literature and the extraction methods we propose. We also present the results of the implementa-tion of three of these methods, as well as the visualization of the 3D reconstructions from anatomicalcolor images of the VHP database.

4.1 Anatomy of the Branching Structures in the Lungs . . . . . . . . . . . . . . . 161

4.1.1 Tree Structure of the Pulmonary Artery. . . . . . . . . . . . . . . . . . 164

4.1.2 The Lungs of the Visible Human Male. . . . . . . . . . . . . . . . . . . 166

4.2 Blood-Vessel Extraction . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 174

4.2.1 Other Imaging Techniques. . . . . . . . . . . . . . . . . . . . . . . . . 174

4.2.2 Detection Methods . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 175

4.2.3 Fusion of Cross-Sectional Segmentation. . . . . . . . . . . . . . . . . . 176

4.2.4 Local Thresholding with Hysteresis .. . . . . . . . . . . . . . . . . . . 179

4.2.5 Region Growing Segmentation. . . . . . . . . . . . . . . . . . . . . . . 185

4.3 Conclusions and Perspectives.. . . . . . . . . . . . . . . . . . . . . . . . . . 187

4.1 Anatomy of the Branching Structures in the Lungs

The lungs in mammals are constituted by three systems of highly branched trees: the blood-vesseltrees of the Pulmonary Artery (PA) and the Pulmonary Veins (PV1 and PV2), and the airways orJ ”PA, PV”bronchial tree which ends in the alveoli, the small air compartments (less than 0.3 mm in diameter)where gas exchanges have place. Under normal conditions, carbon dioxide (CO�) and other gases

162 Extraction of Bronchial Blood-Vessels from the VHP

pass from minute venules to alveoli while oxygen (O�) and other gases (mainly azote (N)) pass fromalveoli to arterioles. Oxygen-rich blood is collected from all capillar vessels up to the PV into theheart during dyastole (dilatation), to be redistributed in the circulation network of the body at systole(contraction). The venous blood, charged withCO� from all the organism, is then pumped to thelungs by the heart. Connective and supporting tissue encase the alveoli, the airways and the bloodvessels. This structuring tissue has a porous texture and is known as thelung parenchyma . A detailedcharacterization of this texture would allow segmentation and analysis of important structures in atexture-based analysis (see for example [Morris88]), but this was not the case of the present study,and we used only the first-order statistics to characterize tissue as the background for blood vesselstructures.

The lungs are inflated and deflated during the respiration cycle to exchange air with the medium.This cycle implies an important change in volume that demands a global deformation of the wholestructure. A first consequence of this dynamical behavior for computer analysis is the impossibility,with present technology (e.g. spiral computer tomography, the fastest available technique), of highresolution digitization of the whole respiratory system in vivo, specially in patients with respiratorydefficiencies. At the same time, deformable models, atlases and a better understanding of the struc-ture of the lungs is needed to study “thick-slice” samples at specific regions of interest, conventionalradiographies, two-dimensional CT axial samples, and three-dimensional reconstructions from se-rial slices or CT scans of post-mortem specimens. In the following subsections we present somedetails of the structure of the lungs and indicate the relevance of this knowledge for segmentation,three-dimensional reconstruction and analysis.

In function of energetic needs, there are high changes in the rythm of the cycle and the volumefraction of the lungs deformation. A large volume of air (up to 3 liters in a health adult) has totraverse and fill the airways in less than half a second during extreme physical efforts. At the restingstate, all the blood of the body passes through the lungs each minute. The fractal structure of thelungs allows this rapid gas exchange by acascading phenomenon [Armin94]. Such a structure isillustrated in Figure 4.1.

Air and blood follow similar pressure patterns during the respiration cycle in each tree, facilitat-ing their fast distribution through the lungs, but also enabling the expulsion ofCO � and the diffusionof O� into the blood. There are predominantly-onlydychotomic ramifications1 (or bifurcations).In fact, Horsfield and Cumming report exclusively bifurcations in their resin cast of the airways[Horsfield68]. Bifurcations are not symetrical in both lung sub-trees (dychotomic asymetry). Con-sidering only bifurcations, a tree node has�n branch daughters at branching ordern. Since the fluxof the three substances (arterious and venous blood and air) remains more or less constant alongthe trees (however, it gradually changes with time), the cross section of each ramification follows amass-conservation law of the form:� � An � �n, whereAn is the cross section at branch generationordern, with n � f�� g for the airways, and� is the flux (mass of fluid (air) per area unit).There exists however up to 23 generations of bronchioles and 28 generations of blood-vessel bifur-cations, but the power law does not take into account molecular diffusion and other gas-transportconsiderations. When experimentally measuring average diameter of bronchial segments and bloodvessels, Weibel and Gomez report an accurate potential law:

d�n� � d� � ��n�� (4.1)

with d� � ��mm, andn � f�� Norderg, up to branch generationNorder � �� for airways andfor blood vessels up to orderNorder � ��, down to the capillary network [Weibel62]. Figure 4.2shows this relation up to ordern � ��.

1Those directly modeled by a binary tree.

4.1 Anatomy of the Branching Structures in the Lungs 163

Figure 4.1:A resin cast model of the bronchial tree in an adult human (reproduced from [Weibel79a]).

From these considerations, a simple a priori estimation of the relations of branch diameter at ordern is available, and can be useful for reconstruction and morphometrical analysis of the lungs, fromanatomical slices. Assuming no partial-surface effects (“one-pixel diameter vessels”), and assumingneither deterioration nor geometric distorsionsof the specimen, then, at pixel resolutionof��mm, branches from orders 16 and 17 (0.42 and 0.33 mm in diameter) indicate the finest vesselresolution (see also Table 4.1). With a knowledge of the minimum detected vessel, and its relationwith resolution, we can generalize these estimations. Letf � R � R an invertible monotonicalfunction relating resolutionL and average diameter:L � f�d�. Usually,f reduces to a multiplicativeconstant. Letd�n� as in Equation 4.1. In this case, the bifurcation ordern at resolutionL is givenby:

n �� log�f��L��d��

log ��(4.2)

whered� can be measured from the slices in a particular specimen. Thus, in a more realistic approx-imation (blur, noise due to texture and partial volume), we can expect to detect 7, 5 or 3 pixel-widevessels. In these cases,L � �d, L � �d, andL � �d, and from Equation 4.2, we obtain the corre-sponding orders: 8, and 10 to 12, and vessels from 2.68 to 1.06mm in diameter. In this conditions,in order to detect the smallest capillars (0.1 to 0.03 mm at orders 22 to 28), an optical-microscopyresolution of less than 0.01 mm (magnification��) would be needed.

Another observation of Horsfield and Cumming, and Weibel and Gomez is that nomonopodialbranches exist, neither in the airways nor in the PA or PV trees, that is, there are no small arteriolesor venules branching from stem or large vessels; thus diameter reduction is gradual and small struc-tures do not connect directly to large segments. A consequence in segmentation methods of vesselcomponents of the lungs is the a priori knowledge thatsmall vessels or airways cannot lay too closeto large branches.


0d = d 2-n/3

bifurcation order n

dve

ssel

ave

rage

dia

met

er

(

mm

)

Figure 4.2:Power law of average diameter d of vessels at bifurcation (or binary branching) order n. Diameterat order n � (pulmonary artery from the heart) may change. The airways follow the power law up ton � � .

In conclusion, anatomical models and their experimental validation provide some quantitativerelations useful to evaluate segmentation and branch extraction at a given resolution, provided thatthe power law of Equation 4.1 holds. In the following paragraphs we review the structure of thevessel and bronchial trees, and examine the particular characteristics of the lungs from the databaseof the Visible Human Project.

4.1.1 Tree Structure of the Pulmonary Artery

Arteries and veins in the lungs extend from bifurcation sites known ashilar points, following adivergence principle, described by Yamashita in [Yamashita78]. Recursive division of the three treestructures proceeds in similar ways: for the airways this hierarchy begins at the trachea, leadingthrough bronchi, bronchioles, acinar airways, respiratory bronchioles, alveolar ducts and alveolarsacas, comprising up to 23 levels or generations (from which only 10 follow closely the powerlaw). Vascular structures begin to run almost parallel, connected by the airway wall but then PAand PV run independently. The PA and the airway tree follow a similar branching pattern (seeFigure 4.4 running more or less parallel, separated by the airway wall, and both follow almost thesame diameter distributiondown to generation 10th. This hierarchical organization enables vessels tocarry and collect blood very efficiently, and constitutes for physicians an a priori information to infervessel localization in any lobe. A particular notation system has been attributed to label main arterieswith a capital “A”, veins with “V”, and bronches with “B”, each one with a superindex indicatingthe segment number, and a lowercase letter for identifying sub-segments. Thus,Ab refers to sub-


(a)

Anterior Posterior

(c)

(e)

Anterior

(f)

(g)

(h)(d)(b)

Left lung Right lung

Figure 4.3:Lateral view of the lungs: costal surfaces of main features. (a) Left upper lobe; (b) left lowerlobe; (c) right upper lobe; (d) middle lobe; (e) right lower lobe; (f) left oblique fissure (also known as leftmajor fissure); (g) horizontal fissure (also known as minor fissure); (h) right oblique fissure (or right majorfissure).

segment b of the fourth segment of the PA. In some anatomical books this notations changes slightly(e.g. A4B forAb).

A tree-like representation in 3D of the pulmonary blood vessel anatomy was first proposed, forthe aid of computer reconstruction from CT scans, by Inaoka et al. (1991) [Inaoka91]. Their modelconsisted of a tree with regular dichotomy, without regard to length and branch diameter. Usinga 3D vector representation, and the 26 combinations of 3D opposite directions (up and down, foreand back, right and left), a knowledge base is encoded as aframe-type representation. An expertphysician validated such a structure in the work of Inaoka et al., where position, direction, and lengthhave been integrated later.

Other anatomical information useful for description and location of branching subsets (sub-trees),is segmental and lobar anatomy, in which segment and lobar organization are taken into account.Lobe organization is shown in Figure 4.3a-e. Internal interfaces of the lobes (see Figure 4.3f-h) alsoserve as landmark structures, and are known as themajor or oblique fissures in the left lobe andthe minor or horizontal fissure. We did not use this information, which needs to be extracted bya semi-automatic method from the VHP images, due to low resolution (1mm per slice) and tissuedegradation. In better conditions, it should be valuable for a specific application in which eachbranch would be labeled in accordance with its lobar location, and identified with a specific bronchialsegment from atlases.

Other structures in the Thorax Between the lungs and over the diafragm membrane and thestomach, lies the esophagus duct, the heart cavities, the Aorta artery, the Cava vein and other im-portant vessels other than the PA and the PV. When extracting the contours of the lobes of the lungsor the airways and blood-vessels, these structures must be identified and separated. As the diafragmarea is concave, a substantial part of the upper liver and its blood-vessel systems accomodates underthe diafragm, as well as a portion of the stomach. Vertebrae, sternum and ribs appear in the bordersof the ROI of the lungs, and occupy several cross sections of the lungs of the VHP database. Astheir volume is large in comparison with the ramified structures, simple mathematical morphologyreconstruction (a conditionned closening of a convenient size) suffices to isolate most of them, inorder to mask them out from analysis. A more difficult task is the identification of the interfaces of


T+2

T+2

T+2

2

2

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Pulmonary Artery

A a

A b

6A b

2A b

A a4 4

A b

2A a

1

A c

A c

A a

A a1

A b

A rec.

A asc.

A a3

A b3

A b5

A a6

A c6

Figure 4.4: Notation system to address primary blood vessels of the Pulmonary Artery (PA), which runsparallel to the airways. The pulmonary vein (PV) is not shown. The labels Anm indicate sub-segment m ofArterial segment n. “Rec.” stands for recumbent, “asc.” for ascendent (labels are not exhaustive). Diagramredrawn from [Inaoka91].

each lobe, since they may appear as unconnected thin structures from one slice to another, in contactwith small vessels.

4.1.2 The Lungs of the Visible Human Male

Tiede et al., reported in [Tiede93] several problems concerning the available information from IRMand CT images of the VHP in general when considering soft tissue, in particular the lungs, whichare particularlly porous, and vulnerable to fast deterioration. Airways are scarcely visible in IRMimages, and only some of the most important blood vessels appear. As resolution was also limited(�� in the CT, and�� in the IRM images when considering the ROI of the lungs), weopted to analyse the color images, and more precisely the blood-vessels which are still much morevisible. At the other hand, only the main airways appear because the specimen lungs were collapsedafter death [Spitzer96]. Moreover, a substantial quantity of blood was drained out, and many smallblood vessels are difficult to track. This is not a disadvantage as it would appear, since withoutcirculation, all blood concentrates in the lower parts of the body, deforming vessels, organs andtissue, and expanding during freezing, with the consequent damage of small structures. Resolutionfor the ROI comprising the lungs is�� in these images. As the VHP database wascreated very recently, we have not found much information in literature about the lungs of the VHP,because we ourselves (Marquez and Schmitt) were in 1996 the only group working in the colorimages of the lungs, discovering the radiometric inhomogeneities reported in [MarquezVHP96].


1297

slice#

1305

13651388

1342

1415

14531430

14831504

Figure 4.5: Coronal view of the thorax in an adult and approximate position of the axial cross sectionscomprising the lungs of the Visible Human Project (VHP)database. A coronal slice of the axial cross sectionsis shown at the right.

Figure 4.6:Color slice number 1408 of the VHP.

Even by using manual segmentation slice by slice, with editing assistance, several groups work-ing with the VHP dataset have reported in [Imielinska96, QuackenbushVHP96, SchiemannVBC96]a very difficult extraction of small branches of blood vessels and other fine structures when workingwith the physical color cross-sections. They also report work only with the red channel. Despiteall mentioned limitations, information from these slices is still very valuable because voxel size is�� , and color resolution is 8 bits per channel.

Local and Global ROI Statistics To approximately locate the axial slices of the thorax in a humanbeing, refer to Figure 4.5, where the position of cross sections are indicated in a coronal view of


the thorax. Figure 4.6 is the red-channel image component of slice number 1408, after correctionof radiometric inhomogeneities with a suitable stretch function for clarity (the original image istoo dark). Details are shown in Figure 4.7. An intensity profile for each color channel is shown

Figure 4.7:Slice number 1408 of the VHP (detail). Note that most cross sections are elliptical. Rectanglesamples with (left) and without (right) vessel cross sections are shown in Figure 4.9.

in Figure 4.8, with the image of the red channel superimposed. Note that useful information (the“valleys” corresponding to vessel cross sections) is found mostly in the red-channel profile. Anintensity histogram including mainly parenchyma and a second one including several blood vesselsare shown in Figure 4.9. An exhaustive exploration of blood vessel characteristics through localhistograms, gave us the following information for each color channelR�G�B. These measures weretaken observing each separated channel image and the color image on three consecutive images ata time, in order to ensure that a thin vessel was present at the selected pixel area. We give thesemeasurements in the form of a vector of components�R�G�B�, rounded to half integer values:

✦ The mean value� u � with components�R�G�B� varies along the volume, in function ofglobal illumination characteristics. After gamma correction and radiometric inhomogeneityfiltering (see Chapter 3), the minimal value found was�� pixels, and themaximal value of�� . The last incertitude is high because manualclassification of what is vessel or texture features is subjective, and partial surface effectsproduce a one-pixel wide “halo” around some “one-pixel” vessels, or afalse vessel (a textureartifact introduces the incertitude; in both cases the feature or noise is of sub-pixel size). Theglobal mean value was found to be� u �ROI� �� . By using this value as a globalthreshold one cannot obtain a ’maximum’ area of segmented valid fragments of blood vessels.Most of the segmented fragments turned to be eitherover-segmented (background frgments arelabeled as blood-vessel foreground) orunder-segmented (blood-vessel fragments are labeledas the background), as shown in a single slice sample in Figure 4.10.


0

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(a)

(b)

(c)

gray

-leve

l int

ensi

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Figure 4.8:Gray-level intensity profiles of the red (a), green (b) and blue (c) channels on a line parallel toaxis X (indicated by arrows). Note the correspondence of valleys in the image across the horizontal line, withvessel cross sections in the red-channel profile (a). Image intensities have been stretched by an exponentialtransfer function (gamma a � �� ) to make details visible.

✦ The local standard deviationof gray-levelsinsideblood-vessels was found to be much moreconsistent:�in � �� . The same value was obtained for windows of3x3 and 5x5 pixels (small vessels).

✦ The local standard deviation of parenchyma regionsaround vessels, was found to be:�out �� . This value was the same for large regions (up to 30x30 pixels),where none or only few small vessels were present. The local mean value has also high intra-slice and inter-slice variations (from 0 to 160, even if the local standard deviation remainslower). Nevertheless, gradients were qualitatively found to be very smooth, after the inho-mogeneity corrections (these corrections filtered discontinuous variations which could be mis-taken as transitions from vessel material to tissue). The only exception is a single slice, number1454, where some illumination problem must have happened during the slicing session, sincesaturation at 0 is attained for channels blue and green at the bottom of the image.

We noted that parenchyma of the lungs has a stronger green and blue component than bloodvessels in most areas, as expected, since it is more flesh-colored. Unfortunally there remain verydark areas where green and blue channels have darker values than red. Figure 4.11 shows the sumof the blue and green channel of one of these slices, in which there is no under-saturation, but red-channel information is much richer and reliable. No information is useful, specially at the bottompart of the image (slice 1451) because parenchyma is darker than blood vessels in the blue and greenchannels (the opposite occurs in most part of the slices).


50

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0 150 20010050 250

54

200150100

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56

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xels

intensity R

pixe

ls

intensity R

Figure 4.9:Intensity histogram of parenchyma ROI (red channel R) without blood vessels (left) and intensityhistogram of a second ROI with cross sections of blood vessels (right). The insets show the ROIs (See alsoFigure 4.7).

For the given reasons, in the present tests we used only the red channel, butPrincipal ComponentAnalysis may be applied in the future, as reported by Schiemann, for bone and skin segmentation ofthe VHP [SchiemannVBC96].

Partial Volume Effects Any time a voxel (pixel) corresponds to the interface (transition) betweentwo (or more) regions with different gray-level information, it will contain information about bothregions (parenchyma tissue and blood-vessels or bone and tissue, air and tissue, tissue with contrast-agent and normal tissue, etc.). In 3D this phenomenon is named thepartial volume effect, and in 2D,the partial surface effect, and it is present in most imaging modalities with discrete representationof spatial information. In some modalities it is easier to determine subvoxel (subpixel) location ofthe interface or boundary, from gray-level contents (e.g. 255 for bone, 0 otherwise). There are othersources of gradual transition between regions, such as blur and the Point Spread Function (PSF) of”PSF”Ithe acquisition system.

In the present study, since cross-section images come fromphysical, anatomic cryosections of1mm-thick, no blur exists (ideally) in the axial (Z) direction, that is, for each voxel in slicezk noinformation is convolved with the adjacent voxels at sliceszk�� zk��. Mechanical image aligne-ment in cross-sections and camera callibration have been warranted to have at most 0.1 mm, whichcorresponds to�� pixel error [Spitzer96], thus a net partial volume error should comprise onlythe PSF of the CCD camera and partial surface effect (i.e., only in 2D). All observed edges and thethree RGB channel alignement show just one to two-pixels wide transitions. This is difficult to verifyquantitatively in lung parenchyma, since texture features are fuzzy by nature, but a key feature is thefractal nature of the lungs. Equation 4.1 predicts the presence of blood vessel sub-trees at scalesmuch smaller than image resolution (0.33 mm). The observed texture of parenchyma is already com-posed of partial-surface pixels (per slice), with assorted contribution from several scales. Figure 4.12


Figure 4.10:Global thresholding of slice number 1408 (red channel). The threshold value (38) was selectedfrom local statistics in a small ROI with vessels and tissue.

illustrates partial volume effects. Compare subfigure (d) with the pixels of the zoomed vessels in theright inset of Figure 4.12.

To estimate the importance of this contribution, let us calculate the average density of sub-pixelblood vessels at ordern, by assuming binary bifurcations and a maximum of 28 bifurcation orders.There are�n branches at ordern, with diameterd given by Equation 4.1. Since the adult normal lungs(parenchyma) occupy a volume ofVlungs=5 liters. (or 5��mm�), the density of blood vessels of

ordern, is: �� n�Vlungs. Densities for ordersn � �� are given in Table 4.1. This is only

a global average estimate and a local measure (per slice) would take into account length betweenbifurcations. It predicts however, that if vessels of ordern are intersected on a given slice, theirexpected number, per cubic millimeter, is of the order of that found in the table.

Considering pre-processing degradations, the radiometric-correction method presented in Chap-ter 3 does not introduce any blur: in difference images, a pixel value coherent with its slice-neighborseither remains equal, or is shifted by a small quantity��R�G�B� to make the mode location of itsimage-neighborhood match the mode location of the neighborhood in the precedent slice (same co-ordinatesx� y). No information is convolved with the neighboring pixels, whether the image neigh-borhoods are large or small. The only possible source of error are very weak-intensity innovations.These innovations must be considered as under-sampled structures and true partial-surface pixels:alveoli, parenchyma texture, nerves, connective tissue or collapsed airways and very small bloodvessels (less than a pixel-wide) traversing in a skewed angle, in order to produce such innovations.

By hand tracing some contours in a sample of ten slices from the ROI, we found an averagesurface area of lung tissue of about�� mm�; which confirms the�-liter volume measure of thelungs used to estimate densities in column 4th of Table 4.1 (recall the ROI is�� ). The


Figure 4.11:Contrast-streched image of slice number 1451: combined information of blue and green chan-nels (R�B). Note that many vessels (but not all) present higher values of blue and green than the surroundingtissue, making the red channel a better choice for segmentation. The original image is almost “black” and astretching with an exponential transfer function (gamma of ��) was applied to allow visualization.

average number of vessels per slice is more difficult to estimate, since we have to take into accountvessel lengths and their natural bending towards the root [Weibel62]. A very rough approximationand lower bound is the vessel density divided by each branch volume (� � �diameter�� length)multiplied by the slice volume. This measure turns out to be approximately�n itself, since lengthsfollow a similar power law.

Just by hand-counting the smallest and darkest visible dots from 3 to 5 pixels-width, correspond-ing (after the branching law) to generations 10 to 12 in the Table 4.1, we found in three samples(slice 1408 shown in Figure 4.6 and slices 1347 and 1450) a much lower number of them:�� and�� . A plausible reason is that vessels are contracted by tissue deformation or colapsed(recall that blood was drained before freezing), and thus the expected diameter corresponds to earlierbranching generations: 7 to 9. This implies an error (from the image capture and partial surfaceeffects) of up to 7 pixels, if the law holds strictly (the difference of diameters between generation7 and 12 is 2.4 mm, and each pixel is�� mm). If this diameter difference holds also forthose corresponding (from the�n law) to generations 16 and 17, then pixel resolution correspondsto generations 13 and 15, and the observed texture includes tens of thousands of subsampled vesselcomponents (deformed by colapsation) of at least one-pixel diameter, and hundreds of thousands ofsmaller capillars.

We stress the fact that many branching generations (a natural fractal), which tend to fill availablespace as already noted, make an important contribution to background nature, making the blood-vessel tree a very difficult structure for segmentation, since there is one “cut-off” generation 10 inwhich foreground is no more resolved and is treated as a textured background. Some of this textureis visible in Figure 4.8.


(a) (b) (c) (d)

Figure 4.12:Partial volume effect. (a) Original object and superposed grid for discrete sampling. Afterdigitization, each voxel contains both, information of foreground (black) and background (white), coded asgray-level densities corresponding to the mixture (c); plus the effect of the Point Spread Function of the acqui-sition system (here represented by a �� convolution). Only the largest branches are noticeable in (c). Notethe partial surface effect(d) on tubular cross sections (b), when diameter is about two pixels.

We did not continued this kind of analysis for other vessel diameters, since the objective wasonly to have a rough idea of how useful could be the�n and the��n�� laws (number and densityof branches and their diameter), and the importance of the fractal structure in partial volume effectsand background texture. We were looking forward to obtain three dimensional reconstructions tomeasure directly diameters and branching orders in the VHP database before realizing, by the latterdisussion, thatbackground is noised by the same structure details we want to extract!

Two different consequences are drawn from all these observations:

✦ Background segmentation of the bronchial vessel tree may be difficult by 2D image analysisand contour extraction. Coherence can be an advantage, against a textured background in a 3Dapproach (adjacent voxels with similar gray-levels possibly belong to the same structure), butthin structures reduce the number of pixels per vessel, per slice. The coherent property wasalready exploited with success for radiometric homogeneization.

✦ Neighborhoods for analysis of local standard deviation�� have to be small, since they mayinclude gray-level information from sub-sampled vessels of several diameters.


Table 4.1:Densities of blood vessels of branching order n for volume of �� mm� and initial diameter ofd � �� mm. orders 16 and 17 correspond to pixel resolution in the VHP images.

Order n Diameter (mm) �n Vessel Density (��mm�)

0 17.000 1 0.000000201 13.493 2 0.000000402 10.709 4 0.000000803 8.500 8 0.000001604 6.746 16 0.000003205 5.355 32 0.000006406 4.250 64 0.000012807 3.373 128 0.000025608 2.677 256 0.000051209 2.125 512 0.00010240

10 1.687 1024 0.0002048011 1.339 2048 0.0004096012 1.062 4096 0.0008192013 0.843 8192 0.0016384014 0.669 16384 0.0032768015 0.531 32768 0.0065536016 0.422 65536 0.0131072017 0.335 131072 0.0262144018 0.266 262144 0.0524288019 0.211 524288 0.1048576020 0.167 1048576 0.2097152021 0.133 2097152 0.4194304022 0.105 4194304 0.8388608023 0.084 8388608 1.6777216024 0.066 16777216 3.3554432025 0.053 33554432 6.7108864026 0.042 67108864 13.4217728027 0.033 134217728 26.8435456028 0.026 268435456 53.68709120

4.2 Blood-Vessel Extraction

4.2.1 Other Imaging Techniques

X-ray Digital Subtraction Angiography and similar techniques rely on vessel identification of ac-quired images before and after intra-arterial injection of a radio-opaque contrast agent. In 2D thesetechniques provide the highest resolution available (from 0.06mm per pixel to 1mm) [Sun89]. Vascu-lar networks in general and bronchial blood vessels in particular pose different problems to classicalsegmentation methods. These are mainly due to poor contrast of very thin structures. Complex vas-cular abnormalities, aneurysms, stenosis andtortuosity demand high resolution 3D imaging, and alarge scale analysis with CT imaging of the lungs cannot be donein vivo, because of the high dose

4.2 Blood-Vessel Extraction 175

of X-rays required, and the movement of the lungs during a long acquisition session. Costello et al.,and Paranjpe and Bergin [Costello92, Paranjpe94] examine and compare spiral CT images for 3Dsegmentation of lung structures and blood vessels at small, local ROIs or very low resolution samples(1.5cm appart), allowing assessment of only transversal segments of blood vessels and the overalldisposition of main arteries.

Magnetic Resonance Imaging in two dimensions is suitable for airway tracking,in vivo, up to thefourth or fifth branching generation [Portman92]. With the introduction of contrast agents,MagneticResonance Angiography provides projections in several planes and allow reconstruction in 3D oflarge arteries. A systematic analysis of the best blood vessel imaging techniques is given by Verdonckin [Verdonck96], but color images from cryosections in high resolution were not available until 1995.

Spiral CT-imaging allows to scan thick sections2 of the thorax for local exploration of the air-ways, in vivo. The main limitation is the maximum allowed dosis of X-ray exposition for non-clinicalstudies, and the short apnea lapses, all of which reduce the amount of CT-scans for 3D reconstruc-tion analysis. Thus, with actual technology, a large structure as the vascular trees of the lungs or thebronchial tree is only accessible for high resolution 3D reconstruction throughpost-mortem analysis,either by CT imaging of the fresh cadaver (first generations of the airways), or from histologicalcross sections from prepared specimens, as those provided by the VHP (blood vessels, mainly).

For the particular 3D segmentation of color images from anatomical slices of the whole vasculartrees, there is no current reported literature on automatic methods. However, general segmentationsapproaches as those applied on CT and IRM imaging can be employed. Using the VHP databaseImielinska et al. [Imielinska96] extracted the bronchial tree by hand segmentation by an expert, forpedagogic purposes, as illustrated in Figure 4.13.

Figure 4.13:Hand segmentation and 3D-rendering with color texture mapping of the bronchial tree of thelungs (same VHP database). Extracted from [Imielinska96b].

4.2.2 Detection Methods

Very thin vessel structure has been successfully detected in 2D images by probabilistic approaches,using Markov random fields [Descombes95]. The extension of this method to 3D and large databases

2Thick sections of about 1-1.5cm in a set of 20 images, after Dr. J. Frija, from theHopital Saint-Louis, Paris.


is very difficult and costly. For example, it relies on the use of look-up-tables of pixel configurationsto be penalized or enhanced during relaxation, in order to preserve those corresponding to fine struc-tures. In 2D images,�� configurations are required (eight nearest neighbors) while in 3D up to��

configurations would be needed to be somehow coded, even if only a fraction is needed. Even ifmemory is not the main concern, the energy calculations would require a huge amount of terms to bepre-calculated.

Many other approaches exist for small-vessel detection in 3D, but we describe in the followingthose we have tested and finally adopted:

✦ A two dimensional approach, slice-by-slice. In this approach a 2D segmentation is done oneach slice of the ROI. Slices can be organized (reformatted or “resampled”) in three orthog-onal directions for 2D-image processing. We tried to use the output of three slice-by-slicesegmentations of the same volume. Segmentation itself was tested using:

– Local thresholding in small ball neighborhoods.

– Refinement by filtering from average noisy pixels. This was accomplished by using theSigma-filter.

– Further refinement could be done withhysteresis thresholding; but at this step, we passedto a 3D approach, and hysteresis was employed for a progressive segmentation method.

✦ A three dimensional approach by progressive segmentation, using first a given segmentationresult (2D or 3D) and then the hysteresis approach of adding more information while scanningthe boundary representation extracted from the segmentation output. This step was combinedwith a full 3D segmentation method of region growing, which gave the best result.

4.2.3 Fusion of Cross-Sectional Segmentation

We started our tests with a 2D segmentation of blood vessels in individual images, slice by slice, andthen combining information from the three segmented volumes (orthogonal slices in directionsX� YandZ of the same ROI), which are described in this Subsection. In Chapter 3 we used the originalsets of serial slices (images) in theZ axis as a setfIng� n � �� Nslices� ��, with Nslices ��. The first imageI� corresponds to original color slice number 1320 in the VHP database. Tosimplify notation, we have droped the subindex indicating gamma correction, homogeneization, orbinarization, and we assume to work with a radiometric-corrected set.

The ROIV can be represented as sets of other orthogonal planes byresampling, or reslicing, thatis, reorganizing 3D information into stacked 2D images (recall Figure 3.2, page 147). To distinguisheach set (the ROI itself, expressed in three different ways), we use the following notation, where� isthe color or gray-level label of the voxelv at point�x� y� z� � V :

V �� Ncols� �� Nrows� �� Nslices� �� (4.3a)

IzXY�� f��x� y�� j v�x� y� z� � Vg� (4.3b)

IxY Z�� f��y� z�� j v�x� y� z� � Vg� (4.3c)

IyZX�� f��z� x�� j v�x� y� z� � Vg� (4.3d)

IXY�� fIzXY j z � �� Nslices� ��g� the Axial set� (4.3e)

IY Z�� fIxY Z j x � �� Ncols� ��g� the Coronal set� (4.3f)

IZX�� fIyZX j y � �� Nrows� ��g� the Saggital set� (4.3g)


whereNslices � �� Nrows � �� andNcols � ��. Note thatIY Z , andIZX are extracted fromIXY , by reading either the rows, or the columns in theX andY directions (respectively) ofIXY toform coronal or saggital images. To differentiate between rows (columns) from one set and another,we denote them in an absolute frame of reference asZ-runs of voxels (fixed coordinatesx� y andvarying z), X-runs andY -runs. Figure 4.14 describes orthogonal planes and orthogonal runs ofvoxels.

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

xz

ZX

YZXY

I

II

y

XY YZ

IZX

II

��

��

��

��

(a)

(c)

(d)

(e)

(b)

��

��

��

��

Z

Y

X

Figure 4.14:Orthogonal planes and runs. (a) A volume I can be processed as a set of image slices fIzXY gin the plane XY (axial representation, axis Z). Similarly for the axis X (saggital representation, set fI xYZg)and Y (coronal representation, set fIyZXg�. (b-e) Given a point in the volume �x� y� z� � I, the linear runs ofvoxels parallel to the main axes are univocally identified as X-runs, with fixed �y� z� and varying x, and Y -runs which form respectively the rows (c) and columns (d) of each image IXY . The Z-runs are the transverseruns of voxels (e), that form the intersections of plane images IY Z and IZX .

We consider now the information extracted by a segmentation operatorS (local thresholding,for example) from each setIXY � IY Z � IZX and combine it with the following criteria, to obtain anew set that we denote generically asJXY , plus a superindex “�”, “ ”, or both, to indicate whichcombination criterion was applied:

Conjunctive combination J�XY

If S is chosen or adjusted to segment only “well-defined” structures (possibly causing under-segmentation, because of a “severe” policy), we define:

J�XY

�� S�IXY � � S�IY Z� � S�IZX�

Disjunctive combination J�XY

If S is chosen or adjusted to segment also “not-well defined” structures (possibly causing over-segmentation under an “indulgent” policy), we define:

J�XY

�� S�IXY � S�IY Z� S�IZX��


Associative combinationJ��XY

If segmentationS is for example indulgent with the axial setIXY , and severe with the sagittaland coronal images, then we define:

J��XY

�� S�IXY � �S�IY Z�� S�IZX��

we may equivalently have two segmentation procedures, a severe oneSunder , and an indulgentoneSover , then:

J��XY

�� Sover�IXY � �Sunder�IY Z� � Sunder�IZX��

In both cases, information not present in the segmentation results overIY Z � IZX may serve to“trim-out” the oversegmented setS�IXY ��

Note that intersection and union of data are the same as boolean AND and OR operations betweenvoxels, with binary labels�� .

Combination of different segmentation sets depends thus on confidence on the extracted infor-mation. Our data volume is not istropic, since voxels sides are�Lx� Ly� Lz� � �� inmm, thus information extracted from each individual imageIkZY or imageIkZX is less reliable thanthat extracted fromIkXY which possesses the highest resolution with square pixels.

The combination criteria depends also on data anisotropy and position in space, even if samplingis isotropic. If fine blood vessels are perfectly perpendicular to each planeI kXY , they may be con-founded with texture or noise. Coherence from slice to slice would allow to overcome this drawback.If fine blood vessels are more or less parallel to each planeIkXY , they may be detected by a linear-structure detector. As they are on average slanted, we may restrict the segmentation operatorS todetect elliptical shapes only, because most noise and artifacts won’t exhibit this particular shape, onaverage. Thus some vessels will be better detected in one set of image planes than another, and anoverall conjunctive combination is expected to give the best results.

II

I

I

I

xy 3

xy 2

xy 1

yzzx

��

��

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��

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��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

x y

z

Figure 4.15:Intersections of orthogonal planes with a ramified object.

In the following section we discuss a particular choice of a segmentation operatorS compatiblewith the cross-sectional approach and with a 3D, boundary-oriented approach.


4.2.4 Local Thresholding with Hysteresis

The segmentation method to obtainSover andSunder was an adaptive-local thresholding to whichwe incorporated the hysteresis linking feature introduced by J. Canny in his PhD thesis about seg-mentation [Canny83]. We first explain the local thresholding, then a sigma-averaging enhancement,and finally recall the hysteresis concept.

Local Average. The local behaviour of our approach consisted in selecting a 2D segmentationthreshold in function of the local average per image, on ball-neighborhoodsBr�p� of eachp � V , withradiusr. This segmentation consists in finding first a threshold by visual examination of some ROIsamples, as described in Section 4.1.2, and then by using the background/foreground proportion for adifferent region where background is darker or lighter (as clearly illustrated in Figure 4.6, page 167).To this end, an accelerated local thresholding is achieved using therolling buffer technique presentedin Section 3.3.3, page 156. A typical result with some oversegmentation is presented in Figure 4.19.

Sigma Averaging. We later refined this first step by using theSigma-average filter, introducedby J. Lee in [Lee83]3. Sigma-averaging consists in enhancing the average estimation inBr�p� bydiscarding from averaging those pixels too different from the central pixelp�. If pk, with k �� KN �, are the neighboring gray level values aroundp�, with KN � cardBr � � (the centralpixel is excluded), we first define the list of selected pixels in the neighborhoodB r�p��:

M� � fk j p� � � � pk � p� �gwith � an empirical parameter used to select the pixels to be averaged. Then, the sigma-estimate�p ofthe mean is obtained as [Lee83]:

�p ��

cardM�

Xk�M�

pk (4.4)

When the listM� coincides withBr, then�p is the usual average.

We tested neighborhood sizes ofr � �� , in order to detect the smallest vessels. Following thesuggestion by Hancock in [Hancock91] from the estimation of hysteresis thresholds, the sigma range� for the gray-level window was selected as half the standard deviation of gray-levels inBr�p�. FromSection 4.1.2 we had, inside small blood-vessels the color values�in � �� , and using thered channel (thus, passing to gray-levels), we obtain� � ��in�R � ��.

Hysteresis. Let two threshold levelsTlow � Thigh. When using a single thresholdT such thatTlow � T � Thigh (no hysteresis), the output of the detection operator fluctuates above and belowT along the edges, breaking their continuity. Canny called this the “streaking problem” [Canny83]and introduced a conditional connectivity constraint: in a second pass on the “half-thresholded”image (gray levels� Thigh are labeled with value 255), thresholded voxels with gray level aboveTlow are further selected or discarded if they are connected to well-detected zones (those thresholdedwith Thigh � T ). Gerig et al. applied this criterion for cerebral vessel tree extraction, for globalthresholding [Gerig93b].

Since voxels to be added or deleted first are boundary voxels,this technique was thus suitablefor implementation with our boundary-based methods in a growing-region-like approach. We

3Here the term “sigma” used by Lee has nothing to do with the local standard deviation� or other. For the sigma-average parameter we use the Greek symbol .


Figure 4.16:Sigma-filter applied to gray-level image of slice 1437. Note that blood-vessel borders are sharp(not changed by smoothing) and low-contrasted texture is smoothed.

Figure 4.17:Local thresholding applied to gray-level image of slice 1437 after sigma-filtering (from Fig-ure 4.16), using a window of radius 7. Blood-vessel borders are sharp (not altered by smoothing) and low-contrasted texture is smoothed.


Figure 4.18:Local thresholding . Undersegmentation arises from considering contrasts proportions, ratherthan absolute differences, thus the texture of parenchyma is mistaken as vessel components. However, thereappears a lot of fragmentation.

further added a gradual lowering of thresholdTlow in a second and third passage of the linkingprocess, that is, while traversing the boundary extracted so far, neighborhoods were examinated tosegment and label new vessel components, and then repeat the boundary tracking operation.

Gerig et al. and Verdonck et al. concluded that hysteresis thresholding, without overfilteringimage data in angiographic studies, do not allow acceptable segmentations [Gerig93a, Verdonck96].The VHP color images are not of the same nature, thus the conclusions of these authors are notnecessarily appliable to color images from anatomical slices. Gerig et al. used moreover globalthresholding under the assumption that angiographic images are homogeneous, but we have alreadyseen that partial volume and background texture depend on the structure of the vessels itself whichis not homogeneous in the lung’s ROI. Both authors chose to enhance blood vessel structures by ap-plying other dedicated filtering techniques, Gerig et al. working on the whole volume, and Verdoncket al. locally, using star-contours in a plane perpendicular to the vessels and dynamic-programmingto link each detected contour.

The latter techniques are suitable for single-vessel tracking on smooth-gradient images, as thoseavailable from spiral CT imaging. However, vessels should be relatively large to have a meaningfulcontour diameter, for which Verdonck reported a much better extraction.

In a 3D approach we continue to use local-thresholding with a relative value�T low but withoutincluding yet the costly sigma-filtering, which proved very useful slice-by-slice, as it can be seencomparing Figure 4.19 (just local threshold) and Figure 4.17.

The algorithm ofprogressive segmentation is described as follows (see Figure 4.22). There areremarkable similitudes with a conditionnal MM dilation.


Figure 4.19:Local thresholding with a window radius of 7 pixels. Oversegmentation arises from consid-ering contrasts proportions, rather than absolute differences, thus the texture of parenchyma is mistaken asvessel components (though texture doesincludes sub-sampled small vessels). As in Figure 4.18, there is highfragmentation, too.

PROGRESSIVE SEGMENTATION ALGORITHM .

1 2D local threshold atThigh, with Sigma-averaging, slice by slice, in all the volume.

2 3D boundary extraction of connected components�Vk.3 Scan (traverse) all boundaries�Vk and examine the main-front voxels of each facet (that is,

the front-voxel boundary �Vk).

4 For eachv � �Vk, examine a 3D 5x5 neighborhood; do a local-segmentationof the originalvolumewith a lowered thresholdTlow � p��p, withp the local average read from the originalvolume, and�p � � (the “lowering” that would produce oversegmentation, if applied to thewhole volume).

5 Segmented voxels are automatically connected to the boundaries (since they are 6-adjacent,belonging to all �Vk) and may produce new inter-connections when considering the first-segmented volume, reducing the number of connected componentsk. Segmented voxels arelabeled as foreground.

6 Repeat as in step 2 the boundary extraction of the hysteresis-thresholded volume.

7 Extract all boundaries on the whole volume. If there are many connected components (morethan 10, for example4) then repeat step 3-6, lowering threshold�Tlow by 1 gray-level to pro-

410 is an arbitrary number to allow easy identification with different colors; fragments may initially be just too many(thousands).


Figure 4.20:Local thresholding applied to gray-level image of slice 1437 with Sigma-filtering (from Fig-ure 4.16). Note that vessels are detected at all regions and the interior of large constrasted structures is wellrespected.

Figure 4.21:Segmentation, surface extraction and 3D-rendering of the bronchial vessel tree of the lungs.Method: associative combination Sover�IXY � � �Sunder�IY Z� Sunder�IZX��, where Sunder is a Sigma-average local threshold, and Sover are local thresholds. The arrow indicates an artifact from the left obliquefissure (see Figure 4.3).


mote inter-connection voxels. Else, stop and analyse the biggest components (the extractedtree and some large branches).

++S (V)

S (V)S (V)

S (V)

S (V)

under over

under

>>

(b)

(c)

(a)

real object

over-segmentation

"well" under-segmented branches k under

over

(d)

Figure 4.22:Progressive segmentation. Solid contours represent the real object and light shaded areas rep-resent large vessel fragments k � �� detected by a coarse but restrictive (or severe) segmentation (left),Sunder. This operator can be for example a local sigma-thresholding with Thigh as in our implementation.Missing detected connected components (a) may be detected by “relaxing” Sunder, while retaining connec-tivity (as with hysteresis thresholding, with threshold T low). In a dual approach, dark shaded areas representoversegmentation by a coarse but permissive (or “indulgent”) operator (right), Sover and progressive seg-mentation consists in eliminating spurious areas (b). Sover may be still unable to extract some areas (c) andcreate topological artifacts (d) (see for example the artifact in Figure 4.21). The connected components of theextracted regions are the starting dataset for a more “indulgent” (left) or “restrictive” (right) operator, un-der connectivity constraints. Note that either conditionned growth (dilation, merging) or contionned pruning(erosion, splitting) may proceed by voxel layers, on S�V�.

This progressive method is suitable for a complementary operation (as if exchanging backgroundand foreground), eliminating voxels from the inner boundary voxels (Vk or even �Vk), if theyare oversegmented voxels. In this case, the first threshold is “permissive” (or indulgent) instead of“constrictive” (or severe), and the second threshold is higher than the first one. In such approachthe advantage is to begin with very few connected components (ideally one), and elimination isnothing else than an adaptive erosion. The connectivity constraint is then similar to that of simple-point elimination (but using a gray-level threshold criterion), and the progressive segmentation is


rather similar to a 3D thinning. However homotopy is not necessarily preserved if threshold analysisimposes to unselect a too low-valued voxel.

A third technique was implemented and tested: it consisted in replacing step 1 with a 3D regiongrowing segmentation. It is explained in the following subsection.

4.2.5 Region Growing Segmentation

A widely used method for segmentation of blood vessels isregion growing (RG). It has been alreadytested for the extraction of the blood-vessel tree of the lungs from MRA and contrasted CT images:Pommert et al. [Pommert93], Pisupati et al. [Pisupati95], and Wood et al. [Wood95] report moreor less satisfactory results of this approach. RG complements an edge-detection approach whereinterior regions are not detected by gradient methods, while RG allows to better identify interiorpoints where statistics is robust but detection is less reliable at boundaries (where growing stops).

There are two attractive features of RG:

✦ It can be combined and implemented with the local-threshold method, in order to get robuststatistics, since well-segmented vessels establish a confident reference for the mean value usedduring RG.

✦ It can also be combined with a boundary based approach, since the growing front is itself aboundary.

Furthermore, the change in cardinality of connected components of this boundary is an indicative ofbranching events, or converging/fusing events, depending on the growing direction and the numberof seeds. When considering locally “fast” growing regions, such as blood-vessel branches, a largevolume is traversed (because of the branching nature of the lungs), and only a very small part of theROI is processed for the segmentation by RG (because of the diameter of vessels), which is thus veryeconomical. If the RG region becomes too compact, then the growing criterion is not applicable (asudden increase in volume or “volume explosion”, indicates a large oversegmentation, including toomany background voxels). This sensitivity regarding traversed volume versus grown volume allowsto adjust region growing parameters to maximize voxel-merging in the region, just before attainingoversegmention.

RG is usually accomplished by seeding and growing several regions, which are merged if theyare connected and sufficiently similar. The bronchial-vessel tree of the pulmonar artery is a singleobject, so we worked on the basis of a single voxel-seed for RG, located at the center of one of themain branches of the lower generations. The seedv� is considered to belong to the object but initialstatistics (average and standard deviation) are measured on a ball-neighborhoodB�r�v��, with r � �.After 5 branching generations the vessel diameter is smaller thanr.

We further take advantage of facet information of each voxel that is a candidate for growth, inorder to implement a fast version of traditional 3D RG as described in the following paragraph,using a queue data structureQ for the new voxels and a second queueQstat for the local statistics.Figure 4.23 illustrates how voxels are incorporated from the starting seed, examinating the next-frontvoxels for each facet (oriented boundary face). We call this implementation afacet-driven regiongrowing .

REGION GROWING SEGMENTATION ALGORITHM .

1 v� � Q Queue the seed voxel,


(b)

growV

(a)

∋

f

k

+f

∋v

v++k

∋f

v∋

kf

^

kf

growVgrow

u

V++

k

∋

f

(c) (d)

(e)

Figure 4.23:Facet-driven region growing in 3D. Neighboring voxels of a seed-voxel are examinated follow-ing the 6 faces of the seed (a). For each one satisfying the growing criteria the process is repeated, except forthe initial face (b). All visited voxels are labeled to avoid multiple tests. Shape boundaries are formed wheregrowing stops (c), and the number of connected components of the front increases after bifurcations (d). Thespherical fronts of the figure are approximative (e); at advanced front generations, their shape become irreg-ular. 6-connected neighbors are automatically visited by the facet-representation offsets (LUT of coordinatesfor each facet fk � �Vgrow, and frontal facet neighbor f�k � ��Vgrow).

2 label(v�)=Lab growth label the seed.Calculate thestatistics�v�� of B�r�v��.statistics�v�� Qstat queue the statistics

3 While(Q �� ∅ )

4 f Q� v Unqueue a voxel,

5 Qstat � statistics�v� un-queue statistics.

Updatestatistics�v) of Br�v�.6 � wi � N �

� � i � ��

7 if( label(wi)=Lab growth AND Growth�statistics�wi))=YES)

8 f wi � Q Queue the grown voxel.

4.3 Conclusions and Perspectives. 187

statistics�wi�� Qstat Queue its statistics.

label�wi� �Lab growth Label the voxel.

g9 g End of growing. Voxels labeled asLab growth form a region connected with seedv�.

In steps 2 and 5 average and standard deviation are calculated for the given candidatewi. Thisoperation can be omitted or done from time to time: statistics are thus calculated for each point (verycoslty in time ressources), or just for regions far from the seed, or during first thousand grown voxelsin order to have several new seeds well distributed in several places. One possibilitywe did not testedis to partition the ROI into large blocks, pre-calculate statistics for each block and use this value.

In step 7 a decisionGrowth�� in function of local (or global) statistics is count (we used athreshold criterion equal to the last local average minus half the standard deviation). The final resultsfrom RG segmentation with the progressive segmentation enhancement (with two iterations) arevisualized in Figures 4.24 to 4.27 for two different points of view.

Even if no local statistics are obtained, RG performs still better than simple thresholding, since asmall part of the total ROI is analyzed. When boundary extraction is later used, RG allows to isolatesingle connected objects, while thresholding and local thresholding leave thousands of ”particles”,noise artifacts or fragments of other organs. Several connected sub-branches extracted by RG re-main to be merged into one single tree, if appartaining to the Pulmonary Artery. At the same time,Pulmonary Vein components are also to be identified and separated, as well as the heart components.

4.3 Conclusions and Perspectives.

We presented in this chapter some of our experiences with the 3D extraction and visualization of thebronchial blood-vessel tree of the lungs in man, from serial images of cryosections from the VHPdatabase. Original data presented a number of problems, such as:

(i) color camera callibration and radiometric discontinuous inhomogeneities (for which Chap-ter 3 is specifically devoted),

(ii) a branching fractal structure that makes difficult the separation of background from fore-ground, and

(iii) the presence of several other objects and noise which makes difficult the boundary extrac-tion from segmentation results, since thousands of complex objects remain in the ROI. Localthresholding was useful by selecting pixels for the average to be classified as background orforeground. Selection is based on similarity to the central pixel to be examinated (such slec-tion constitutes the Sigma filter). Combination of results from sets of orthogonal sections gavesome preliminar results, but a full 3D approach gave much better results. The 3D methodconsisted in a preliminar segmentation by region growing, boundary extraction and furtherrefinement by hysteresis thresholding on voxel neighborhoods of the voxel-based boundary.Further tests must be performed to obtain a satisfying validation of the methods and a lateranatomical validation by an expert morphologist would also be valuable.

Besides the reported work, we explored several ideas for designing and implementing tubularextraction methods; some of them using our boundary-based approach, as well as rolling buffers andother concepts described in the precedent chapters. Time limitations did not allow us to present here


Figure 4.24:Best obtained segmentation, surface extraction and 3D-rendering of the bronchial blood-vesseltree of the lungs (region growing and merging of connected voxels). Observe some correspondences with thediagram in Figure 4.4.

in detail all these methods and the preliminar results. However, we give a draft of the main ideas inAppendix B, page 239.

4.3 Conclusions and Perspectives. 189

Figure 4.25:Same segmentation result as in Figure 4.24. Visualization of the depth buffer (gray-levels codedistance to the point of view).

Figure 4.26:Same segmentation result as in Figure 4.24; another point of view.


Figure 4.27:Same as in Figure 4.26; visualization of the depth buffer.

Chapter 5

Analysis and Visualization of ComputerSimulations of AmorphousPhoto-Deposition

Abstract

As an aid to understand models and experimental data of ultraviolet induced chemical vaporphoto-deposition (UV CVD), this chapter explores the use of quantitative and qualitative toolsand methods to analyze numerical volumes from computer simulations of UV CVD produced at theCNET-Bagneux laboratories. We give an account of different discrete-geometry techniques orientedto extract morphological characteristics of bulk (mainly porosity and atom-species mixture), andsurface (mainly roughness) of discrete volume data corresponding to computer-simulated deposits.Emphasis is given on measures and features that can be obtained or estimated through experimentalvalidation, which allows evaluation of the simulator and the physical model. A choice of methodsand specific measures is proposed and some of them were implemented. We finally present someresults from our collaboration with the CNET-Bagneux. The proposed and tested techniques can beapplied in a great variety of microstructural analysis in material sciences and other fields.

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

5.2 Objectives of the Study . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 193

5.3 Monte-Carlo Simulation of Amorphous Deposition . . . . . . . . . . . . . . . 194

5.3.1 Model and Simulation Parameters . .. . . . . . . . . . . . . . . . . . . 195

5.3.2 Morphological and Topological Dynamics. . . . . . . . . . . . . . . . . 196

5.4 Characterization of Deposition Morphology. . . . . . . . . . . . . . . . . . . 197

5.4.1 Surface Features and Bulk Features .. . . . . . . . . . . . . . . . . . . 197

5.4.2 Complex Internal Features. . . . . . . . . . . . . . . . . . . . . . . . . 199

5.5 3D Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

5.6 Analysis methods and feature extraction .. . . . . . . . . . . . . . . . . . . 203

5.6.1 Other Possible Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . 212

5.6.2 Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

192 Analysis and Visualization of Computer Simulations of Amorphous Photo-Deposition

5.7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

5.7.1 Porosity, Roughness, Connectivity, and Anisotropy. . . . . . . . . . . . 213

5.7.2 Surface Density, Interfaces Surface .. . . . . . . . . . . . . . . . . . . 213

5.7.3 Interpenetration Quality and Segregation of a Three-Species Deposit . . . 215

5.7.4 Experimental Validation. . . . . . . . . . . . . . . . . . . . . . . . . . 217

5.8 Conclusion and Perspectives. . . . . . . . . . . . . . . . . . . . . . . . . . . 217

5.1 Introduction

This chapter is based on a draft proposal for a collaboration between theGroupe Modelisation etSimulation de Procedes pour l’Optoelectronique (GMSPO) of the Centre National d’Etudes en Téle-communications (CNET)1, at Bagneux, and theTraitement du Signal et des Images (TSI) Departmentof theEcole Nationale Superieure de Telecommunications2 (ENST) at Paris.

The GMSPO group has developed models and computer simulations of cold, ultraviolet-induced,chemical vapor photo-deposition mechanisms (UV CVD ) in three-dimensional discrete space.”UV CVD” ILaboratory experiments are very expensive and complex, with a high number of parameters to test.Hence,computer simulation of amorphous or crystalline deposition becomes a valuable method tohelp understanding optoelectronic devices, their modelling and fabrication, and for the morphologi-cal study of surface phenomena (reflectivity, conductivity, layer interfaces, etc.), required in appliedresearch of thin films, semiconductor interfaces and the design of photonic devices.

Understanding and evaluating models of physical phenomena, as well as their computer simula-tions, require in general to study the effect of model parameter variations on the numerical outcome ofthe simulation. In the case of UV induced photo-deposition, simulations produce three-dimensionaldiscrete samples, in aface-centered cubic (fcc) grid, where each site (voxel) information may bebinary or includes a gray-level label, for instance. A binary value indicates a vacant or occupiedsite, while a gray-level value encodes information about one or more parameters, such as the type ofatomspecies, or the elapsed time in relaxation cycles units. In this document, the termspecies referto single atoms, ions, molecules, atom clusters, or the substrate chemical composition. The termadatoms refers to adsorbed3 atoms or ions.

Complexity of per-site information depends on what is recorded during simulations (i.e., theinsitu conditions). In our collaboration with the CNET, each site corresponds to the particular discreteposition in space where calculations were done for a single atom or molecule, and microstructureis studied at the nanometer scale. Other deposition or aggregation phenomena may consider big-ger entities (atomic clusters, molecules or particles), and structural features span on many scales[Armin96].

This chapter is organized as follows.

✦ In Section 5.2 we mention the original objectives of our collaboration with the CNET-Bagneuxlaboratories and the expected results.

✦ In Section 5.3 we describe the computer simulation of deposition phenomena, in order tounderstand the origin of structural features of the data to be analyzed.

1http://www.cnet.fr2http://www.tsi.enst.fr3Not “absorbed”, but accumulated on a surface (cf.Webster dictionary).

5.2 Objectives of the Study 193

✦ Section 5.4 presents a summary of physical characteristics such as roughness and porosity,which can be generalized to other kinds of complex volume data.

✦ The 3D methods to represent and quantify the physical features of simulated deposits are ex-plained in Section 5.5 and Section 5.6; we also show in these sections how we took advantageof the boundary-based approach to 3D-analysis, as presented in Chapter 2.

✦ In Subsection 5.6.2 we discuss the utilityof the 3D visualizationof the boundary representationof the volume data, and particular details concerning the present application.

✦ Results so far obtained are presented and discussed in Subsection 5.7.

✦ Some validation issues are commented in Section 5.7.4.

✦ After the conclusions and perspectives, we present in Appendix C an implementation of aDiffusion Limited Aggregation process, capable of generating complex structures useful asphantoms for the development and validation of analytical tools.

5.2 Objectives of the Study

The aim of our collaboration is to develop and test quantitative and qualitative tools and methodsto analyze numerical volumes obtained by computer simulation ofamorphous photo-deposition ofNitride-Silicium. These simulations are obtained at the CNET-Bagneux. Details of the model andthe specific Monte-Carlo simulator of the CNET may be found elsewhere [Flicstein97a, Pata97,Flicstein97b].

Our contribution (ENST) focuses on analyzing the morphology of computer simulated depositsand extracting morphological parameters which will be interpreted and related by the CNET re-searchers to simulation parameters and experimental observations. This kind of correlations havealready been performed, as for example the theoretical, experimental and model/simulation curvesof bulk features (v.g. porosity) and deposition surface features (v.g. RMS roughness) versus temper-ature [Armin91, Flicstein97a].

The specific goals comprise the following:

1. Extraction of morphometric data for each simulation to characterize parameter dependence,mainly porosity and roughness versus temperature, luminance, or incidence rate.

2. Characterization of species mixture quality versus diverse deposition conditions.

3. Visualization (surface renderings) of 3D reconstructions of the simulation data, for differentexperimental conditions. The objective of this visualization is a qualitative display of data,with colors and gray-levels representing data features (e.g., species, connected components,etc.). A selective display of information should provide a quick overview of particular dataand their structural organization.

The CNET will independently validate these correlations and compare them with experimentalestimations on real deposits.


5.3 Monte-Carlo Simulation of Amorphous Deposition

We present in this section a summary of the amorphous deposition models and relevant details of theMonte-Carlo simulation of UV CVD.

The Monte-Carlo character of numerical simulations consists in the computer “throwing” ofrandomized events: random displacement of particles represented by memory locations, particleinteractions, substrate absorption and other. The asignment of event probabilities is based on theoret-ical considerations (more precisely, from statistical-mechanics and physico-chemical models) whichenhance the approximation. After some number of simulated events (or cycles), the outcome ap-pears as a complex distribution of occupied sites in computer memory. The structural characteristicsof this distribution reflects physical features of the model: shape, texture, roughness, anisotropies,clustering, and so for.

The models of amorphous deposition are elaborated cases of a Diffusion Limited Aggregation(DLA) process, as described in Chapter 1, Section 1.4.3. There are however several physico-”DLA” Ichemical constraints in the interaction of aggregated particles (adsorbed atoms oradatoms in theatom-by-atom vapor deposition). An important stage isrelaxation after particle aggregation (atomadsorption). In this stage further interaction occurs along the deposit surface. We are not concernedwith energetic-bond calculations and experimental realization, and limit our attention to the origin ofstructural features, having in mind their quantitative characterization.

Several simple models of deposition exist, giving rise to different aggregation phenomena. Themain models, in increasing order of sophistication are:random deposition, random deposition withsurface diffusion, ballistic deposition, restricted solid-on-solid deposition and spatially correlateddeposition. In random deposition particles fall down at random positions to form a deposit whichexhibits some bulk and surface features, but no lateral movement or interactions are allowed (seeFigure 5.1a-b). Particles stick to the cluster when they touch the top of the last deposited layer or bytouching the side of a particle.

More elaborated Physics-based models exist. A first improvement is lateral displacements mod-eled by brownian-motion. Vulgarization literature often compares this improved model to the computer-game “Tetris” when variously-shapedparticles are considered, and refers to such deposition modalityas “a la Tetris”. Further improvements to the models consist in introducing particle (ion or atom)interaction and surface dynamics in function of the several external and internal parameters, such astemperature, luminance, adatom species and substrate, as described later. This interaction is simplycalled “surface diffusion” or relaxation. With all these model characteristics, real physical depositshave been accurately modeled and simulated, but analytical tools as those presented in this chapterhave not been reported.

The deposition process simulated by the CNET is illustrated in Figure 5.1, in terms of discretesites (voxels). Following the color labels (a-j) in the figure, we explain the stages and events duringdeposition. In therandom deposition and theballistic deposition models the particles fall vertically(a) into a substrate (b) where other particles already form a deposition cluster (c). In the extendedmodels, each particle (d) arrives at growth sites (e), by random walks. Then the particle interactswith the top layer (g), sticking or migrating to an equilibrium position. The final local configurationcreates new growth sites (h), and forbids the access to other sites. Further interactions with next-nearest neighbors fill gaps (i), but there remain inaccessible, stabilized voids as for example those in(j), forming pores andmeanders (see Figure 5.4h and Section 5.6-5d).

In Figure 5.2 a step-by-step description of the computer simulation of vapor deposition processis presented. It corresponds to different simulators mentioned in literature [Eden60, LAFEM92,Merriman98] and to that designed by the CNET. Physical formulations and other details, includ-

5.3 Monte-Carlo Simulation of Amorphous Deposition 195

(h)

(g)

(f)

(e)

(d)

(c)

(i)(b)

(a)

(j)

��

��

x

x

Figure 5.1:Cross section of a deposition model (each cube represents a site). A Physics-based real-istic model comprises lateral displacements, particle interactions and surface dynamics. In randomdepositionand the ballistic depositionmodels, the particles fall vertically (a) towards a substrate (b)where other particles already form a deposit (c). In the extended models, each particle (d) arrives atgrowth sites (e), by random walks (f), interacts with the upper layer (g) and creates new growth sites(h). Further dynamics fills gaps (i), but there remain inaccessible, stabilized voids as for example (j),forming pores and meanders.

ing the particular implementation of the CNET are given in [Pata97, Flicstein97a, Flicstein97b,Armin91]. The parameters listed in Figure 5.2 control the deposition process; they are briefly de-scribed in the following paragraph.

5.3.1 Model and Simulation Parameters

Regarding UV vapor photo-deposition, we considermodel parameters, andsimulation parameters.The model parameters comprise: atom speciesa, substrate speciess, temperatureT , UV irradiationluminanceL, and incidence rate of atomsIr. The simulation parameters include: adatom-adatomdissociation energyEaa, adatom-substratum dissociation energyEas, total precursor ratePr, de-posit efficacyDeff and the neighborhoodcharacteristics for atomic interactions [Pata97, Flicstein97b].In the data-set provided by the CNET-Bagneux, up to three deposited species are being studied: sili-con (Si), nitrogen (N) and hydrogen (H), and there are three substrate species: Indium phosphate (InP), Si, and silene or silicium oxyde SiO�. The participating species determine the dissociation ener-gies, the temporal evolution of photonucleation (the early substrate coverage) and photodeposition.


r aa(a, s, T, L, I , E , E , P , D , neighborhood)rasInitial conditions

(b)

(c)

(d)

(e)

(f)

(g)

(a)

Nucleation center creation (precursors -defects)

fcc (face centered cubic) grid

Amorphous Photo-Deposition

Random deposition on the substratum

Repeat for M cycles

Random adsorption in function of sticking coefficients

Relaxation (diffusion or surface dynamics)

Repeat for N relaxations

eff

Crystalline or amorphous structure

Migration, stability or random desorption ofprobabilites in function of energy barriers

adatoms

Monte-Carlo Simulation

Figure 5.2:Standard model of amorphous deposition with relaxation by Monte-Carlo simulation.Initial conditions (a) consist in: atom species a, substrate species s, temperature T , UV luminanceL, incidence rate of atoms Ir , adatom-adatom dissociation energyEaa, adatom-substratum dissoci-ation energyEas, total precursor ratePr, deposit efficacyDeff and the neighborhood characteristicsfor atomic interactions. See text for details.

5.3.2 Morphological and Topological Dynamics

The effect of the aforementioned parameters on the simulation outcome includesmorphologicalandtopologicalchanges in the deposition bulk and its top-layer surface. Morphological changes includeshape, size, texture, and spatial distribution of features. Topological changes refer to number ofcomponents, connectivity relationships, neighborhood configuration, and holes4. All these featurescharacterize physical or chemical properties in laboratory experiments, and relate to mechanical,optoelectronic and surface phenomena in general.

The termstopology andmorphology bear a different meaning in surface Physics. The first refersto the geometric shape of the substratum, and the second to surface shape and roughness. Internalshape is namedmicrostructure, nanostructure, or ultrastructure when fine details are considered. In

4these features are related to formal definitions of topology properties as those “invariant under continuous transforma-tions”, and are also related tohomotopy.

5.4 Characterization of Deposition Morphology 197

this document,morphology andtopology of the numerical deposit will preserve their mathematicalusage: morphology will refer to shape and structure of one or more objects and their boundaries in 2or 3 dimensions, and topology will refer to cardinality, connectivity and neighborhoodconfigurations.The termssubstratum geometry andmicrostructure or nanostructure, will be used with their surfacePhysics meanings. We use the termsurface with two connotations: (1) as in surface Physics, it is thethree-dimensional top boundary of the deposit (Figure 5.11) which may be sometimes representedby a level-set, that is, a single-valued function of 2D coordinates,z � f�x� y�, and (2), as surfacearea, referring to the bidimensional metric quantity, expressed in squared-length units. The contextshould clarify which meaning, (1) or (2), is intended.

Since morphology and topology features change in function of simulated or real experimentalconditions (sets of parameters), there exists an evolution or dynamics associated to time-varyingconditions. The interest in knowing their characteristics is the possibility of controlling such struc-tural features, with some particular goal. This includes the design of solid-state or opto-electronicdevices, or the control of particular physical properties of interfaces, from mechanical stability todielectric and refraction coefficients. As in any dynamical system, researchers of the CNET expectto identify phase-transitions, and extremal numerical values (e.g., minimal porosity, maximal mix-ture of components), corresponding to particular parameter-settings. Figure 5.20 in page 220 showsa graphical example of transitions in the distribution characteristics of pore shape and size.

In the following sections we describe in more detail the main structural features and their discreterepresentation, in order to make clearer how a quantitative analysis can be done.

5.4 Characterization of Deposition Morphology

We examine first the most relevant and global characteristics of microstructure in amorphous depositsand then the more complex features arising from connectivity inside the bulk. The latter are closerto three-dimensional textural features and our analysis applies to other kind of complex sets (particleclusters, sponge-like and networked structures).

5.4.1 Surface Features and Bulk Features

Morphology features of a deposit are broadly classified in “external” or surface features and “inter-nal” or bulk features.

External or surface features are often represented byroughnessof the external surface (the lastdeposited monolayer). In Image Processing,intensity may be interpreted asheight, and con-versely. Thus, the surface texture may be viewed as the projection to infinity of each top-layersite height (see below Equation 5.6), giving a one-to-one function of coordinates x,y, whichmay be subject to wave or periodicity characterization, that is, by frequency domain analysis,in particular by examining thepower density autocorrelation function.

Internal or bulk features are exemplified byporosity of the material, which may include not onlythe volume densities of material/void, but also the size distribution, or other features of inter-stices and cavities.

Surface roughness and bulk porosity have been the most common studied features, mainly be-causetheoretical modelsandexperimental validationenable their estimation on real data. For ex-ample, roughness relates to surface refraction index, measurable by ellipsometry techniques [Guenther84],


or by laser speckle metrology [Briers93]. Porosity is in turn estimated by backscattering analysis,or X-ray density measurement. In amorphous deposition models by the CNET, roughness is simplyrepresented by the root mean squared (RMS) deviation from the mean height� h � of the lastdeposited layer (see Figure 5.3):

h�RMS ��

NxNy

Nx��Xi��

Ny��Xj��

�hij� � h �� (5.1)

where � h ��

NxNy

Nx��Xi��

Ny��Xj��

hij � (5.2)

wherehij is the top-surface height at growth site�x� y� � �i� j�, Nx � Ny � �� Nz � �� arethe deposit space dimensions, andNxNy � �� is the number of substrate growth sites in asquare grid. In the simulations of the CNET, energy calculations restricted simulations to very lowresolutions in the three axes, but mean height (directionZ) was allowed to attain up to 90 monolayersin some simulations. Due to discretization and scale, this low resolution makes very imprecise theFourier, multiresolution and scaling-behavior analysis of surface characteristics, which are moremeaningful for resolutions starting from�� .

hij

xi

z

y

j

Figure 5.3:Top-surface heigth z � hij at growth site with coordinates �x� y� � �i� j�. Mean height� h � is the average of all hij in a deposit of �� growth sites. Root-mean-square heightdeviation from � h � defines the RMS roughness(see also Figure 5.13). Simulation data providedby the CNET, visualization by the author.

5.4 Characterization of Deposition Morphology 199

An example of a theoretical model to predicthRMS evolution (sometimes namedRMS rough-ness), is theKPZ nonlinear equation (Equation C.1 in Appendix C, page 250), where parameters areestimated from the deposition model (random deposition with relaxation, etc). A short exposition isgiven in Appendix C.

In a broader sense, roughness is a much more complex notion, closely related to surface texture.Concerning porosity, the CNET uses in its experimental protocols the standard definition:porosityis the ratio of vacant-sites volume to total deposit volume. As for roughness, “porosity” also encom-passes more complex structural features of bulk, as explained in the following paragraphs. Withthe simple definitions of RMS-roughness value and void/material volume ratios, dependence on op-erating conditions of surface roughness and bulk porosity can be predicted [Flicstein97b]. To ourknowledge, more complex morphological characteristics have not been studied until now in amor-phous deposition simulations and similar volume data.

5.4.2 Complex Internal Features

Complex relations of internal features arise when connectivity is taken into account. Connectiv-ity may be ’weak’ or ’strong’, giving rise to ’closed’ pores or ’open’ pores, respectively (see Fig-ure 5.4). Closed pores orbubbles, are more or less round in shape (’compact’), while open poresmay have more complex shapes, may contact the space around the deposit, or even may form anintrincate network through the deposit. Open pores are sometimes considered as interconnected bub-bles [Armin91], and the size of theconnection neck determines a degree of ’openness’5.

(f)

(e)

z(b) (c)

(f)

(a)

(g)

(d)

(h)

Figure 5.4:Cross section of amorphous photo-deposit bulk of a single atom species: growing axisZ (a) in monolayer units; closed pore or bubble (b); external wavy surface (c); surface overhangs(d), substrate (e); ”open” pores (f) and (h) with a connection neck (g); surface meanders (h) (also”open” pores). Notice that meanders and overhangs imply one another.

More complex features may become meaningful when considering several atom species (or sitelabels in general) in order to characterize species segregation and inter-species mixture quality (Fig-ure 5.5). Such complex features include the contact surface between different bulk components, themean local neighborhood composition of a site, size or shape of homogeneous bulk components,

5A formal treatment is possible usingMathematical Morphology (MM), if we identify the minimum detectable size ofthe neck with thestructuring element size of MM operators.


etc. Other global attributes can be induced by simulation conditions, for example: the anisotropyof shape characteristics, and three-dimensional “textures”. Many features of the deposition modelsare similar to random-field model realizations, which may producecoherent structures, as for exam-ple: filamentary, trabecular or column structures, granular, flake or layered structures (see Chapter 1,Section 1.2.5 and Section 1.6).

A practical consequence of all these structural relations and features is that even if rough esti-mations of porosity and roughness can be made by simple volume scans, there are new structuralcharacteristics that must be taken into account, such as connectivity relations6. Thus, connectedcomponents need to be extracted, and this is where we have participated using the discrete geometryapproach described in Chapter 2.

Bulk internal properties are very difficult to study or to isolate by simple slice-by-slice analysisor sampling tools. However, three-dimensional reconstruction makes available a set of methods todirectly measure some of those 3D internal features, and to estimate or just make evident (i.e. visible)other characteristics. These methods include volume processing, three-dimensional visualization,shape morphometry and Mathematical Morphology (MM) filtering (“shape filtering”).

Quantitative analysis frequently requires a processing stage, which may include some filtering,using for example MM operations. An example is the decomposition of the bulk species or theporous phase into ’weak’ connected components, whose number and shape distribution can thenbe analyzed. Another example of MM analysis isGranulometry, as detailed in Section 5.6. Thus,MM-based processing can be applied to simplify (by separating, or regrouping), select or discardspecific shape components, before analysis. Some shape modifications may also be performed (shapesmoothing, for example).

(b)

(a) species 2species 1

Figure 5.5: Amorphous photo-deposition of two atom species: low-quality mixture (a) is charac-terized by species segregation, small simple interfaces and large volume clusters of each species;high-quality mixture (b) is characterized by high species inter-penetration, large complex interfacesand small volume clusters of each species.

We explain in the next section the application of our boundary-based approach to analyze mor-phologic characteristics of amorphous depositions resulting from computer simulations.

6In percolation aggregates connectivity is the most important feature.

5.5 3D Methodology 201

5.5 3D Methodology

Besides traditional 3D analysis, we use a discrete-geometry approach for surface extraction, 3Dimage processing, morphometry and visualization of volume data with any degree of complexity. InChapter 2 we have described how external surfaces orboundaries of detected objects are representedby lists of facets (voxel faces). An important step of the 3D image processing consists in analyzingand modifying these lists and their associated data structures, by examination of neighboring facetsand voxels in order to perform the desired operation or measurement.

Let us consider foreground and background boundaries in amorphous deposits. The first bound-aries comprise particle clusters and the second all voids or pore structures. With our methods, theseboundaries are extracted by traversal of adirected graph associated with the discrete surface of eachobject, and a voxel representation is replaced and simplified to a face representation, which preservesall connectivity relations, since all connected components are identified. Neighborhood relationshipsare taken into account by coordinate offsets from each facet, combining the 6-, 18- and 26-adjacentvoxel neighbors (see Subsection 2.3.5). These neighbors correspond to thefirst, second and thirdnext-neighbors or also thenearest, next-nearest and next-to-next nearest neighbors in the terminol-ogy of Crystallography, solid and surface Physics (see Figure 5.6).

Figure 5.6:First-order, K-adjacent voxel neighborhoods, with (a) K � �� in 2D, and (b) K � � �� in3D (same as Figure 2.7). (a) Voxels of E� are named the nearest neighborsof the central site, while (b)-(a),i.e., E��nE� are the next-nearest neighborsand (c)-(b)-(a) (i.e., those sharing a vertex) are the next-to-nextnearest neghbors.

The three following examples illustrate the interest of our approach:

1) Filtering: a morphological conditional erosion (or background dilation) becomes a selectiveelimination of voxels only at the object’s boundary�V ;

2) Morphometric calculations: the volume (or any higher-order moment) of an objet is calculatedfrom its boundary, using the discrete Gauss Theorem (or Divergence Theorem), given in Equa-tion 2.12. In general, instead of a single object, we may also consider a particle distribution,and study its bounding volume, centroid, of other morphometric features.

3) Local analysis: Neighborhood analysis or processing operations can be done exclusively onthe object boundaries. See Figure 5.7.

4) Visualization: to render realistic shadings the local normal is estimated from the facet neigh-borhood configuration, and only the external surface components are displayed.


Interactive visualization allows to perform complex analysis tasks by proper chaining of bodyselection criteria (size, shape):

✦ regrouping,

✦ shape-filtering by MM operators,

✦ boundary traversals for labeling of specific configurations,

✦ other boundary-based morphometry or processing.

As noted in Chapter 2, our software includes some standard volume processing and volume mor-phometry tools. The format of volumic data is also compatible with the software package TIVOLI(Traitement d’Images VOLumIques) of the TSI departement, at the ENST [Geraud98].

��

��

��

|| w-v || < r }

��

v

��

��

rB ∋N | round( )

��

(v) = { w 3

��

��

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��

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��

��

��

��

��

��

��

��

��

V

V

(c)

��

(a)

(b)��

��

��

��

discrete sphere:

��

��

��

��

��

E6E (v)

��

��

(v)18 (v)26

. . .

∋

��

��

E

Figure 5.7:Neighborhood operations on geodesic discrete boundaries. (a) Local analysis or processing (e.g.MM operations) follow the voxel-based boundary V (only a 2D profile is shown). Euclidean neighborhoodsinclude from (b) single voxels, to K-adjacent voxel neighborhoodsEK , with K � � �� , and even r-balls(discrete spheres) Br�v�� withv � V.

Quantitative analysis of species segregation (mixture or inter-penetration quality) is a good exam-ple for which our boundary-based approach has useful properties. In Section 2.3.6 we have describedvoxel and face neighborhoods to explore on the boundary interfaces of each species, in order to mea-sure the local composition of atom species (See Figure 2.8). Figure 5.8 illustrates in a 2D slicevarious shape and structure configurations: top-surface boundary, pore boundaries and interfaces be-tween al least two speciesA�B. Two (or more) species give rise to sets of deposition clustersVA�VBconstituting the bulk. Inter-species boundary characteristics in 3D are shown in Figure 5.9.

Notice in Figure 5.9 that the internal intersection (e)VB VC is the interface (d) between bothspecies. Also note that in voxel representation�VB VC� �� V , and in a face representation�VB VC� � �VB �VC .

Given for example 3 species, 100% of background neighbors correspond to no mixture at all, anda 25% of each other species labels per voxel ideally corresponds to perfect mixture. Other kind of

5.6 Analysis methods and feature extraction 203

(a) (f)

(c)

��

(d)(e)

(g)

��

(f)

(h)

��

��

��

(b)

��

��

��

��

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��

��

��

��

��

��

��

��

��

��

B

VA

Lay

ers

z

zero

last

VV

��

c

Figure 5.8:Top-surface-boundary topology and pore-boundary topology. Upper layer of the depositas a 3D face-boundary �V cjzlast (a), where meanders and overhangs (b) are taken into account.Last-layer voxels (solid boundary) are in a lighter shade. The entire boundary of the depositionbulk includes the closed pores (c). In this case, the background V c in (f) (i.e., the complement ofthe foreground V) is said to have several connected components, with N�c boundaries �kVc� k �� N�c. Also, two or more atom species a (labels A (d), and labels B (e)) give rise to a partitionof the bulk V (the foreground) into several connected components (V �

� VA � VB�. This partitioncreates boundary interfaces (g), constituted by all intersection components of ��VA� ��VB�. Shapeof the substrate (h) (”geometry” in surface Physics) can also be taken into account as an additionalboundary, the boundary of the ”zero” or null layer �Vjz��, which constitutes part of the initialconditions.

local information is obtained from geodesic neighborhoods which are calculated in a system of ori-ented faces, as described also in Section 2.3.6 (See Figure 2.9 in page 107 which shows first, secondand third order geodesic-neighborhoods). Table 2.1 and Table 1.4.1 from Section 2.5.1 summarizessome morphometric measures and operations obtained with our boundary-based approach.

5.6 Analysis methods and feature extraction

In the next paragraphs we list the simplest and most representative features that may be studied tocharacterize bulk and surface morphology of a discrete scene (volume), once the connected com-ponents extraction and representation has been performed. All features assume ’closed’ pores, but’openness’ may be taken into account. The main items describe the general feature to be character-ized, and their following sub-items the specific measure or analytic method to be employed. A shortdiscussion is given for some of them.

1. Bulk porosity in a deposit with one or several species.


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��

��

��

��

��

��

��

��

�� B C

C

B

U

A

(e)

(a)

(d)

(c)(b)

Figure 5.9:Discrete inter-species boundaries (interfaces). Surface (a) with labels corresponding to3 species A, B, and C. The boundary (b) of the B/C inter-species boundary VB VC (or interface)is constituted by polygonal contours ��VA VB�� VA VC� and ��VB VC�. A voxel (or site)can have only one associated label (c), but interfaces are implicitly defined by two voxels with twodifferent labels sharing the same face (e) (a single site may thus share up to six different interfaces).

(1.a) LetVa denote the atom-species aggregates, witha � f�� g corresponding to atomspecies Si, N, and H. We proposed and implemented the following measures:Npores number of pores,NVa number of aggregates,Vpores volume of pores,Va volume of aggregates of atom speciesa,S surface area of�Va andS�V surface/volume ratios of pores or species aggregatesVa

We further proposed to compute histograms and observe the correlation factor as functionof a simulation parameter (T , L, etc.; we actually did it forT ). Other studies may bedesigned, such as a histogram analysis of the maximum diameter of balls contained inthe bulk, or the porous phase, and so for.

(1.b) The same as above, taking into account the “openness” of the pores. A differential com-parison may help to evaluate validity of the “closed pore” approximation.

2. Morphometry of species aggregates and pores.

(2.a) Surface density per volume unit and volume density. These measures can be calcu-lated for pores, or clusters of one species in relation to another (species density). The”surface” for surface density here refers to theN�a boundary components�jVa, with


j � �� N�a anda � f�� g. Inter-species boundaries (interfaces) are defined asthe intersection sets:�V� �V�� V� �V�, and�V� �V�, in face representation,where�Va �

�S�j �jVa. When the objects are the pores themselves (a �� f�� g),

we useN�c to denote the number of background components (the complement of fore-ground, the bulkV � V� � V� � V�). Pore boundaries are thus denoted by�kVc, withk � �� N�c, where inter-species boundaries are not included, since the foregroundis all the bulkV . Pores boundaries and inter-species interfaces are illustrated in 2D slicesin Figure 5.8 and 3D interfaces in 3D are shown in Figure 5.9.

(2.b) Voxel-based boundaries give rise to similar measures (e.g.,V� V�), which is a closedsurface, from the point of view of facet boundaries, while�V� �V� is a boundary patch.Furthermore, what is measured as “solid surface” (card Vj) has a different meaning,which can not be directly validated by physical experiments (these are internal surfacesmaking up the bulk porous structures). What turns out to be more meaningful, is toanalyze some average internal surface density� V , interpreted assite densityusing Eu-clidean neighborhoods ofv � V , Equation 2.9:

� V �� card�Er �v�� card�Nr�v� �� (5.3)

(2.c) Local topology (number of components, neighborhood connectivity). These measuresshould provide information about two opposite qualities:granularity andpercolation,that is, to which extend the bulk aggregates are structured as a network.

(2.d) Granulometry for a fixed parameter set [Coster89]. The MM operation of “opening”7

is applied several times, and the size and number of residual objects is recorded aftereach opening. The curve of size distribution should be characteristic of the aggregationprocess (small or high population of big, medium and small sized components). A similaranalysis can be done with the complementary MM operation of “closing”.

(2.e) Vertical distribution of pores. The growth axis of the deposit is a privileged direction,and distribution of pores in this axis can be expected to be non-uniform, in functionof the number of cycles,T , and other parameters. Thez component of the Center ofMass (CM z) of each pore should provide a simple, meaningful measure of pore spatialdistribution.

(2.f) Pore anisotropy. Orientation of pores is expected to be isotropic at least in theX andY directions. Vertical-elongated or flat- horizontal pores may be important if anisotropictrends are present in theZ axis. Simple measures of pore anisotropy can be obtained fromthe eigenvalues of each pore’s tensor of inertia; for example the ratio:�max��min� �min��, or those introduced in Section 1.2.1. The corresponding eigenvectors are theobject’s principal axes.

3. Mixture quality estimation with several species.An “inter-penetration” measure of two or more species may be estimated in several ways:

(3.a) By contact area or area of the inter-species boundaries (i.e., the area of the interfacesbetween different-species components): a high contact area corresponds to high inter-penetration and low segregation.

7An erosion operation thins objects, shrinking them into points, filaments or membranes, whiledilation thicks objectsand fills gaps. Topology changes usually arise after combination of these operations. For example, erosion-then-dilation(or opening) splits weak connected objects, widens concavities and filter small particles, while the dilation-then-erosion(or closing) connects close objects, and remove small holes.


(3.b) By local inter-species topology characterization; for example, the average number ofneighbors� NN � of different species. With 3 species and a fcc grid (6 faces), anoptimum value of� NN � would be close to 2 (in average), which corresponds to“perfect” inter-penetration through the bulk. These measures may be taken only at thespecies interfaces, or through the whole bulk volume.

(3.c) By granulometry analysis per species, followed by comparison of residual size distri-butions for different sets of parameters: very sparse and small residuals (or none) corre-spond to low segregation and high mixture quality; big residuals indicate high segregationand poor mixing of the species.

4. Deposit characteristics evolution, as a function of one specific parameter.Deposit simulation of a single species may include a label per site, as noted in the Introduc-tion. This label can be an integer value in the interval�� for each site, representingan in situ variation of some varying parameter recorded during one single simulation, or formany accumulated simulations with the same initial conditions. An input look-up-table wouldmap the integer valuesf�� g to the real physical model values for a later correlationanalysis. In this particular kind of labeling, each isovalued bulk layer is analyzed and visual-ized separately, allowing to make a sequence of “level sets”, in function of the time-varyingparameter (see Figures 5.10 and C.1). While studying the effect on bulk features of particu-lar time-dependent variations (of temperature, precursor rate, or anyorder parameter), phasetransitions and critical states, if present, may be detected and further studied.

Lz2Lz1

Figure 5.10:Deposit evolution (one species), as a function of the changing parameter L (e.g., lumi-nance). In this example, gray-level labels encode at each site a parameter value fromLz� to Lz�, withz�� z� as functions of height, in monolayer units. Early nucleation dynamics is expected to shapebulk and surface characteristics (compare early (light) and advanced (dark) profiles after growth).

5. Top-surface roughness of the deposit.There are two posible definitions for the external or top surface of a deposit: thelast-layer topsurface , and theprojected top surface .

The last-layer top surfacetakes into account the meanders and overhangs and is constitutedby the last deposited layer of adatoms. It is denoted as:

�Vjzlast (5.4)

and is an open surface (see Figure 5.11). For a precise definition, we consider the background,constituted by a closed surface including the scene boundaries in contact with the deposit, andincluding the last-layer top surface; we denote it as�V cjzlast (see Figure 5.8a). Let�U be the


box bounding volume of the scene (aNx�Ny�Nz parallelepiped). The last-layer top surfaceis then defined by:

�Vjzlast

�� opface��Vcjzlast n �U� (5.5)

where theopface�� is theopposite face operator, which swaps the face orientation interpreta-tion fromu� to u�, andu� to u�, with u � fx� y� zg.Theprojected top surfaceof the depositV is the set of surface points seen from infinity, andsimply denoted by heightz � f�x� y� (or alternatively, the intensity imagez � I�x� y�)), andin discrete coordinates,x � i � �� Nx � ��; y � j � �� Ny � ��. In a formaldefinition off�x� y�, let �a� b� denote the open set froma to b, then:

f�x� y�� fz j �x� y� z� � �V and �z�� Vc g (5.6)

In practice, projected top surfaces are easily calculated from projecting a vertical line fromthe upper bounding planez � Nz down to the deposition volume, for each pairx� y, andreading the first intersection with�V to build an arrayff�i� j�g � fhijg (see Figures 5.3,and 5.11). We also denote�Vjz�� the face-based representation of the projected top surface.Notice that in generalcard��Vjz�� card��Vjzlast�, and if meanders are important,�Vjzlast

occludes itself and parts of it are invisible fromz � �. Using these definitions, we proposethe following roughness measurements, which include the simplest measurements that can beobtained using our 3D boundary-based approach:

(5.a) Surface roughness analyzed as image-intensity texture,

(5.b) surface density,

(5.c) average extrema width at mean height,

(5.d) mean width-to-height ratio and

(5.e) density of line intersections.

These measurements are examinated in the following paragraphs.

(a) (b)

x

z = f(x,y)

x

z = f(x,y)

Figure 5.11:Three-dimensional renderings of the z � f �x� y� projected top surface of a deposit. (a)discrete surface rendering (facets of the fcc-lattice); (b) triangulated surface rendering. Simulationdata provided by the CNET; same simulation as in Figures 5.3, and 5.12.


(5.a) Surface Roughness Analyzed as Image-Intensity TextureInterpretation of heightz � hij as gray-level intensity values for pixel�i� j� leads toimage representations of the projected top surface of deposits (see Figure 5.12). In orderto use frequency domain analysis tools (power density autocorrelation), a higher sub-stratum size is desirable (at least�� sites) to obtain more meaningful resultsand reduce low-resolution artifacts (e.g., the Gibbs effect at the borders). Neverthe-less, the roughness of low-resolution substratums may be studied with features otherthan the RMS value of the deposit thickness (Figure 5.13). The most simple “texture-from-roughness” features in classic image processing include:edgeness, obtained fromthe image Laplacian, thus depicting slope properties,density surface per unit of vol-ume, relative extrema density (see Figure 5.14) [Haralick92]. Gray-level co-occurrencematrix (GLCM) features may also prove useful, as they depict second order statisticsof spatial correlations between pairs of separated pixel values (with height interpretedas intensity). Two-dimensional extrema characteristics can also be estimated from one-dimensional profile statistics throughIntegral Geometry tools (also known asStereolog-ical quantification [Weibel79a]). Some of these tools, applied widely in texture analysis[Jain89, Chu90], includerun-length statistics, density of line intersections, average ex-trema width at mean height, and mean width-to-height ratio [Corkidi98].

Figure 5.12:Two-dimensional image display of the z � f �x� y� projected top surface of a deposit,as visible from axis Z. Gray-level intensity represents height, and roughness details turn into textureand shape features. Data provided by the CNET; same simulation as in Figure 5.3.

(5.b) Surface DensityIn our discrete-boundary approach, several estimates of discrete surface are available,with different accuracy/performance ratios. For comparison purposes, we are concernedwith relative measures. The simplest local surface-area measure is the number of bound-ary facesNbf �p�N � in a 3D neighborhoodN �p�, for each pointp � �Vjzlast, withp � �i� j� hij� lying on the last-layer top surface (see Figure 5.3, at page 198). It isthus defined as:

Nbf�p�N �� card� �Vjzlast N �p�� (5.7)


As the deposit is small (�� sites), we have considered the 26-connected neighbor-hoodN �p�

�� E�� consisting of� � � � � voxels. The average surface density is

the ratio� Nbf � �V �E�� Nbf �, calculated on allp � �Vjzlast. Otherpossible first-order neighborhoods are shown in Figure 5.6, where each voxel representsa site from the fcc-grid. By varying the size of neighborhoodN , a fractal dimensiondescribing roughness could be estimated from the log-log correlation between size andsurface density. We only estimated surface densities of the last-layer top surfaces with afixedN .

(5.c) Average Extrema Width at Mean Height

(e)

(c)(b)

(a)(d)

Figure 5.13:(a-d) Some profile measures related to surface roughness (any cross section throughaxis Z). (a) mean height � h � in monolayer units; (b) peak width at mean height; (c) valley widthat mean height; (d) RMS height-deviations from � h �; (e) mean free path at height h. Similarmeasures can be obtained by exchanging the roles of deposit and pores as foreground/background.

(5.d) Mean width-to-height ratio of extrema. Corkidi, Marquez et al. recently introduced aroughness quantification measure in surface-intensity images, themean depth-width ra-tio of extrema (MDWRE) based on a stereological approach [Corkidi98]. This parameterJ ”MDWRE”proved to be very useful as textural feature for the classification of metaphase images oflimphocytes in optical microscopy images. In the present work we changed the notationand name convention of MDWRE, since we deal with physical heights, instead of inten-sities, and also adopt the notation used in surface Physics. We thus introduce it as theMean Height-to-Width Ratio of Extrema (MHWRE). J ”MHWRE”

Haralick and Shapiro [Haralick92] have analyzed a simple and effective texture featureproposed by Rosenfeld and Troy [RosenfeldVTA70]: the number of extrema per unitarea in row scans for a texture measure (Relative Extrema Density (RED)). We studied amore robust feature sensitive to differences in the spatial distribution of specific extrema,adapted for noisy and very low-resolution images, which appear as sub-sampled textures.This feature is well adapted to very low-resolution surfaces from deposition simulations(�� sites), where Fourier analysis is poorly suited, due to quantization artifacts.

Let us consider one-dimensional scans or profiles from an image; that is, 1D signals.A single rough or wrinkle feature on a profile can be characterised by depth or height(grey-level intensity) and width (separation between extrema). Instead of counting se-lected peaks as the RED-like feature, we considered height-to-width ratios of all valleysin all scans. Heights weighted by the inverse width of each valley permits selective fil-tering of specific ratios, making MHWRE tunable to structural-induced texture at very


low resolutions [Corkidi98]. These lengths are also known in surface Physics aschar-acteristic width �k andcharacteristic height �. Thus, we measured the horizontal (forwidth) and vertical (for height) projections of vectors� and� at each valley of the profile(Figure 5.14), and summed their ratios in all scans. Notice that peak (or valley) heighti� i � �� N � � is the average�� i � i��, and the peak (or valley) widthi isthe sum�k i �k i, withN the total number of one-dimensional extrema (roughs) of theobject (either valleys or peaks).

The height selection ”window” consisted in testing the following conditions:a) the ratiobetween the min/max vertical peak projections should be greater than 0.5 (approximatevalue where the human eye distinguishes a roughness feature or wrinkle in textural im-ages) and,b) the peak height (the average of the two vertical projections) should begreater than a preestablished threshold of noise� (smallest wrinkles):

MHWRE � ��N

N�Xi��

wi i

where

i �� i � i��

�k i �k i

and

wi �

�� if min�� i�� i�

max�� i�� i��

and �� i � i��

� otherwise

The windowwi allows to measure only certain peaks (or valleys): those which are narrowand depth enough.

ζξ ||

ξ

||

ζζ ξξξ ζζ

(a)

ξ T

(b)

(c)

Figure 5.14:Characterization of roughness by relative extrema in cross sections through axis Z.Valley left-side height (a), and Valley left-side width (b) are also the coordinates of vectors � (leftarrow) and � (right arrow) joining minima and maxima in scanline profiles. (c) Similar vectorsdefining peaks. Simple features of roughness are derived from properly combining these measures.

(5.e) Meander Analysis.Traditionally, pores are more likely to be treated as internal cavitiesor bubbles separated from the exterior of the bulk. In a closer look, many open pores usu-ally contact the external surface, and all “true” surface features and pore features shouldtake into account interconnections that may tunnel deeply inside the bulk, themeanders


of the external surface are not visible in a vertical projection of the deposit (Figure 5.15b),and are then often mistaken as pores. A meander is accompanied by a correspondingoverhang, the bulk portion above (Figure 5.15c). A complete, unbiased, morphologicalcharacterization is possible by analyzing a true, “internal-and-external” surface, whichincludes the projected top surface (the one visible fromz � �) and the non-visiblemeander surface (which we called the last-layer top surface). Both approaches providetwo sets of measurements of porosity and roughness. Differential comparisons betweenboth sets should provide a measure of “meander-ness”, indicating how important areinterconnections between open pores and external surface.

(b) (d)

z(a)

(f)(e)

(c)

Figure 5.15:Dashed contour (a) represents a section profile of the projected top surface f�x� y�,as visible from z � �; open pores (b) form surface meandersunder overhangs(c) that may bemisinterpreted as closed pores (d) when the projected top surface (a) is extracted. When using af�x� y� representation, the observed deposit (e)�(f) has neither meanders nor overhands (f), whichcan lead to a misunderstanding of the underlying physics.

Table 5.1 summarizes a choice of:

Column 1: The physical properties to study.

Column 2: The simulation methods or protocols such as required samples or sequences in function ofT ,L, simulation parameters, etc.;

Column 3: The main features and measures to characterize them (there may be others).

All tests were done on a basis of data sequences in function of one parameter (T ) of about 8 volumesamples. Results were returned in sorted tables, and color GIF images. Analysis, interpretation and


presentation of morphometric and visualization results, in terms of the Monte-Carlo simulation andthe physico-chemical models, were performed by the CNET, with participation of the ENST. De-sign of complex analysis, as granulometry, meander-ness, and percolation shall be further discussed.Mixture quality analysis includes measures of species segregation and interpenetration. Custom orin-detail studies such as specific visualization settings, granulometric sequences, or feature evolutionfor in-situ variations, may be considered, after careful design of the simulation/analysis protocol.

Table 5.1:Table of microstructural features, method, morphometrical parameters and calculationtime. Bold names indicate those features actually measured on deposition simulations from theCNET, in function of temperature T.

Physical feature Method/Protocol Parameters

Porosity(T) sequence 1 V , S, NumberOpen porosity (T) idem V , S, NumberPore distrib. Z(T) idem CMZ

Pore anisotropy (T) idem �max��min� �min��

Granularity(T) idem HistogramsOpen granul.(T) idem HistogramsPercolation samples Local topologyOpen percolation samples Local topologyRoughness(T) sequence 1 surf. dens.:�� Nbf �

Open roughness(T) idem surf. dens., MHWREMeanders(T) idem differences with/without TSGranulometry samples Hist. openings1 species visualiz. sequence 1 pore distrib.Mixt.Qual.2 sp.(T) sequence 2 bulk distrib.Mix.Qual.3 sp.(T) sequence 3 bulk distrib.3 species visualiz. sequence 3 bulk distrib.

5.6.1 Other Possible Analysis

Scale invariance properties, described by fractal dimensions and critical exponents, have foundmuch recent attention in porous media and surface-Physics literature [Armin91, Armin94, Wong86,Jacquin87, Ruffet91]. A pertinent example is a structure with a large surface-to-volume ratio, pro-duced by intricate foldings found in networked pores or interfaces between different species. Owingto the space-filling properties of fractals, a high fractional dimension of contact surfaces betweenspecies phases imply high species interpenetration, hence, a high mixture quality. A second applica-tion of fractals is the scaling behavior analysis of in pore structures, and features such aslacunarity,defined in Section 1.7.1 would be employed. Nevertheless, the same as with frequency domain(Fourier) analysis, in order to estimate fractal dimensions, self-similarity properties and other mul-tiresolution features, a larger volume simulation is required, say�� voxels. There aremany other measures useful in microstructural characterization, and some of them may eventually beconsidered in our collaboration, but previous experience must be acquired before exploring furtherpossibilities. It has been noticed by the researchers of the CNET that, no matter how descriptive arethese morphological and structural features, interest in some of them depends on feasibility of theirexperimental validation. This concern would require the development of stereological techniques for

5.7 Results 213

parameter estimation on slice samples, when it is not possible to obtain full experimental equivalentsof computer simulations of volumic data.

5.6.2 Visualization

Three-dimensional color renderings of internal structures allow to assess at once many of the afore-mentioned features and their spatial distribution, and help to better choose which measures shall prop-erly characterize the observed features. Visualization with the programs we developed may be highlyselective, allowing to render visible only certain specific components or slanted slices (stereotomies)or sub-volumes, to make evident anisotropic trends or internal relations. With a sensible design ofthe display settings, assigning color maps and intensity cues may quantitatively represent numericalvalues (a local size density, for example) to convey information in a way that no table of numericalmeasures or correlation curves may be able to do. In the long term, a simple visualization sequence(animated, or in a mosaic display) may replace the trial of a high number of combinations of morpho-logical/model parameter correlations. For example, a phase transition may be more easily detectedby visualizing in sequence the bulk simulations for different luminance sets: transition of one deposi-tion mode to another may be gradual or discontinuous, being characterized by specific trends in poredistribution, even if porosity remains more or less constant. In this example, simple rendering of thepores themselves, in a set of simulations with a progressive luminance scale may allow to visuallydetect at what value a transition occurs and what kind of morphological features it characterizes.

5.7 Results

We present in this section some preliminar results representative from our current collaboration withthe CNET. These include 3D renderings and data correlations with temperature variations. Somesimulation runs have been very time-demanding, and some features of the simulator of the CNET arestill in the development stage. At present, we have tested a small subset of the proposed measures toevaluate bulk porosity (Figure 5.16), surface roughness (Figure 5.16), and inter-penetration quality(mixture) of multiple-species simulations (Figure 5.16b).

Detailed reports of results concerning visualization, porosity distribution, roughness and theirinterpretation in terms of the simulator can be found in [Pata97, Flicstein97, Guillonneau98].

5.7.1 Porosity, Roughness, Connectivity, and Anisotropy

5.7.2 Surface Density, Interfaces Surface

Note the lost of meaning of the classical RMS roughness measure in the last-layer surface of Fig-ure 5.18a. Surface density describes better meanderness, but experimental validation works on atop-projected-surface interpretation as in Figure 5.19b and c, presenting a ’visibility’ effect, if pho-tons or electrons strike the surface perpendicularily to its plane. Note other characteristics of poredistribution.


a bc d

Figure 5.16:3D-reconstruction of selected pores. Empty space and pores are renderedin opaque colors, and the bulk (the deposition material) is transparent in the figure.Color labels identify each pore. (a) All pores present in the amorphous photo-depositionof a single species, at T � ��K, and 20 monolayers. (b) Same as (a), with selectionof pores smaller than 5 atomic sites. (c) Rendering of the 20 biggest pores. (d) Idem,from another point of view.

Numerical simulations from the CNET were obtained for varying temperaturesTn � f�� Kg, with n � �� . A transition or critical temperatureTC from high tolow porosity is easily located at the minimum of the distribution of porosity against temperature:

TC�� argmin

Tn� porosity�Tn� ��

It is shown in Figure 5.20c, that is, atTC � ��K � ��, at a resolution of 7 samples on a��Ktemperature interval (only four samples are shown).

There is also a “structural” transition betweenT � ��K andT � ��K (See Figures 5.20aand b); located at the temperature which maximizes the mean volume (or other size measurement)of pores:

Tvol�� argmax

Tn�V olk�Tn� �� with k � �� Npores�

5.7 Results 215

o700 K400 450 500 550 600 650

*

*

**

*

100

*

125

150

75

25

50

0

Temperature T

Por

e nu

mbe

r (v

olum

e <

20

site

s)

o700 K

µm3

400 450 500 550 600 650

0

Temperature T

Por

osity

with

out m

eand

ers

200

100

300

400

500

600

*

*

* * *

*

o700 K

µm3

400 450 500 550 600 650

0

Temperature T

Por

osity

with

mea

nder

s

200

100

300

400

500

600

** *

*

*

*

700 Ko

0.02

0.04

0.08

0.06

0.20µ

0.10

-1m

400 450 500 550 600 650

0

Temperature T

Sur

face

den

sity

rou

ghne

ss

**

*

**

*

a bc d

Figure 5.17:Porosity and roughness measures obtained from deposition simulations fortemperatures from T � ��K to T � ��K. (a) Number of pores having a volume ofmore than 20 sites; (b) global porosity, without meanders; (c) global porosity, includingmeanders; (d) surface-density roughness in function of temperature.

Other state transitions could be analyzed in a similar fashion for other morphological or structuralparameters, and different simulation conditions. It is in particular interesting the rough to smoothtransition, which can be observed also experimentally by atomic force microscopy and refractionindex ellipsometry.

5.7.3 Interpenetration Quality and Segregation of a Three-Species Deposit

The mixture of three absorbed species in amorphous photo-deposition presented interesting featuresduring simulations, difficult to assess without a three-dimensional discrete representation. Once seg-mented with one color label per species, each connected network of deposit material (one species)can be visualized in isolation or together with the other species cluster. The results illustrate ourwork with mixed species to show interpenetration and interface between different labels. Visualiza-


ab

Figure 5.18:Top rough surface of amorphous deposition data. (a) Projected surfaceat T � ��K, with several holes corresponding to meanders and overhangs. Somepores are shown. (b) Projected surface at T � ��K, with holes blocked out to forma solid boundary. Empty space and pores are rendered in opaque colors, and the bulk(the deposition material) is transparent in the figure. (Simulation data provided by theCNET).

5.8 Conclusion and Perspectives 217

tions consist in cross sections of 3D segmented volumes and 3D renderings of them (bulk with anycombination of labels ”on” or ”off”). These results are summarized in Figure 5.21, for a 20-layerphoto-deposit and Figure 5.22. Porosity is rendered as voids, whose cardinality and morphology isnow relative to each species or combination of species, and is thus ”multiple-defined”, since 0-voxels(voids) share neighboring sites with labels from different species.

5.7.4 Experimental Validation

There are two aspects concerning validation, the analytical methodsper se, independent of data, andthe morphometrical data themselves (simulator and modelling).

Validation of the analytical methods on numerical data has already been done for many morpho-metrical parameters extracted by our 3D software, using synthetic shapes orphantoms in discretegrids. More sophisticated computer phantoms, for texture and complex growth modeling, have beenrecently tested for binary or multi-valued labels given by Monte-Carlo simulations ofDiffusion Lim-ited Aggregation (DLA) with relaxation [Witten81, Armin91, Armin94]. A brief exposition is givenin Appendix C. Even with very simple particle interactions, they can emulate some features ofcomplex morphology similar to many real aggregation processes (Figure C.1). Their morphologicalcharacteristicsper se have been controlled to a certain degree, first without reference to a particu-lar physical phenomenon. With a proper integration of physico-chemical constraints, they may beeventually be related to the amorphous deposition models and simulations from the CNET. We havealso noticed that aggregation and deposition process, 3D texture simulations and DLA model char-acteristics relate very closely and could be modeled with those ofrandom Markov fields, at leastqualitatively. It remains to investigate more closely this relationship.

For the data series, experimental validation techniques exist in some cases. In particular,RMS-roughnessandporosity can be measured from ellipsometric analysis and other techniques duringa real deposition. This provided recent validation of the simulator and the theoretical model behind,for both characteristics [Guillonneau98, Flicstein99]. In order to apply all other analytic methods toreal, experimental data (mainly to validate the physico-chemical models and their simulation imple-mentation), three-dimensional digital samples should be extracted. For example, a per-slice imageacquisition should be first implemented from X-ray diffraction, Rutherford Back-Scattering (RBS) orTEM/SEM techniques [LAFEM92], then a 3D alignment of image slices and a 3D-image segmenta-tion process should all be performed, in order to attain the same starting condition as with computersimulation data (ie. digital volumes in a fcc grid). Otherwise, if no direct comparison of simula-tion and real experimental results is possible, some real-deposition features can still be estimated bystereological analysis of slice samples, or other indirect measurements.

5.8 Conclusion and Perspectives

Quantification performed on simulation results of amorphous or crystal photo-deposition will al-low researchers of the GMSPO-CNET or similar groups to assess, on a selective basis, the sizeand shape of pores and deposition clusters, the porosity and feature distribution in function of tem-perature or other parameters, and the inter-penetration quality of multi-atom species simulations.Visualization of bulk and parameter distribution in space should also provide efficient, global qual-itative evaluations of complex structures and phase transition phenomena, in response to different


parameter settings of simulations. On an interactive basis, visual cues may guide the selection of themorphological analysis to be applied, help as a feedback of the simulation conditions, or assist incomputer implementations of new improvements to the deposition models. In short, the final goalsof 3D quantification of computer simulations are: (1) to reduce the time and cost of experiments,(2) to help understanding of simulation process and their parameter dependences and (3) to providenumerical means for testing, through simulations and experiments, the theoretical models.


a b

c d

Figure 5.19:True rough surface and complement of the bulk of a �� deposit atT � ��K. (a) Last-layer surface, with meanders and overhangs. No pores are shown.(b) Projected surface without meander voids (similar to Figure 5.18b). (c) Pores andmeander voids interpreted as “open” pores. Subfigures (b) and (c) constitute togetherthe complement of the bulk. (d) First 100 biggest pores, when identifying and separatinglarge meanders (not shown). (Simulation data provided by the CNET).


a bc d

Figure 5.20:3D reconstructions of amorphous deposition in function of temperature.(a) Surface and pore distribution at T � ��K, with several connected componentsand 100 Kcycles (about 100 monolayers). (b) Surface shape and pores formed at T ��K, (c) at T � ��K, and (d) at T � ��K. Empty space and pores are renderedin opaque colors, and the bulk (the deposition material) is transparent in the figure.(Simulation data provided by the CNET).


a bc d

Figure 5.21:Mixed species. Axial sections of a photo-deposit at heights (a) z � �, (b)z � ��, (c) z � �� and (d) z � ��. Contours of level-set (regions in black) account forsurface waviness. Color represents each species: Si in yellow, N in red and H in blue.See also Figure 5.22. (Simulation data provided by the CNET).


Figure 5.22: Mixed species. (a) Clusters of amorphous photo-deposition of three species (Si inyellow, N in red and H in blue.), and one single connected component by each species. Occupiedsites (the bulk) are now rendered in opaque colors and empty space and pores are transparent. Seealso Figure 5.21. (Simulation data provided by the CNET).

Conclusions and Perspectives

The present work comprise an exploration of complex-shapes and complex-structure character-istics in 3D, and a development of several methods of representation, processing, visualization andmorphometric analysis based mainly on discrete boundaries. These methods can take into account thepotentially arbitrary nature of complex shapes. We have further applied some of them in two specificapplications, one in Biomedicine and another in surface Physics where complex-structure featuresare present: the extraction of the blood-vessels of the lungs from serial slices at high resolution, andthe analysis of 3D data from simulations of amorphous photo-deposition.

Previous personal experience in other applications, mainly biomedical, guided some of the topicspresented in chapter 1 and 2. Since literature abounds in multiple meanings of “complexity”, and nu-merous shape features exist, it was difficult to choose or to build a guiding thread from the beginning.Boundary representations themselves (the “main thread”) may not be well-defined for fractal objectsandR�. We noted that several definitions or calculations must be adapted or radically changed whenpassing to discrete spaces (e.g., connectivity, solid interior, normals, surface area, etc.).

From the conclusions already drawn in each chapter, we summarize here our most relevant con-tributions:

✦ To our knowledge, very few works have tried an exhaustive exploration of the concepts charac-terizing structural complexity. Chapter 1 constitutes an overview of literature from many fieldswhere we identified common properties of complex features, without attempting any indepthdescription (which can be found in the cited references). Number of components, features,local coherence, dimensions and lenght of descriptions are key characteristics, together withthe relative absence of one or many of several invariances (scale, topologic, geometric, etc.).We linked several of these concepts with our boundary-based approach (Chapter 2) and thesolution of specific problems in Chapters 3, 4 and 5.

✦ Boundary-based descriptions (orb-reps) constitute another key concept in our work, and sev-eral features can be defined or mesured from boundary information. We implemented someprocessing and analytic methods based on b-reps, and made use of both, the face-based bound-ary �O of an objectO and its voxel-based boundaryO. Facets allow to define and obtainestimations of the local normal, which was used for visualization, and has a high potential foranalysis in the normal directions of the local surface boundary. In the other hand, voxel-basedboundaries allow for example to define and perform mathematical-morphology operations.

✦ Regional-based descriptions were not explored enough by us to make a useful comparisonwith b-reps, but we used a hybrid approach in the segmentation of the vascular (blood-vessel)pulmonary tree, by applying region growing and boundary extraction, from which a refinedsegmentation was obtained.

✦ A tool related to regional analysis is the moving-discrete neighborhood known asrolling bufferor sliding window. We used a 2D-implementation of this tool to scan adjacent images for local

226 Conclusions and Perspectives

color-histogram analysis. A 3D-version was employed in a preliminar implementation for aCentral-Axis detector in tubular structures.

✦ We were the first group reporting the radiometric inhomogeneities of the Visible HumanProject database, and proposed an original method of correction, by analysis of locally differ-ential images. The method itself exploites a feature examined in Chapter 1, that ofstructuralcoherence. We also deviced several color visualization methods to make evident the inhomo-geneities in the serial images.

✦ We were also the first group to obtain automatic extraction of a substantial part of the vas-cular pulmonary tree of the VHP. There are several reasons for the lack of bibliography: (i)the radiometric inhomogeneities that make the use of global and local segmentation methodsunadapted, and (ii) the size of the volume and the nature of the data which are only avail-able since 1995. Our results remain to be validated, since no measurements and comparablesegmentations of the same dataset exist until now in literature.

✦ The amorphous-depositionapplication in collaboration with the CNET gave us the opportunityto demonstrate the pertinence of some features and means to quantify them; these includeboundary-interfaces, and the taking into account of meanders and overhangs, which are nottraditionally studied, due to lack of analytical tools (existing literature in this kind of analysisdo not report comparable methods). We exploited local connectivity properties at the atomic-site level to extract large-scale, topological features of voids (pores). We proposed severalways to characterize porosity and roughness, and tested some of them with success. Finnaly,3D-visualizations of labelled volumes further allowed to make evident critical transitions withtemperature variations in the bulk morphology and the pore distribution of several simulationruns.

To mention further perspectives of our work, we require an understanding of our failures and therecognition of the weak points of the present work:

✦ Since a great quantity of topics were treated or approached at the same time, some lack ofdeph and organization resulted. A more natural strategy for such a general subject would bethe constitution of a line of research in 3D analysis and visualization of complex objects. Thiswas, as stated in the Preface, our personal objective, but the lack of an effective collaborationin various fields and the difficult in obtaining numerical data reflects how ambitious the taskwas.

✦ The Geometry of Digital Spaces (as termed by Herman et al., [Herman98]) is a very recentfield of developpement in which discrete Mathematics play a fundamental role. We did notpresented nor used some important concepts (e.g. Jordan paths and graph representations), anddid not give enough emphasis to topological notions inN

� (e.g., simple point remotion), whichare timely notions in complex shape modeling and manipulation. Further work is required inthis direction to fully exploit b-reps, processing, analysis and visualization using them. Someother concepts were only drafted or used scarcely (e.g. geodesic neighborhoods and the run-lenght compression nature of the face-based boundaries).

✦ We presented only few quantitative results (data tables and graphics), and chose rather to havefirst a qualitative evaluation of various algorithms, through 3D renderings and other visualiza-tion modes. Other numerical tests and performance were indeed done to validate and evaluateboundary extraction, processing and morphometrical methods, but not presented here due totime limitations.

Conclusions and Perspectives 227

✦ We did not exploit all the color information present in the VHP images, and we did not demon-strated that the red channel indeed is the only useful information in this case.Principal Com-ponent Analysis and adaptedColor Space representations could be employed for this set ofdata. A 3D morphological approach is feasible, extracting and examinating the structures incolor space of the clouds of color points.

✦ Some quantitative work remains also to be done, such as comparing time processing of a wholescene against processing of only the boundaries of the objects of interest (the superiority of thelatter seems evident but overheads in boundary traversal have to be carefully estimated, forexample).

✦ Computer implementation of the user interface in particular requires improvements and somework remains for a full exploitation of our software and their integration within the ressourcesof the TSI Department.

Topics that represent other interesting tracks to explore in future works include the following:

✦ Many of the concepts characterizing complex shapes, such as fractals, shape factors, Kol-mogorov Complexity and other constitute an unexplored field of potential morphological fea-tures in which analytical methods based on discrete boundaries may prove very valuable.

✦ Strong relationships between object structure and texture were indicated at several points ofthis thesis, e.g. the coherence principle used for radiometric homogenization, the componentsof the bronchial blood-vessels that become “background” texture at a given resolution, and thepore distribution in amorphous photo-deposits. It would be valuable to explore the transitionbetween one and another, and study the relations between textural features and shape featuresin rough surfaces, for example.

✦ Various ideas for tubular-structure detection and Central-Axis-Tree extraction were proposedand preliminar implementations are on course. These should allow a throughout explorationof branching structures in the lungs and their morphometrical analysis (diameters, lenghtsand bifurcation laws). Some of the presented methods (see Appendix B) are complementaryapproaches to contour tracking and snake approaches, and interesting comparisons could bedone.

✦ Local neighborhoods combined with either regional representations (Central Axis Tree, or Me-dial Axis Transform in the case of a tubular branching structure), or boundary representationsconstitute a subject with high potential, in view of geometry-driven analysis and scale-spacerepresentations which were not treated in the present work.

✦ Due to the lack of experimental data we explored very briefly the synthesis of complex objectsby Monte Carlo methods (such as those produced byDiffusion-Limited Aggregation simula-tions in 2D and 3D), in order to obtain various complex-structure situations in which analyticaltools may be tested (see Appendix C). In such a case, b-reps serve not only for analysis of databut for simulation purposes. This research track is however a subject belonging to model-ing objectives, in which physical information has to be included, in order to obtain realisticmodels.

Along this manuscript we saw how discrete boundary representations are present and timelyin several fields in science and engineering, and we believe that our contributions can be furtherextended and exploited in the characterization, analysis and visualization of complex data.

228 Conclusions and Perspectives

Appendix A

Further Definitions of Object Subsetsand Boundary-Related Subsets

In this section we introduce further concepts and definitions concerning interiors, sub-boundaries,cavities and their organization. Some of them are used to describe complex features, specially inChapter 1. We propose also some equivalent concepts in discrete spaces, in Chapter 2. As with withthe preliminar notions of Section 1.1, general references are [Weisstein98, Kiyosi96a, Kiyosi96b,Iyanaga68, Koenderink90, Coster89, Barnsley88, Herman98, Klette98]. We examine first the notionof Solid Interior of an objectO.

Solid Interior of an Object

In this work we are in particular interested in boundary representations and other definitions that usethem. Some difficulties arise when trying to define what is the interior of an objectO � R

n, ex-pressed in terms of its boundary, even when it has one-single connected component (no cavities). Wechoose to avoid altogether a formal definition ofSolid Interior in the continuous case, assuming thatit is equivalent to the objectO itself, as a set of points which define it,given only the information ofits boundary�O, when such representation is possible. In the discrete case, several “filling” methodsexist to find points inside�O, and we give Definition 2.15, page 96. When cavities are present, butthe largest boundary is considered, then the solid interior is the objectplus its “filled” cavities, includ-ing thus all voids bounded by sub-boundaries geometrically bounded by some connected componentof �O. Figure A.1 illustrates solid interiors, cavities, fragments and nested boundaries, introducedlater.

We denote such a representation ofO, its “solid interior”, and use the following notation:

interior��O� (A.1)

To be more precise, and firmly set the foundations of some concepts later introduced, we wouldrequire anoperational definition, which does not exist, to our knowledge, for the general case of anyobjectO � Rn. If O has one single boundary, we may try to express such a solid interior as:

interior��O� �� fp � O j � p T � Rn� R

n� q � �O� such that p � T �q� g(A.2)

230 APPENDIX A. Further Definitions ofO Subsets

This presuposes the existence of a method or functionT which describes (and allows to find) aninfinite set of pointsp � O, given someq � �O. This makes better sense as a limit-process of adiscrete representation, but also has several problems.

An example of suchT in 2D is the classical test of a pointp as belonging or not to the interiorof O, by considering the integral of all subtended angles bewteenp and all contour pointsq � �O.If this sum is 0, thenp � O, and such test may take into account voids and nested boundaries,iforiented contours are defined.

Definitions in terms ofCauchy sequences, accumulation points and other entities would be alsopossible, but several discontinuous, fractal objects pose an operational-definition problem which isto-date an open issue, not treated here. To be strictly formal, we may restrict our study for objectswhere such aT can be found even in implicit form (v.g. most smooth objects, deterministic fractalsand other sets), and take equation A.2 as an operational definition of solid interior.

For an object with several boundary components�O, the latter definition is extended to includepointsp � Oc, inside the cavities ofO, and inside the main, largest boundary component ofO (seeFigure A.1). This solid interior blocks out the cavities and gives a set of points spatially containing,but different fromO itself, if we take only one connected component of�O, containing all other.This will be precised later with “nested boundaries” and the “outermost boundary”.

O

OkOk

interior( )

(b)(a)

Figure A.1: (a) An object O with cavities; its boundary has several components, such as �kO. (b) Solidinteriorof �kO; cavities are “filled”. See also Figure A.2.

Even if no cavities are present (one-boundary component), a distintion betweenO and its solidinterior arises in discrete representations where discrete boundaries are sets of facets or voxels, etc.,which have a finite extension. A useful operational definition is given in Chapter 2 (see Defini-tion 2.15), in terms of local discrete orientations, and will allow to extract the solid interior of anobjectO, given only its boundary representation�O. This definitions can be extended for a multiply-defined boundary and negative boundaries.

We finnaly note that, for feature extraction, something equivalent to an operatorT ��O� do existsin differential geometry: boundaries that can be described by analytic functions may be used toextract some morphometric features of an object, using for example the Green Theorem in 2D andGauss theorem in 3D (which expresses a volume integral in terms of a surface integral defined byboundary of the region in question). Thus, information from the solid interior of suchO may beobtained from its boundary specification.

APPENDIX A. Further Definitions of O Subsets 231

Nested sets, Cavities and Solids

We introduce in this section nesting relationships, which allow to describe several ways in whichboundary sets may be organized to form cavities and unconnected components, and other elementsof a solid.

DEFINITION A.1 (Sub-Boundary)Let O be an object whose boundary is composed by N� connected components that we denote as�jO, with j � �� N�� Thus, �O �

�SN�

j�� jO. The subsets �jO are called a boundarycomponentor sub-boundary of �O iff

��i� j � �� N�� i �� j� ��iO �jO� � ∅ � ��jO is a closed surface��

Note that not any subset of�O is a sub-boundary. We also expressN� as the cardinality of theboundary, that is,N�

�� card��O�. The solid interior of�O is then defined also as:

interior��O� ��

N��j��

interior��jO � (A.3)

We will describe in Section 2.4.4 an explicit method to find the discrete solid interior of a givenboundary, using horizontal run-lengths. The method is also equivalent to discretevolume traversal,volume filling, or 3D flood of region boundary. Such a boundary may be that of an objectO, or anyof its fragments or cavities, as defined later.

DEFINITION A.2 (Nested Boundary)Let �O �

�SN�

j�� jO, a boundary consisting of multiple-connected, closed components. A bound-ary component �iO is a nested boundaryof �jO iff interior��iO� � interior��jO�, for somei� j � �� N��, with i �� j.

We also say that�iO is embedded ornested within�jO (for somei� j with i �� j), or that�iO is achild of �jO and�jO is theparent of �iO. Figure A.2a and Figure A.3b show an example of nestedboundaries.

NOTATION A.1 (Symbol “is nested within”) We introduce the symbol “b” to represent the ex-pression “�iO is nested within�jO”:

��iO b �jO� �� interior��iO� � interior��jO� �

Remarks. All the properties of the relation symbol “�” are inherited by “b”. The symbol�bmeans“not nested within” or “not a child of”, etc. Symbol “c” applies in reverse order, meaning the relation“is parent of”.

DEFINITION A.3 (Direct Child)Let O an object with N� boundary components �iO, with i � �� N��. Let �iO b �jO . We say thata �iO is a direct child of �jO iff

� � k � �� N�� iO b �kO b �jO� (A.4)

otherwise, �iO is called an indirect child of �jO, and �kO is an intermediary child between�iOand �jO. Similarily we define direct and indirect parent.


In the example for 2D boundaries of Figure A.3b-f boundaries 20,24 and 26 are indirect children of��O, while the 32,33,34 and 35 are direct children of��O.

DEFINITION A.4 (Negative Boundary)A boundary component�kO � �O is a negative boundaryif it has no solid interior, i.e, �interior��kO�n�kO� � ∅ .

Remarks. An analytic formulation of oriented boundaries could also be employed for definingsigned boundaries, if certain regularity conditions are satisfied (e.g., to be a regular and continuous-differentiable mapping). In that case we could define�O as “negative” if a certain surface integralsatisfies:

V �O� ��

ZZ�O

f�s� � n ds � ��

with n the outer surface normal vector at eachp � �O, with knk � �; ds is the surface differentialelement, andf�s� a vector function such thatjV �O�j is the volume ofO.

A non-empty and closed boundary�kO � �O is called apositive boundary iff it is not anegative boundary (i.e.,�interior��O� n �O� �� ∅ ). We also call the solid interior of a positiveboundary itspositive interior , and use notation:interior� �O �

� interior �O. Important remark:we reserve notation�� and�� for other purposes. To indicate that a subset�kO is a positive (+) ora negative (-) boundary, we use the notation “�kO�” and “�kO�”, respectively, which is consistentwith the following definitions.

DEFINITION A.5 (Fragment of an Object)Let �kO� � �O be a positive boundary, with k � �� N� indexing all boundary components of�O. The set O�

k�� interior �kO� is called a fragment of O.

Remarks. Fragments of an objectO are connected components of the foregroundO, and if frag-ments have no nested boundaries,O � �N�

k��O�k . Otherwise, we need to account for the cavities to

be defined in the following paragraphs. For negative boundaries we denote first theirnegative solidinterior, which allows to define cavities ofO. We again define it here as the set of points, foundfrom the boundary information�kO�. An operational definition, as with solid interior, is possible indiscrete sets.

NOTATION A.2 (Negative Solid Interior)Let �kO� b �O be a negative boundary component nested within �O. The negative solid interiorof �kO� is denoted by:

interior��kO�� (A.5)

Note that it is different from the positive solid interior ofO�, which may correspond to allp � Oc if �O has a single connected component. Together with the boundary componenet itself,the negative solid interior of�kO� constitutes a cavity ofO:

DEFINITION A.6 (Cavity)Let �kO� b �O. The set O�k

�� interior��kO�� kO� is called a cavity of O.

Cavities of an objectO are connected components of the backgroundOc (not necessarily all ofthem) including negative boundaries of the object. We also define thesolid exterior of �O as thecomplement of its solid interior1:

1Recall that no formal definition was given for solid interior for continuous sets.


)

+

out

out

O

O2 out

-

O6

6

+

12O

19

5O

+O =19

O =

interior ( )

p’

O

2

5

interior(

O13

1

p

][1

1 O

19O

p

O19

+

O

interior ( )

interior ( )6

O19

O-

O =

O = 6 O

(d)(c)

q

(f)(e)

q’p

(a) (b)

O =

q

c

p’

q’q

6 exterior( )

Figure A.2:Nested boundaries and interior sets in 2D. (a) Object O with multiple-connected, closed bound-aries: �O �

SN�

k �kO, where N� � card��O� � ��. Shaded area represents the solid interior of �O;boundary �� O is nested within ��O, which is in turn nested within ��O. Boundaries 6, 12, 13 are examplesof negative boundaries, and boundaries 1,2,5 and 19 are examples of positive ones. Their solid interiors formrespectively some cavities and fragments of O. (b) Outmost boundary �Ojout , composed of five connectedcomponents, which are non-nested within other boundaries. (c) Solid interior of �� O, (d) Interior of bound-ary ��O. (e) Solid exterior of boundary ��O. (f) Note that Oc

� � exterior��O�. Notice the intersectionproperties of points s � �pq � O�, for some p� q � �� O, for example. These intersections may characterizesmooth interiors, but may not hold for irregular objects. (f) Note that an “exterior” of a sub-boundary may bethe complement of its interior, such as Oc

� � exterior��O�.

NOTATION A.3 (Solid Exterior)Let �O be a 3D boundary composed by one connected component. The solid exterior of �O isdenoted by:

exterior��O� (A.6)

DEFINITION A.7 (Absolute Solid Interior)Let �kO b �O be a boundary nested within �O, with k � �� N��. The absolute solid interior of�kO is the set:

jO j � j interior j��kO � �� (A.7)�

� p�q��kO��

p ��q

fs j s � pq �interior��kO � � interior��kO ��g


42

5O

3

O19

1

38O

16

12O

40

O6

54

86

28

2

30O

7

O2

18

O1

17

13O

151413

26 29

121110

30

3632

+

312725

9

37 3834

23

33

O

41

24

3935

19 20

32

43

O

O23

2221

O

(f)

(c)

(d)

(e)

(a)

(b)

(g)

Figure A.3:Tree structure representation of nested boundaries in 2D. (a) Object O with multiple-connectedclosed boundaries: �O �

SN�

k �kO, where N� � card��O� � ��. (b) Subset organizationas a containmenttree structure; nodes represent each boundary (or component O�); for clarity only indexes are shown. (c)First branching level corresponds to outmost boundaries1-5; levels (c),(e) and (g) contain positive boundaries(fragments) and levels (d) and (f) contain negative boundaries (cavities).

Remark 1. The exterior of�O excludes the boundary itself, and it is the complement of the solidinterior. For boundaries without cavities,interior��O� � O andexterior��O� � Oc.

Remark 2. In union-set operations, the solid interior of a negative boundary is to be subtracted.These definitions allow to treat in a consistent fashion objects with nested cavities and non-connectedcomponents. A consistent connectivity remains to be included for the discrete case.

Remark 3. Operational definitions of discrete volume and discrete surface are given in Chapter 2.We only note here that we define the volume of a negative solid interior to be a negative quantityV �O�k � � �, wherejV j is the volume contents of the absolute solid interiorjO�

k j.A boundary consisting ofN� multiple-connected and closed components can be partitioned into

N�� positive andN�

� negative boundaries, withN�� N�

� � N�:

�O �� N��

i�� interior��iO�n�iO��∅

�iO�� N��

j�� interior��jO�n�jO� ��∅

�jO�

(A.8)

or using notation with superindexes (N �� N

�� O��O��:

�O ��N�

��i��

�iO�� N�

��j��

�jO��

(A.9)


When considering the corresponding solid interiorsO i�Oj (positive or negative) of nested bound-aries, that is, fragments or cavities ofO, we have alsonested objects and say that “Oi isnested withinOj”, and represent this property by the expressionOi b Oj , which is a slightly stronger conditionthanOi � Oj if we consider cavities and fragments as belonging or not to each object. In practicewe consider negative boundaries nested within a positive boundary (nested cavities), or conversely(which define solids within cavities). That is,O�

i b O j .

Remarks. We formulate remaining definitions using condition “Oi � Oj ”, rather than “Oi b Oj”,which needs further study.

DEFINITION A.8 (Empty Fragment and Empty Cavity) A fragment (or cavity)O i � interior ��iO�,with �iO � �O is called empty fragment (or empty cavity) if it has no children (no nested subsets).That is, ��j � �� N�� j �� i�� Oj �� O i .

Remarks. In R� andR� a containment tree structure representation off�iO� i � �� N�g,empty fragments (cavities) areleaves (terminal nodes).

We define also the external oroutmost boundary as the union of connected boundary componentsin whose solid interior lay all other nested boundaries; i.e, those boundaries which have no parents(see Figure A.2b and Figure A.3c):

DEFINITION A.9 (Outmost Boundary)

Let �O ��

SN��

i�� iO�� SN�

�

j�� jO�� be a boundary consisting of multiple-connected closedcomponents (fragments and cavities which may be nested). The outmost boundary ofO is the set:

�Ojout�� f�kO� � �O j k � �� N�

� � (A.10)

�� j � �� N�� j �� k� �kO �� interior ��jO�g

That is,�Ojout is the set of all boundary components that are not children of (i.e., not nested within)other boundary components (we restricted the definition positive outmost boundaries). The outmostboundary is also defined as the boundary of the union of all solid interiors, excluding the cavities(which possess negative interior):

�Ojout��

�N��

i��

interior��iO��n� N�

��j��

interior��jO��

(A.11)

DEFINITION A.10 (Filled Interior)The filled interior of O is the set: �O� �

� interior� �Ojout �.

DEFINITION A.11 (Principal Background)The principal background , or principal exterior of O (or principal component of Oc) is the set�O�c �

� �exterior� �Ojout ��.

Shaded areas of Figure A.2b represent�O� � interior� �Ojout �. Note that�O� describes an objectwith all its cavities “filled” or blocked out; that is,�O� c is a connected set (one connected-componentobject). Porosity of an object can be characterized by studying the filled interior and the solid interiorof its boundary. We can define for example a porosity measure as the ratio:

porosity��

volume�interior��O��volume�interior� �Ojout ��

(A.12)


In Chapter 5.1 we have done some work with discrete open boundaries (interfaces), but we justgive a provisional and intuitive definition for the continuous case:

DEFINITION A.12 (Open Surface)Let Q � �O, and let C� �

� �Q be the set of border points of Q. The set Q is a connected opensurfaceor a boundary patch ofO iff C� �� ∅ .

In physical objects, basically in region transitions,C� consists usually of closed contours. Surfacephenomena, textures and surface patterns may produceC� consisting also of clouds of points, fila-ments or paths on a surface boundary.

From the notion of open surfaces, we are able to give a definition forinterface boundaries,through the notion ofrestricted boundary, but first introduce some notation conventions.

NOTATION A.4 (Set Restricted to “Condition”)Let A any set. The set Ajcondition � A is called setA restricted to “condition” , where “condition”is a mnemonic for a rule to select PA elements p � A or PA sets Ak � A, with k � ��MA�, forsome PA�MA � N.

Remarks. For example,R�jz�z� can denote the set of points in the planez � z�, for somez� � R.

NOTATION A.5 (Restricted Cardinality)Let A be a finite set; let NA

�� card�A�, its cardinality. The number NAjcondition � NA is called

cardinality NA restricted to “condition” , where “condition” is a mnemonic for a rule to select PAsubsetsAk � A, with k � ��MA�, for some PA�MA � N, such that NAjcondition � card�

SPAk��Ak�.

Remarks. Depending on how is it formulated, the “condition”, finite setsA�Ajcondition, and thenumbersNA � card�A�, andNAjcondition in general satisfyNAjcondition � card�Ajcondition�. Forexample, letA � �� , andAjk��, thenNA � �, andNAjk��k�A � card�Ajk�� .

NOTATION A.6 (Restricted Boundary) The set �Ojcondition � �O is called boundary �O re-stricted to “condition” , where “condition” is a mnemonic for a rule to select p � �O or boundarycomponents �kO, with k � �� N��.

When�Ojcondition is an open surface, we call it aboundary patch. The outmost boundary�Ojout isan example of restricted boundary that remains closed. Empty cavities and fragments are restrictionsto a “no-children” condition. Examples of open-restricted boundaries (boundary patches) are theboundary interface components between foreground regionsOA andOB from Definition 1.7:

�OAjOB� �OBjOA

�� OA�B (A.13)

To identify the case of two labelsA�B on a single object, we shall use notation�OA�B. For inter-faces between two objectsV �W , we use either�VjW , or�WjV .

From the above definitions it is clear that multiple objectsO�P �Q, etc., or different labelsA�B�C� etc., affecting several regions of a single bodyO �

Sk�A�B�C��Ok, give rise to much

more complex combinations of sub-boundaries, nesting, interior and exterior relations. Other con-cepts inNn (solid or voxel boundary, face boundary, internal and external boundaries, etc.) areintroduced in Chapter 2.


For a static physical object with one of its sub-boundaries��O nested within another sub-boundary��O, we have��O� � O � , which express that cavities can only contain fragments astheir direct children (and conversely).

We may however consider nested boundaries of the same sign when comparing two objects, orwhen comparing the same object and its boundaries before and after successive transformations. Tothat end we define:

DEFINITION A.13 (Evolving Objects, Evolving Boundaries, Evolution sets)Let O�t��P be two objects, and let � �t�P , where O�t� and ��t� depend of a parameter t � N (i.e.,O�t�� O�t��, and ��t��P �� t��P , for some t� �� t�). In either case, O�t�, �O�t� and ��t�P aresaid to be evolving in function of t. Let t�� tn� n � N. The sequence

En� ��P� �� f��t��P � � � � � ��tn�Pg

is called an evolution setof boundary �P from t� to tn. The sequences f�O�t�� O�tn�g andfO�t�� O�tn�g also form evolution sets.

Note that the objectO�t� itself can be the result of a transformation:O�t� � T �O�t��, for someT � R� � R�. For an evolving boundary��t�P , its evolution comes from the way the boundary (orboundaries) ofP is (are) extracted; for example, as isodensity surfaces whenP is a scalar field. Inthis case the parametert is a scalar threshold for density values. Note thatE n� ��P� is an example oflevel set functions [Sethian96].

DEFINITION A.14 (Shrinking and Expanding Boundary)Let P an object, and En� ��P� an evolution set of its boundary. Let t � �� n� �� n � N,

✦ �P is called a shrinking boundary iff ��t��P � interior��t�P�, � t � �� n� ��.

✦ �P is called a expanding boundary iff ��t�P � interior��t��P�, � t � �� n� ��.

✦ Strict subset� in the above definitions, instead of “�” correspond to monotonically or strictlyshrinking boundaries and strictly expanding boundaries. Similar definitions apply for theboundaries of an evolving object O �t�.

We introduce a notion of weak and non-local homotopy under boundary transformations.

DEFINITION A.15 (Boundary Homotopy of Order 1)Let an object P and an evolution set E n� ��P�. Let Ttk � R� � R

�, for some k � N be a transfor-mation such that � �tk��P � Ttk��

�tk�P�. Let card��tk�P� � � be a one-component boundary. Wesay that:

✦ Ttk is boundary-homotopic of order 1 iff card��tk��P� � card��tk�P� � �.

✦ Otherwise, we say that there has been a global change of topology of order 1under Ttk .

✦ En� ��P� is boundary-homotopic of order 1iff � k � �� n�� card��tk��P� � card��tk�P� ��.

Other notions of topology require to consider local transformations and topological spaces. We dealhere with very coarse properties of boundary transformations.

Symbolic notation and most definitions used along this work are summarized in page 11.


Appendix B

Tubular Extraction Methods andBranching Analysis

The extraction problem of the blood-vessels of the lungs from color cryo-sections of the VHPdatabase (Chapter 4) prompted us to device and test several methods for tubular-structure detec-tion. Some of them take advantage of our boundary-based approach, as well as rolling buffers, andother concepts described in the precedent chapters. One of the objectives is to guide the segmentationprocess presented in Section 4.2.5 by a preliminar central axis-based description. We give in thesepages a draft of the main ideas, introducing first a local characterization of tubular structures andthen some notes about a classicalregion representation, using theMedial Axis Transform.

From Boundary to Regional Representations

Boundary representations (b-reps) use a set of elements derived or equal to the set of boundaryelements or features of an object. In principle, there is no explicit reference to internal informationin the b-rep. We discussed some characteristics of tubular structures using b-reps in Section 1.6.1,page 64.

The dual paradigm of b-reps areregional representations, where feature elements (used to de-

scribe shape) are now internal (i.e., in�O) and derived from other features in the object, including

�O itself. These internal features are most oftengeometric loci, that is, sets of points satisfying ageometric relationship with boundary points.1

The duality with b-reps appears for example in the importance and complementarity of charac-teristics of the normal behaviour. Let�n� t�tk� be a local orthogonal coordinate system on a 3Dsurface; this set is for example theDarboux trihedron , where basis vectors are thenormal, binormalandtangent, andt � tk�n (see Figure B.1b). Consider a 3D objectO possessing a smooth surfaceboundary�O. In such a case a well-defined normal vectorn exists for eachp � �O and a slowly“varying” tangent plane with directionstk�t can be assigned to each pointp.

A skeleton orMedial Axis Transform (MAT) of the same objectO will be in general constitutedby 2D interfaces for which no stablen can be assigned, nor a single tangent plane exists. At thesame time, a dominant tangent plane with directionstk�t do exists for this regional representation,following the orientation of the linear elements.

1See for example the operational Definition A.2 of “solid interior”, page 229.

240 APPENDIX B. Tubular Extraction Methods

>>0

||t

Φ

T

n

t

||t

n

n?

||t

Φ= 0

(b) n

t T?

k

(d)(c)

Φ

(e)

Φ= 8

(a)

Figure B.1: Behavior of a local reference system for a tubular structure according the size of itsmean radius �. (a) On a plane the normal n is constant, and no tangent directions tk, t arepredominant (� � ��. (b) For smooth curved surfaces or (c) “fat” cylinders, n varies slowly andtk, t correspond to the direction of the principal curvatures of the surface (� � � � � � �). (d)In a tubular object a coherent direction tk appears when n�t change fast in small neighborhoods

on the surface (� � �� ). (e) In a thin filament only vector tk characterizes the object (� � �).

Discrete Tubular Structures (II) and the MAT

The most common regional representation of an objetO is its skeleton denoted by MAT�O�. As-suming a voxel connectivity choice in a othogonal discrete representation, the MAT�O� of O is con-stituted by a one-voxel-wide tree or graph, in its simplest implementation. Let us consider boundarypointsp � �O and their nearest pointsq � MAT�O�, that is,

q � argminq ��MAT�O�

kp� q � k

and let us define the average distanced�� MAT �O� �� kp� qk �, and the standard deviation

�� MAT �O� ��Xp��O

�p� d�� MAT �O��

These two parameters offer several morphological possibilities, but we just mention the following:

✦ d�� MAT �O� � � and�� MAT �O� � �. The object is identical to its skeleton. Furthermore,�O � MAT�O� and the object is constituted only by thin or filamentary structures.

✦ � � d�� MAT �O� � �, �� MAT �O� � � for some small integers�� . In general, the object maybe a thin membrane or convoluted surface, a tubular structure, or a branching one; the size�determine the “bulkyness” of the object, and� its roughness.

✦ �� MAT �O� � �, andd�� MAT �O� � � is a constant. If the MAT�O� is a one-voxel-wide treeor graph, thenO is generalized cylinder, or a cylindrical ramified structure.

APPENDIX B. Tubular Extraction Methods 241

Thus, tubular ramified objects such as botanic trees and blood-vessel trees are well characterizedas being very similar2 to their own regional representation, constituted by a one-voxel-wide tree orgraph. A local slow varying set of orientationsftkg along MAT�O� is also a more representativefeature than the distribution of their surface normalsfng, which is rather perturbated or even un-defined for thin filaments. This behavior is illustrated in Figure B.1, where the predominance of aparticular direction vector changes when passing from a smooth surface to a filament. Letp � �O,andN��r� p� a geodesic neigborhood ofp, as in Definition 1.8. with a numberNgeo of pointsqi,with i � �� Ngeo. Note that smooth tubular surfaces presentlocal directional coherence (we

avoid the term “linear”); recalling Definition 1.17, page 42: the average vectortk�� tk i �, with

i � �� Ngeo is constant over small and medium-sized neighborhoods, and its standard deviation

�v�� kvi � vk� � remains small (all tangents point to the same direction at eachp).

Tubular structures and generalized cylinders can be thus represented by a central axis and a setof radial functions. In many cases, just a constant radius may be specified for each branch. It maybe specified as a function of the branching order according to flux conservation laws [Kalda93]. Therelevant features are coded along the central axis and the most successful models of complex ramifiedobjects are represented by L-systems, as described in Section 1.6.2

A similar observation can be made about thin convoluted surfaces and sheets, where a regionalrepresentation is close to the object itself, and the behaviour ofn can be locally described by planarcoherence. Figure 1.16, in page 69, shows an example of a convoluted surface.

Extraction of the Central Axis Tree by Local Centroid Tracking

A first problem to deal with the extraction of the MAT of complex objects is the simple-point identi-fication and the conversion of the resulting set of voxels into a list of vertices that should be traversedin a determined exhaustive way. A second problem is the noise sensibility of a MAT, which has beensolved by regularization, and hierarchical approaches [Ogniewicz94].

There are several methods in the litterature for simple-point identification, but most of themrely on some kind of assumption, such as the unimportance of the two-dimensional character of theextracted skeleton, from the fact that the analyzed object (e.g. the brain cortex) is two-dimensional innature [Mangin95]. Graph representations require the reduction of the 3D skeleton to linear chainsof single voxels, and to record only the most important branches.

Tubular structures should ideally produce robust unidimensional skeletons, but they present thesame problems that any bulky object. The final set of MAT points may be almost as difficult to ma-nipulate as the initial object itself. Less precise, but more robust representations have been proposed,specially for branching objects and tubular structures. Time limitations do not allow us to reviewthese definitions and the rich bibliography on the MAT and the more general concept ofmedial-ness or centerlines andcores (see for example [Morgenthaler81, Lee82, Matheron88a, Schmitt94]for the MAT definitions and [Niblack90, BurbeckPizer94b, Fritsch94, Fritsch95] for medialness andcores). Moreover, scalespace representations allow for example to account for the roughest features[Morse94]. We mention those issues related to our work.

Since average operations are robust to small deviations, the local Center of Mass (CM) or Cen-troid is also used in some approaches. This is the case of theCentral Axis Tree (CAT), which isdefined as a simplified graph representing a branching tree structure, where nodes are found at thecenter of branching regions or curvature points in such a way that polygonal arcs always lay inside

2Close to MAT�O�, up to a characteristic radius�.


��

��

��

��

r

π 3 r (2/3 (2 -1) (8 +1) +1 )1/221/2

rB

RBB

U

r

B

\Vol (T (B B )) =

R

R = 2 r1/2

R

Vol (T B B ) = Vol (B B ) Vol (T (B B ))

rR r. .

U

c

Vol (B B ) = rR

TT

Tc

---2r---

π 3 r 4/3 (2 -1)3/2\

R

U

\

U

\\ r R rRr

Figure B.2:Minimal and maximal radii and intersected volume relationships for the coupled balls (see text)

the branches. A local CM is obtained bymoving average calculations, given a neighborhood of atest-point inside the tubular structure.

The region growing (RG) segmentation described in Section 4.2.5 provides the means to track thecenter point of sliding neighborhoods required for a moving average calculation, in order to updateregion statistics during growth. The 3D-extension of the rolling buffer described in Section 3.3.3accounts for an incremental approach. In [Pisupati95], Pisupati et al. took advantage of a simi-lar process preserving the voxel coordinates of the RG process (usually these coordinates are justpopped-out from a queue), and they calculate a local centroid of the extracted tree, when the neigh-borhood under analysis is not fragmented (that is, the RG-wave has passed a branching point, asillustrated below, in Figure B.4-c). Pisupati et al. do not describe how they are able to know howmany components of the queue are or not connected; these components are not generally found atadjacent locations in memory and tracking them requires a more sofisticated data structure than aparallel buffer.

However, the idea of Pisupati et al. prompted us to device a similar approach, keeping track of thelocal centroid, in order to be able to detect a medial-axis approximation by usingcoupled spheroidalsensors, which are a dynamic extension of local annular-neighborhoodsB�r� R� p� � BR�p�nBr�p�,wherep is a moving point, tracking the centerline of a vessel, in function of gray-level statisticsof Br�p� andBR�p�. The main problem we encountered, in a feasability study, was the quantityof parameters to control, such asR�r ratios, and size in function of vessel diameter and the manycases to deal during detection. Figure B.2 shows details of annular neighborhoods inside a tubularsegmentT in the case of a centered pointp and a radiusR��

p�. Figure B.3 illustrates several

situations to be analyzed and resolved by the annular detector, in which complex decisions dependon combinations of the valuesR� r� R�r� p and the foreground and background proportions insideregionsBr�p��BR�p� andB�r� R� p�.


��

��

rB B

U

(c).

R.

��

��

��

��

��

��

.

.

.

.

.

.

. ..(a)(b)

.

.

(h)

(f)(e)

(d)

(g)

(i)

(k)

(j)

Figure B.3:Some of the problematic cases to be analyzed by a tubular detector based on coupled balls.

Detection of Branching Points.

Large ball-neighborhoods of a test-pointp � T could also be analyzed in terms of their intersectionswith a tubular structureT , a branching point being characterised by the number of intersectionsbetween the ball and the tubular boundaries (see Figure B.6).

Separation of branch components.

In a region-based approach, the MAT provides a graph representation of a branching structure, inwhich morphology analysis and measurement can be done to extract, for example, the branch lengths,average diameters (or the radii distribution itself). A boundary-based approach can be devised by an-alyzing information from boundary representations. Figure B.7 illustrates this approach by extractinga local depth map using the estimated normal of the facets of�T . Gray-level labels are then assignedto boundary points (facets) and branches separated in terms of length and diameter transitions. Errordue to normal estimation, vessel deformation, and natural variations do not warranty that separationwill occur exactly at branching points (see Figure B.7f).

Detection of Tubular Components by Local Tensor of Inertia Analysis

We have already mentioned the importance of local coherence in complex structures. The bronchial-vessel tree provides an example, since vessels follow relatively straight paths for short distances3.Even some tortuous vessels preserve a global orientation in the sense of having a global principalaxis in neighborhoods not much larger than the largest diameter. An ellipsoid of inertia can beassigned to each point in space, as illustrated in Figure B.8. Hence, Principal Component Analysis

3The neighborhood of the branching nodes poses some problems.


(b)

(d)(a)

(c)

(e)

(f)

Figure B.4: Central-axis tracking by growing-ball intersection. (a) The intersection between the tubularstructure T and a sequence of R discrete balls Br of radius r � �� R (with R � N) is tested todetermine the number of connected components kc of its boundary (kc � card�Y �, where Y �� Br � T ).After the first step (a), subsequent ball sequences produce one new intersection at branch segments (b), andtwo or more after bifurcations (c). The green dot (a) is a recalculated Center of Mass of the tubular segmentdefined by BR � T , and is saved for later use. At each sequence (R steps) the CM of each intersection Ykc iscalculated and a new sequence is initiated (b,c). At the left (d) the approximated cross sections of T are shown,corresponding to contours defined by ��BR � T �. Inset (e) shows the corrresponding graph tree formed bythe CM at each intersection step nR� k � �� False intersections (f) constitute one of problems to dealwith.

(PCA) may provide information of the presence of a tubular structure as well as a particular case oforientation trends and anisotropy, as presented in Section 1.2.1.

The remaining figures further illustrate these and other ideas for tubular structure detection.


maxrmin

U

T

(b)

(a)

(b)r < ( B T)

maxr > (T)

Figure B.5:Initial and final radius rmin� rmax for ball growing determine the tracking step (the approximatedistance between CM nodes). The smallest ball (a) before restarting (or maximum growing ball), Brmax

,shoud not be smaller than the average local diameter of the tubular structure, ��T �. The largest minimal ball(b) should not exceed the average diameter of the intersection of the last ball Brmax

with T .


r’

rB (p)

6,7,8...p

r

U

T {

B (q)

B (p) } = 3

T

U

{ B (q)r’ } = 2

p

q

(a)

TT 4

p

p

p

p12

3

p5

T

(c)

T

(b) (d)

card

card

Figure B.6:Branching points are characterized in several ways. (a) Their ball-neighborhoods ideally havea defined number of intersection with the tubular components. Analysis is done at interior points p� q � T ,ideally near the medial axis. The number of connected-component intersections define the branching degreeequal to card(intersections)-2, with 0 indicating no branching, and -1 indicating a terminal point. (b-d)Saddle zones also locate branching points in a boundary-based analysis where pi � �T . Terminal points areless evident to characterize.


order

r

n

(c)

��

��

��

��

��

��

��

��

(a) (d)

(f)

(e)(b) p

d

Figure B.7:Local thickness (or depth) analysis, applied to branching structures T . (a) a normal is estimatedat each p � �T , and distance to the opposite site is measured and used for (c) a depth label on p whichcodes local diameter. (d) Rough branch-components segmentation of the depth-labeled boundary, using anaverage-diameter criterion.


(a)

(b) (d)

(c)

(e)Object

Principal local axes

Test points

Artifacts

Figure B.8:Local tensor of inertia analysis (or PCA of the local scatter matrix) for geometry-driven detectionof oriented fine structures. (a) A proper local neighborhood (dotted circle) allows to detect tubular regions.(b) At isotropic regions, interior and exterior points present uniform eigenvalues (much larger than zero forforeground and very small or zero for background) in small neighborhoods. (c) Boundary or tubular-nearpoints define anisotropic regions. (d) elongated artifacts are also detected. (e) “missing” segments may becompleted by directional filtering (e.g. oriented closennigs).

Appendix C

Synthesis of 3-D Phantoms of ComplexPorous Structures by Diffusion-LimitedAggregation with Relaxation

We discuss in this appendix some considerations of simulated data using the Diffusion-Limited Ag-gregation (DLA) model. During the collaboration with the CNET we recognized the richness of theDLA growth and random deposition models as versatile generators of complex structures, havingfractal (scaling) features and various morphological behaviours (evolution) in function of simula-tion parameters. Data generation to develop and test our methods posed one particular problem:physico-chemical calculations are very expensive for computer simulations of real phenomena, lim-iting resolution and parameter ranges. In consequence, only some short ranges of parameters werestudied on a this first approach.

In order to understand and characterize morphological configurations of 3D data, from a moreabstract point of view, we found ourselves concerned with theper-se study of complex structuregeneration, and considered the DLA model incorporating relaxation (thus, “DLA+R”) in a similarway that deposition models, but reducing physical interactions to random controled behavior. Sim-ilar paradigms for complex structure generation (phantoms) are Markov Random Field realizationsfor texture synthesis, auto-regressive models, and fractal recursive generators with randomization(fractional brownian motion and Fourier synthesis are common examples).

A second goal is to simulate data for the development and test of morpho-analytic tools for threedimensional data. In order to analyze different complex configurations and 3D-textures (branching,multiple connectedness, porosity, mixed and convoluted surfaces), we developed a DLA+R processsimulator in 2D and 3D, incorporating many features to allow for semi-controlled structure variation.The simulator’s features include initial geometry configuration, topology constraining, displacementthrough potential fields and barriers, neighborhood interaction (which allows to emulate relaxationand surface dynamics after cluster coalescence), anisotropic flux of the clustering particles, stickingprobabilities, temperature-like and reaction behaviour, configuration-specific penalization or favour-ing, and otheremulating real phenomena. Instead of calculating precise energy balances to favor aspecific event, random decisions are taken on the basis of pre-calculated probabilities correspondingto particular neighborhood configurations.

Interest in diffusion process relies in the possibility of mathematical modeling of important fea-tures that may be verified by experiments. In surface growth, DLA process takes place on a substratelayer (for example, a plane). The expected roughness of the projected top surface, defined as the

250 APPENDIX C. Synthesis of 3-D Phantoms by DLA+R

RMS deviation from a mean heighth can be predicted by the diffusion (Kardar-Parisi-Zheng) KPZ-equation [Vicsek92], which is a non-linear version of the Langevin equation used in brownian motionmodels:

�hRMS�x� t�

�t� F �r�hRMS �x� t� � �rhRMS �x� t��

� � �x� t� � (C.1)

where hRMS � h � hhi�

and:

✦ hRMS is the RMS roughness of the projected top surface,h the continuous height as functionof x, the vector position in the substrate, andt, the particle diffusion cycles. The discretedefinition ofhRMS was given in Equation 5.2.

✦ F is a “drift” offset.

✦ � is the coefficient associated to surface tension (or viscosity), and the Laplacian term isknown as the relaxation, or diffusion term.

✦ � is the coefficient for the lateral growth contribution, which accounts for neighborhoodreaction interactions. It is also known as bias term.

✦ � is the “input signal”, which usually is constituted by gaussian noise.

(a)

(b)

Figure C.1: 2D deposition emulated by Diffusion-Limited Aggregationprocess with relaxation(DLA+R). (a) Deposit with complex porous structures (porosity is higher than ��); the arrowpoints at a single layer, in light gray. (b) Column-like structuring with deep vertical meanders anda very rough surface (RMS-value: �z � ��). Gray-levels correspond to time, in 10K depositioncycles per level. Instance (a) is qualitatively similar to a real physical phenomenon: surface growingin microgravity conditions. Simulation program developed at ENST.

APPENDIX C. Synthesis of 3-D Phantoms by DLA+R 251

The terms in this equation code the steps of several growth and deposition process. When� �� , the KPZ equation corresponds to random deposition with surface diffusion, as describedin Figure 5.2. Using this and other parameter-evolution equations, the CNET has predicted some ofthe observed results for roughness and porosity, in computer simulation as well as in experimentalvalidations.

At the same time, the KPZ equation codes the main components of the 2D DLA+R simulator wedeveloped for complex micro-structure generation. Some examples are presented in Figure C.1 andin Chapter 1, in Figure 1.12, page 58. The instance of Figure C.1a is similar to a real physical phe-nomenon:surface growing in microgravity conditions (remark made by Dr. Flicstein, of the CNET).The instance of Figure C.1b is qualitatively similar to viscous fingering phenomena. A 3D imple-mentation of the DLA+R generator of complex structures is in work at present. This generator couldprovide 3D phantoms to test and develop new analytical methods as those presented in Chapter 5.1.

Resume long

Ce resume presente brievement les differentes parties successives de la these, les idees principales,nos contributions et les resultats les plus importants.

Objectifs et organisation de la these

Le but general de ce travail est le développement de méthodes de segmentation, morphométrie etvisualisationde structures complexes en trois dimensions, en particulier des microstructures poreuseset des vaisseaux sanguins qui présentent des ramifications compliquées.

Dans la première partie de la thèse nous avons étudie les differents problèmes posés par l’analysede ce type de données volumiques, comprenant les définitions desattributs de complexite structuraleexaminees dans le chapitre 1, et lesrepresentations par frontieres discretes, decrites au chapitre 2.Nous proposons aussi quelques outils exploitant ces représentations.

La deuxieme partie concerne des applications sur deux ensembles de données volumiques cor-respondant à deux problématiques différentes respectivement en biomédecine (chapitre 3 et 4) et enphysique (chapitre 5). La géneralite et la souplesse du modèle de représentation par frontières nousa permis d’utiliser les mêmes outils informatiques pour les deux applications. La notation utiliséedans cette thèse est présentee en détail a la page 11.

Caracterisation des structures complexes dansR�

Nous avons d’abord identifié et caractérise les attributs morphologiques intrinsèques et extrinsèquesdes differentes structures complexes, tels que : les axes principaux, la compacité, la coherence lo-cale et globale, les caractéristiques des structures tubulaires et le comportement de la normale (sielle existe), la représentation des ramifications, convexités et cavités, les interfaces et membranes,la porosite, la rugosité et d’autres. Pour ce faire, nous avons d’abord fait une revue des concepts degeometrie dansR�, tels que : voisinage, voisinage géodesique, boule euclidienne, frontière (contour3D), interieur, cavites et d’autres, qui permettent une description géometrique des diverses propriétesdes formes irrégulieres et complexes. Le sujet de la complexité aete aussi abordé brievement, etnous avons constaté qu’il y a peu des cadres théoriques pour sa description ; ces cadres consistentprincipalement en : (1) la complexité algorithmique de Kolmogorov et (2) les dimensions fractales etles systemes dynamiques non linéaires. Le premier cadre théorique permet de mesurer la complexitépar la longueur de la description la plus courte possible. Le deuxième cadre prend compte des répe-titivit es sur plusieurs échelles, mais s’avère peu génerale, lorsque les irrégularites ne présentent pas

254 Resume long

la propriete d’auto-similarite. D’ailleurs, des objets fractals soi-disant complexes, dont l’ensemblede Mandelbrot, peuvent avoir une description mathématique très courte, en termes des systèmes dy-namiques, mais ils peuvent présenter des détails geometriques tres irreguliers et à tous les niveauxde l’analyse. Nous faisons dans ce chapitre une exploration systématique des définitions existanteset les liens particuliers entre plusieurs attributs morphologiques et les applications dont nous avonsconnaissance ou une expérience personnelle. Un objet s’avère complexe lorsque sa description esttres longue et présente des régularites tres specifiques, mais aussi lorsqu’elle fait appel à une cardi-nalite eleve d’elements structuraux ou de relations internes.

Les modeles dansN� : une approche fondee sur les frontieres

Pour l’etude des objets de structure complexe, nous avons choisi une approche en géometrie discretepour l’extraction des surfaces, le traitement et la visualisation des données volumiques. Dans cetteapproche, les surfaces extérieures ou frontières des objets détectes sont représentees par des listes defacettes. Dans le chapitre 2 nous étudions des élements dansN� qui permettent la formulation d’unerepresentation complète, et la possibilité d’etendre les définitions du cas continu au cas discret, pourun objet ou sous-ensembleV � N

�. Ceselements comprennent : une maille discrète, des voxelset leurs faces ordonnées, une affectation cohérente des orientations des faces, et finalement deuxensembles d’élements : l’ensemble des faces orientées de la frontière (facettes) donnant lieu à unefrontiere par facettes �V , et l’ensemble des voxels correspondant à unefrontiere par voxels V . Avecun choix specifique d’orientation de la surface, et des connexités compatibles (parmi les cas 6-,18-,ou 26-connexes), cette double représentation comporte des avantages pour l’analyse et le traitementdes ensembles de voxels : la première representation (�V) est liee au parcours et l’extraction dessurfaces par suivi de contour 3D, et les paramètres volumiques que l’on en peut déduire ; la deuxième( V) est liee aux notions de morphologie mathématique classique dansN�. Nous formulons aussi lepassage d’une représentation à l’autre.

Une partie importante des traitements 3D avec ces représentations consiste à analyser et modifierles listes des facettes qui constituent�V et les structures de données associées, en examinant lesfacettes et les voxels voisins pour effectuer les opérations ou mesures nécessaires. L’extraction detelles frontieres se fonde sur une méthode (existante dans la littérature) de parcours du graphe dirigéassocie a la surface de chaque objet (en fait, de chaque composante connexe). Diverses structureset outils sont alors introduits, tels que : les voisinages des facettes, ceux des voxels et des frontières(��V � ��V , �V , ...), les voisinages géodesiques, et des implantations de filtres de morphologiemathematique fondées sur la représentation de frontière par voxels. Certaines opérations sont acti-vees durant le parcours du graphe dirigé, dont par exemple l’estimation de la normale à la surfacepour chaque facette, et l’élement d’aire surfacique par facette, à partir de la configuration locale desfacettes voisines. La méthode de parcours de�V et un choix d’orientation permettent aussi d’utiliser

une methode novatrice de parcours de l’intérieur geometrique�V d’un objet discretV , etant donnée

seulement sa représentation�V structuree comme une liste de couples de facettes opposées. Cettemethode est liée aussi à une compression par plages de voxels (run-length coding).

Les trois exemples suivants permettent d’illustrer l’intéret de notre approche :

(1) Filtrage : une érosion morphologique conditionnée revient à uneelimination selective desvoxels situes sur la frontière V de l’objet ;

Resume long 255

(2) calcul morphométrique : le volume d’un objet est calculéa partir de sa frontière�V , en utilisant

le theoreme de Gauss discret, et notre méthode de parcours de l’intérieur geometrique “�V” de

l’objet ;

(3) visualisation : pour calculer l’ombrage d’une facette à afficher, la normale locale est estiméeapartir des configurations des facettes voisines.

Nous montrons aussi quelques attributs morphométriques dérives ou formules en termes des éle-

ments introduits, notamment à partir de :�V , V , la methode de parcours de�V et les voisinages de

voxels et des frontières. Ces attributs, introduits déja au chapitre 1 pour le cas continu, permettentl’ etude quantitative de formes très complexes dans le cas de données volumiques discrètes. Quelquesexemples de segmentation des composantes faiblement connexes et de visualisation réaliste illustrentles possibilites de notre démarche. D’autres exemples sont présentes dans les chapitres suivants.

Homogeneisation radiometrique des images des cryo-sectionsen couleur du VHP

La premiere application a éte developpee sur des données biomédicales correspondant aux poumonset provenant de la base de données du Visible Human Project (VHP) de la National Library of Medi-cine et du National Institutes of Health (Bethesda, EU). Les coupes anatomiques en couleur du VHPposent des problèmes particuliers pour la segmentation et la visualisation des structures internes etnous avons ainsi développe une technique novatrice pour la correction d’héterogeneites radiomé-triques regionales. Notre démarche a consistéa exploiter le principe decoherence locale structurale(introduit au chapitre 1) pour chaque pixel, dans des images de différence entre deux coupes adja-centes. Plusieurs techniques de visualisation des héterogeneites ontete proposées, en particulier lesimages entrelacées en échiquier et les histogrammes couleur empilées (stacked color histograms),qui mettent en évidence le caractère regional et irregulier des discontinuités qui genent l’analyseen 3D. Notre technique, utilisant un modèle auto-regressif de première ordre, consiste à decalerachaque pixel le profil de l’histogramme local pour les images différentielles, de sorte que les maximade deux histogrammes de coupes consécutives soient alignés. Cette correction est propagée au longdu volume dans les deux sens, une seule fois. Une telle correction n’introduit pas de bruit, ni deflou, car elle dépend des discontinuités radiométriques locales et ne se fait pas si les pics des his-togrammes sont alignés. Les techniques de visualisation en échiquier des coupes adjacentes et lescoupes ré-echantillonnees axiales et sagittales ont démontre l’efficacite de cette correction.

La description de notre technique et les résultats obtenus ont fait l’objet de deux travaux présen-tes en congrès internationaux, et une publication internationale avec comité de lecture. Les imagescorrigees ont éte utilisees pour l’extraction des structures vasculaires, présentee au chapitre suivant.

Extraction des vaisseaux sanguinsa partir des cryo-sectionsen couleur du VHP

Dans ce chapitre nous examinons les données du VHP en vue d’une segmentation de l’arbre vas-culaire des poumons, ainsi que les techniques que nous avons testées ou proposées et les résultatsobtenus.

256 Resume long

Les sections des vaisseaux plus fins forment une “texture” dans les images des coupes coronales.La nature fractale de cette texture rend difficile et inefficace l’utilisation des approches classiques desegmentation, car les données (la forme) et le fond (le tissu plus des vaisseaux sous-échantillonnes)se trouvent fortement mélanges. Nous avons néanmoins proposé une methode hybride, qui com-bine le seuillage par hystéresis et le filtrage par moyenne “Sigma”. Cette combinaison compensepartiellement le problème de discrimination entre fond et forme. Nous avons combiné aussi cesdeux approches avec une approche de segmentation par croissance de régions pour une segmenta-tion prealable, et la segmentation par hystéresis est effectuéea l’aide des frontières voisines (�V ,��V) de la frontiere principale, définie par la croissance de régions. Cette dernière exploite aussi

la representation par facettes et leurs voxels voisins.

A partir des resultats de segmentation, nous avons fait des visualisations réalistes de l’arbre vas-culaire pulmonaire obtenu à la suite des différentes étapes de la segmentation et du traitement 3D,toujours en utilisant la représentation par frontières. La qualité des rendus surfaciques est suffisam-ment bonne pour permettre d’apprécier la complexité des structures extraites, leurs interrelationsspatiales, et l’effet sur ces structures de tout traitement spécifique.

D’autres methodes possibles pour l’extraction des structures tubulaires sont décrites tres brieve-ment, mais en raison des limitations de temps, notre étude sur ces méthodes est restée preliminaire.

Analyse et visualisation des simulations numeriquesde photo-depot amorphe

La deuxieme application, en physique, a pu être abordée dans le cadre d’une collaboration avec lelaboratoire de Physique des Surfaces du Centre Nationale d’Etudes en Télecommunications (CNET),a Bagneux. Le problème a consisté a mesurer les propriétes de porosité, rugosite, connexité et dis-tribution des paramètres morphométriques extraits des résultats obtenus par simulation numériqued’un photo-depot amorphe de Nitride-Silicium. Nous présentons d’abord les caractéristiques mor-phologiques de ces simulations, et leur dépendance des paramètres de température, pression, champelectromagnétique et les conditions initiales.

Les structures poreuses des dépots amorphes et la nature de la ségregation du dépot avec deuxou trois espèces d’atome (Si, N, et H), nous ont amenésa considerer les définitions de connexité, lesinterfaces discrètes et la pertinence des outils introduits dans la première partie du travail de thèse.La collaboration avec le CNET a permis de proposer et tester des outils d’analyse supplémentairesaux outils traditionnels (porosité volumique globale et rugosité RMS, correspondant à l’ecart-type dela hauteur moyenne), qui sont mal adaptés au cas des surfaces qui présentent des “méandres” (pourla rugosite), et lorsque les pores présentent une distribution de tailles et de formes très diverses (pourla porosite). Nous faisons la distinction entre une surface projetéea l’infini (qui peut etre etudieecomme une image d’intensité ou de hauteurz � f�x� y� ), et la surface réelle�V du depot, qui inclutdes composantes surfaciques à l’interieur des amas. De telles composantes constituent les méandreset repliements cachés dans une simple projection dans l’axeZ.

Nous proposons aussi différentes techniques et paramètres pour étudier la rugosité d’une tellesurface complexe, dont la densité de surface locale, et les densités relatives à chaque espèce ato-mique. Une hauteur et une largeur caractéristiques permettent aussi d’étudier la densité moyennedes rapports entre la profondeur et la largeur des extrema locaux. Ces mesures sont aussi liées auxnotions des textures surfaciques et de rugosité revues au chapitre 1.

Resume long 257

Nous avons effectué des reconstructions 3D des données de simulations, l’identification des com-posantes connexes pour les pores ou les amas d’espèces differentes, et l’extraction des paramètresmorphometriques. Les visualisations et les mesures obtenues ont permis aux chercheurs du CNETde connaˆıtre : la taille et la forme des pores, la taille et la forme des amas du dépot, la distributionde la porosité par rapport à la temperature ainsi que la qualité d’interpenetration de plusieurs espècesatomiques. En outre, les rendus 3D permettront de repérer des transitions critiques de la morphologiedes amas (ou de la distribution de pores), identifiés clairement par des étiquettes en couleur.

Vers la fin de cette collaboration, une validation expérimentale a pu confirmer les mesures de po-rosite fournies au groupe du CNET-Bangeux, en mesurant la porosité par des méthodes d’ellipsométriein situ, correspondant aux paramètres de simulations étudiees par ordinateur. Ces résultats et unapercu de notre contribution ont fait récemment l’objet de deux publications supplémentaires ac-ceptes en revue avec comité de lecture, ce qui complète les travaux de deux étudiants de DEA duCNET-Bangeux pendant 1997 et 1998, et deux travaux présentes en congrès (memes dates).

Conclusions et perspectives

Nous avons identifié etetudie plusieurs concepts relatifs à la complexite structurale des objets typi-quement présents en biologie, médecine et en physique de surfaces. Cette exploration est loin d’êtreachevee totalement, et constitue désormais une partie importante de nos axes de recherche.

La methode de parcours de surface, existant déja dans la litterature, a éte exploitee systéma-tiquement et enrichie avec notre démarche et les différentselements pour l’analyse 3D que nousavons apportés, en utilisant des frontières par facettes et par voxels. Il nous reste encore à raffinerces methodes, et par exemple tenir compte du théoreme de Jordan qui permet de résoudre certainsproblemes de topologie discrète, eta les mettre en valeur dans des publications à venir.

Notre groupe a éte le premier à signaler les héterogeneites presentes au niveau des poumonsdans les coupes anatomiques de la base des données du VHP, et nous avons proposé une methode decorrection efficace que nous a permis d’extraire une partie considérable de l’arbre vasculaire pulmo-naire. Les méthodes d’analyse et segmentation de structures tubulaires que nous avons commencé aexplorer sont aussi une voie intéressante à suivre dans le futur.

Dans la collaboration avec le CNET-Bagneux nous avons tenu compte des caractéristiques réellesde la surface et de l’intérieur des dépots. Ces caractéristiques sont normalement ignorées dans les mé-thodes traditionnelles, faute des moyens pour leur évaluation. D’ailleurs, deux des techniques testéesont puetre validees de fa¸con experimentale, ce qui a permis de mettre en valeur notre contribution.Une voie future de collaboration pourrait consister à mettre en relation des techniques expérimentalespour pouvoir valider les outils et mesures proposés (densité de surface, quantification des méandres,granulometrie des pores,...), ainsi que d’autres caractéristiques de la simulation numérique realiseepar le CNET-Bagneux.

258 Resume long

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Index

K-adjacency, 97K-connectivity, 97

adaptive homogenization, 150adatom, 192airways, 161amorphous deposition, 192, 194anatomical cryosections, 144

b-reps, 81, 239back-face boundary, 110back-facet, 109back-voxel, 109back-voxel boundary, 110background, 25ballistic deposition, 194body-centered cubic, 88bronchial tree, 161

cardinality, 27Central Axis Tree, 241characteristic

height, 45, 210length, 34, 44width, 45, 210

checkerboard display, 147chemical vapor photo-deposition, 192CNET, 192co-occurrence images, 146coherence

directional, 42local, 41, 153spatial, 41

colorhistograms, 146

computer simulation, 192connected components, 27connected set, 26contention wall, 121continuous boundary, 92convoluted surfaces, 68

coronal cross-section, 145

Darboux trihedron, 239Diffusion Limited Aggregation, 56, 194digraph, 119dimension

capacity, 50embedding, 48Euclidian, 48Hausdorff, 49information, 50Lyapunov, 51mathematical-morphology, 51Minkowski, 51similarity, 49spectral, 51topological, 49

distance, 98DLA, see Diffusion Limited Aggregation

ellipsoid of inertia, 243

face-based boundary, 110face-centered cubic, 88, 192facet, 90facet boundary, 92facet wall, 121facet-driven region growing, 185facet-to-voxel operator, 90fading memory constant, 154Fat-boundary neighborhood, 112foreground, 25front-face boundary, 110front-facet, 109front-voxel, 109front-voxel boundary, 110

gamma correction, 144geodesic neighborhood, 24, 104geodesic path, 98geometric interior, 93

278 INDEX

Hamiltonian circuits, 119hysteresis thresholding, 176

immediateK-neighborhood, 103innovations, 150interfaces, 25, 68

Kolmogorov Complexity, 58

laminar structures, 68last-layer top surface, 206lateral directed circuits, 105, 117Lindenmayer formal grammars, 66lung parenchyma, 162

maximum intensity projection, 157MDWRE, see mean depth-width ratiomean depth-width ratio, 209meanders, 194Medial Axis Transform, 239microstructure, 196MIP, see maximum intensity projectionMonte-Carlo simulation, 194morphological boundary, 128moving average, 242

neighborhoodannular , 24discrete, 24geodesic, 24, 26spherical, 24

next-front-face boundary, 110next-front-voxel boundary, 110

PCA, see Principal Component AnalysisPoint-Spread-Function, 70, 170pores, 194porosity, 193, 197, 199, 213, 217, 235Principal Component Analysis, 29, 159projected top surface, 206PSF, see Point Spread Functionpulmonary artery, 161pulmonary veins, 161

radiometric inhomogeneities, 144region growing, 185regional representation, 239RMS roughness, 193, 198, 208, 213, 217ROI or Region Of Interest, 62, 83, 86rolling buffer, 156

sagittal cross-section, 145Sigma-average filter, 179solid boundary, 93solid interior, 96species, atom, 192Stereology, 63structuring elements, 103surface density, 108surface diffusion, 194surface tracking, 113surface-intensity images, 209

tensor of inertia, 29, 248Tetris deposition, 194top layer, 194traversal circuits, 91tubular detector, 243tubular structure, 36, 64, 241–244

UV CVD, see chemical vapor photo-deposition

Visible Human Project, 144, 164volume traversal, 96, 123

watershed segmentation, 84, 131weakly-connected components, 84, 130

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