compactly supported radial basis functions...

Numéro d’ordre : 2006-ISAL-0095 Année 2006

THÈSEprésenté devant

L’Institut National des Sciences Appliquées de Lyon

pour obtenir

LE GRADE DE DOCTEUR

ÉCOLE DOCTORALE : ÉLECTRONIQUE, ÉLECTROTECHNIQUE, AUTOMATIQUEFORMATION DOCTORALE : SCIENCES DE L’INFORMATION, DES DISPOSITIFS ET DES

SYSTÈMES

par

Arnaud GELASIngénieur de l’INSA de Lyon

Compactly Supported Radial Basis Functions:Multidimensional Reconstruction and Applications

Soutenue le 27 Novembre 2006 devant la commission d’examen

Jury :

Michel BARLAUD Professeur RapporteurJean-Marc CHASSERY Directeur de Recherche Président du JuryOlivier DEVILLERS Directeur de Recherche RapporteurDenis FRIBOULET Professeur ExaminateurTakashi KANAI Associate Professor Co-encadrantYutaka OHTAKE PhD ExaminateurRémy PROST Professeur Directeur de Thèse

SIGLE ECOLE DOCTORALE NOM ETCOORDONNEES DU RESPONSABLE

CHIMIE DE LYON M. Denis SINOUUniversité Claude Bernard Lyon 1Lab Synthèse AsymétriqueUMR UCB/CNRS 5622Bât 308

M. Denis SINOU 2ème étage43 bd du 11 novembre 191869622 VILLEURBANNE CedexTél : 04.72.44.81.83 Fax : [email protected]

E2MC ECONOMIE, ESPACE ET M. Alain BONNAFOUSMODELISATION DES COMPORTEMENTS Université Lyon 2

14 avenue BerthelotMRASH M. Alain BONNAFOUS

M. Alain BONNAFOUS Laboratoire d’Economie des Transports69363 LYON Cedex 07Tél : [email protected]

E.E.A. ELECTRONIQUE, ELECTROTECHNIQUE, M. Daniel BARBIERAUTOMATIQUE INSA DE LYON

Laboratoire Physique de la MatièreM. Daniel BARBIER Bâtiment Blaise Pascal

69621 VILLEURBANNE CedexTél : 04.72.43.64.43 Fax : [email protected]

E2M2 EVOLUTION, ECOSYSTEME, M. Jean-Pierre FLANDROISMICROBIOLOGIE, MODELISATION UMR 5558 Biométrie et Biologie Evolutivehttp://biomserv.univ-lyon1.fr/E2M2 Equipe Dynamique des Populations Bactériennes

Faculté de Médecine Lyon-SudM. Jean-Pierre FLANDROIS Laboratoire de Bactériologie BP

1269600 OULLINSTél : 04.78.86.31.50 Fax : [email protected]

EDIIS INFORMATIQUE ET INFORMATION POUR M. Lionel BRUNIELA SOCIETE INSA DE LYONhttp://www.insa-lyon.fr/ediis EDIIS

Bâtiment Blaise PascalM. Lionel BRUNIE 69621 VILLEURBANNE Cedex

Tél : 04.72.43.60.55 Fax : [email protected]

EDISS INTERDISCIPLINAIRE SCIENCES-SANTE M. Alain Jean COZZONEhttp://www.ibcp.fr/ediss IBCP (UCBL1)

7 passage du VercorsM. Alain Jean COZZONE 69367 LYON Cedex 07

Tél : 04.72.72.26.75 Fax : [email protected]

MATERIAUX DE LYON M. Jacques JOSEPHEcole Centrale de LyonBât F7 Lab. Sciences et Techniques des Matériaux et des Surfaces

M. Jacques JOSEPH 36 Avenue Guy de Collongue BP 16369131 ECULLY CedexTél : 04.72.18.62.51 Fax : [email protected]

Math IF MATHEMATIQUES ET INFORMATIQUE M. Franck WAGNERFONDAMENTALE Université Claude Bernard Lyon1http://www.ens-lyon.fr/MathIS Institut Girard Desargues

UMR 5028 MATHEMATIQUESBâtiment Doyen Jean Braconnier

M. Franck WAGNER Bureau 101 Bis, 1er étage69622 VILLEURBANNE CedexTél : 04.72.43.27.86 Fax : 04.72.43.16.87

MEGA MECANIQUE, ENERGETIQUE, GENIE M. François SIDOROFFCIVIL, ACOUSTIQUE Ecole Centrale de Lyonhttp://www.lmfa.ec- Lab. Tribologie et Dynamique des Systêmeslyon.fr/autres/MEGA/index.html Bât G8, 36 avenue Guy de Collongue, BP 163

69131 ECULLY CedexM. François SIDOROFF Tél : 04.72.18.62.14 Fax : 04.72.18.65.37

[email protected]

Résumé

Cette thèse traite l’application des fonctions de base radiale, à support compact (CSRBF), pourla reconstruction multidimensionnelle de signaux et de surfaces à partir d’échantillons, ainsi quepour la segmentation des images. En particulier, cette approche permet de considérer l’échan-tillonnage irrégulier.

Dans chacune de ces applications nous proposons de nouvelles méthodes. Pour la reconstruc-tion multidimensionnelle de signaux nous proposons une approximation multirésolution basée surune classification hiérarchique des échantillons, puis une approximation adaptative avec calcul lo-cal du support de chaque CSRBF. Pour la reconstruction de surfaces implicites, nous proposonsune méthode composite qui associe la partition de l’unité et les CSRBF, puis une approximationadaptative où le support de chaque CSRBF est déterminé à partir d’une approximation de l’axemédian. Enfin, nous proposons un formalisme de collocation pour la résolution de l’équation depropagation des ensembles de niveaux en segmentation, par représentation de la fonction implicitesur une base de fonctions. En particulier, nous illustrons cette méthode avec les CSRBF en seg-mentation d’images médicales.

Mots-clés

Reconstruction, Approximation, Interpolation, Régularisation, Multirésolution, Partition del’unité, Surface implicite, Axe médian, Segmentation d’image, Ensemble de niveaux, Collocation,Fonctions de base radiale.

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Abstract

This thesis deals with the application of Compactly Supported Radial Basis Functions (CS-RBFs) for reconstruction and image segmentation. Reconstruction consists in finding a (contin-uous) function which fits good to some given samples. Image segmentation consists in findinglimits of relevant areas in images. These general problems are essential for the study, analysis,further processing of data, such as multidimensional signals, images, or surfaces.

In each listed application, we propose new methods. For multidimensional signal reconstruc-tion from non-uniform samples, we propose a new multiresolution scheme based on a hierarchicaldata clustering and an adaptive approximation method, where the support of each CSRBF is calcu-lated locally. For implicit surface based reconstruction, we propose a composite approach whichsolves locally the reconstruction problem and blends all solutions together by partition of unity,and an adaptive approximation method, where the support of each CSRBF is locally computedfrom an approximation of the Medial axis. Finally, we propose a new collocation formalism forsolving the level set propagation equations in image segmentation, by expanding the evolvingimplicit function on a basis functions. In particular, we illustrate this method with CSRBFs inmedical image segmentation.

Keywords

Reconstruction, Approximation, Interpolation, Regularization, Multiresolution, Partition ofunity, Implicit Surface, Medial axis, Image segmentation, Level sets, Collocation, Radial basisfunctions.

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Acknowledgments

Ce travail de thèse a été réalisé au sein du Centre de Recherche Et d’Application en Traite-ment de l’Image et du Signal (CREATIS), UMR CNRS 5515, Inserm U620, Institut Nationaldes Sciences Appliquées de Lyon (INSA Lyon), Université Claude Bernard Lyon I. Je remercieIsabelle MAGNIN, Directeur de recherche INSERM et du laboratoire, ainsi que le Professeur Di-dier REVEL pour m’avoir accueilli au sein de cette unité.

Que Monsieur Rémy PROST, directeur de thèse trouve ici mes plus vifs remerciements pouravoir accepté de m’encadrer, pour son aide précieuse, ses conseils éclairés et pour toute la confi-ance qu’il m’a accordée pendant ces trois années.

I would like to thank Takashi KANAI to have given me the opportunity to work in his labo-ratory, to follow my work since I went to Japan, to make me discover Japan and japanese culture,and to accept to be one examiner of my defense.

I would like also to thank Yutaka OHTAKE for all interesting conversations we had while Iwas in Japan, for all advices, comments he gave me during this work.

J’adresse mes remerciements à Monsieur Michel BARLAUD, Professeur à l’Ecole Polytech-nique de l’Université de Nice-Sophia Antipolis, et Monsieur Olivier DEVILLERS, Directeur derecherche INRIA Sophia Antipolis, de me faire l’honneur d’être mes rapporteurs de ce travail,pour le temps qu’ils ont consacré à ce manuscrit.

Mes remerciements s’adressent aussi à Monsieur Jean-Marc CHASSERY d’avoir accepté departiciper et de présider ce jury, et à Denis FRIBOULET d’avoir participé à ce jury.

Je tiens aussi à remercier Dassault Sytèmes, et plus particulièrement David BONNER, in-génieur R&D Dassault Systèmes Suresnes, pour m’avoir accueilli et avoir participé à mon en-cadrement pendant les six premiers mois de ma thèse. Je tiens à remercier Olivier BERNARDet Denis FRIBOULET pour la fructueuse collaboration, les bons moments que nous avons passésensemble dans d’interminables réunions...

Je tiens à remercier le ministère de l’éducation du Japon (MEXT), la Fondation de la RechercheMédicale (FRM), la région Rhône Alpes, qui m’ont soutenu financièrement pendant ces trois an-nées.

Je tiens à remercier tous les membres du laboratoire CREATIS pour les bons moments quel’on a pu partager pendant mon DEA et ma dernière année de thèse. Je voulais aussi remerciertoutes les personnes de CREATIS qui m’ont toujours soutenu, et même dans les moments diffi-ciles: Alex, Chris, Yougz...; et puis je voulais enfin envoyer un gros big-up pour mes acolytes de

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dernières années: les membres du secteur G (Léo, Joël) qui m’ont converti, au zboud’ avec qui j’aipassé mon été a rédigé ce document, à tous les doctorants de CREATIS qui m’ont supporté cesderniers mois...

I also wanted to thank the brazilian band of Tokyo: Marcos, Sydeney (Junior), Felipe, andtheir unbelievable girls, who showed me the way through the dark side of the force; Penny forher kindness and supports; Georges for ”Hey Ya!!!” and these long Karaoke parties; et puis tousles francophones que j’ai cottoyés à Tokyo: Sylvain (Merci pour les stats), Guillaume (arrête demagasiner), FX (te fais pas mal), Leilu, Arthur, Nicolas...

Enfin ces derniers mots iront enfin pour mes parents, ma famille, Luc, Sylvain, mes amis, etLydie qui m’ont toujours soutenu pendant toutes ces années d’étude...

Contents

I General Introduction 1

Introduction (français) 3

Introduction 5

Notation 7

II Multidimensional Scalar Reconstruction 9

Résumé 11

1 Multidimensional Reconstruction 171.1 Global Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.1.1 Multivariate Polynomial Methods . . . . . . . . . . . . . . . . . . . . . 181.1.2 Band-Limited Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2 Moving Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.3 Partition of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.4 Reconstruction in Shift Invariance Spaces . . . . . . . . . . . . . . . . . . . . . 211.5 Variational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.6 Radial Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Radial Basis Functions 252.1 SPD-RBF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.1.1 Positive Definite Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . 262.1.2 Positive Definite Radial Basis Functions . . . . . . . . . . . . . . . . . . 26

2.2 CPD-RBF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3 From Variational Principle to Radial Basis Functions . . . . . . . . . . . . . . . 292.4 In Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.5 Partition of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.6 Compactly Supported Radial Basis Functions . . . . . . . . . . . . . . . . . . . 32

2.6.1 Generality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.6.2 Use of CSRBF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.7 Multilevel or Multiresolution Approach . . . . . . . . . . . . . . . . . . . . . . 35

3 Our contributions 373.1 Multiresolution Signal Reconstruction . . . . . . . . . . . . . . . . . . . . . . . 39

3.1.1 Clustering Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1.2 Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

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3.1.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 Adaptive Signals Approximation with Fixed Budget . . . . . . . . . . . . . . . . 48

3.2.1 Algorithm Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2.2 Determinating Centers . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2.3 Support Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2.4 Coefficients computation . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3 Adaptive Signals Multiresolution Approximation . . . . . . . . . . . . . . . . . 533.3.1 Determining Center Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 533.3.2 Support Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.3.3 Coefficients Computation . . . . . . . . . . . . . . . . . . . . . . . . . 543.3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Conclusion 57Bilan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

III Surface Reconstruction 59

Résumé 61

Introduction 63

4 Implicit Surface 674.1 Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.1.1 Normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.1.2 Curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.1.3 Ridges And Valleys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2 Constructive Solid Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.3 Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3.1 Ray Tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.3.2 Polygonization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5 Implicit Surface Based Reconstruction : State of the art 755.1 Blobby Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.2 Signed Distance Function Estimation . . . . . . . . . . . . . . . . . . . . . . . . 765.3 Trivariate Implicit Polynomial Fitting . . . . . . . . . . . . . . . . . . . . . . . 775.4 Moving Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.5 Partition Of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.6 Radial Basis Function based Implicit Surface Reconstruction . . . . . . . . . . . 79

5.6.1 Global Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.6.2 Fast reconstruction and evaluation methods . . . . . . . . . . . . . . . . 80

6 Our Contributions 836.1 Reconstruction With Composite Implicit Surface . . . . . . . . . . . . . . . . . 85

6.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.1.2 Support Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.1.3 Local Polynomial Approximation . . . . . . . . . . . . . . . . . . . . . 866.1.4 Least Square Approximation . . . . . . . . . . . . . . . . . . . . . . . . 86

CONTENTS xi

6.1.5 Constructing Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.1.6 Computational Complexity . . . . . . . . . . . . . . . . . . . . . . . . . 886.1.7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.2 Support Adapted to the Medial Axis . . . . . . . . . . . . . . . . . . . . . . . . 936.2.1 Support Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.2.2 Trivariate Quadric Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . 966.2.3 CSRBF Coefficients Computation . . . . . . . . . . . . . . . . . . . . . 966.2.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976.2.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98


IV Segmentation 107

Résumé 109

Introduction 111

7 Active Contours 1137.1 Active Contours Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1137.2 Level Set Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

8 Our Contribution : Level Set with CSRBF 1218.1 Implicit Function Decomposition in the Level Set Framework . . . . . . . . . . . 1228.2 Level Sets with CSRBF Collocation . . . . . . . . . . . . . . . . . . . . . . . . 122

8.2.1 Unified representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1228.2.2 Solving Level Set With RBF Collocation . . . . . . . . . . . . . . . . . 1238.2.3 Compactly Supported Radial Basis Functions . . . . . . . . . . . . . . . 1248.2.4 RBF Centers Distribution and Velocity Sampling . . . . . . . . . . . . . 1248.2.5 Resolution of the ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . 1258.2.6 Bounded Implicit Function . . . . . . . . . . . . . . . . . . . . . . . . . 126

8.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1278.3.1 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1288.3.2 Experimental Medical Data . . . . . . . . . . . . . . . . . . . . . . . . 1328.3.3 Computation Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136


V Conclusion And Perspectives 139

Bilan et Perspectives 141

Conclusion and Perspectives 143

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Appendix 147

A From Variational Problem to RBF 147

B kd-tree 151

C 3D Partition of Unity 153C.1 General formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153C.2 Weight Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

C.2.1 Spherical Weight Function . . . . . . . . . . . . . . . . . . . . . . . . . 154C.2.2 Cubical Weight Function . . . . . . . . . . . . . . . . . . . . . . . . . . 154

D Convergence proof elements 155

Bibliography 159

Bibliographie Personnelle 159

Part I

General Introduction

1

Introduction

Au 20e siècle, les avancées en informatique, télécommunications, systèmes d’acquisitionont profondément changé notre vie. En imagerie médicale, l’imagerie ultrasonore, l’ima-gerie à résonnance magnétique (IRM), la tomographie X (Computed Tomography, CT),

sont maintenant devenues des modalités courantes d’imagerie. Au départ, les radiographies àrayons X étaient utilisées pour produire des projections 2D des organes, principalement pour visu-aliser les os. Avec le développement de l’IRM et de la CT, il est maintenant courant de visualiserdirectement des images tridimensionnelles et même des séquences d’images tridimensionnelles:3D+t. A présent, les praticiens demandent plus d’informations pour leur diagnostic, et il est deplus en plus courant d’utiliser la tomographie à émission de positons (TEP) couplée à l’IRM. Parconséquent, avec le développement de ces technologies d’imagerie, des données de haute résolu-tion spatiale, en 2, 3, 4 dimensions, ou plus, doivent être traîtées, étudiées, modélisées.

Un formalisme nD a été introduit par les méthodes dites meshless, c’est à dire ne nécessitantpas de maillages sous jacents [Shepard, 1968,Hardy, 1971,Duchon, 1976,Duchon, 1977,Lancasterand Salkauskas, 1981, Wendland, 1995]. Ces méthodes font l’objet de nombreuses recherchesen mathématiques appliquées. Elles ont été, à l’origine, développées pour des applications engéodésie [Hardy, 1976], géophysique [Hardy, 1971], cartographie [Shepard, 1968]. Maintenant,on retrouve des applications dans beaucoup de domaines, tels que la théorie de l’apprentissage[Schwenker et al., 2002], réseau de neurones [Broomhead and Lowe, 1988], statistiques [Hastieet al., 2001], finance [Hutchinson, 1995], optimisation [Gutmann, 2001], résolution d’équationsaux dérivées partielles (EDP) [Fasshauer, 2006]... A notre connaissance, ces méthodes sont mé-connues en imagerie et en traitement d’image médicale, c’est pour cette raison que nous avonsdécidé d’explorer cette voie.

Il nous a semblé intéressant de donner un bref historique des méthodes dites meshless, etplus particulièrement d’une de ces méthodes qui utilise des bases de fonctions bien particulières:les fonctions de base radiale (RBF). Au début des années 70, Hardy développa les bases multi-quadriques et mutliquadriques inverses pour des applications en géodésie [Hardy, 1971]. A lamême époque, Duchon formula une approche variationnelle qui l’amena aux thin plate splinesou splines polyharmoniques [Duchon, 1976, Duchon, 1977]. Dans les années 80, de nombreuxtravaux plus théoriques ont permis la caractérisation de ces fonctions et d’arriver à d’importantsrésultats concernant leur utilisation [Madych and Nelson, 1983, Micchelli, 1986]. Malgré tous cestravaux, ce n’est qu’avec les récents travaux de Wendland [Wendland, 1995] que les premièresméthodes à faible coût opérationnel virent le jour, avec les RBF à support compact (CSRBF).

Dans ce manuscrit, nous nous intéressons à l’utilisation des CSRBF pour la reconstruction etla segmentation d’images. La reconstruction consiste à trouver une fonction (continue) qui est une(bonne) approximation d’une fonction inconnue dont on possède uniquement des échantillons.La segmentation d’images consiste à déterminer les frontières de régions d’intérêt dans les im-

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4

ages. Ces deux problèmes sont essentiels pour l’étude, la compréhension, le traitement, l’analysed’informations contenues dans les signaux multidimensionnels, images ou surfaces.

Ce manuscrit est composé de trois parties, correspondant aux trois applications des CSRBFque nous avons retenues: la reconstruction multidimensionnelle de signaux, la reconstruction desurfaces et la segmentation d’images.

Dans la partie II, nous étudions la reconstruction de signaux, et de fonctions, multidimension-nels à partir d’échantillons à répartition spatiale non-uniforme, c’est à dire à partir d’échantillonsqui ne sont pas sur grille régulière. Nous commençons par un bref tour d’horizon des méthodes ex-istantes (chapitre 1). Ensuite dans le chapitre 2, nous résumons les propriétés principales des RBF,qui sont utiles pour la compréhension du manuscrit, et présentons des méthodes de la littérature.Par la suite nous introduisons deux nouvelles méthodes d’approximation. Plus exactement, dansle chapitre 3, nous présentons une nouvelle méthode d’approximation multirésolution, utilisantune classification hiérarchique des données, et une méthode adaptative où le support de chaqueCSRBF est calculé localement. Nous illustrons ces propositions en traitant le cas de signaux 1Det 2D avec un échantillonnage non-régulier.

Dans la partie III, nous nous intéressons à la reconstruction de surfaces. Les méthodes de re-construction de surfaces peuvent être classées en fonction de la représentation souhaitée: surfacemaillée, surface paramétrique et surface implicite. Parmi ces trois modélisations, nous avons dé-cidé de nous consacrer à la reconstruction de surfaces implicites à partir de nuages de points. Unesurface implicite est définie comme le niveau 0 d’une fonction implicite 3D. Dans le chapitre 4,nous présentons le formalisme et les avantages de cette représentation. Ensuite nous faisons un breftour d’horizon des méthodes existantes (dans le chapitre 5). Dans le chapitre 6, nous proposonsdeux nouvelles méthodes. La première méthode consiste à résoudre localement des problèmes dereconstruction à partir de CSRBF et de construire la solution globale par le principe de partition del’unité. Dans notre deuxième méthode, nous adaptons localement le support de chaque CSRBF àl’axe médian et nous approximons le nuage de point en minimisant une fonctionnelle quadratiquerégularisée.

Dans la partie IV, nous nous intéressons à la segmentation d’images. Parmi toutes les mé-thodes proposées dans la littérature, nous avons décidé de nous consacrer aux contours actifs quiconsistent à approximer les frontières des objets par des interfaces. Pour cela, l’évolution des in-terfaces, contours en 2D, surfaces en 3D, est régie par la solution d’une EDP. Dans le chapitre 7,nous présentons rapidement les deux implémentations principales de contours actifs, qui dépen-dent du type de représentation de l’interface: représentation explicite telle que les snakes [Kasset al., 1988], ou implicite, comme dans le formalisme des ensembles de niveaux (level sets enanglais) [Osher and Sethian, 1988]. Compte tenu des avantages des représentations implicites etde connaissances acquises dans la reconstruction de surfaces implicites, il nous est apparu nor-mal de nous concentrer sur les méthodes d’ensembles de niveaux. Jusqu’à présent, ces méthodesutilisaient des différences finies sur une grille régulière pour la recherche de la solution de l’EDP.Ici, dans le chapitre 8, nous proposons un nouveau formalisme dans lequel nous décomposons lafonction implicite sur une base de fonctions. Nous démontrons la validité de notre proposition, enutilisant des CSRBF qui assurent l’existence d’une solution pour toute dimension.

Introduction

In the 20th century, advances in computer science, telecommunications, and acquisition tech-nologies, have deeply changed our life. In medical imaging, Ultrasound images, MagneticResonance Imaging (MRI), Computed Tomography (CT), are now common modalities. At

first, X-ray radiographs were used to produce 2D projections of parts of the human body, mainlyused for bones. With the development of MRI and CT, it is now common to visualize directly 3Dimages and even spatio-temporal 3D+t images. Now, in order to use the more data that physicianscan get for diagnostic purpose, some modalities are used in conjunction, e.g. Positron EmissionTomography (PET) scan is currently associated to MRI. Thus, with the development of these tech-nologies high spatial resolution data, in dimension 1, 2, 3, 4 or more, must be processed, studied,modeled.

In order to work in any dimension, meshless methods have been introduced [Shepard, 1968,Hardy, 1971, Duchon, 1976, Duchon, 1977, Lancaster and Salkauskas, 1981, Wendland, 1995].Meshless methods are the topic of recent research in many area of computational science. Orig-inally, the motivation of these methods came from applications in geodesy [Hardy, 1976], geo-physics [Hardy, 1971], mapping [Shepard, 1968]. Now applications are found in many areas suchas in learning theory [Schwenker et al., 2002], neural networks [Broomhead and Lowe, 1988],statistics [Hastie et al., 2001], finance [Hutchinson, 1995], optimization [Gutmann, 2001], numer-ical solution of partial differential equations (PDE) [Fasshauer, 2006]... To our knowledge, inmedical imaging these methods are misknown, for this reason, we decide to explore this domain.

Let us give a brief history of meshfree methods over the past few decades, and more specif-ically about kind of methods based on some particular basis functions: radial basis functions(RBFs). In the early 1970s, Hardy introduced multiquadric and inverse multiquadric for geodesyapplications [Hardy, 1971]. Roughly at the same time, Duchon formulated a variational approachthat led to thin plate splines or polyharmonic splines [Duchon, 1976,Duchon, 1977]. In the 1980s,many theoretical works have been done about the use of RBFs [Madych and Nelson, 1983, Mic-chelli, 1986]. Despite these studies, it is only with the recent work of Wendland [Wendland, 1995],that RBFs become computationally efficient. Wendland presented a new class of RBFs, with com-pact support, that is generally referred as Compactly Supported Radial Basis Functions (CSRBFs).

In this thesis, we will study the application of the CSRBFs for reconstruction and image seg-mentation. Reconstruction consists in finding a (continuous) function which fits good to somegiven samples, measurements, at given distinct locations. Image segmentation consists in findinglimits of relevant areas in images. These general problems are fundamental for the study, under-standing, analysis of information contained in data, such as in multidimensional signals, images,or surfaces.

The manuscript is composed of three main parts, corresponding to the three target applications

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for the use of CSRBFs: multidimensional signal reconstruction, surface reconstruction and imagesegmentation.

In Part II, we study multidimensional signal, and scalar function, reconstruction from non-uniform samples, i.e. the location of data are not on a regular grid. At first, we give a briefoverview of existing methods for multidimensional reconstruction (in Chapter 1). Then in Chapter2, we briefly summarize main properties of RBFs, that will be useful for the comprehension of thewhole document, and present some RBF based methods of the literature. In Chapter 3, we presenta new multiresolution approximation method, based on a data clustering method, an adaptive ap-proximation method, in which the support size, i.e. the influence domain, of one CSRBF is locallyadapted. We present results on non uniform sampled 1D, 2D signals.

In Part III, we focus on surface reconstruction. Basically there are three approaches to rep-resent a (2D) surface embedded in 3D space: meshes, parametric surfaces and implicit surfaces.Among these three representations, we choose to reconstruct implicit surfaces from 3D point sets.An implicit surface is defined as the zero set of a 3D implicit function. In Chapter 4, we presentgeneralities and advantages of such representation. In Chapter 5, we discuss previous works aboutimplicit surface reconstruction. In Chapter 6, we introduce two new methods. The first one com-bines locally reconstructed CSRBF implicit based surfaces together by applying a partition ofunity blending. The second one locally adapts the support size of each CSRBF to the medial axis,and approximates given point set.

In Part IV, we focus on image segmentation. Among all the work done in image segmentation,we decide to study active contours which consists in fitting interfaces to object borders. To thisend, interfaces evolve according to PDE. In Chapter 7, we present the two main implementationsof active contours principle, which depends on interface representations. Indeed interfaces canbe modelled by an explicit representation, like snakes [Kass et al., 1988], or by an implicit one,like in the level set formalism [Osher and Sethian, 1988]. With all the advantages of implicitrepresentation, and using our knowledge about implicit interfaces, we decide to focus on level setinterface evolution. Existing methods solve interface evolution by using finite difference on a grid.In Chapter 8, we propose a new formalism in which we expand the implicit function on somebasis functions. We demonstrate the effectiveness of this decomposition by using CSRBFs, whichensure the existence of basis in any dimension.

Notation

Following Wendland [Wendland, 2005b], we fix now some notations that will be used through-out this document.

We denote

• Πm(Rd) the space of d-variate polynomials of absolute degree at most m.

• 〈f, g〉 the usual inner product of two functions f, g : Ω→ R, which is given by

〈f, g〉 =∫

Ωf(p) · g(p) dp

• ‖ · ‖Lp(Ω) the Lp-norm on Ω, which is given by

‖f‖pLp(Ω) =∫

Ω|f(p)|p dp

for 1 6 p <∞ and by ‖f‖L∞(Ω) = ess supp∈Ω |f(p)|.

• Ck(Ω) the set of k times continuously differentiable functions on Ω, where Ω ⊆ Rd shouldbe measurable, i.e. they consist of all measurable functions f having finite Lp-norm. In thisdocument, by abuse of language, sometimes we omit to the domain definition where thisproperty is satisfied.

• ˆ indicates Fourier transform in d dimension.

Since we are working with several points in Rd, we should fix some convention about indices.For a point p ∈ Rd, its components will be given as p0, . . . , pd−1, whereas p0, . . . ,pN−1 denoteN points in Rd. The components of pi are thus denoted pi,j with 0 6 j 6 d.

As usual we denote ‖u‖p is the discrete p-norm

(‖u‖p)p =d−1∑

j=0

|pj |p

for 1 6 p 6 ∞ and ‖u‖∞ = max |uj |. In this document, by abuse of language, we write ‖u‖ todenote the Euclidean norm, i.e. the L2 norm. As for functions, we also define the standard innerproduct as follows 〈u,v〉.

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Part II

Multidimensional ScalarReconstruction

9

Résumé

Le traitement numérique des signaux utilise principalement des signaux discrets multidi-mensionnels, représentés par des fonctions f : Ω ⊆ Rd → R, que l’on obtient par uneopération d’échantillonnage de ce signal. Cet espace d’échantillonnage peut être temporel,

en 1D, spatial en 2 ou 3D.

Dans de nombreux domaines d’application, il est essentiel de pouvoir trouver le signal (la fonc-tion) d’origine, ou son approximation, à partir d’échantillons (données). Le processus, l’état d’unsystème, peut alors être étudié, modélisé à d’autres instants, dans le cas temporel, d’autres posi-tions, dans le cas spatial, que les échantillons eux-mêmes... La reconstruction de signaux consistea résoudre ce problème, c’est à dire trouver un signal (fonction) qui est une bonne approxima-tion des échantillons. Dans un cadre plus général, pour n’importe quelle dimension d de l’espaced’échantillonnage, nous appelons ce problème: reconstruction de signaux multidimensionnels.

Depuis l’antiquité, la reconstruction a fait l’objet d’un grand intérêt par les mathématiciens etscientifiques, mais aussi pour des besoins pratiques. En effet, les paysans basaient leur stratégie deplantation sur la position des corps célestes, soleil, lune, planètes... A cette époque, ils utilisaientla reconstruction pour prédire la position des astres dès qu’ils ne pouvaient les localiser dans leciel, à cause de mauvaises conditions atmosphériques ou d’un masquage de certains corps célestespendant certaines périodes. Au fil des siècles, le problème de la reconstruction a été étudié par bonnombre de mathématiciens et scientifiques, tels que Galilée, Newton, Lagrange, Gauss... Maisc’est au 20e siecle, avec l’avènement des technologies numériques et les travaux fondateurs deShannon que le problème de la reconstruction de signaux a reçu un intérêt encore plus important.

Pour certaines applications, comme en imagerie numérique, on peut considérer un échantillon-nage uniforme, les échantillons étant placés sur une grille cartésienne en d-dimension (voir Fig.1(a)). Cependant, dans de nombreux autres cas, on ne peut considerer de tels échantillonnages,on parle alors d’échantillonnage non-régulier ou irrégulier. Par exemple, lors d’envoi de donnéessur un réseau, des données peuvent être perdues, ou subir une telle altération qu’on ne puisse plusconsidérer tous ces échantillons (voir Fig. 2). En astronomie, les mesures de luminosité d’uneétoile ne peuvent être prises à intervalles de temps réguliers (voir Fig. 3), à cause des perturba-tions atmosphériques, de la rotation de la terre et de phénomènes parasites. En imagerie médicale,la tomographie (CT) et l’imagerie à résonance magnétique utilisent parfois des échantillonnagesnon-uniformes, sur une grille polaire, dans l’espace de Fourier (voir Fig. 1(b)). Il existe encore bonnombre d’applications, que nous ne présenterons pas ici, mais l’on comprend bien l’importancede la reconstruction de signaux multidimensionnels à partir d’échantillons non-uniformes.

Dans cette partie, nous allons donc étudier le problème de la reconstruction de signaux multi-dimensionel à partir d’échantillons non-uniformes, c’est à dire un signal qui donne une ”bonne”approximation de mesures, ou échantillons. Comment définir une ”bonne” approximation? De

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façon générale, on différenciera deux cas: l’interpolation et l’approximation pour déterminer cequ’est une bonne approximation. Dans le cas de l’interpolation, nous cherchons une fonction quipasse exactement par les données; alors que pour l’approximation, nous cherchons une fonctionqui approxime les données. Malheureusement ces deux problèmes sont mal-posés, il existe uneinfinité de solutions, et par conséquent résoudre ces problèmes nécessitent des connaissances apriori. Dans cette partie, nous considèrons uniquement les méthodes ne nécessitant pas de mail-lages sous-jacents, dites méthodes meshless.

Dans le chapitre 1, nous voyons que l’une des hypothèses forte généralement utilisée, pourrendre ce problème bien posé, est de considérer que la fonction peut se décomposer sur une basede fonctions. Nous décrivons brièvement les différentes méthodes permettant de reconstruire unefonction continue à partir d’échantillons, en utilisant des bases polynomiales, en considérant dessignaux à bande de fréquence limitée, ou encore des moindres carrés glissants, en utilisant leprincipe de partition de l’unité, les approches variationnelles, et enfin les fonctions de base radi-ale.

Dans le chapitre 2, nous nous attardons plus en détails sur les fonctions de base radiale, endonnant leurs principales propriétés, des explications et remarques sur leur utilisation, qui sonttrès utiles pour la compréhension de ce document.

Enfin dans le chapitre 3, nous présentons de nouvelles méthodes de reconstruction de signauxmultidimensionnels à partir d’échantillonnage non-uniforme. Ces méthodes s’appuient sur unesous famille de fonctions de base radiale: les fonctions de base radiale à support compact (CSRBF)introduites par Wendland [Wendland, 1995].

Nous décrivons une méthode d’approximation multirésolution qui s’appuie sur une méthode declassification, où par l’on ajoute de l’information seulement là où elle est requise. Nous illustronscette méthode par des résultats sur des exemples 1D et 2D avec des échantillonnages aléatoires.

Nous présentons une méthode d’approximation adaptative avec un budget fixé de CSRBF,dans laquelle le support de chaque fonction de base est adapté localement à la distribution descentres. Nous illustrons cette méthode sur un exemple d’image avec un échantillonnage aléatoire.

Nous présentons une modification de la méthode précédente pour traîter le cas d’une recon-struction à erreur fixé. Pour celà, nous utilisons à la fois une méthode adaptative et une méthodemultirésolution.

Introduction

Modern digital processing always use a discrete version of one original multidimensionalsignal, represented by a function f : Rd → R obtained by sampling. For example, thesampling set could be in the time domain, in 1D, e.g. one can measure at different time

some relevant characteristics of one system; the sampling could also be spatial, in 2 or 3D, bytaking a picture of a scene with a digital camera.

In many practical applications, it is important to find the original signal from these samples,or to approximate it, in order to model or to study the concerning process in a way that requiresinformations not included explicitly in the samples. This process is generally referred as signalreconstruction, and on the most general case, i.e. for any dimension sample space, we recall thisproblem multidimensional signal reconstruction.

Since the antiquity, reconstruction has received a lot of interests for first important practicalneeds. Indeed, farmers, for example, would base their planting strategies according to some pre-dictions of the positions of the sun, moon, and planets. By that time, reconstruction was used topredict these positions whenever it was not possible to locate them, due to either some bad atmo-spheric conditions or the impossibility to see all celestial bodies during certain periods. With thepassing centuries, reconstruction has been studied by many famous scientists, e.g. Galileo, New-ton, Lagrange, Gauss... In the 20th century, reconstruction problem received even more attentionwith the advent of digital technologies and pioneer’s works of Shannon.

In some applications, e.g. digital imaging, it is justified to assume a uniform sampling, i.e.samples form a d-dimensional Cartesian grid (see Fig. 1(a)). However in many realistic situations,data are known only on a non-uniformly spaced sampling set, thus the uniformity assumption cannot be made. For example, in communication when data from a uniformly sampled signal are lost(see Fig. 2); in astronomy the measurement of star luminosity (radiation) provides extremely non-uniformly samples time series (see Fig. 3); or in medical imaging Computed Tomography (CT)or Magnetic Resonance Imaging (MRI) sometimes use the non-uniform polar samples in Fourierdomain (see Fig. 1(b)). Thus non-uniform multidimensional reconstruction is an important keyproblem.

In this part, we focus on meshless methods for multidimensional signal, function, reconstruc-tion. Consequently, certain methods, which generally require an underlying mesh, such as tensorialproduct of splines, finite elements, etc. would not be considered in this part, even if these methodscan succeed in the multidimensional reconstruction context. Thus, the objective of this part is togive an overview of existing methods, and to develop some new ones to recover multidimensionalsignal from such non-uniform samples.

More precisely, we consider no restriction on data location distribution, thus this problem

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14

(a) (b)

Fig. 1. Uniform samples in a 2D space (a). Polar sampling in the Fourier space sometimes used in MRIand CT reconstructions (b).

(a) (b)

Fig. 2. Middle sagittal slice of human brain (a), image with 75% randomly missing samples (b).

Fig. 3. Measurements taken at regularly with a period T when it is possible (similar problem with starluminosity measurement).

15

is obviously ill-posed, i.e. there is an infinity of solutions, we use a specific data dependent basiscalled Compactly Support Radial Basis Functions which transforms this problem into a well-posedone, i.e. there exist only one solution.

In this part, we start by reviewing popular methods for multidimensional signal reconstructionin Chapter 1, and give more details about Radial Basis Functions in Chapter 2, before presentingsome new methods that we applied on some practical cases in Chapter 3.

Chapter 1

Multidimensional Reconstruction

Contents1.1 Global Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.1.1 Multivariate Polynomial Methods . . . . . . . . . . . . . . . . . . . . 18

1.1.2 Band-Limited Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2 Moving Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.3 Partition of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.4 Reconstruction in Shift Invariance Spaces . . . . . . . . . . . . . . . . . . . 211.5 Variational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.6 Radial Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

We have a set of data, i.e. measurements and locations at which these measurements wereobtained, and we want to find a rule which allows us to deduce information about the process weare studying also at locations different from those we got measurements. Thus, we aim to find afunction which fits ”good” the given data. There are many ways to decide what means ”good”.In the case of interpolation, we want the reconstructed function to match exactly the given mea-surements at their corresponding locations. Whereas in the case of approximation, we want thereconstructed function to approximate the given measurements at their corresponding locations. Ifthe locations at which these measurements have been obtained do not lie on a uniform, or regular,grid, those problems are often recalled scattered data interpolation or approximation.

More precisely, we consider a finite set of known values, or measurements, siN−1i=0 ∈ R at

locations P : pi ∈ ΩN−1i=0 , and we aim to find a continuous function f : Ω ⊆ Rd −→ R, which

interpolates (see Eq. 1.1), or approximates the given data (see Eq. 1.2).

f(pi) = si, ∀ i ∈ 0, . . . , N − 1 (1.1)

f(pi) ≈ si, ∀ i ∈ 0, . . . , N − 1 (1.2)

Since we do not fix the dimension d of the space we are working in, this problem allows us toconsider many types of problem, as we will see later.

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18 CHAPTER 1. MULTIDIMENSIONAL RECONSTRUCTION

Unfortunately, interpolation or approximation problems are clearly ill-posed, i.e. there are in-finite of solutions, and some a priori knowledge about the function to be reconstructed is required.

A common approach to solve these problems is to make the assumption that the function f isa linear combination of certain basis functions uk, i.e.

f(p) =N−1∑

k=0

ck · uk(p), p ∈ Rd (1.3)

where ck are some real coefficients.Under this assumption, solving

• the interpolation problem leads to a linear system of equations of the form

A · c = s

where the interpolation matrix A is given by Aij = uj(pi), i, j = 0, . . . , N − 1, c =[c0, . . . , cN−1]

T and s = [s0, . . . , sN−1]T .

• the approximation problem is commonly done by considering a discrete least-squaresmethod, where the number of basis functions M is lower than the number of samples N(M 6 N ), we minimize the Mean Square Error given by:

E[f ] =N−1∑

i=0

(f(pi)− si)2 (1.4)

This problem can be solved with aQR-decomposition of the approximation matrix B givenby Bij = uj(pi), i = 0, . . . , N − 1, j = 0, . . . ,M − 1, or by solving the following linearsystem:

B ·BT · c = BT · s

where c = [c0, . . . , cM−1]T and s = [s0, . . . , sN−1]

T .

Note that if M = N , A = B, thus discrete least-squares method leads to an interpolatingsolution.

These problems are clearly well-posed if the space V spanned by the basis uk is a Haar space,i.e. for any distinct points p0, . . . ,pN−1 ∈ Ω and any basis u0, . . . , uN−1 of V , det(uj(pi)) 6= 0.

In this chapter, we briefly review common methods for multidimensional reconstruction [Wend-land, 2005b], which consider various kinds of basis, or various principles.

1.1 Global Fitting

1.1.1 Multivariate Polynomial Methods

The reconstruction problem is completely understood in the univariate case, i.e. d = 1. Indeedit is well known that one can interpolate, or approximate, arbitrary data at N distinct locationsusing a polynomial form of degree N − 1, in the interpolation case, or of degree lower than N − 1for the approximation case. It is important to notice that the admissible solution space ΠN−1(R1)

1.2. MOVING LEAST SQUARES 19

does not depend neither on the data location, nor on the values, but only on the number of data N .It means that ΠN−1(R1) is an N -dimensional Haar space for subset Ω of R that contains at leastN distinct points.

Mairhuber [Mairhuber, 1956] shows that Haar spaces, in higher dimensions, only exist in someconfigurations concerning the data locations. More precisely, if Ω ⊆ Rd, for d > 2, contains oneinterior point, then there exists no Haar space on Ω of dimension N > 2. Hence, it is impossibleto interpolate, or approximate, all kinds of data at any set of data locations P = p0, . . . ,pQ−1,with Q = dim Πm(Rd), by polynomials from Πm(Rd). So multivariate polynomial interpolation,or approximation, are considered only in some restricted configuration, i.e. when the data loca-tions are Πm(Rd)-unisolvent.

Definition 1.1. We call a set of points P = p0, . . . ,pN−1 ⊆ Rd Πm(Rd)-unisolvent, withN > Q = dim Πm(Rd), if the zero polynomial is the only polynomial from Πm(Rd) that vanisheson all of them.

1.1.2 Band-Limited Signal

The space of band-limited functions is the space of function whose Fourier transform is suchthat f(ω) = 0, ∀ ω /∈ Ω = [−W,W ]d.

Shannon shows in [Shannon, 1948, Shannon, 1949] that ”there is one and only one functionwhose spectrum is limited to a band W and which passes through given values at sampling pointsseparated 1/2W seconds apart”, and this function is represented by

f(x) =∞∑

k=−∞sk · sinπ(2Wx− k)

π(2Wx− k) (1.5)

It has been shown that this formula is equivalent to says that the setej2πkω

forms an or-

thonormal basis called the harmonic Fourier basis. This equivalence has been extended to treatsome particular case of non-uniformly samples data [Kadec, 1964].

In [Grishin and Strohmer, 2004], authors propose to determine coefficients by a discreteweighted least-squares method, which is solved by combining a fast Discrete Cosinus Transform(DCT) based algorithm and a conjugate gradient method.

1.2 Moving Least Squares

The idea of the moving least squares (MLS) approximation [Lancaster and Salkauskas, 1981]is to solve for every point p a locally weighted least squares problem. More precisely, for p ∈Ω ⊆ Rd, the value of the f(p), i.e. the MLS approximant, is given by f(p) = g∗(p) where g∗ isthe solution of

min

N−1∑

i=0

ω(p,pi) · (si − g(pi))2 : g ∈ Πm(Rd)

(1.6)

where ω : Ω× Ω→ R is a weight function which becomes smaller the further away its argumentare from each other, and g is a multivariate polynomial form. Ideally, ω vanishes for argumentsp,q ∈ Ω with ‖p− q‖ greater than a certain threshold. Such behavior is generally modeledby using translation-invariant or rotation-invariant weight function, with using a scaled version acompactly supported weight function.


The coefficients of g∗ depend on the evaluation point p, and thus for every evaluation of f alinear system needs to be solved, i.e.

G(p) · c(p) = sg(p) (1.7)

where c(p) are the coefficients in the basis formed by gjm−1j=0 , the matrix G(p) has entries

Gjk(p) =N−1∑

i=0

ω(p,pi) · gj(pi) · gk(pi) (1.8)

and the right-hand side vector consists of the projections of the data onto the basis functions, i.e.

[sg(p)]j =N−1∑

i=0

ω(p,pi) · gj(pi) · si (1.9)

These methods apparently seem difficult to use, due to the fact that for every evaluation pointp a linear system needs to be solved. However, by using compactly supported weight functiononly a few terms are considered in the matrix computation (see Eq. 1.8), or even completely avoidsolving the linear system [Fasshauer, 2002].

Note that MLS method in the case m = 1 with g0(p) = 1 is known to yield Shepard’s

method [Shepard, 1968], where there is only one coefficient c0(p) =

N−1Pi=0

si·ω(p,pi)

N−1Pi=0

ω(p,pi)

.

1.3 Partition of Unity

The main idea of these methods is to decompose global domain into overlapping sub-domainswhere we can find a local solution fi(p). Then in order to get one continuous function on theglobal domain, some positive weight functions θi(x) are used for blending all local solutions to-gether. Partition of unity methods differ in the expression of the local solution fi (see Fig.1.1).

More precisely consider a global bounded domain Ω and divide it into M overlapping sub-

domains ΩiM−1i=0 with

M−1⋃i=0

Ωi ⊆ Ω. Then we associate a set of non negative compactly sup-

ported function θiwithM−1∑i=0

θi = 1 on the entire domain Ω and with limited support supp(θi) ⊆Ωi.

For each sub-domain Ωi, a set of points Pi = p ∈ Pi | p ∈ Ωi is constructed, and a localapproximation fi is computed. Then the global function f can be defined as a combination of

the local functions ponderated by θi. The conditionN∑iθi = 1 is obtained by normalizing the

contribution of each function ωi.

f(p) =M−1∑

i=0

ωi(p)M−1∑j=0

ωj(p)fi(p) =

M−1∑

i=0

θi(p) fi(p) (1.10)

Note that the partition of unity process respects an interpolation property, i.e. if all fi areinterpolants at P ∩ Ωi then f is an interpolant at the entire point set P .

1.4. RECONSTRUCTION IN SHIFT INVARIANCE SPACES 21

(a) (b)

(c) (d)

Fig. 1.1. Illustration of partition of unity principle. Data piN−1i=0 and a global domain Ω (a) are first

divided here with some spherical sub-domains ΩjM−1j=0 with its associated weight function ωj (b).

On each sub-domains an approximation or an interpolation is computed (c). The global solution isfinally obtained by blending all local solutions following Eq. 1.10 (d).

Moreover, even in this general situation, we can bound the error:

|s(p)− f(p)| =∣∣∣∣∣M−1∑

i=0

(s(p)− fi(p)) · θi(p)

∣∣∣∣∣

6M−1∑

i=0

|s(p)− fi(p)| · θi(p)

6 max06i6M−1

‖s− fi‖L∞(Ωi)

It shows that the global approximation error is governed by the worst local approximationerror. It means, if the local approximation provides good approximations so will the global one.

1.4 Reconstruction in Shift Invariance Spaces

Definition 1.2 (kernel). Consider a domain Ω ⊆ Rd, we call a function Φ : Ω×Ω −→ R a kernel.

A shift-invariance space SI(Φ), with a so-called generator Φ, is defined as the space of fun-ctions of the form

f(p) =∑

k∈Zd

ck · Φ(p− k) such that∑

k∈Zd

c2k <∞. (1.11)

It is important to notice that, according to this definition, the space of band-limited signal isidentical to a shift-invariant space. However, since sinc function (in Eq. 1.5), i.e. sin x

x , has infinitesupport and slow decay, many coefficients contribute to one value of f . That is the reason whyauthors focus on non band limited shift-invariant space reconstruction method.


This problem has been addressed by many researchers, see [Aldroubi and Gröchenig, 2001]for a recent survey. Here we briefly present the method described in [Aldroubi and Gröchenig,2001] with some simplifications.

For the spaces SI(Φ) to be well-defined, the following condition must be imposed on theFourier transform of Φ:

0 < m 6∑

k∈Zd

∣∣∣Φ(ω + k · 2π)∣∣∣2

6 M <∞, (1.12)

for some constants m > 0 and M > 0 [Mallat, 1989]. If this condition is satisfied, then the setΦ(· − k) forms a Riesz basis for SI(Φ). It also guarantees the existence of a dual generatorΦ satisfying

⟨Φ(·), Φ(· − k)

⟩= δ(k), where δ the multivariate Kronecker function. It can be

shown that Φ(·−k) also forms a Riesz basis. Since the dual generator belongs to SI(Φ), it canbe expressed as follows

Φ(p) =∑

k∈Zd

bk · Φ(p− k) (1.13)

where the coefficients bk are the discrete inverse Fourier transform of(∑

k

∣∣∣Φ(ω + 2πk)∣∣∣2)−1

The reconstruction of a function f from samples in SI(Φ) relies on the existence of the so-called reproducing kernel.

The reproducing kernel Kp, i.e. K(p, ·), associated with the location p is defined by

f(p) = 〈f,Kp〉 =∫

Rd

f(q) ·Kp(q) dq, ∀ f ∈ SI(Φ) (1.14)

A sufficient set of conditions for the existence of a reproducing kernel for every p is thatthe generator Φ is continuous and satisfies

∑k∈Zd

ess sup|Φ(p + k)|2;p ∈ [0, 1]d

< ∞. Conse-

quently, the reproducing kernel can be expressed as

Kp(q) =∑

k∈Zd

Φ(p− k) · Φ(q− k) (1.15)

Now we suppose that, for each p, a unique reproducing kernel exists. Then, any functionf ∈ SI(Φ) can be perfectly reconstructed from samples, if the associated reproducing kernelsKpi

N−1i=0 are such that

A ‖f‖2 6N−1∑

i=0

∣∣⟨f,Kpi

⟩∣∣2 =N−1∑

i=0

|s|2i 6 B ‖f‖2 (1.16)

where A and B are some positive constant independent of f . If this condition is satisfied, thereexist a dual frame Kpi

,pi ∈ P and the function f can be recovered as

f(p) =N−1∑

i=0

⟨f,Kpi

⟩Kpi

=N−1∑

i=0

si · Kpi(1.17)

However, a dual frame is difficult to find in general and this method for recovering f from datais often not practical.

1.5. VARIATIONAL METHODS 23

An indirect way to recover the function using frame formalism is to invert the frame operator

Tf(p) = f0(p) =N−1∑

i=0

⟨f,Kpi

⟩Kpi

(p) (1.18)

The inverse operator T can be expressed as

T−1 =2

A+B

∞∑

n=0

(I − 2

A+B·T

)n

(1.19)

This define an iterative frame reconstruction algorithm which is made up an initialization f0

(see Eq. 1.18) and iteration

fn =2

A+Bf0 +

(I − 2

A+B·T

)fn−1 (1.20)

and converges to f∞ = T−1f0 = T−1Tf = f .In [Aldroubi and Gröchenig, 2001], authors show that the reconstruction can also be obtained

as the least-squares fit to the data instead of interpolation.

1.5 Variational Methods

The solution of these ill-posed problems can be obtained, from the regularization theory, byvariational principle containing data closeness and smoothness information. Following this prin-ciple, solving the reconstruction problem (approximation, or interpolation) consists in finding thefunction f which minimizes the following general functional:

H[f ] =N−1∑

i=0

(f(pi)− si)2 + λ · (R[f ]− r0) (1.21)

where λ is a Lagrange Multiplier, generally called regularization parameter, R[f ] is a smoothnessterm or regularization functional which has the property of a (semi) norm and generally penalizesthe lack of smoothness, and r0 is related to the wanted smoothness. It is important to notice herethat if λ is null, the reconstructed function f will interpolate exactly samples si at their location pi.

For some particular regularization functionalR[f ], the general solution of this variational prob-lem leads to Radial Basis Functions (see section 2.3 for more details). However anyone can usea discretization of this variational problem into any basis, e.g. Arigovindan et al. discretize on atensorial product of B-spline basis [Arigovindan et al., 2005].

1.6 Radial Basis Functions

In the Radial Basis Functions (RBFs) based methods, depending on the used basis, we aim tofind a continuous function f of the form:

f(p) =N−1∑

i=0

αi · φ (‖pi − p‖) (1.22)

f(p) =N−1∑

i=0

αi · φ (‖pi − p‖) +Q−1∑

j=0

πj · gj(p) (1.23)


where α and π coefficients are computed by solving a linear system (see next chapter for moredetails).

Fig. 1.2. Illustration of interpolation with RBFs in 1D. Blue disks are data given at distinct locations. Bluecurves represent the contribution of each RBF. Red curve is the resulting curve interpolating samples.

Chapter 2

Radial Basis Functions

Contents2.1 SPD-RBF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.1.1 Positive Definite Kernels . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.1.2 Positive Definite Radial Basis Functions . . . . . . . . . . . . . . . . . 26

2.2 CPD-RBF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3 From Variational Principle to Radial Basis Functions . . . . . . . . . . . . 29

2.4 In Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5 Partition of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.6 Compactly Supported Radial Basis Functions . . . . . . . . . . . . . . . . . 32

2.6.1 Generality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.6.2 Use of CSRBF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.7 Multilevel or Multiresolution Approach . . . . . . . . . . . . . . . . . . . . 35

Since the pioneer works of Hardy in the early 1970s [Hardy, 1971], and Duchon in the late1970s [Duchon, 1976, Duchon, 1977], radial basis functions have received a lots of interest frommathematicians to understand properties of these basis functions, such as positiveness, construc-tion, relation to variational problems... RBFs are still now an important field of research in math-ematics.

In this Chapter, we briefly summarize main properties of RBFs, that can be found in [Buh-mann, 2003,Wendland, 2005b,Fasshauer, 2006], that is useful for the comprehension of the wholedocument.

First we define and explain how to use the two main types of RBFs: strictly positive definite(SPD) and conditionally positive definite (CPD) RBFs. Then we show the correspondence be-tween the choice of one type of RBF and one variational formulation. Despite the fact that thereconstruction problem is always well-posed with using RBF, the main problem of such methodsresides in the high computational cost due to the global support of most of RBFs. For this rea-son, we present some methods from the literature to decrease the computational cost, and givesome formal details for the particular case of partition of unity, compactly supported radial basisfunctions and multiresolution approaches.

25

26 CHAPTER 2. RADIAL BASIS FUNCTIONS

2.1 Strictly Positive Definite Radial Basis Functions

2.1.1 Positive Definite Kernels

Definition 2.1 (symmetric kernel). A kernelΦ (see Definition 1.2) is symmetric, if Φ(x,y) =Φ(y,x) holds for all x,y ∈ Ω.

If Ω has some additional geometric structure, kernels may take a simplified form, making theminvariant under geometric transformations on Ω.

For instance, kernels of the form:

• Φ(x− y) are translation-invariant on Abelian groups,

• Φ(xTy) are zonal on multivariate spheres,

• Φ(‖x− y‖2) are radial on Rd

Definition 2.2 (radial basis functions). A kernel Φ : Rd → R is called radial basis functionprovided there exists a univariate function φ : R+ → R such that Φ(p) = φ(r), with r = ‖p‖,and ‖ · ‖ is the Euclidean norm on Rd, i.e. the L2-norm.

Radial functions have the nice property that they are invariant under (rigid-body) Euclideantransformations, i.e. translations, rotations, reflections. This is an immediate consequence of thefact that Euclidean transformations are characterized by orthogonal transformation matrices andare therefore norm-invariant.

Definition 2.3 (Positive Definite Kernel). A symmetric kernel Φ is positive definite on Rd if andonly if for all finite subsets P =

p0, . . . ,pN−1

of distinct points of Ω the matrix AΦ,P with

entries Φ(pi,pj), 1 6 i, j 6 N is positive definite, i.e.

cT ·AΦ,P · c =N−1∑

i=0

N−1∑

j=0

ci · cj · Φ(pi,pj) > 0, ∀ c ∈ RN . (2.1)

Whereas if the only vector c that turns the previous equation into an equality is the zero vector, thekernel Φ is strictly positive definite.

From the previous definition, it is obvious to check that all eigenvalues of a positive definitematrix are positive, and thus a positive definite matrix is non singular. Therefore, this suggest thatwe should use strictly positive definite kernels as basis functions, i.e. uk(p) = Φ(p,pk), thusrewrite the Eq. 1.3 as follows:

f(p) =N−1∑

i=0

ci Φ(p,pi), p ∈ Rd. (2.2)

2.1.2 Positive Definite Radial Basis Functions

Most of strictly positive definite kernels used are radial, but some use a tensor product to createmultivariate positive definite functions from univariate ones.

From now, we make the assumption that there is no privileged direction in data, i.e. that thereconstructed function should be independent under, at least, rotation. This assumption leads touse radial kernel on Rd, that it is traditionally referred as Radial Basis Functions (RBFs).

2.1. SPD-RBF 27

So, we express the radial basis function as follows:

Φ(p,q) = φ (‖p− q‖) , p,q ∈ Rd (2.3)

with φ : R+ → R a positive definite functions.

There are several ways to demonstrate, or create, positive definite radial basis functions, weinvite interesting readers to directly refer to [Wendland, 2005b] for details. In Table 2.1, we givecommon strictly positive RBFs (SPD-RBFs).

TABLE 2.1 COMMON STRICTLY POSITIVE DEFINITE RADIAL BASIS FUNCTIONS (r = ri = ‖p− pi‖AND u IS THE HEAVISIDE FUNCTION).

Name φ(r) ParametersGaussian e−β·r2

β > 0

Sobolev SplinesrνKν(r) ν > 0

Kν spherical Bessel functionInverse Multiquadrics (r2 + c2)−β/2 β > d/2

Compact Support u(1− r)l · P (r) l > bd/2c+ 1

(a) (b) (c) (d)

Fig. 2.1. Gaussian RBF represented in 1D (a), and in 2D (b). Inverse multiquadric RBF represented in 1D(c), and in 2D (d).

With using SPD-RBFs, the function f can be decomposed as follows:

f(p) =M−1∑

i=0

αi · φ (‖p− ci‖) (2.4)

where α are called the RBF coefficients, ci are generally called centers, and M is the numberof elements in the basis functions. In the interpolation case the number of basis is equal to thenumber of data, i.e. M = N , and data locations are used as centers, i.e. ci = pi, whereas in theapproximation case M 6 N and center could be a subset of data locations or a set of M distinctpoints different from data locations.

In the interpolation case, the RBF coefficients are calculated by solving the following linearsystem

Aφ,P ·α = s (2.5)

with Aφ,P(i, j) = φ(∥∥pi − pj

∥∥).


2.2 Conditionally Positive Definite Radial Basis Functions

The interpolation problem leads to the idea of using strictly positive definite RBFs. But notall popular choice of RBF that are fit into this scheme. In this section, we define and give detailsabout the other type of RBFs: the Conditionally Positive Definite ones (CPD-RBFq).

Definition 2.4. A real-valued even kernel φ is called conditionally positive definite of order m onRd if, for every set of distinct points piN−1

i=0 ∈ Rd, and for every set of real numbers αiN−1i=0

satisfyingN−1∑i=0

αi · gj(pi) = 0, ∀ j ∈ [0, . . . , Q− 1], the quadratic form

N−1∑

i=0

N−1∑

j=0

αi · αj · Φ(pi,pj)

is positive, where gj0,...,Q−1, gjj=0,··· ,Q−1 is a basis in m-dimensional null space con-taining all real-valued polynomials in d variables and of order at most m (i.e. Πm(Rd)) hence

Q =(m− 1 + d

d

).

As for SPD-RBFs, we can define Conditionally Positive Definite RBFs (CPD-RBFs), by usinga radial kernel Φ. In Table 2.2, we present common CPD-RBF.

TABLE 2.2 COMMON CONDITIONALLY POSITIVE DEFINITE RADIAL BASIS FUNCTIONS (r = ri =‖p− pi‖).

Name φ(r) Parameters Order mMultiquadrics (−1)dβe

(r2 + c2

)ββ > 0, β /∈ N dβe

Polyharmonic splines (−1)dβ/2erβ β > 0, β /∈ 2N dβ/2ePolyharmonic splines (−1)k+1r2k log r k ∈ N k + 1

(a) (b) (c) (d)

Fig. 2.2. Multiquadric RBF in 1D (a), in 2D (b). Thin plate RBF, i.e. polyharmonic splines, in 1D (c) and2D.

In order to make the interpolation well-posed, we requireN > Q as well as the side conditionswhich ensures orthogonality:

N−1∑

i=0

αi =N−1∑

i=0

αiπ1 = · · · =N−1∑

i=0

αiπQ = 0 (2.6)

Finally the reconstructed function f is expressed as follows

f(p) =N−1∑

i=0

αi · φ (‖p− ci‖) +Q−1∑

j=0

πj · gj(p) (2.7)

2.3. FROM VARIATIONAL PRINCIPLE TO RADIAL BASIS FUNCTIONS 29

RBF and polynomial coefficients, α and π respectively, are calculated by solving the followinglinear system:

Aφ,P GT

G 0

·

α

π

=

s

0

(2.8)

where Aφ,P(i, j) = φ(∥∥pi − pj

∥∥)and G(i, j) = gj(pi).

2.3 From Variational Principle to Radial Basis Functions

In section 1.5, we briefly explain variational principle. Now, in this section, we give somedetails about the variational aspect of RBFs.

We say that a function is smoother than another one, if it has less energy in high frequenciesdomain. The high frequency content of a function can be measured by first high-pass filtering thefunction, then measuring the energy. This suggests the use of the following smoothness term inEq. 1.21:

R[f ] =∫

Rd

|f(ω)|2Φ(ω)

dω (2.9)

φ is some positive function that tends to 0, when ‖ω‖ → ∞.

It can be shown [Girosi et al., 1993] (or see in Appendix A), that the function which minimizesthe functional (Eq. 1.21) with the smoothness functional introduced above (Eq. 2.9) has the form:

f(p) =N−1∑

i=0

αi · Φ(p− pi) +Q−1∑

j=0

πj · gj(p) (2.10)

where Φ is a definite, or conditionally definite, positive kernel, gjj=0,··· ,Q−1 is a basis in m-dimensional null space containing all real-valued polynomials in d variables and of order at most

m hence Q =(m− 1 + d

d

), and we require N > Q as well as the side conditions which

ensures orthogonality:

N−1∑

i=0

αi =N−1∑

i=0

αi · π1 = · · · =N−1∑

i=0

αi · πQ = 0 (2.11)

If we make the assumption that there is no privileged direction, we should consider a kernelwhich a radial symmetry in the considering domain Ω. Therefore, under this consideration, thislead to RBFs.

However, there is one difference in the way to calculate the RBF coefficients, instead of solvingEq. 2.5 or Eq. 2.8, we should now solve the corresponding linear system:

(Aφ,P + λ · I) ·α = s, for SPD-RBFs (2.12)Aφ,P + λ · I GT

G 0

·

α

π

=

s

0

, for CPD-RBFs (2.13)


where I is the N ×N identity matrix, and λ has already been defined in Eq. 1.21.

It has been shown that such regularization functional corresponds to a norm in the native spacecorresponding to one family of RBF (for more details refer to [Wendland, 2005b, chapter 10]).

For example, the most widely used regularization functional belongs to Duchon’s family ofsemi-norms [Duchon, 1976, Duchon, 1977]. A Duchon’s semi-norm is parameterized by its orderm and has a simple expression in the Fourier domain given by

R[f ] =∫

Rd

‖ω‖2m ·∣∣∣f(ω)

∣∣∣2dω (2.14)

What makes this regularization functional popular is the duality of this expression in the spatialdomain. Indeed this functional can be rewritten in the spatial domain, as follows:

R[f ] = Dm[f ] =∫

Rd

‖hmf(p)‖2 dp (2.15)

where hm is the vector of all partial derivative of the form√

m!µ! ∂

µ such that |µ| = m. Byusing this regularization functional, this lead to express the reconstructed function in terms ofpolyharmonic splines (see Table 2.2).

In dimension 2, this yields to the so-called thin-plate spline, i.e. φ(r) = r2 ln r, inspired fromthe following analogy: by taking a thin sheet of metal, laying horizontally and bending it in a waythat it touches the tips of the vertical poles at height si at the positions pi, the metal resists to thebending so that it smoothly changes its height between the positions pi. This resistance is modeledby the smoothness term, for d = 2, R[f ] in Eq. 1.21

R[f ] =∫

Ω

(∂2f

∂x2(p)

)2

+ 2(∂2f

∂x∂y(p)

)2

+(∂2f

∂y2(p)

)2

dp

In dimension 3, the corresponding smoothness term R[f ] is

R[f ] =∫

Ω

(∂2f

∂x2(p)

)2

+(∂2f

∂y2(p)

)2

+(∂2f

∂z2(p)

)2

+ 2(∂2f

∂x∂y(p)

)2

+ 2(∂2f

∂x∂z(p)

)2

+ 2(∂2f

∂y∂z(p)

)2

dp

2.4 In Practice

Despite the fact that traditional RBF methods are elegant for solving reconstruction problem,in practice they are difficult to use for large problems. Indeed naive approach to solve the linearsystem associated to the interpolation, or approximation, problem leads to a complexity ofO(N3)time and O(N2) space. Furthermore, every evaluation of the function needs another O(N) time.For a large number N of data this is unacceptable. For these reasons, many authors consideredradial basis functions only for small problems, where the number of processed data is about fewthousands.

Thus more efficient methods are necessary to make RBFs more attractive. Many methods havebeen proposed in the literature to reduce the computational cost:

2.5. PARTITION OF UNITY 31

Fast Multipole Method (or Multipole Expansions method) , in order to get sufficiently fast eva-luation for any evaluation point, splits the sum in two terms corresponding to a near-fieldand far-field.

Domain Decomposition method , introduced in [Beatson et al., 2000], is an iterative method forinterpolation. The idea is to subdivide the original data set into several smaller data sets andto iteratively solve the interpolation equation and to form residuals.

Partition of Unity method decomposes global problem into several local problems and the globalsolution is generated by blending all local solutions as we already explained in section 1.3.More details will be given in section 2.5 for RBFs.

Compactly Supported RBF allows defining local influence for a single RBF. Many required op-erations can be speeded up thanks to the local influence of one RBF. For example, the func-tion evaluation does not require to evaluate the contribution of every centers. More detailswill be given in section 2.6.

Multilevel or Multiresolution method allows solving the reconstruction problem with differentscale for the support size, and overcome some difficulties addressed by using CompactlySupported RBF.

For more details about these methods, please refer directly to the excellent survey of Wendland[Wendland, 2005a], or [Wendland, 2005b, chapter 15]. However, in the next sections we givedetails about few of these methods which interest us more particularly.

2.5 Partition of Unity

Partition of unity approaches with RBFs, decompose data sites, or point set, and the entiredomain Ω into overlapping sub-domains Ωi. In this case, the local approximants are formed assolution of the interpolation, or approximation, problem where the global data set, or point set, Pis replaced by a local one Pi = P ∩ Ωi.

According to [Wendland, 2002], the complexity of this approach is governed by the followingassumptions:

• it requires a data structure for the point set such that we can find points contained in sub-domains Ωi efficiently,

• it requires a sub-domains structure such that

– each sub-domains contains only a small number of point,

– each p ∈ Ω is contained only in a small number of sub-domains,

– sub-domains can be found efficiently.

Wendland’s assumptions lead to the requirement that the number of sub-domainsM is propor-tional to the number of points N . This means in particular that the interpolation or approximationproblem can be solved in O(N) time.

For data structure, Wendland [Wendland, 2002] chose tree like decomposition which providesgood results for both cases: the operation can be performed in O(logN) time with an additionalpreprocessing step to build the data structure in O(N logN) time.


2.6 Compactly Supported Radial Basis Functions

2.6.1 Generality

Recently authors have been searching on some new basis with compact support [Wu, 1995,Wendland, 1995, Buhmann, 2001].

As a simple consequence of a theorem of Schoenberg [Schoenberg, 1938] a compactly sup-ported univariate function φ : R+ → R can not be positive definite for all space dimensions.Therefore we have to accept the dependence of φ on the space dimension d as soon as we workwith Compactly Supported Radial Basis Functions (CSRBFs).

However, due to the compact support and properties of radial basis functions, it ensures thatφ is strictly positive definite [Buhmann, 2003]. CSRBFs can be strictly positive on Rd only for afixed maximal dimension d.

Therefore authors focus their attention on the characterization and construction of functionsthat are compactly supported, strictly positive definite and radial on Rd for some fixed dimensiond.

With using CSRBFs, due to the positiveness, the reconstructed function f can be expressed asfollows:

f(p) =N−1∑

i=0

αi · φσ (‖p− pi‖) (2.16)

with φσ(r) = φ(

rσ

), σ is the support size.

In [Wendland, 1995], Wendland constructs, by dimension walk, a popular family φd,k of CS-RBFs, expressed with a polynomial form whose degree is minimal for a given dimension spaced and whose continuity is C2k. Wendland gives a recursive formula for the functions φd,k for alld and k, we instead list the explicit formulas in Theorem 2.1 from [Wendland, 2005b] (see Table2.3, and Fig. 2.3 for some representations).

Theorem 2.1. The functions φd,k, k = 0, 1, 2, 3 have the form

φd,0(r) = (1− r)l+ ,

φd,1(r)·= (1− r)l+1

+ [(l + 1)r + 1] ,

φd,2(r)·= (1− r)l+2

+

[(l2 + 4l + 1)r2 + (3l + 6)r + 3

],

φd,3(r)·= (1− r)l+3

+

[(l3 + 9l2 + 23l + 15)r3 + (6l2 + 36l + 45)r2 + (15l + 45)r + 15

]

where l = bd/2c + k + 1, and the symbol ·= denotes equality up to a multiplicative positiveconstant.

Whereas in [Wu, 1995], Wu presents another way, by convolution, to construct similar CSRBF,but provides higher polynomial degree for a prescribed smoothness and dimension. In [Wendland,2005b], Wendland gives a general formulation, which includes the Buhmann’s ones [Buhmann,2001], and allows anyone constructing his own CSRBF.

2.6. COMPACTLY SUPPORTED RADIAL BASIS FUNCTIONS 33

TABLE 2.3 WENDLAND’S RBFS [Wendland, 1995] FOR VARIOUS DIMENSION d AND CONTINUITYC2k . THE SYMBOL

·= DENOTES EQUALITY UP TO A MULTIPLICATIVE POSITIVE CONSTANTAND u IS THE HEAVISIDE FUNCTION.

Dimension d φ(r) Continuityd = 1 φ1,0(r) = u(1− r) C0

φ2,1(r)·= u(1− r)3 (3r + 1) C2

φ3,2(r)·= u(1− r)5 (38r2 + 5r + 1) C4

d = 3 φ2,0(r) = u(1− r)2 C0

φ3,1(r)·= u(1− r)4 (4r + 1) C2

φ4,2(r)·= u(1− r)6 (35r2 + 18r + 3) C4

d = 5 φ3,0(r) = u(1− r)3 C0

φ4,1(r)·= u(1− r)5(5r + 1) C2

φ5,2(r)·= u(1− r)7(16r2 + 7r + 1) C4

(a) C0 (b) C2 (c) C4

Fig. 2.3. Wendland’s CSRBFs for dimension d = 3 and for various continuity order C2k.


2.6.2 Use of CSRBF

2.6.2.1 Data Structure

Because CSRBFs have finite support, many procedures (function evaluation, construct of theinterpolation or approximation matrix) can be speeded up by using efficient data structure for rangequery searches. Due to this compactness, only few centers should be considered for evaluating thefunction at a point p, or for computing each term of the interpolation matrix Aφ,P . In our setting,given a region of space R, this amounts to determining the set of CSRBF centers P contained inR.

There has been a considerable amount of work on devising efficient data structures for rangequery search [Arya et al., 1998, Bentley, 1990, Clarkson, 1997, Kleinberg, 1997, Preparata andShamos, 1985, Sproull, 1991]. According to Wendland [Wendland, 2005b] kd-tree data structureseems to be one of the most efficient data structure for the general case. kd-tree data structure[Bentley, 1990, Friedman et al., 1977] is based on a recursive subdivision of space into disjointaxis aligned boxes. The principle of the kd-tree may be then briefly described as follows. The rootnode of the kd-tree is a box which contains all data points P and the whole domain Ω. Consideran arbitrary node in the tree, with its associated box B and points X . As long as the number ofpoints contained in this node is greater than a prescribed quantity, the bucket size, this node is splitinto two new nodes. There are several splitting method to determine the hyperplane which splitsthe box and the points into two. We use the standard method, i.e. the splitting dimension is thedimension of the maximum spread of point set X , and the splitting value is the median value ofthe coordinates of X along the splitting dimension.

We present the detailed procedure to build the kd-tree in the algorithm 5. By following thisalgorithm, kd-tree can be constructed inO(dN logN) time and requiresO(dN) space in memory.With such data structure, range query search can be computed in O(logN) (see algorithm 6).

2.6.2.2 Computational Complexity

In Table 2.4, we provide the complexity of the various computation stages. In order to have areference, it is compared to the complexity corresponding to the use of globally supported RBF.

The figures related to globally supported RBFs are provided by Morse [Morse et al., 2001]and rely on the following features:

• the interpolation matrix Aφ,P is dense and its computation is O(N2)1;

• a LU decomposition is used for the computation the inverse of the symmetric matrix Aφ,P ,which yields an O(N3) algorithm;

• The evaluation of the function requires O(N) operations.

In the case of the proposed compactly supported RBF,

• using a kd-tree data structure for the center set, theAφ,P matrix computation isO(N logN)and the implicit function f evaluation is O(logN),

• the computational complexity of sparse Cholesky factorization cannot be given in generalsince it depends on how sparse is Aφ,P . However, according to [Botsch et al., 2005], thefactorization can be considered to be in O(nzf), where nzf is the number of non zerofactors, which depends on the CSRBF centers distribution and on the CSRBF support sizeσ. In practice, we choose sufficiently small support, so nzf will be much smaller than N3.

1Due to the symmetry of H , complexity can be decreased in half, however it is still in O(N2).

2.7. MULTILEVEL OR MULTIRESOLUTION APPROACH 35

TABLE 2.4 COMPLEXITY OF THE VARIOUS COMPUTATION STAGES FOR GLOBALLY AND COMPACTLYSUPPORTED RBFS (nzf IS THE NUMBER OF NON ZERO FACTORS IN THE SPARSE MATRIX).

Global Supported RBFs CSRBFsComputation to build Aφ,P matrix O(N2) O(N logN)Computation to solve O(N3) O(nzf)Computation to evaluate Eq. 2.4 or Eq. 2.7 O(N) O(logN)Effect of a single center global local

2.6.2.3 Support Size

A serious difficulty arises with using CSRBFs. If the support size σ is too small, the recon-structed signal will not be continuous; on the other hand, the matrix Aφ,P becomes dense if thesupport size σ is too large (see Fig. 2.4).

(a) (b)

Fig. 2.4. Too small supports lead to uncontinuous reconstructed function (a). Too large supports leads tosmooth reconstructed function (b), at high computational cost (interpolation or approximation matrixbecomes dense).

2.7 Multilevel or Multiresolution Approach

With single scale methods, it is difficult to efficiently reconstruct a signal from irregular sam-ples with CSRBFs (see the previous section). Multiresolution recursive schemes [Floater and Iske,1996, Ohtake et al., 2003b, Iske and Levesley, 2005] have been introduced to overcome this diffi-culty. Differences between these approaches reside in the way to determine a hierarchy of centerscl

i and corresponding coefficients αli.

Here, all these approaches are unified in the following equations

f0(p) =M0−1∑

i=0

α0i · φ

(‖p− c0i ‖

σ0

)(2.17)

f l(p) = f l−1(p) + dl(p) (2.18)

dl(p) =M l−1∑

i=0

αli · φ

(‖p− cli‖

σl

)(2.19)

where f l and dl are, the approximation or interpolation at the resolution level l, and details addedto the approximation at the resolution level l − 1, respectively.

Chapter 3

Our contributions

Contents3.1 Multiresolution Signal Reconstruction . . . . . . . . . . . . . . . . . . . . . 39

3.1.1 Clustering Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1.2 Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 Adaptive Signals Approximation with Fixed Budget . . . . . . . . . . . . . 483.2.1 Algorithm Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.2 Determinating Centers . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.3 Support Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.4 Coefficients computation . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3 Adaptive Signals Multiresolution Approximation . . . . . . . . . . . . . . . 533.3.1 Determining Center Sets . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3.2 Support Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3.3 Coefficients Computation . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

As we saw in the previous chapter, multidimensional scalar function reconstruction, withoutany restriction on the samples distribution, is clearly ill-posed. The most common way to solvethis problem is to consider variational approach, where RBF are general solutions, or directly RBFwithout taking into account the variational aspect. Unfortunately, most known RBFs are globallysupported, thus the linear system to be solved is dense. In such a case, due to the complexity tosolve the linear system, we can only consider a low amount of samples.

CSRBFs have been introduced to reduce the computational complexity, e.g. by making the lin-ear system sparse. However, in the case of highly irregular samples, higher supports are required,and denser the matrix is.

In this chapter, we propose some methods, largely based on our papers [Gelas and Prost,2006a, Gelas and Prost, 2006b], to overcome this problem:

37

38 CHAPTER 3. OUR CONTRIBUTIONS

A multiresolution method where the center set is computed by a principal component analysisbased clustering, and RBF coefficients by a discrete least-squares approximation.

Adaptive approximation method where the reconstruction is achieved for a fixed amount of pa-rameters, i.e. CSRBFs, with a locally adaptive support size.

Adaptive multiresolution method where the reconstruction is achieved for a fixed error bound.

3.1. MULTIRESOLUTION SIGNAL RECONSTRUCTION 39

3.1 Multiresolution Reconstruction of Irregularly Sampled Signalswith Compactly Supported Radial Basis Functions

In this section, we propose a clustering based hierarchy, rather than tuning the decimationprocess for defining the center set. Our clustering method of samples is based on a principalcomponent analysis, where a cluster’s centroid will be used as CSRBF centers. Due to the noisein the data, we will consider the approximation problem instead of interpolation. In our method,we focus on adding centers only where it is required; i.e. where the local error is the largest.

3.1.1 Clustering Method

Here, we use a binary space partition, based on principal component analysis (PCA) for clus-tering data, in the spirit of [Pauly et al., 2002].

From a given cluster Kk, we compute the centroid ck and its associated covariance matrixCovk (see Eq. 3.1). Then a hyperplane, which splits the cluster into two, is defined by the centroidck and the eigenvectors v0 associated to the largest eigenvalue µ0 of the covariance matrix (seeFig. 3.1).

Covk =|Kk|∑

j=0

(pj − ck

) · (pj − ck

)T (3.1)

By considering at its lowest level, the whole point set P as a cluster, we can compute a tree,where the leaves would be samples. It is possible to define various strategies to split one cluster: thenumber of samples contained some statistical information, or geometrical information, dependingon the application.

During the initialization phase we will only consider the number of samples contained in thecluster Kk, i.e. if |Kk| > m, then Kk is split, where m is a predefined value, then we split eachcluster when it is required (see Section 3.1.3).

3.1.2 Approximation

Consider the level l, with a list of clustersKl

j

M l−1

j=0and their respective centroid

cl

j

M l−1

j=0,

M l is the number of clusters at level l, and the approximation at the previous level f l−1.Due to the fact that all CSRBFs are strictly definite positive, our framework is large enough to

take into account any CSRBF (see section 2.6). However, we choose Wendland’s CSRBFs [Wend-land, 1995], according to the dimension d and to the desired continuity order k of the problem (seeTable 2.3).

We compute the approximation f l by minimizing the Mean Square Error MSEl:

MSEl =N−1∑

i=0

(si − f l(pi)

)2

=N−1∑

i=0

si − f l−1(pi)−

M l−1∑

j=0

αlj · φ

∥∥∥pi − clj

∥∥∥σl

2

Finally, it is easy to calculate the coefficients αl =αl

j

M l−1

j=0:

αl =(BPφ,Cl

·BPφ,Cl

T)−1 ·BPφ,Cl

T · rl,


(a) (b)

(c) (d)

(e)

Fig. 3.1. The Binary Space Partition Clustering Method in the 2D case. Starting from samples representedby disks (a), we first compute a PCA, define centroid and splitting hyperplane of root node of the tree(b). Then at the next level, we compute PCA, define new centroids and splitting hyperplanes, for eachsub-cluster, i.e. children nodes (c). We can repeat this process (d)-(e), until each cluster, or node,contains only one sample.


where

BPφ,Cl(i, j) = φ

∥∥∥pi − clj

∥∥∥σl

(3.2)

andrl(i) = si − f l−1(pi) (3.3)

3.1.3 Algorithm

At first, we shall compute a list of clusters K0j , so that they contain at least fewer than m

samples, and we compute the first approximation f0. Then iteratively for a given level l, for allclusters Kl

j we will compute the local error ηlj , and compare it with a user defined threshold ε.

ηlj = max

pi∈Klj

∣∣∣si − f l(pi)∣∣∣ (3.4)

If the error ηlj is higher than ε, the cluster Kl

j is split into two, and at the next level two newCSRBFs will be added. Then we compute a global least square approximation as described above.In the Algorithm 1, we summarize our proposed method.

Algorithm 1 PCA Based Multiresolution Reconstruction1: l← 02: σ0 ← R3: Compute initial clusters

Kl

j

j, such |Kl

j | < m

4: repeat5: σl ← 1

2 · σl−1

6: Compute coefficients αl

7: Compute for all clusters Klj the local error η0

j

8: if ηlj < ε then

9: Cluster Klj is split into two clusters

10: end if11: l← l + 112: until M l = ∅

Note that we add functions, only where the local error ηlj is the largest (see Fig. 3.2). Some

readers could see here some similarity between this process and Hard Thresholding with waveletcoefficients [Weaver et al., 1991]. It is now well-known in wavelet theory, i.e. multiresolutionframework, that hard and soft thresholding [Donoho, 1995] can be used for denoising or regular-ization. We suppose that it could be the same in our own framework (see section 3.1.4 for someresults about regularization with this method).

3.1.4 Results

We demonstrate here the effectiveness of our method on irregular sampled signals of dimensiond = 1, 2.

3.1.4.1 1D Signal

In order to illustrate our approach, we shall consider the signal withN = 100 random samplesuniformly distributed on [0, 1] as follows (see Fig. 3.3):


(a) (b)

Fig. 3.2. On the left picture (a), at a given level of resolution l for all cluster Klj we compute the corre-

sponding error ηlj (see Eq. 3.4). On the right picture, we add CSRBF at the next level l+1 for clusters

whose error is above a given threshold, i.e. if ηlj is above a given threshold, otherwise no CSRBF is

added.

s(x) = 2 sinc(10x) + 0.75 e−2.5|x−0.75| sin (50x+ 0.1)− 2 e−10x cos (55x) , x ∈ [0, 1] (3.5)

Fig. 3.3. 1D signal to be reconstructed (see Eq. 3.5).

Fig. 3.4(a) shows the initial approximation with a cluster containing less thanm points (|K0j | <

m = 40). Then other figures present result at different resolutions, l = 2, 3, 4, for a given thresholdε = 0.1.

Fig. 3.5 illustrates the influence of the user defined threshold ε with noisy data. As expected,if ε is too low (see Fig. 3.5(a)), the reconstructed signal captures noise, and if it is too large (seeFig. 3.5(c)), the reconstructed signal loses the information from the original signal. In Fig. 3.6,we present the error according to the threshold value ε, for different SNRs. Clearly there is alwaysan optimal value ε where the error is the lowest.

3.1.4.2 2D Signal

We have tested our algorithm on the peaks function (see Fig. 3.7) of Matlab R© with N = 500random samples uniformly distributed (see Fig. 3.8). Fig. 3.9 shows both resulting function forvarious resolutions, i.e. 1 to 6, and the corresponding error foriginal − freconstructed, for an initialclustering stage with m = 100 samples. As expected, with higher resolution the error is highlydecreasing until being almost null.


(a) (b)

(c) (d)

Fig. 3.4. Approximation of irregularly sampled signal at different resolutions : the lowest resolution 1 (a),resolution 2 (b), resolution 3 (c), and the highest resolution 4 (d).


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.5

0

0.5

1

1.5

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.5

0

0.5

1

1.5

(c)

Fig. 3.5. Influence of the threshold ε on the reconstructed signal, with noisy data (SNR=30 dB). With toolow ε = 0.2 (a), optimal ε = 0.37 (b), for too high ε = 0.7 (c).

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

threshold

L2 e

rror

Fig. 3.6. l2 error according to the threshold ε, for different SNR: 27 dB(x), 30 dB(o), 33 dB(.)


(a) (b)

Fig. 3.7. peaks function of Matlab R©.

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

Fig. 3.8. Sample locations in a top view.


(a) Level 1 (b) Level 1

(c) Level 2 (d) Level 2

(e) Level 3 (f) Level 3


(g) Level 4 (h) Level 4

(i) Level 5 (j) Level 5

(k) Level 6 (l) Level 6

Fig. 3.9. Approximation of an irregularly sampled surface. On left column resulting signal represented asa surface at different resolutions. On right column resulting error at different resolutions.


3.2 Adaptive Approximation of Multidimensional Irregularly Sam-pled Signals with Fixed Budget

In this section, we propose another approach with a fixed amount of centers L, or CSRBFs,and with locally adaptive support size.

In the previous chapter, we saw that traditional approaches using RBFs consider as much cen-ters as data leading to large linear systems. Moreover, it is difficult to deal with irregular sampleswith single scale CSRBF, the resulting support should be large enough to fill all gaps contained indata. One way to handle this problem is to use a multiresolution procedure, like in the previoussection. In this section, we decide to work with local adaptive support, i.e. the support of eachCSRBF is locally adapted to an internal scale invariant parameter σi. This parameter given bycomputing a Bounded Voronoï Diagram, is directly related to the center distribution. In order toavoid numerical instabilities, it is preferable to deal with a center as regular as possible.

This section is organized as follows, we first give an overview of the proposed method (seesection 3.2.1) and present each step in the next section, i.e. how to compute the center set (seesection 3.2.2), how to compute the support size (see section 3.2.3), and finally how to computeCSRBF coefficients (see section 3.2.4).

3.2.1 Algorithm Overview

We first compute the center set

Algorithm 2 ICSP Fixed amount of CSRBF1: Compute Center Set C, with |C| = L2: Compute Bounded Voronoi Diagram BVC3: for ci ∈ C do4: Compute Support Size σi

5: end for6: Compute CSRBF coefficients αi

3.2.2 Determinating Centers

There are several ways to choose centers from a given point-set P , i.e. to down-sample P intoC. Ohtake et al. [Ohtake et al., 2003b], Iske and Levesley [Iske and Levesley, 2005] used an octreebased strategy; Floater and Iske [Floater and Iske, 1996] decimate the given point set P accordingto a criterion. Here, we decide to use a furthest point algorithm (see Fig. 3.10), cause it tends toproduce regular center set C, for |C| ¿ N .

At first, we choose the closest point p0 to the bounding box of P , and set C0 = p0. Theniteratively until we reach the given number of centers L, we add the point pn+1 with the maximaldistance to Cn. Thus, we define Cn+1 = Cn∪pn+1. For commodity, we recall one center ci oneelement of C.

3.2.3 Support Size

We propose here a definition of the support size σi independent under any Euclidean transfor-mations or scaling, for this purpose we introduce a scale parameter satisfying these constraints,

3.2. ADAPTIVE SIGNALS APPROXIMATION WITH FIXED BUDGET 49

(a) (b) (c)

(d) (e) (f)

Fig. 3.10. Illustration of the furthest point algorithm (see section 3.2.2). Disks are data points, and bluesquares are centers.

i.e. a scale parameter σi. The support size is thus related to the scale parameter by the followingrelationship:

σi = ρ · σi, (3.6)

where ρ is a fixed global parameter.In the case of single scale CSRBF, it is obvious that the support size should be connected to

the fill distance hC,Ω (see Eq. 3.7) in order to fill the whole domain, i.e. every point p ∈ Ω isat least contained by one CSRBF. The fill distance can be interpreted as the radius of the largestball which is completely contained in Ω and which does not contain any center site ci; or as thedistance to the furthest point contained in the bounded Voronoï cell of ci.

hC,Ω = maxp∈Ω

minci∈C‖p− ci‖ (3.7)

In the following, we begin by introducing the background necessary to the comprehension ofour proposal. Then we explicit clearly how to define this scale parameter σi.

3.2.3.1 Preliminaries

Let P be a finite set of distinct points in Rd.

Voronoï DiagramThe Voronoi cell of the site p ∈ P is given as follows:

Vp =x ∈ Rd : ∀ q ∈ P − p, ‖x− p‖ 6 ‖x− q‖

(3.8)

The sets Vp are convex polyhedra. Closed faces shared by two Voronoi cells are called Voronoifaces, and edges shared by d Voronoi cells are called Voronoi edges. The points shared by d + 1or more Voronoi cells are called Voronoi vertices. The Voronoi diagram VP of P is the collectionof all Voronoi cells, faces, edges and vertices.

If pi represents an element of the convex hull of P , Vpiis unbounded.


(a) (b) (c)

Fig. 3.11. Random samples P , its associated Voronoi Diagram VP , and its associated Bounded VoronoiDiagram BVΩ

P on a bounded domain Ω represented by the rectangle.

Bounded Voronoï DiagramFor a finite domain Ω, we recall BVΩ

pithe bounded Voronoï cell which is defined as the intersection

of the Voronoï cell Vpiwith the domain Ω. Thus, we define the Bounded Voronoï diagram BVP

as BVΩP by BVΩ

P =⋃

pi∈PBVΩ

pi.

The computation of the Voronoï diagram, hence bounded Voronoï diagram, requires some nu-merical predicates. Indeed with conventional floating point precision, in some ugly configurations,the Voronoï diagram computation can fail. To overcome this problem, one can use some librarywith adaptive numerical precision like CGAL [Fabri et al., 2000], or QHULL [Barber et al.,1996].

3.2.3.2 Proposal

In order to fill the Domain Ω with hypersphere, i.e. all points should be included at least byone hypersphere (see Fig. 3.12). This can be fulfilled by choosing the internal scale parameter σi

as follows:σi = max

p∈BVΩci

‖ci − p‖ (3.9)

Hence, the local support size σi for center ci, is directly related to the furthest Voronoï verticesfrom the bounded Voronoï diagram BVΩ

C , corresponding to a local definition of the fill distance.

3.2.4 Coefficients computation

In order to compute CSRBF coefficients, we minimize the Mean Square Error given by thefollowing equation:

MSE =N−1∑

i=0

(f(pi)− si)2

=N−1∑

i=0

M−1∑

j=0

αj · φj (pi)− si

2

where φj(p) = φ(‖p−cj‖

σj

).

3.2. ADAPTIVE SIGNALS APPROXIMATION WITH FIXED BUDGET 51

Fig. 3.12. Adaptive support size represented in the 2D case with ρ = 1.

(a) (b)

Fig. 3.13. Middle sagittal slice of human brain (39 060 pixels) (a), random samples (11 718 samples) (b).

Finally, it is easy to get the coefficients α = αjM−1j=0 :

α =(BPφ,C

T ·BPφ,C)−1 ·BPφ,C

T · sBPφ,C(i, j) = φj (pi) , si = si

3.2.5 Results

We present here some results for the 2D case, i.e. d = 2. For our experiments, we consider onemedical image (see Fig. 3.13(a)), with some random sampling, we keep only 25% of the numberof pixels of the original image (see Fig. 3.13(b)).

First of all, we introduce some error measures that we will use in all our experiments. For eachpixel pi, we compute the absolute error η(pi) between the original image I0 and the reconstructedone Ir, i.e.

η(pi) = |I0(pi)− Ir(pi)| (3.10)


We also define the reconstructed error err = ‖I0−Ir‖‖I0‖ which takes into account all pixel from

both images. Note that the reconstructed error err obviously underestimate the performance ofthe method, since it does not correspond to the functional we aim to minimize in our method andwe can not add information that as not contained into the data. For this purpose, we introduce thereconstructed error restricted to the data pixels errP , i.e.

errP =‖I0(pi)− Ir(pi)‖

‖I0(pi)‖, pi ∈ P

(3.11)

In the first column of Fig. 3.14, we show the reconstructed image for various amounts ofCSRBFs. It demonstrates the ability of the method to reconstruct an approximative image forvarious amounts of CSRBFs. In the second column of Fig. 3.14, one can see both the reconstructedimage and the absolute error distribution in the image.

(a) (b) (c)

(d) (e) (f)

Fig. 3.14. Top row: reconstruction image Ir for various budget of CSRBF: 1 172 (a), 5 859 (b), 11 718 (c).Bottom row: corresponding absolute error image |I0 − Ir|.

Fig. 3.15, gives both reconstruction error, i.e. err and errP , versus the number of CSRBFs.As expected both error are decreasing with increasing number of CSRBFs.

3.3. ADAPTIVE SIGNALS MULTIRESOLUTION APPROXIMATION 53

(a) (b)

Fig. 3.15. Error err (a) (see Eq. 3.10) and errP (b) (see Eq. 3.11) versus the number of CSRBFs, moreprecisely versus the ratio between the number of CSRBFs and the number of samples (here 11 718).

3.3 Adaptive Multiresolution Approximation of Multidimensional Ir-regularly Sampled Signals

In this section, we propose to reconstruct a multidimensional signal for a fixed error bound ε,with locally adaptive support size.

It is difficult to know directly how many CSRBFs are required to reach a prescribed errorbound. For this purpose, we use a multiresolution approach in which we add information whereverit is required.

3.3.1 Determining Center Sets

We use here a slightly different version of the method described in section 3.2.2. Indeed insteadof considering the number of centers as stopping criterion, we rather use one on the separationdistance qC (qC = min

ci∈Cmin

cj∈C−ci‖ci − cj‖), i.e. qC 6 d.

More precisely, we first start by determining a center set C0 and a coarse approximation func-tion f0, by using the method described in section 3.2 with a fixed budget. To compute the centerset at the next level, we use the same algorithm described above, with the stopping criterion on theseparation distance, i.e. qCl+1 6 µ · qCl , with µ < 1. From this new center set Cl+1, we only keepcenter cl+1

i whose error at level l, ηli(c

l+1i ), is higher than ε (see Eq. 3.12).

ηli(c

l+1i ) =

∣∣∣si − f l(cl+1i )

∣∣∣ (3.12)

With this method, we only add CSRBF where it is necessary according to the error bound ε(see Fig. 3.2).

3.3.2 Support Size

For the lowest resolution signal, we compute the support size for each center c0i ∈ C0 by the

method described in section 3.2.3.Whereas at higher resolution level l, in order to avoid the computation of a bounded Voronoï


diagram at each level, we rather compute the support size for each center cli ∈ Cl as follows:

σli = K · min

clj∈Cl−cl

i‖cl

i − clj‖,with ρ > 1. (3.13)

where K is a positive constant much higher than ρ.

3.3.3 Coefficients Computation

For each level, we minimize the Mean Square Error:

MSEl =N−1∑

i=0

M−1∑

j=0

αlj · φl

j (pi) + f l−1 (pi)− si

2

where φlj (p) = φ

(‖p−cl

j‖σl

j

).

Finally, we get the coefficients at each level as follows:

αl =(BPφ,Cl

t ·BPφ,Cl

)−1 ·BPφ,Cl

t · rl

BPφ,Cl(i, j) = φl

j (pi) , rli = si − f l−1(pi)

3.3.4 Results

We reconstructed the image from random samples (Fig. 3.13(b)) with a fixed error of 2%,from the level 0 corresponding to the image reconstructed with a fixed budget of 1 172 samples(Fig. 3.16(a)), and we add details information only where the local error at the previous level ishigher than the prescribed error (Fig. 3.16). The objective of a local error less than 2% is reachedfor the level 6 (Fig. 3.16(l)).

3.3. ADAPTIVE SIGNALS MULTIRESOLUTION APPROXIMATION 55

(a) Level 1 (b) Level 1

(c) Level 2 (d) Level 2

(e) Level 3 (f) Level 3


(g) Level 4 (h) Level 4

(i) Level 5 (j) Level 5

(k) Level 6 (l) Level 6

Fig. 3.16. Left column: Reconstruction for various resolutions. Right column: Corresponding error image.

Conclusion

Bilan

Dans cette partie, nous avons proposé plusieurs méthodes d’approximation de signaux à partird’échantillonnages irréguliers, sans restriction sur leur distribution, et en utilisant des CSRBF.

Plus précisément, nous avons présenté

• une méthode d’approximation multiresolution de signaux multidimensionnels. Les échan-tillons sont tout d’abord classés suivant un arbre binaire construit à partir d’une analyseen composante principale, où le centre de chaque classe est utilisé comme centre de CS-RBF. L’erreur quadratique est ensuite minimisée en ne prenant en compte que les centres oùl’erreur locale, par rapport au niveau de résolution précédent, est la plus significative. Nousavons testé notre algorithme en 1D et 2D. Les résultats expérimentaux prouvent l’efficacitéde cette méthode pour traiter les données bruitées, en choisissant un seuil approprié pour lala sélection des centres significatifs.

• une méthode avec un budget fixé de CSRBF, où le support est localement adapté à la ré-partition des centres dans le domaine d’étude. L’erreur quadratique est minimisée par unmoindre carré discret. Nous avons illustré notre méthode sur une image 2D avec un échan-tillonnage aléatoire.

• une variante de la méthode précédente, pour laquelle nous considérons, non plus le nombrede CSRBF comme contrainte, mais une borne sur l’erreur. Pour cela, nous utilisons uneméthode adaptative multirésolution dans laquelle les CSRBF sont ajoutées uniquement sielles sont nécessaires pour atteindre cet objectif. Nous avons testé notre méthode sur uneimage 2D échantillonnée aléatoirement.

Conclusion

In this part, we have proposed several methods using CSRBFs for multidimensional signalsapproximation from irregular samples without any restriction on their distribution.

More precisely, we have presented

• a new multiresolution scheme for multidimensional signals approximation. Samples are firstclustered using principal component analysis and their centroids define CSRBF centers. Themean square error is minimized by selecting centers where the largest local error at the previ-ous level is. We demonstrate the effectiveness of our algorithm in one- and two-dimensional

57


cases. Experimental results prove the effectiveness of this approach for processing noisydata by using the appropriate threshold for CSRBF center selection.

• a new approximation scheme with a fixed number of CSRBFs, where the support size is lo-cally adapted to the center distribution. The mean square error is minimized by consideringa discrete least-squares approximation. We illustrate our method for 2D image randomlysampled.

• a revisited variant of the previous method, where the reconstruction is performed with aprescribed error bound. To this end, we use an adaptive multiresolution scheme in which weonly add CSRBFs on centers where it is required to reach the fixed objective. We illustrateour method for 2D image randomly sampled.

Part III

Surface Reconstruction

59

Résumé

Avec les récents développements de technologies d’acquisition tridimensionelle, et en vi-sion par ordinateur, la reconstruction de surface a fait l’objet d’un grand intérêt cesdernières années. En effet, les procédés d’acquisition procurent un grand nombre de

points, ou échantillons, et il est nécessaire de reconstruire une surface représentative de ces ob-jets.

En imagerie médicale, bon nombre d’algorithmes de segmentation peuvent être accélérés parl’utilisation d’approches basées modèle. La construction de modèles d’organe est une étape im-portante. Un clinicien, grâce à son expertise en anatomie, sélectionne des points sur des coupes(scanner ou IRM) à partir desquels un model surfacique 3D doit être reconstruit (voir Fig. 3.18).

Le problème de la reconstruction de surface peut être classé en trois catégories, correspondantaux trois principales représentations de surface.

Maillage: La surface de la reconstruction est représenter par un maillage 3D surfacique, constituéde polygones, tels que des triangles, quadrilatères, etc...

Pour la reconstruction de surface, on peut différencier deux types d’approches qui constru-isent un maillage polygonal:

• Approximation globaleLe problème de reconstruction est résolu par l’utilisation d’un modèle déformable.Un modèle de base est déformé pour qu’il représente de façon optimale les pointsdonnés [Terzopoulos and Vasilescu, 1991, Chen and Medioni, 1995]. Ces approchesconsidèrent la forme de base comme une collection de points reliés par des ressorts.En ajustant la raideur et la position de ces points, ces méthodes déforment le maillagede base pour approximer le nuage de points.

• Méthodes basées Delaunay (ou Voronoï )La surface résultante de ces méthodes [Amenta et al., 1998, Amenta et al., 2001, Deyand Goswami, 2003] est un maillage triangulaire, que l’on obtient en calculant toutd’abord une triangulation de Delaunay 3D (volumique, constituéz de tétraèdres), ouson dual un diagramme de Voronoï volumique. Ces méthodes utilisent ensuite lescellules de ces structures pour définir une connectivité entre les échantillons.

Surface paramétrique: La surface est représentée par une fonction Π : R2 → R3 qui donneune correspondance entre un espace parmétrique 2D (u, v) et l’espace 3D (x, y, z). Cettefonction Π est souvent décomposée en terme de produit tensoriel de splines, ou de surfaceslimites de processus de subdivisions.

61

62

Les méthodes d’approximation de surfaces explicites [Lee et al., 1997, Suzuki et al., 1999,Jeong and Kim, 2002] fixent tout d’abord un ou un ensemble de domaines paramétriques,puis modifient la fonction Π pour obtenir une approximation optimale du nuage de points.

Surface implicite: La surface est représentée par la 0-isosurface d’une fonction f : R3 → R.

Les méthodes de reconstruction de surface implicites [Muraki, 1991, Hoppe et al., 1992,Turk and O’Brien, 1999, Carr et al., 2001, Morse et al., 2001, Ohtake et al., 2003b, Ohtakeet al., 2003a, Tobor et al., 2004b, Ohtake et al., 2004a], approximent, ou interpolent, dansun premier temps le nuage de points par une fonction implicite 3D, puis extraient le niveau0 de cette fonction.

Les principales difficultés des méthodes donnant des maillages ou des surfaces paramétriquessont la topologie de la surface et la distribution des échantillons. En effet, la plupart de ces mé-thodes ont besoin de connaître la topologie de la surface, ou bien contraignent cette dernière à unetopologie simple. D’autre part, certaines méthodes, comme celles basées Delaunay ou Voronoï ,rencontrent de certaines difficultés à traiter des surfaces échantillonnées irrégulièrement.

Dans cette partie, nous nous sommes focalisés sur les méthodes donnant des surfaces im-plicites, car il n’y a pas de restriction sur la topologie, peu de contraintes sur l’échantillonnage, etelles procurent de nombreux avantages en comparaison des autres modélisations:

• elles garantissent que la surface soit manifold, sans auto-intersection,

• flexibilité topologique,

• le test d’appartenance à l’objet est simple,

• détection de collisions simple à calculer,

• représentation indépendante de maillages,

• coût de stockage faible des données utilisées pour représenter la surface,

• calcul simple des caractéristiques géométriques différentielles.

Dans le chapitre 4, nous définissons plus précisément les surfaces implicites, ainsi que leursprincipales propriétés. Dans le chapitre 5, nous faisons un rapide état de l’art des méthodes dereconstruction de surface implicite. Dans le chapitre 6, nous présentons les méthodes de recon-struction de surface que nous avons développées. Ces méthodes s’appuient, elles aussi, sur lesfonctions de base radiale à support compact (CSRBF).

Introduction

With the recent developments of three dimensional acquisition technologies, such asrange scanner (see Fig. 3.17), light detection and ranging (LIDAR), mechanical touchprobes, and computer vision techniques such as depth stereo, surface reconstruction

receives a lot of interest in the last years. Indeed these technologies generally provide huge num-ber of points, i.e. samples, and it is desirable to reconstruct a surface which will be used for furtherprocessing, for applications in computer graphics, or in reverse engineering.

(a) (b)

Fig. 3.17. David sculpture of Michelangelo scanned by a laser rangefinder. These images are courtesy ofThe Digital Michelangelo Project and the Soprintendenza ai beni artistici e storici per le provincedi Firenze, Pistoia, e Prato.

In medical image processing, many segmentation algorithms can be speeded up by using modelbased approaches. So constructing model of organs is an important step. A physician, using hisknowledge of anatomy, selects characteristic points on sliced images, from which a 3D surfacemodel should be reconstructed (see Fig. 3.18).

The surface reconstruction problem can be roughly classified into three main categories, cor-responding to the three main surface representations for the resulting surface.

Mesh: The resulting surface is represented as a 3D polygonal mesh, constitued by a collection ofpolygons, such as triangles, quadrilaterals, etc...

For surface reconstruction, we differentiate two kind of approaches in order to provide apolygonal mesh as an output:

63

64

(a) (b) (c) (d) (e)

Fig. 3.18. Short axis slides of a Magnetical Resonance Image (MRI) of human heart (a-d). Hand-madesegmentation of the epicard on short axis slides (e). These images are courtesy of Helsinki Univer-sity of Technology, Jyrki Lötjönen from VTT Technical Research Centre of Finland and PatrickClarysse from CREATIS, Lyon, France.

• Global fittingThe reconstruction problem is solved by using a deformable model, i.e. by deforming abase model to optimally fit the input point set [Terzopoulos and Vasilescu, 1991, Chenand Medioni, 1995]. These approaches represent the base shape as a collection ofpoints with springs between them and adapt the shape by adjusting spring stiffness andpoint locations according to the surface samples.

• Delaunay based (or Voronoï based) methodsThe resulting surface of such methods [Amenta et al., 1998, Amenta et al., 2001, Deyand Goswami, 2003] is a triangular mesh obtained by first computing volume De-launay complexes or dual Voronoï diagram structures, then by using cells of thesestructures to define connectivity between samples.

Parametric surface: The surface is represented as a function Π : R2 → R3 that provides amapping between a 2D parametric domain (u, v) into the 3D space (x, y, z) in which it isembedded. Generally the mapping function Π is expressed in terms of bivariate tensorialsplines product, or as the limit surface of some subdivision processes.

Explicit surface fitting methods [Lee et al., 1997, Suzuki et al., 1999, Jeong and Kim, 2002]first determine one or a collection of parametric domains. Then the mapping function is thenmodified to optimally fit point set.

Implicit surface: The surface is represented as an iso-surface of a function f : R3 → R, to bemore precise, as the zero level set of the implicit function f .

Implicit surface based reconstruction methods [Muraki, 1991, Hoppe et al., 1992, Turk andO’Brien, 1999, Carr et al., 2001, Morse et al., 2001, Ohtake et al., 2003b, Ohtake et al.,2003a, Tobor et al., 2004b, Ohtake et al., 2004a], first approximate, or interpolate, samplepoints by a 3D implicit function, then extract the reconstructed surface as the zero level setof this function.

Main difficulties with mesh and parametric surface based reconstruction are about the topologyand the distribution of the samples. Indeed for most of these methods, some knowledge about thetopology of the surface to be reconstructed is required, or is constrained to simple topology. Onthe other hand, some methods, such as Delaunay or Voronoï based can not easily process irregularsampled surfaces.

In this thesis, we focus on implicit surface based methods, because they can reconstruct sur-faces of any topology, with almost any sampling distribution, and on the other hand, as we will

65

see in the next section, such surface representation for objects and their surfaces has a number ofadvantages:

• guarantee for the surface to be manifold, with no self-intersection,

• topological flexibility,

• efficient membership classification test,

• efficient collision detection,

• mesh independent representation,

• efficient memory storage,

• simplicity to compute geometric differential characteristics

Chapter 4

Implicit Surface

Contents4.1 Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.1.1 Normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.1.2 Curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.1.3 Ridges And Valleys . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2 Constructive Solid Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 694.3 Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3.1 Ray Tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.3.2 Polygonization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

The implicit surface is defined as S = p ∈ Ω ⊆ R3|f(p) = 0, sometimes also called the0-isosurface, isosurface at the value 0, zero level-set of a 3D implicit function f . By convention,we define the interior of the surface to be all points p where the value of the implicit function fis strictly positive, i.e. f(p) > 0, and the exterior of the surface all points p such the value of theimplicit function f is strictly negative, i.e. f(p) < 0 (see Fig. 4.1).

(a) (b)

Fig. 4.1. Representation of the implicit function with axis oriented planes (a) corresponding to a humanhand-made segmented epicard (see Fig. 3.18), and the implicit surface (b).

According to the definition of an implicit surface given above, it is obvious that such modellingoffers an efficient membership classification test, efficient collision detection, by checking the signof corresponding implicit functions.

67

68 CHAPTER 4. IMPLICIT SURFACE

4.1 Differential Geometry

If the surface S is regular, i.e. ∀ p ∈ S,∇f 6= 0, and the second partial derivatives arecontinuous, then S will have continuous curvature (i.e. , the surface is G2 continuous), and willbe manifold1.

In such case, implicit surfaces allow user to compute analytically differential geometric prop-erties, such as normals, curvatures, to determine ridges and valleys point.

4.1.1 Normal

The normal n can directly be related to the gradient ∇f of the implicit function. We decide toconsider the orientation of the normal from the interior to the exterior, thus we get the followingrelationships:

∇f(p) =[∂f

∂x(p),

∂f

∂y(p),

∂f

∂z(p)

]T

(4.1)

n(p) = − ∇f(p)‖∇f(p)‖ (4.2)

4.1.2 Curvatures

Following [Goldman, 2005], in order to define normal curvature, principal curvatures and theirdirections, we should first compute the matrix ∇n, where n is the unit normal vector as describedabove.

−∇n =1

‖∇f‖(I − n · nT

) · Hessf (4.3)

This formula stands for any implicit function f : Rd → R, with I is the d× d identity matrix,Hessf is the Hessian of f , i.e.

Hessf (p) =

∂2f(p)∂x2

0. . . ∂2f(p)

∂x0∂xd−1

.... . .

...∂2f(p)

∂xd−1∂x0. . . ∂2f(p)

∂x2d−1

(4.4)

Principal curvatures and associated directions are respectively defined as the eigenvalues andeigenvectors of ∇n. More precisely, in our case, i.e. in 3D, (0, κmin, κmax) are theeigenvalues, and associated eigenvectors are (n, emin, emax).

Gaussian curvature defined by K = κmin · κmax is thus defined for implicit surface [Goldman,2005] as follows:

K = −

∣∣∣∣Hessf ∇fT

∇f 0

∣∣∣∣‖∇f‖4 (4.5)

1The 2-manifoldness is a fundamental concept from algebraic and differential topology. It is a surface embedded inR3 such that the infinitesimal nieghborhood around any point on the surface is topologically equivalent to a disk.

4.2. CONSTRUCTIVE SOLID GEOMETRY 69

Mean curvature defined byH = κmin+κmax2 , or by 1

2 ·Trace(−∇n), is thus defined for implicitsurface [Goldman, 2005] as follows:

H =∇f · Hessf ·∇fT − ‖∇f‖2 · Trace (Hessf )

‖∇f‖3 (4.6)

4.1.3 Ridges And Valleys

Following [Belyaev et al., 1998,Ohtake et al., 2004b], consider the derivatives of the principalcurvatures along their corresponding curvatures directions umax = ∂κmax

∂emaxand umin = ∂κmin

∂emin.

Notice that umin and umax are defined locally in neighborhood of non-umbilical points2.The extrema of the principal curvatures along their curvature directions are given by the zero-

crossings of umax and umin, and the ridges and valleys are characterized by

umax = 0,∂umax

∂emax< 0, κmax > |κmin| for ridges

umin = 0,∂umin

∂emin< 0, κmin < − |κmax| for valleys

4.2 Constructive Solid Geometry

Non-manifold surfaces can be implicitly represented by extending the definition of f to theseparation between arbitrary regions of space [Rossignac and O’Connor, 1990]. Thus, primitivesin constructive solid geometry (CSG) may be represented implicitly and combined by set-theoricboolean operations or blending.

We should first define boolean operation for the implicitly defined shapes F and G (f and gare their respective corresponding implicit function).

Complement F = −f

Union F ∪G = min(f, g)

Intersection F ∩G = max(f, g)

Difference F \G = F ∩ G = max(f,−g)

However since min and max are not differentiable everywhere, Ricci [Ricci, 1973] introducesanalytical expressions which approximate union and intersection, i.e.

Unionα (F0, . . . , FN−1) =

(N−1∑

i=0

f−αi

)−1/α

Intersectionα (F0, . . . , FN−1) =

(N−1∑

i=0

fαi

)1/α

2An umbilical point is a point on a surface at which the curvature is the same in any direction.


where α is a positive constant. It is important to notice that the limit of these two functions is minand max respectively, when α tends to 0.

Many blending operation have been used in the literature. Here we only present some basicblending, i.e.

linear BL(F0, . . . , FN−1) = −1 +N−1∑i=0

fi

super-elliptic BS(F0, . . . , FN−1) = 1−(

N−1∑i=0

max(0, 1− fi)β

)1/β

4.3 Visualization

4.3.1 Ray Tracing

Ray Tracing is a global illumination based rendering method, it enables to directly visualizeimplicit surfaces without any additional storage overhead while creating high quality images. Forevery pixel of the output image, a single ray is traced from the view point into the scene. Then therays are tested against the implicit surface contained in the scene to determine if they intersect ornot. If the ray misses the implicit surface, then that pixel is shaded by the background color.

Here we discuss various method to intersect a ray defined parametrically, p(t) = o + t · d,where o and d are the origin and the direction of the ray, respectively and an implicit surface Sfrom the implicit function f . Whereas analytical solutions exist for low degree algebraic implicitsurfaces (the implicit function f can be written as a trivariate polynomial form), we concentratehere on the more general case of any implicit function f .

4.3.1.1 Interval analysis

Mitchell [Mitchell, 1990] used the first interval analyis to find the intersection of a ray withan implicit surface. First, zero values of the function f(p(t)) are isolated by finding intervalst ∈ [t1, t2] containing one and only one solution of the equation f(p(t)) = 0. Then a refinementtechnique based on Newton’s method and regula falsi is applied to these intervals until the raysurface intersection is located as accurately required.

4.3.1.2 Lipschitz methods

These methods make the assumption that the implicit function f is Lipschitz continuous, i.e.there exists a positive constant L such that for all points p1 and p2 the following inequality holds

|f(p1)− f(p2)| < L · ‖p1 − p2‖ (4.7)

The Lipschitz bound is the smallest valueL that satisfies Eq. 4.7. This bound can be interpretedas the upper bound for the distance between p1 and the implicit surface S (if p2 is on the surfaceS).

Sphere tracingSphere tracing method [Hart, 1996] is a ray-tracing method for implicit surfaces which marchesalong the ray from the origin o (t = 0), with irregular steps f(p(t))

L towards the first intersection.This step definition guarantees not to intersect the implicit surface.

4.3. VISUALIZATION 71

LG SurfacesBesides the existence of a Lipschitz bound L, LG Surfaces [Kalra and Barr, 1989] require theexistence of directional bound G of the gradient along the ray:

∣∣∣∣∂f(p(t1))

∂t− ∂f(p(t2))

∂t

∣∣∣∣ < G |t1 − t2| (4.8)

The Kalra and Barr’s ray tracing algorithm relies on the determination of an interval [t1, t2]nearest to the origin of the ray with exactly one intersection of the ray with the implicit surface.This interval can be refined by using Newton iterations.

In order to determine the number of intersections on the interval [t1, t2], they use the meanparameter value tm = t1+t2

2 , the half range parameter value d = t2−t12 and the directional bound

G. If f(p(t1)) and f(p(t2)) are of opposite signs and∣∣∣∂f

∂t (p(tm))∣∣∣ > G · d, there is exactly

one intersection in this interval. Otherwise, if f(p(t1)) and f(p(t2)) have the same sign and∣∣∣∂f∂t (p(tm))

∣∣∣ > G · d, there is no intersection in this interval.To speed up their algorithm, Kalra and Barr first search for regions in space that do not contain

any intersection with the implicit surface using the Lipschitz bound L of the implicit function fand then prune them away.

4.3.2 Polygonization

Here, we present several methods to represent an implicit surface S defined by a 3D implicitfunction f : R3 → R by discrete polygonal meshes, this conversion is called polygonization.

A major advantage in comparison to ray tracing techniques is that the polygonal mesh haveto be computed only once for a static surface, and it is totally independent of the point of view.However it may require a considerable amount of time and may induce storage overhead.

The resulting polygonal mesh should be homeomorphic to the implicit surface and shouldaccurately describe it. Furthermore, this polygonal mesh should be topologically consistent, i.e.without any disconnected vertices, edges or faces, and distorted faces should be avoided.

There have been considerable amount of work about this subject, so we will not review all thiswork here since it is not the purpose. However, we will describe briefly the three main categoriesof polygonization.

4.3.2.1 Grid based

These techniques subdivide the 3D domain Ω into cells, generally cubes or tetaedra, and searchfor the cells that intersect the implicit surface S.

The reference poygonization method is the Marching Cubes algorithm [Lorensen and Cline,1987], which divides the domain into cubic cells. At each vertex of the cube the implicit functionf is evaluated, and it determines the intersection of the surface with the edges of the cube. Thenthe marching cubes provides a lookup table index by the sign configuration at the eight corners thatyields the topology of the mesh inside of this cube. After processing each cube, the meshed surfaceis complete. However some configurations in this lookup table are ambiguous, different trianglescan be created, and it may lead to some topological inconsistencies for the wrong choice. With theassumptions that f is trilinear over each cube, it is possible to resolve these ambiguities [Nielsonand Hamann, 1991, Natarajan, 1994, Chernyaev, 1995, Lewiner et al., 2003].

Marching Tetrahedra algorithms, such as [Shirley and Tuchman, 1990, Guéziec and Hummel,1995, Treece et al., 1999], further divide the cubic cells into tetaedra, and for each tetraedra, thereare less possible configurations, than in Marching Cubes algorithm, and without any ambiguous


cases resulting in topologically consistent polygonal meshes. Nevertheless, marching tetraedraalgorithms create numerous, and often over distorted triangles.

There have been many attempts to extend Marching cubes algorithm from uniform to adaptivegrids such as octrees. However these methods all require sort of patching cubes of different res-olution in order to avoid some cracks in the resulting polygonal mesh [Shu et al., 1995, Shekharet al., 1996].

Some dual methods, e.g. [Gibson, 1998, Ju et al., 2002], have been proposed that effectivelyeliminate this patching problem. In these methods, each cube that intersects the implicit surfaceis given a representative vertex. In order to represent well features [Kobbelt et al., 2001, Ju et al.,2002], this vertex is generally placed at the location which minimizes a quadratic error func-tion [Lindstrom, 2000] which takes into account normals information provided by gradient of theimplicit function.

Recently Ho et al. [Ho et al., 2005] introduce a promising method which preserves sharpfeatures, maintain consistent topology and generates surface adaptively without crack patching.Their method resolve those problems by converting 3D Marching Cubes into 2D cubical marchingsquares, resolving topology ambiguity with sharp features and eliminating inter-cell dependencyby sampling face sharp features.

4.3.2.2 Surface fitting

These techniques create a base mesh that coarsely approximate the implicit surface and pro-gressively subdivide, deform, adapt it to fit the implicit surface.

For example, Wood et al. [Wood et al., 2000] first create a base mesh with the same globaltopology as the implicit surface, by slicing the surface along contours that capture the overalltopology. Whereas Labsik et al. [Labsik et al., 2002] first create a base mesh by using March-ing Cubes algorithm on a coarse grid. Then after applying a mesh decimation, they adaptivelysubdivide, project new vertices on the implicit surface and improve the vertices distribution byrelaxation.

The main difficulty of such methods resides in how to create the right coarse mesh whichcaptures the right topological and geometrical information of the implicit surface.

4.3.2.3 Surface tracking

Surface tracking techniques start from a seed element on the surface and iteratively grow apolygonal mesh that approximates the implicit surface.

Bloomenthal polygonization algorithm [Bloomenthal, 1994] start from a volumic cell that in-tersects the implicit surface and iteratively find all intersecting cells among its neighbors. Themain drawback of this method is the difficulty to determine a seed cell.

Particle system can be used to distribute samples over the implicit surface like in [Witkin andHeckbert, 1994], and a Delaunay triangulation is computed for creating the resulting mesh. Suchmethod requires dense enough samples on the surface in order to create a topologically correctmesh.

Hilton et al. [Hilton et al., 1996] start from a seed triangle, and iteratively triangles are createdfrom boundary edges by using a relaxed Delaunay constraint. This method have been further im-proved in [Akkouche and Galin, 2001, Karkanis and Stewart, 2001].

4.3. VISUALIZATION 73

4.3.2.4 Post-processing

After polygonization of the implicit surfaces, the resulting polygonal mesh can further beoptimized in a post process, e.g. to represent accurately sharp features [Ohtake et al., 2003c].

Chapter 5

Implicit Surface Based Reconstruction :State of the art

Contents5.1 Blobby Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.2 Signed Distance Function Estimation . . . . . . . . . . . . . . . . . . . . . 76

5.3 Trivariate Implicit Polynomial Fitting . . . . . . . . . . . . . . . . . . . . . 77

5.4 Moving Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.5 Partition Of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.6 Radial Basis Function based Implicit Surface Reconstruction . . . . . . . . 79

5.6.1 Global Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.6.2 Fast reconstruction and evaluation methods . . . . . . . . . . . . . . . 80

In this section, we present several methods to reconstruct implicit surfaces from large pointsets. We start by giving some formal definitions. Let P = p0, . . . ,pN−1 be a point set consist-ing of N distinct points pi ∈ R3 without any connectivity information that could be provided in amesh representation.

From the point set P , we can consider two problems: interpolating or approximating implicitsurface reconstruction, i.e. depending if we want the resulting surface to pass through all points ofP (∀ pi ∈ P, f(pi) = 0), or next to them (∀ pi ∈ P, f(pi) ≈ 0).

Of course like any reconstruction problem, the problem of reconstructing implicit surfacesfrom a given point set is clearly ill-posed, and the most common approach to solve this problem,as we saw in the previous part, is to make the assumption that the implicit function f is a linearcombination of some basis functions. Thus many methods from the previous part can be appliedfor implicit surface reconstruction. However due to the fact, that wished implicit function val-ues si = 0, it leads to the trivial solution of all coefficients are null, i.e. α = 0. Therefore,methods described in the previous part can be considered with some additional constraints or littlemodifications.

In this chapter, we will review some previous works about implicit surface reconstruction asan introduction to next one in which we will present our contribution to implicit surface recon-struction.

75

76 CHAPTER 5. IMPLICIT SURFACE BASED RECONSTRUCTION : STATE OF THE ART

5.1 Blobby Models

To our knowledge, Muraki [Muraki, 1991] was a kind of pioneer, by reconstructing complexshapes defined as implicit surfaces from a given 3D point set. The resulting implicit function f isexpressed as blending of simple 3D based primitives, Blobby primitives, together. Blinn [Blinn,1982] introduced Blobby primitive Bi for a 3D shape function di (spherical, superquadric [Barr,1981]. . . ) as a scalar field, or an implicit function, fi defined as follows:

fi(p) = bi · exp (−ai · di (p)) (5.1)

To form an implicit function f , summing the contribution of each scalar field fi and fixing athreshold T are required:

f(p) = T −M−1∑

i=0

bi · exp (−ai · di (p)) (5.2)

In his study, Muraki limited his study to spherical blobby models, radially symmetric shapesaround a set of skeleton points ciM−1

i=0 .First, he started with a single primitive B0 centered of the mass of the data c0, with a0 =

b0 = T e (e is the Euler constant), which is added to a list. Iteratively, an arbitrary primitive Bi

is selected and removed from the list, which is split into two new primitives B′i and B′′i added tothe list. Then by minimizing an energy functional, parameters of the new primitives B′i and B′′iare computed. The energy to minimize is composed of a data closeness term, a normal deviationterm, and a shrinking term which minimizes the influence of each primitive. This procedure is re-peated until the energy function becomes sufficiently small, i.e. falls below a fixed threshold value.

Unfortunately, reconstructing implicit surfaces with this method is computationally expensive.To tackle this problem, Tsingos et al. [Tsingos et al., 1995] allowed to interactively specify theinitial skeleton points, and it was further enhanced to an automated method by Bittar et al. [Bittaret al., 1995].

5.2 Signed Distance Function Estimation

Hoppe et al. [Hoppe et al., 1992] reconstructed a signed geometric distance function d : R3 →R between point p (x, y, z) ∈ R3 and the unknown surface S, such that the zero set of the signedfunction d is the reconstructed surface. The surface S is defined as local linear approximations ofpoint set P , with using an estimation of the tangent planes. The signed distance function is definedas the distance to the tangent plane of the closest point pi from P , multiplied by ±1 depending onwhich sides of the tangent plane lies p.

In the first step, normals are estimated by a local principal component analysis for all pointspi ∈ P , in a k−neighborhood of pi. More precisely, for every point pi, they compute the covari-ance matrix Ci on the k−neighborhood (see Eq. 5.3), and the eigenvector corresponding to thesmallest eigenvalue is considered as the normal to the point pi.

Ci =k∑

j=0

(pj − oi

) (pj − oi

)T (5.3)

In the second step, the orientation of the normal is modified in order to ensure the consistencyof the surface orientation, i.e. two close points pi,pj ∈ P2 have a similar orientation. Ideally,

5.3. TRIVARIATE IMPLICIT POLYNOMIAL FITTING 77

when the point set P is dense, the sampled surface is smooth, Hoppe et al. consider that the cor-responding tangent planes are nearly parallel, and share the same orientation, i.e. ni · nj ≈ +1.The problem can be modeled as an optimization problem using a graph with one node per tangentplane, or normal, with an edge (i, j) if either oi is in the k− neighborhood of oj , or if oj is in thek− neighborhood of oi, associated with a cost ni · nj . The solution is given by selecting orien-tations which maximizes the total cost of the graph. Since this problem is NP complete, Hoppeet al. introduce an approximation algorithm as follows. By assigning the cost 1 − |ni · nj | forevery edge (i, j), an initial orientation is propagated by a depth first order traversal of the minimalspanning tree of the resulting graph. This order largely avoids ambiguous situations since it tendsto propagate along direction of low curvature in P .

The resulting surface is described by the zero-set of the signed distance function d. Hoppe etal. demonstrated the efficiency of their algorithm on several examples, from various sources, withdifferent topological complexity. However, reconstructed surface has only C0 continuity, and inpractice this method is not used to reconstruct smooth surfaces.

Nevertheless, Hoppe et al.’s method to estimate consistently normals from a given point set isstill used in many implicit surface reconstruction methods.

5.3 Trivariate Implicit Polynomial Fitting

Trivariate implicit polynomial are generally referred as Algebraic Surfaces. Here we onlyfocus on trivariate quadratic implicit surface, i.e. Π2(R3) (the degree of the polynomial form is 2).

Different methods have been proposed in the literature to fit a 3D quadratic surface on data, byusing non linear least squares methods [Taubin, 1991,Taubin et al., 1994], or least squares method.Due to the complexity of non linear least squares methods, and the difficulty to reach the absoluteminimum, we focus here only on linear least squares methods.

The main problem of such approaches resides in the choice of additional constraints. Indeed

applying a simple least squares minimization on data (minN∑

i=0(g(pi))

2) leads to the trivial solu-

tion g = 0. So various constraints have been introduced.

Ohtake et al. compute a local frame, then fit a local bivariate quadratic polynomial [Ohtakeet al., 2003b]. In [Ohtake et al., 2003a], they add some off-surface constraints placed on a regu-lar grid and compute an approximated signed distance to these constraints points. The 3L fittingalgorithm [Blane et al., 2000] adds for each point two off-surface constraints points (internal andexternal). The gradient-one fitting algorithm [Tasdizen et al., 2000] provides improvement by re-placing added constraint data sets (obtained in the 3L fitting algorithm [Blane et al., 2000]), byexplicit differentiation and by constraining the gradient vector to have a fixed norm. Recentlyin [Helzer et al., 2004], Helzer et al. improve the performance and the stability of the gradient-onealgorithm [Tasdizen et al., 2000] by constraining the norm of the gradient vector with a data de-pendent value. They present two different algorithms [Helzer et al., 2004], i.e. with two differentvalues for the gradient vector norm: Min-Max fitting algorithm, which is aiming to obtain a uni-form maximal deviation along the zero-set; Min-Var fitting algorithm, which is aiming to obtain auniform variance of the error at each point of the data set.


5.4 Moving Least Squares

Alexa et al. [Alexa et al., 2001, Alexa et al., 2003] introduced a purely local method to recon-struct implicit surfaces, based on work of Levin [Levin, 2003], and they call the resulting surfacea point set surface.

The reconstruction procedure can be divided into two steps:

• For a point r close to the surface, a local plane H = p| 〈n, r〉 −D = 0,p ∈ R3, with Dis a constant, n ∈ R3 and ‖n‖ = 1 is computed so as to minimize a local weighted sum ofsquares distance of the points pi to the plane

N−1∑

i=0

(〈n,pi〉 −D)2 ω (‖pi − q‖) (5.4)

where q is the orthogonal projection of r on H , and ω is a radially symmetric, positive, andmonotically decreasing weight function.

• the reference domain for r, given by the orthonormal coordinate system on H with q asorigin, is used to compute a local bivariate polynomial approximation of degree d of thesurface in a neighborhood of r. Let qi be the projection of pi onto H , and the heighthi = 〈n, (pi − q)〉, and the coefficients of the polynomial approximation g are computedby minimizing the weighted least squares error

N−1∑

i=0

(g(ui, vi)− hi)2 ω (‖pi − q‖) (5.5)

where (ui, vi) is the representation of qi in the local reference domain.

This projection operator appears really useful for defining the surface and providing tool togenerate points on that surface. Unfortunately, the projection operation is comptutationally ex-pensive because of the associated non-linear optimization problem, and the approximating tangentframe is not tangent to the surface.

To overcome these problems, Adamson and Alexa [Adamson and Alexa, 2003] introduce animplicit definition of a smooth surface from points, first presented as an attempts to simplify thecomputation of [Alexa et al., 2003]. The authors express the implicit function f as follows

f(p) = n(p)T · (p− a(p)) (5.6)

where a is a weighted average of points at a location p (see Eq. 5.7), and n is the normal directionwhich establishes an approximating tangent frame to the surface. This method can be seen as aprojection of a point p onto the plane defined by a(p) and n(p).

a(p) =

N−1∑i=0

ω (‖p− pi‖) · pi

N−1∑i=0

ω (‖p− pi‖)(5.7)

5.5. PARTITION OF UNITY 79

The normal direction n(p) in each location p can be defined by a weighted average of inputnormals, whenever normals are provided, as follows

n(p) =

N−1∑i=0

ω (‖p− pi‖) · ni

∥∥∥∥N−1∑i=0

ω (‖p− pi‖) · ni

∥∥∥∥(5.8)

otherwise n(p) is determined by the normal of the weighted least-squares fitting plane of thepoints pi.

Recently it has been shown [Dey and Sun, 2005, Kolluri, 2005] that given an ε-sample of asurface (sampling density is proportional to the medial axis of the surface, see section 6.2.1.1) theMLS surface reconstructs an isotopic surface.

5.5 Partition Of Unity

Partition of unity approaches have shown their efficiency for solving global problems of ap-proximation, or interpolation. Principle has already been explained in section 1.3. In practice,these methods differ by the way to define sub-domains Ωi, and on the local solution (approxima-tion or interpolation) fi authors use.

Ohtake et al. [Ohtake et al., 2003a] introduce a multilevel version of standard partition of unitymethods. They first rescale the point set into a bouding cube and create an octree-based subdivisionof this cube. For each cell of the octree, a sub-domain Ωi is defined and a local shape approxima-tion is calculated. Depending on the desired accuracy, the cell is subdivided recursively. Ohtake etal. use trivariate quadratic implicit functions and use different fitting strategies depending on thenumber and the configuration of point set contained in the cell.

Tobor et al. [Tobor et al., 2004a,Tobor et al., 2004b] use polyharmonic splines as local solutionand define each sub-domains according to Wendland statements [Wendland, 2002], by using anoverlapping kd-tree (the definition of each sub-domain will be discussed in the next chapter).

5.6 Radial Basis Function based Implicit Surface Reconstruction

5.6.1 Global Support

The pioneering work to reconstruct implicit surfaces from given point sets using RBFs, with avariational technique, can be attributed to Savchenko et al. [Savchenko et al., 1995]. The implicitsurface f : R3 → R is reconstructed by introducing a carrier solid with a defining implicitfunction fc : R3 → R. By using a carrier function fc, the authors avoid the trivial solution α = 0.Indeed the authors assign the carrier function values fc(pi) at all points pi, and transform theimplicit surface reconstruction problem into a volume reconstruction problem. More precisely,they search for a volume function u : R3 → R that interpolates all carrier function values fc(pi).

Finally the implicit function f is computed as

f(p) = u(p)− fc(p) (5.9)

Savchenko et al. describe the function u as a linear combination of Duchon’s basis functions,or triharmonic basis functions, i.e. φ(r) = r3 (see Table 2.2).

To solve the dense linear system (Eq. 2.8), authors use Householder method which complexityis in O(N3). Thus this method can not be applied to large point set, and the choice of the carriersolid has a significant impact on the shape of the reconstructed surface, since it is isomorphic to


the carrier solid.

Turk and O’Brien [Turk and O’Brien, 1999] add some additional off-surface constraints pointsqkK−1

k=0 points to the original point set, and assign to them some non null constraints valueshk 6= 0, i.e.

f(qk) = hk 6= 0, k = 0, . . . ,K − 1 (5.10)

Fig. 5.1. Off-surface constraints points qk.

More precisely, based on common convention concerning the sign of the implicit function f ,they use Normal off-surface points which are placed at positions qi = pi ± κ · ni (see Fig. 5.1),with κ a positive constant and the corresponding value hk = ±1. Finally, in order to computethe implicit function, they consider a multi-dimensional reconstruction problem with an enlargeddata set, i.e. P∗ = P ∪ qk, with associated values si, such that ∀ pi ∈ P, si = 0 and ∀ qk ∈P∗ − P, si = hk 6= 0. To solve this problem they use triharmonic basis function, i.e. φ(r) = r3,and solve the linear system (Eq. 2.8) by using a LU decomposition.

In order to represent sharp edges, Dinh et al. [Dinh et al., 2001] consider that there is someprivileged directions in data, and thus use some anisotropic basis functions.

5.6.2 Fast reconstruction and evaluation methods

Unfortunately previous methods can not reconstruct implicit surface from large point sets,since the computational cost of O(N3) to solve the linear system becomes prohibitively expen-sive, and the huge amount of required memory in O(N2) rapidly grows above available memory.Furthermore, the evaluation of the resulting implicit function is in O(N).

For these reasons, like in the multi-dimensional reconstruction case, authors use methods toreduce the computational complexity.

5.6.2.1 Fast Multipole Method

Carr et al. [Carr et al., 2001] use a Fast Multipole Method (see section 2.4) which approximatesthe evaluation of the implicit function f at a location p by dividing, for each RBF, into a far anda near field expansion. The far field, for evaluation point p far away from centers that are close toeach other, is approximated by a truncated Laurent expansion. The length of the truncated Laurentexpansion can be preset according to the desired precision. In order to determine near and farfields, data have to be structured hierarchically or with an efficient range query data structure.

Using the greedy evaluation technique, the evaluation cost drops fromO(N) toO(1) after con-structing the range query data structure which is done in O(N logN), and calculating a matrix-vector product drops from O(N2) to O(N). Matrix-vector product is involved while the linear

5.6. RADIAL BASIS FUNCTION BASED IMPLICIT SURFACE RECONSTRUCTION 81

system is solved iteratively, by a pre-conditioned conjugate gradient methods, preconditionedGeneralized Minimal RESidual iterations [Saad and Schultz, 1986], or domain decompositionmethods. The computational cost to solve the linear system drops from O(N3) to O(N logN).Furthermore, since the involved matrix of the linear system never has to be computed explicitly,storage requirements are also greatly improved, by reducing to O(N).

Unfortunately the development of the near and far fields is very complex to implement and itsdeviation has to be done for every radial basis functions separately.

5.6.2.2 Partition of unity

Tobor et al. [Tobor et al., 2004a, Tobor et al., 2004b] use a partition of unity based approach(see section 2.5), in order to decompose the global reconstruction problem into several small prob-lems. Authors consider enlarged point set as center set, like in standard method described in [Turkand O’Brien, 1999]. In [Tobor et al., 2004a] authors described two kinds of domain decomposi-tion bow and ellipsoid axis aligned based on overlapping octree decomposition; while in [Toboret al., 2004b] they use a box axis aligned domain decomposition based on an overlapping kd-treedecomposition.

In the partition of unity approach, they consider that the global domain Ω is divided into Moverlapping sub-domains. The assembly of matrices requires O(M(N/M)2) operations, and tosolve all linear systems requires O(M(N/M)3) operations. Concerning the evaluation, the num-ber of operations required is in O(M +N/M). Using an appropriate range query data structure,as in [Tobor et al., 2004a, Tobor et al., 2004b], the evaluation can be reduced to O(logN).

5.6.2.3 Compact Support

Morse et al. [Morse et al., 2001] use Wendland’s CSRBF [Wendland, 1995] with single scaleσ. The matrix in the linear system is thus sparse, and as mentioned in section 2.6, many proceduresin the reconstruction process can be speeded up. Due to the use of a single global support σ, itis impossible to reconstruct sparse or irregularly samples surface, or at high computational cost.Kojekine et al. [Kojekine et al., 2003] improve the performance of this method by organizing theinvolved sparse matrix into a band-diagonal sparse matrix with using an octree data structure. Theresulting linear system can be solved more efficiently with using a combination of block Gaussiansolution and Cholesky decomposition. For computational complexity, please refer to Table 2.4 insection 2.6.2.2.

Major drawback of this method is the choice of the support size σ as we have already seen inthe previous part, and it may be difficult to reconstruct efficiently highly irregular sampled surfaces(see Fig. 5.2).

5.6.2.4 Mutiresolution Approach

Ohtake et al. [Ohtake et al., 2003b] use a multiresolution approach with Wendland’s CSRBF.They construct a hierarchy of center sets C0, . . . , Cl by using an octree based clustering. The initialcenter set C0, i.e. the center set at the coarsest level, is constructed by splitting the bounding boxof P into eight cells and computing the eight centroids with average normals of the points of Pcontained in the corresponding cell. By iteratively repeating this process, the hierarchy of centerset is performed. Then a coarse function f0(p) = −1 is defined, and the function at the next levelis composed of a local quadratic form, determined by weighted least squares minimization thatapproximates the shape of Cl in the vicinity of one center, and a CSRBF term whose coefficientis determined in order to interpolate exactly centers cl

i of Cl. Using a local approximation createslocally an offset, so the value to interpolate is not anymore null, and it avoids the trivial solution


(a) (b)

Fig. 5.2. Irregularly samples on the belly part of the Stanford Buddha (a). Implicit Surface generatedwith single scale CSRBF with too small support (b). These images are courtesy of Yutaka Ohtake,Alexander Belyaev and Hans-Peter Seidel [Ohtake et al., 2003b].

α = 0. The support size between two consecutive level is divided by 2, which allows to fill largeholes and repair incomplete data.

However, since for the finest levels of the hierarchy a large and sparse linear system has tobe solved, this method is as limited as the compactly supported method. On the other hand, asmentioned by authors, this method may generate some undesirable extra zero-sets parts, when thesurface has thin part (see Fig. 5.3).

(a)

Fig. 5.3. With this methods, some extra zero level-sets are generated. These images are courtesy of YutakaOhtake, Alexander Belyaev and Hans-Peter Seidel [Ohtake et al., 2003b].

5.6.2.5 Adaptive Method

Ohtake et al. [Ohtake et al., 2004a] present an adaptive method, i.e. support sizes are thusdefined locally and independently. Centers are chosen randomly from the point set according tothe density and geometry of samples. The support size for one CSRBF is determined by findingthe size for which the local quadratic approximation provides the lowest error. Finally CSRBFcoefficients are computed by a discrete least-squares minimization.

A major drawback of this method is the lack of control concerning the number of requiredCSRBF to reconstruct the surface, and this method can not fill too large holes.

Chapter 6

Our Contributions

Contents6.1 Reconstruction With Composite Implicit Surface . . . . . . . . . . . . . . . 85

6.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.1.2 Support Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.1.3 Local Polynomial Approximation . . . . . . . . . . . . . . . . . . . . 86

6.1.4 Least Square Approximation . . . . . . . . . . . . . . . . . . . . . . . 86

6.1.5 Constructing Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.1.6 Computational Complexity . . . . . . . . . . . . . . . . . . . . . . . . 88

6.1.7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.2 Support Adapted to the Medial Axis . . . . . . . . . . . . . . . . . . . . . . 936.2.1 Support Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.2.2 Trivariate Quadric Fitting . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.2.3 CSRBF Coefficients Computation . . . . . . . . . . . . . . . . . . . . 96

6.2.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.2.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

As we saw in the previous chapter, traditional RBF based approaches use an enlarged cen-ter set C constituted of the whole point set P and some additional off constraints points (seeFig. 5.1). In such case, the number of centers is much higher than the number of samples, i.e.Card(C) = |C| > N , generally |C| = 3N . With using globally supported RBF, e.g. polyhar-monic splines, the computational complexity to assembly interpolation matrix Aφ,C is O(|C|2), tosolve the linear system is O(|C|3) with LU or SVD decomposition, and the evaluation is O(|C|).Moreover, the choice of the support is a difficult task, which could lead to smooth solution, or touncontinuous implicit surface. In addition the computational cost of some methods can be pro-hibitive.

Here we propose methods which ensure a low computational cost for both the reconstructionand the evaluation of the resulting implicit function f , with a fixed amount of parameters or witha fixed error bound for the reconstructed surface. The proposed methods should be able to processirregular point sets.

83


For this purpose, we introduce new reconstruction methods, largely based on our papers [Gelaset al., 2006, Gelas et al., 2007b], in which we assume that the normals are provided during theacquisition process or can be estimated by using the approach proposed in [Hoppe et al., 1992].The proposed methods are the following:

A method with composite implicit surface where the number of centers is much lower than thenumber of points, and by using CSRBFs and partition of unity in order to get a very lowcomputational cost.

A method using RBF with support adapted to the medial axis where the number of centers iscontrolled, and the support size of each CSRBF is locally adapted to the medial axis.

6.1. RECONSTRUCTION WITH COMPOSITE IMPLICIT SURFACE 85

6.1 Reconstruction With Composite Implicit Surface

6.1.1 Overview

Here, we decide to work with a reduced and controlled center set C, i.e. |C| ¿ N . Unfortu-nately, to solve the reconstruction problem and evaluate the resulting implicit function f remainsexpensive. With using CSRBFs (see section 2.6), implicit function evaluation, matrix computationand the linear system to be solved can be done in much lower complexity than common methods(see table 2.4). Moreover using a divide and conquer method, such as partition of unity method,which splits large problems into small ones, leads to a very low complexity.

We briefly present the proposed method: we first compute the center set C (see section 3.2.2),we define sub-domains Ωi (see section 6.1.5), a local approximation fi which has the same formas in the Eq. 6.2, and finally we blend together all local approximations fi on the whole domain Ωby a partition of unity (Eq. 6.1).

We propose to decompose the resulting implicit function f as follows:

f(p) =

M−1∑i=0

ωi(p) · fi(p)

M−1∑j=0

ωj(p), (6.1)

where M is the number of sub-domains Ωi (defined below), ωi is a positive weight function (de-fined below), and fi is expressed as follows:

fi(p) = gi(p) +|Ci|−1∑

j=0

αij · φσ

(‖p− cij‖

), (6.2)

where gi is a local polynomial form of Π2(R3), Ci = cij|Ci|−1

j=0 is the center set restricted to theith sub-domain Ωi, σ is the support size defined for all sub-domains Ωi, and φσ = φ(r/σ) is aCSRBF (see section 2.6).

Finally, the successive steps are described in the following algorithm:

Algorithm 3 Composite Implicit Surface Reconstruction(P, N, TLeaf , q)procedure COMPOSITE IMPLICIT SURFACE RECONSTRUCTION(P , N , TLeaf , q)

Determinate center set C from P , such that |C| = NCompute support size σCompute sub-domains ΩiM−1

i=0 according to TLeaf and qClassify samples points P into partitions Ωi

Compute local polynomial approximation gi

Compute RBF parameters αij

end procedure

6.1.2 Support Size

We should fix the support size of all CSRBFs. In order to fill holes, we should use a supportsize which is large enough. We propose to use a support size directly connected to the center setdistribution and independent of sub-domains definition, by choosing σ = R×hC,Ω, where hC,Ω isthe fill distance hC,Ω = max

p∈Ωminci∈C‖p− ci‖, where R is a positive constant.


6.1.3 Local Polynomial Approximation

For each partition Ωi, with its associated center set Ci = cij and respective normals ni

j,we fit a trivariate quadratic implicit surface gi, given by the following equation:

gi(p) = pT ·Ai · p + bTi · p + ci = 0, (6.3)

where Ai is a symmetric 3 × 3 matrix, bi is a vector of 3 components, and ci is a scalar. In orderto determine Ai, bi, and ci, several methods have been proposed in the literature (see section 5.3).

Here, we use a modified version of weighted form of the Min-Max algorithm proposed byHelzer et al. [Helzer et al., 2004]. We consider for each center ci the energy functional Ei to beminimized (Eq. 6.4).

Ei =|Ci|−1∑

j=0

φi(cij) ·

(

gi(cij)− sj

)2 + ρ ·(

nij −

∇gi(cij)

‖∇gi(cij)‖

)2 (6.4)

where φi(p) = φ(‖p−ci‖

σ

)is the CSRBF associated to the center ci, and ρ is a positive constant

parameter.In such case, following [Helzer et al., 2004], we should first rewrite (Eq. 6.3) as

gi(p) = ai · uT (p) (6.5)

where u(p) is the monomial vector of size 10 defined in (Eq. 6.6), and a = aj9j=0 is theunknown coefficients vector.

u(p) = u(x, y, z) = [xl0 ym0 zn0 , ..., xl9 ym9 zn9 : li +mi + ni 6 2] (6.6)

Then following [Helzer et al., 2004] minimizing, for each center ci, the functionnal Ei (Eq.6.4) can be reduced to solve the linear system Mi · ai

T = bi, where Mi and bi are defined asfollows

Mi =|Ci|−1∑

k=0

φi(cik) ·

uT (ci

k) · u(cik) + ρ · Ju

T (cik) · Ju(ci

k)9∑

l=0

|ul(cik)|

bi = ρN−1∑

k=0

φi(cik) ·

JuT (ci

k) · nikT

9∑l=0

|ul(cik)|

where Ju(p) is the Jacobian matrix of u(p), of size 10× 3.

6.1.4 Least Square Approximation

To take into account the normal given for each point, we enlarge the data set Pi to Qi (|Qi| =3|Pi|) by adding some off-surface constraint points on both sides of the surface (see Fig. 5.1):

qk = pk sk = 0 (6.7)

qN+k = pk + κ · nk, sN+k = −κ (6.8)

q2N+k = pk − κ · nk, s2N+k = κ (6.9)


where k = 0, ..., N − 1, nk is the associated normal to the point pk, and κ is a small positivevalue.

One possible and common way to compute CSRBF coefficients would be to consider interpo-lation problem on the center set C, i.e. by constraining the resulting implicit function f to satisfyfor all centers cj ∈ C: f(cj) = 0. Unfortunatley, interpolation is too much noise sensitive: themore centers, samples from the given point set, the more noise will be captured. It is important tonotice that this affirmation is also true in the case of approximation. For this reason, we decide toadopt an approximation scheme, with a least square term and regularization term (see Fig. 6.1).

(a) (b) (c)

Fig. 6.1. Noisy data (a). Resulting interpolation (b) and resulting approximation with a regularizationfunctional (c).

Here we use a simple regularization functionalR[f ] directly connected to the coefficients αi,i.e. R[f ] =

M−1∑j=0

(αj)2. It is well known in the regularization theory that only adding terms on

the diagonal constrains the smallest eigenvalues of the approximation matrix to be at least equalto added term. If the set of CSRBF were orthogonal, the regularization term would correspond tothe energy of the implicit function.

Here we minimize an energy E, composed by a (weighted) data-closeness term and a regular-ization term:

E =3N i∑

k=0

ωi(qk) · (fi(qk)− sk)2 + λ · (R[f ]− r0) (6.10)

Finally we get the coefficients αi =αi

j

j=0,...,M i−1

:

αi =(BPφ,Ci+ λ I

)−1 ·(QPi

φ,Ci

)T· ri

BPφ,Ci(l,m) =3(N i−1)∑

k=0

ωi(qik) · φi

l

(qi

k

) · φim

(qi

k

)

QPiφ,Ci

(l) =3(N i−1)∑

k=0

ωi(qik) · φi

l

(pi

q

),

ri(j) = sj − gi(qij)

where φij(p) = φ

(‖ci

j−p‖σ

).


6.1.5 Constructing Partitions

We now introduce how we compute the domain partitioning, which respects the statement de-scibed in [Wendland, 2002] (see section 2.5).

We choose to construct an adaptive overlapping kd-tree based domain decomposition [Toboret al., 2004b] (see Fig. 6.2). To construct all partitions, we fix the superior bound TLeaf for thenumber of centers contained in each sub-domains Ωi, and an overlapping rate q.

In this case, each sub-domain Ωi is a parallelepipedic domain (see Fig. 6.2), following [To-bor et al., 2004b], we associate the weight function ωi as a composition of a distance functiondi : R3 → R (see Eq. 6.11 - 6.14) and a compactly supported function ψ : [0, 1] → [0, 1] (seeEq. 6.15), i.e. ωi(p) = ψ (di(p)). Depending on the choice of ωi, we can reconstruct implicitfunction f with various continuity orders.

We use polynomial decay function for the the compactly supported function ψ which satisfythe conditions ψ(0) = 1, ψ(1) = 0 and depending on the desired continuity order k we imposeψ(l)(0) = 0 and ψ(l)(1) = 0 for l 6 k. Finally we get the following decay functions (see Fig.6.3):

ψ(d) =

1− d, for C0 continuity (6.11)

2d3 − 3d2 + 1, for C1 continuity (6.12)

(d2 − 1)2, for C2 continuity (6.13)

−6d5 + 15d4 − 10d3 + 1, for C3 continuity (6.14)

For the distance function, we propose to use the following function di [Tobor et al., 2004b]:

di(p) = 1− 642∏

r=0

(pr − Sr) (pr −Tr)(Tr − Sr)2

(6.15)

with S and T are the opposite corners of the considered sub-domain.

Now, let us give some explanation about how to subdivide a domain Ωl, with its center set Cl,into two overlapping sub-domains Ωl+1

1 and Ωl+12 , with their associated center set Cl+1

1 and Cl+12 ,

such that |Cl+11 | = |Cl+1

2 | (see Algorithm 4).First, centers are sorted according to the longest axis of Ωl. Then, we collect first center set

Cl+11 containing the nl+1 =

⌈(1+q)nl

2

⌉centers with the lowest values, and the second one Cl+1

2

with respectively nl+1 centers with the highest values.This procedure is iterated until each leaf domain contains less than a predefined number of

centers per domain, Tleaf .

6.1.6 Computational Complexity

With our method, assembling the matrices requiresO(M(N ·|C|/M2)) operations1 and solvingthe linear systems requiresO(M ·(µ|C|)) operations, where µ corresponds to the sparseness factorof the matrix. µ is thus related to the center set distribution and to the support size. Concerning theimplicit function evaluation, the number of required operations is in O(logM), by using the treedecomposition. Due to the partition definition, we can make the assumption that |C|/M is constantand thus previous computational complexities are thus much lower than what we considered.

1We consider that we try to fill every elements of the matrix.


(a) (b)

(c) (d)

(e)

Fig. 6.2. PU process on CSRBF centers, represented by circles (a). Construction of the first level partition(b) and the second level partition (c). Local approximations fi are computed (d), and construction ofthe global function f is performed by blending local approximations together (e).


(a) (b) (c) (d)

Fig. 6.3. Polynomial decay function ψ for various continuity order: C0 (a), C1 (b), C2 (c), C3 (d).

Algorithm 4 Partition(Cl, q, TLeaf

)[Tobor et al., 2004b]

procedure PARTITION(Cl, q, TLeaf )Set Ωl is the bounding box of Cl

if nl > TLeaf thenDeterminate the longest axis of Ωl

Sort Cl according to the longest axis|Cl+1| =

⌈(1+q)|Cl|

2

⌉

Determinate Cl+11 and Cl+1

2

PARTITION(Cl+11 , q, TLeaf )

PARTITION(Cl+12 , q, TLeaf )

end ifend procedure

6.1.7 Results

Reconstruction

In order to show the effectiveness of our method, we have tested our algorithm on severalmodels, with different points distributions (see Fig. 6.4, 6.5).

Center determination

To evaluate the error, we introduce two kinds of errors:

• Offset error. This error is given by

ε(p) = |f(p)| (6.16)

• Taubin error. This error is a first order approximation of the euclidean distance to the implicitsurface [Taubin et al., 1994]

η(p) =|f(p)|‖∇f(p)‖ (6.17)

On Fig. 6.6 and Fig. 6.7, we compare several methods to choose centers: randomly cho-sen, pca-based clustering methods with considering cluster centroid as CSRBF center (see section3.1.1), k-means [MacQueen, 1967], and the furthest point algorithm. We demsontrate here theefficiency of the furthest point algorithm described in section 3.2.2, and we show that k-meansalgorithm provides quite similar results.


(a) (b)

Fig. 6.4. Bunny model with samples only on ridge and valley lines (a), and the corresponding reconstructionwith a fixed budget of 500 centers.

(a) (b) (c) (d)

Fig. 6.5. Venus head model (a), reconstruction for different budget of CSRBF |C| = 15 (b), 100 (c), 700 (d).

(a) (b)

Fig. 6.6. Average of the offset error ε (a) and Root-mean-square of the offset error εrms (b) on the Bunnymodel in function of the number of centers, depending on the method to determinate center, i.e. random(green), pca-based (black), k-means (red), furthest point algorithm (blue).


(a) (b)

Fig. 6.7. Average of the Taubin error η (a) and Root-mean-square of the Taubin error (ηrms) (b) on theBunny model in function of the number of centers, depending on the method to determinate center, i.e.pca-based (black), k-means (red), furthest point algorithm (blue).

Regularization

With noisy data, we study the influence of the regularization parameter λ (see Eq. 6.10) onεrms (see Fig. 6.8). We can observe that there is one optimal value for λ which could be evaluatedby cross-validation.

Fig. 6.8. Root-mean-square error εrms depending on the regularization parameter λ for 1000 centers.

6.2. SUPPORT ADAPTED TO THE MEDIAL AXIS 93

6.2 Surface Reconstruction Using Radial Basis Functions With Sup-port Adapted to the Medial Axis

Here too, we decide to work with a reduced and controlled center set C, i.e. |C| ¿ N . Tothis end, we determine centers following the furthest point algorithm (see section 3.2.2). Nowwe propose to locally adapt the support size of each CSRBF to the centers distribution (see sec-tion 6.2.1), to the geometrical and topological informations, about the surface, they contain. Tocompute the support size of each CSRBF, we first compute an approximation of the medial axis,corresponding to center set, and link it to the distance between one center and the medial axis. Theresulting implicit function is finally expressed as a combination of some local trivariate quadraticapproximations and CSRBFs. All coefficients are calculated by minimizing mean square errorterm and another one which takes into account normal or gradient information of the resultingimplicit function.

More precisely, we propose to find an implicit function f which approximates a given pointset P with a fixed number of parameters M ¿ N . The reconstructed implicit function f is given,from a center set C = c0, . . . , cM−1 with M ¿ N as follows:

f(p) =M−1∑

i=0

[gi (p) + αi] · φσi(‖ci − p‖) (6.18)

where φσi(r) = φ(r/σi), φ(r) is a Wendland’s CSRBFs [Wendland, 1995], gi(p) are an unknown

trivariate quadric functions (approximation ofP in σi-neighborhood of ci), and αi are coefficients,which have to be determined respectively.

6.2.1 Support Size

We propose here to locally adapt the support size σi to both the geometry and the toplogy ofthe sampled surface. The definition of the support size should be independent under any Euclideantransformations or scaling. For this purpose we introduce a scale parameter satisfying these con-straints, i.e. a scale invariant parameter σi. The support size is thus related to the scale parameterby the following relationship:

σi = ρ · σi, (6.19)

where ρ is a fixed global parameter.In the following, we begin by introducing the background necessary to the comprehension of

our proposal. Then we explicit how to define this scale parameter.

6.2.1.1 Preliminaries

Let P be a finite set of points in R3. We give here some formal definitions, from [Attali et al.,2006, Cazals and Giesen, 2007], useful for the comprehension of our proposal

Voronoï DiagramThe Voronoi cell of the site p ∈ P is given as Vp =

x ∈ R3 : ∀ q ∈ P − p, ‖x − p‖

6 ‖x− q‖. The sets Vp are convex polyhedra. Closed faces shared by two Voronoi cells arecalled Voronoi faces, and edges shared by three Voronoi cells are called Voronoi edges. The pointsshared by four or more Voronoi cells are called Voronoi vertices. The Voronoi diagram VP of P isthe collection of all Voronoi cells, faces, edges and vertices.


(a) (b) (c)

Fig. 6.9. Voronoï Diagram (a), corresponding Delaunay triangulation (b).

Delaunay TriangulationThe Delaunay Triangulation DP of a set of points P is dual to the Voronoi diagram VP . Theconvex hull of four or more points in P defines a Delaunay cell if the the intersection the cor-responding Voronoi cells is not empty and there exists no superset of points in P with the sameproperty. Analogously, the convex hull of 1 6 k 6 3 points define a (k − 1)-dimensional De-launay face if the intersection of their corresponding Voronoi cells is not empty. A 0-, 1-, and2-dimensional Delaunay face is also called a Delaunay vertex, Delaunay edge, Delaunay trianglerespectively. The collection of Delaunay cells and their faces defines a decomposition of the con-vex hull of all points in P . This decomposition is a triangulation where the Delaunay cells aretetraedra if the points are not aligned.

PolesAmenta et al. [Amenta et al., 1998] introduced poles as follows. The furthest Voronoi vertex p+

in Vp, from the site p, is called the positive pole of p; and the negative pole p− in Vp is thefurthest Voronoi vertex, from the site p, such that the angle ∠(p−pp+) is more than π

2 . If Vp isunbounded, p+ is taken at infinity, and the direction of vp is taken as the average of all directionsgiven by unbounded Voronoi edges.

Medial AxisLet us consider a smooth surface S embedded in R3. According to [Attali et al., 2006], the medialaxis M [S] is the set of points that have at least two closest points in the complement of S.

The distance of a point p ∈ S to the medial axis M(S) is its local feature size. We define theLocal Feature Size Function LFS : S → R which assigns the local feature size to a point p asfollows:

LFS(p) = infq∈M(S)

‖q− p‖ (6.20)

ε-samplesAn ε-sample of S is a subset P ⊆ S such that every point p ∈ S has a point x ∈ P at distanceat most ε · LFS(p). An ε-sample is uniform, if every point has a point in P at distance at mostinfx∈S LFS(x).


In 2D, it has been shown that if samples are dense enough, all Voronoï vertices are closed tothe medial axis. However, it is not the case in 3D, indeed some Voronoï vertices may be locatedclose to the surface, close to the medial axis, and some others can be located very far away. Inthe case of ε-samples, Amenta et al. [Amenta et al., 2001] show that poles are located close to themedial axis to the sampled surface M(S).

6.2.1.2 Adaptive support to the medial axis

We shall now specify the internal scale parameter σi for a given center ci. We define σi as thedistance from ci to the approximated medial axis computed from the center set C. A natural wayto define σi would be

σi = ‖ci − c−i ‖ (6.21)

Clearly for a low amounts of CSRBFs, center set does not satisfy ε-sample condition, andcorresponding poles may be far from the real medial axis. By using such definition of the supportsize may explode locally (see first row of Fig. 6.10). To avoid such problems, we will not considerthe distance ‖ci− c−i ‖ as the feature size. Indeed, we first consider the whole set of poles C−, andthen we compute for each center ci the distance to the closest point in the set of poles C−. So, wecan rewrite the support size as follows

σi = mincj∈C−

‖ci − c−j ‖ (6.22)

(a) 500 (b) 1 000 (c) 2 000 (d) 5 000

(e) 500 (f) 1 000 (g) 2 000 (h) 5 000

Fig. 6.10. Support size σi = σi for the Standford bunny model for various amount of CSRBF |C|. Firstrow of images shows the resulting support size with using Eq. 6.21 and second row of images showresulting support size with our proposal (see Eq. 6.22).


6.2.2 Trivariate Quadric Fitting

We also proposed to approximate the point set P in a vicinity of ci by a trivariate quadraticform gi

gi(p) = pT ·Ai · p +BiT · p + Ci (6.23)

where Ai is a 3× 3 matrix, Bi is a column vector of size 3, and Ci is a scalar value.Here, like in section 6.1.3, we use a modified version of weighted form of the Min-Max algo-

rithm proposed by Helzer et al. [Helzer et al., 2004]. We consider for each center ci the energyfunctional Ei to be minimized (Eq. 6.24).

Ei =N−1∑

j=0

φi(pj) ·(

(gi(pj)− sj

)2 + κ ·(nj −

∇gi(pj)‖∇gi(pj)‖

)2)

(6.24)

where φi(p) = φ(‖p−ci‖

σi

)is the CSRBF associated to the center ci, and κ is a positive constant

parameter.Then following [Helzer et al., 2004] minimizing, for each center ci, the functionnal Ei (Eq.

6.24) can be reduced to solve the linear system Mi · aiT = bi, where Mi and bi are defined as

follows

Mi =N−1∑

k=0

φi(pk) ·

uT (pk) · u(pk) + κ · Ju

T (pk) · Ju(pk)9∑

l=0

|ul(p)|

(6.25)

bi = κN−1∑

k=0

φi(pk) ·Ju

T (pk) · nkT

9∑l=0

|ul(pk)|

where Ju(p) is the Jacobian matrix of u(p), of size 10× 3.

6.2.3 CSRBF Coefficients Computation

One possible and common way to compute CSRBF coefficients would be to consider theinterpolation problem on the center set C, i.e. by constraining the resulting implicit function f tosatisfy for all centers cj ∈ C f(cj) = 0. Unfortunately, interpolation is too much noise sensitive:the more centers, samples from the given point set, the more noise will be captured. It is importantto notice that this affirmation is also right in the case of approximation. For this reason, we decideto adopt an approximation scheme, with a least square term and regularization term.

When there is few CSRBFs, we do not need to use a regularization term, since the least squareterm will give a smooth enough solution; whereas with high number of CSRBFs, we can notassure that centers, we add, represent well the wanted surface. Though, we should consider aregularization term when the number of CSRBFs becomes large.

For this purpose, we search for minimize the following functional:

E =N−1∑

i=0

(f(pi)− si)2 + κ · (R[f ]− r0) , (6.26)

where si = 0, ∀ pi ∈ P and r0 the wanted smoothness.


Huge variations of the implicit function f , next to it zero set, make f sensitive to noise on thegiven point set. One way to handle this problem is to constrain the norm of the gradient next tothe zero set. However, we should not consider the influence of both the primal function and theCSRBF support size. So we aim to find an implicit function which satisfies:

‖f − s‖ 6 ε, ε > 0 (6.27)

‖η‖ 6 t, t > 0 (6.28)

with

fk = f(pk), (6.29)

ηk =

∥∥∥∥∥M−1∑

i=0

σi

K ′ · αi ·∇φi(pk)

∥∥∥∥∥ (6.30)

φi(p) = φ

(‖p− ci‖σi

)(6.31)

where K ′ = maxr∈[0,+∞[

|φ′(r)|.In the case of the Wendland’s CSRBF (for d = 3 and C2 continuity order) we use K ′ = 135

64 .

Following Miller [Miller, 1970], instead of dealing with the two constraints separately, wecombine them quadratically into a single constraints as follows:

‖f − s‖2 +(εt

)2‖η‖2 6 2 ε2

This problem is the solution of the normal equations:[BT B +

(εt

)2DT D

]α = BT b (6.32)

where B is a M × N matrix, D is a M × N matrix whose elements are vectors, b is a columnvector of size N are defined as follows:

Bij = φi(pj), (6.33)

Dij =σi

K ′ ·∇φi(pj), (6.34)

bk =M−1∑

j=0

(sk − f(pk)|α=0) . (6.35)

with sk = 0 in the context of implicit surface reconstruction.

6.2.4 Implementation

The Voronoï diagram and pole extraction are computed using qhull library [Barber et al.,1996]. To visualize implicit surfaces, we use Marching cubes polygonizer [Lorensen and Cline,1987].

To compute the trivariate quadratic fitting (see section 6.2.2), we use the singular value de-composition of the matrix Mi (see Eq. 6.25).

Since CSRBFs have finite support, many procedures of our method can be speeded up by usingefficient data structure range query searches. Due to this compactness, only few centers should beconsidered for evaluating the implicit function f at a point p, or for computing each term ofB and


D matrices, and BT · b vector in Eq. 6.32. To this end, as soon as the center set C is computed, westore it in a kd-tree data structure [Friedman et al., 1977].

We finally solve the linear system (Eq. 6.32) with using a sparse Cholesky decomposition[Amestoy et al., 2006].

6.2.5 Results

Support size

Fig. 6.11, Fig. 6.12 anf Fig. 6.13 show some resulting support size for various models.

(a) 2K (b) 1K (c) 1K

Fig. 6.11. Support size σi = σi for various models and amount of CSRBF: vascular tree with aneuvrism(a), knot (b), squirrel (c).

(a) 2K (b) 5K

Fig. 6.12. Support size σi = σi for the Dragon model for various amount of CSRBFs.


(a) 500 (b) 1K (c) 2K

Fig. 6.13. Support size σi = σi for the Venus half model for various amount of CSRBFs.

Fixed Amount of CSRBFs

To evaluate the fitting accuracy, we use the Taubin distance [Taubin et al., 1994] which is afirst order approximation of the Euclidean distance η(p) between one point p and the zero set ofan implicit function f , i.e.

η(p) =|f(p)|‖∇f(p)‖ (6.36)

Fig. 6.14, Fig. 6.15 and Fig. 6.16 shows several reconstructions of various models withincreasing number of centers. As expected, more centers there are, more details appear on thereconstructed surface. For example, in the case of the anevrism for low numbers of centers thesupport size is too large to efficiently represent two disconnected tubular part. For higher numberof CSRBFs, tubular parts are well-disconnected.

Fig. 6.17 show the error versus the number of centers |C|, for various models.

Irregular point sets

We have tested our proposed method on some irregular point sets (see Fig. 6.18, 6.19, 6.20).These examples show that our method can fill holes for almost any size. However in Fig. 6.19,some problems, some holes, appears into the reconstructed surface. This is due to too small supportsize for this area, and this may result of our approximation of the distance to the medial axis.

Regularization

Here, we study the influence of the regularization parameter(

εt

)2 on synthetic noisy data (seeFig. 6.21(a)). Fig. 6.21 and Fig. 6.22 show the reconstructed surface for various regularizationparameter.

Fig. 6.23 illustrates how the regularization parameter improves the condition number of thematrix B · BT . We observe that if the regularization parameter is higher than 1, the conditionnumber becomes acceptable. Clearly this parameter appears to be model dependant.


(a) 500 (b) 1K (c) 2K

(d) 5K (e) 10K

Fig. 6.14. Reconstruction of the Lucy model with an increasing number of centers.


(a) 500 (b) 1K (c) 2K

Fig. 6.15. Reconstruction of a vascular tree with aneuvrism model with an increasing number of centers.

(a) 500 (b) 1K (c) 2.5K

(d) 5K (e) 10K

Fig. 6.16. Reconstruction of Dragon model with increasing number of centers.


0

0.002

0.004

0.006

0.008

0.01

0.012

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

RM

S T

aubi

n D

ista

nce

Number Of Center

dragon

(a)

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

RM

S T

aubi

n D

ista

nce

Number Of Center

bunny

(b)

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0.011

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

RM

S T

aubi

n D

ista

nce

Number Of Center

squirrel

(c)

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

RM

S T

aubi

n D

ista

nce

Number Of Center

anevrisme

(d)

Fig. 6.17. rms η(pi) (Eq. 6.36) versus the number of centers |C| for various models, i.e. Dragon (a), Bunny(b), Squirrel (c), vascular tree with aneuvrisme (d).

(a) (b) 0.5K (c) 1K

Fig. 6.18. Dragon model where some samples have been deleted (a). Corresponding resulting implicitsurfaces for 500 (b) and 1000 CSRBFs (c).


(a) (b) 0.5K (c) 1K (d) 2K

Fig. 6.19. Bunny model with samples only on ridge and valley lines (a), corresponding reconstruction withfixed budgets of CSRBFs (b-d). Red circles point some artifact in the reconstruction due to a toosmall support size estimation.

(a) (b) 1K

Fig. 6.20. Knot model where some points have been deleted (a), and the resulting reconstruction for a fixedbudget of 1K CSRBFs.

(a) (b) (c)

Fig. 6.21. Noisy Squirrel model (a). Reconstruction with 500 CSRBFs without any regularization (b), andfor a regularization parameter equal to 1 (c).


(a) (b)

Fig. 6.22. Reconstruction of noisy Squirrel model (see Fig. 6.21(a)) with 1 000 CSRBFs without any regu-larization (a), regularization parameter equal to 1 (b).

0

20000

40000

60000

80000

100000

120000

1e-04 0.001 0.01 0.1 1 10 100 1000

Con

ditio

n N

umbe

r

Regularization Parameter

500 Centers750 Centers

1000 Centers

Fig. 6.23. Condition number versus the regularization parameter(

εt

)2.

Conclusion

Bilan

Dans cette partie, nous avons proposé deux nouvelles méthodes de reconstruction de surfaces,où la surface résultante est une surface implicite, c’est à dire la 0-isosurface d’une fonction im-plicite, exprimée sous la forme d’une somme pondérée de CSRBF.

Plus précisément, nous avons proposé

• une nouvelle approche composite avec un budget fixé de paramètres. Tout d’abord, nouscalculons des approximations locales, composées de formes polynomiales et de CSRBF,puis nous obtenons la solution globale en faisant une somme pondérée de chacune de cesapproximations, en suivant le principe de partition de l’unité. Dans notre méthode, nouscommençons par sélectionner les centres des CSRBF, puis nous construisons chaque parti-tion. Dans chacune de ces partitions nous calculons les paramètres qui minimisent l’erreurquadratique régularisée. Nous avons testé notre approche sur différents modèles avec dif-férentes distributions d’échantillons.

• une approche adaptative avec un nombre fixé de CSRBF, dans laquelle le support de chaqueCSRBF est déterminé localement. Le support de chaque CSRBF est adapté localement à lagéométrie et ”topologie” des centres, en déterminant au préalable l’axe médian correspon-dant. Les coefficients des CSRBFs sont ensuite calculés en minimisant une fonctionnellerégularisante. Nous illustrons l’efficacité de notre méthode sur plusieurs modèles, et mon-trons l’intérêt d’utiliser de notre fonctionnelle quand les échantillons sont bruités.

Conclusion

In this part, we have proposed new methods for surface reconstruction. The resulting surfaceis expressed as the zero set of an implicit function, i.e. an implicit surface, and is constituted ofCSRBFs.

More precisely, we have presented

• a new composite approach for surface reconstruction from any unorganized point set, with afixed amount of parameters. First, we consider some local approximations made of trivari-ate quadratic polynomial form and CSRBFs, then blend them by a partition of unity method(PU). In our scheme, we start by selecting CSRBFs centers, then construct partitions, finallyparameters are calculated in order to minimize, on each sub-domain, the mean square errorwith a regularization constraint. We demonstrate the effectiveness of our approach on sev-eral models with different point distributions.

105


• a new adaptive method with a fixed amount of CSRBFs. The support size of each CSRBFis locally adapted to the geometry and ”topology” of the centers, by first extracting thecorresponding medial axis. Then coefficients are calculated by minimizing a regularizingfunctional. We demonstrate the effectiveness of our algorithm on several models. We showthe benefit of our regularization in the presence of noisy data.

Part IV

Segmentation

107

Résumé

La segmentation, qui consiste à séparer les structures d’intérêt, est une étape essentielle entraitement d’image. L’objectif principal de la segmentation est le partitionnement d’uneimage en régions, classes, homogèmes par rapport à une ou plusieurs caractéristiques.

En traitement d’image, la segmentation est souvent une étape préliminaire pour des traitementsultérieurs, tels que le suivi, la détection, la quantification. Par exemple, en imagerie médicale, lasegmentation est utilisée pour l’analyse de structure anatomique, de certains types de tissus, pouranalyser le comportement d’un organe d’intérêt, tel que le coeur. La segmentation est utilisée pourla quantification de sténose artérielle ou reinale (voir Fig. 6.24), pour la détection de tumeur (voirFig. 6.25), pour la mesure du volume de tumeurs et sa réponse aux thérapies, pour l’étude dudéveloppement du cerveau, la détection de microcalcification sur des mamogrammes, etc... Deplus, la segmentation d’image peut être aussi utilisée dans une étape préliminaire pour la visuali-sation et la compression.

Depuis les débuts de l’imagerie médicale, la segmentation est faite par les cliniciens, qui, àpartir de leur connaissance en anatomie, sélectionnent des points, dessinent des contours représen-tant les interfaces des organes à segmenter. On se rend bien compte du temps que cette opérationpeut prendre, et par conséquent que la segmentation automatique est indispensable. Cependantc’est un objectif ambitieux du fait de ses difficultés.

Le nombre de méthodes de segmentation est difficile à dénombrer, voir [Fu and Mui, 1981,Haralick and Shapiro, 1985, Pal and Pal, 1993, Zouagui, 2004], car il n’existe pas de solutionsuniverselles, efficaces pour toutes les modalités d’imagerie. La problème de la segmentation estgénéralement résolu par une méthode spécifique. Différents auteurs ont proposé des classificationsdes méthodes de segmentation, par exemple [Fu and Mui, 1981, Haralick and Shapiro, 1985, Paland Pal, 1993, Zouagui, 2004]. Trois grandes familles de méthodes apparaissent les méthodes deseuillage, les méthodes basées région, les méthodes basées contour.

Nous nous intéressons aux méthodes basés contour, et plus particulièrement aux contours act-ifs qui font évoluer une ou plusieurs interfaces afin de séparer les différentes régions.

Dans le chapitre 7, nous définissons le principe des contours actifs, qui consistent à faireévoluer une ou des interfaces à partir d’une équation aux dérivées partielles. Nous expliquonsrapidement les deux approches pour résoudre ce problème, à savoir les ”snakes” [Kass et al.,1988] et les ”level sets” [Osher and Sethian, 1988]. Les ”snakes” utilisent des représentationsexplicites de l’interface, l’évolution se fait par le déplacement de points caractéristiques: sommetsde l’interface discrète ou points de contrôle. L’approche ”level sets” utilise une représentationimplicite de l’interface. Comme nous le montrons, cette dernière approche permet des change-ments de topologie durant l’évolution, ne dépend pas d’un échantillonnage de l’interface, peut être

109

110

facilement implémenté pour n’importe quelle dimension, ne nécessite pas d’initialisation prochede la solution.

Dans le chapitre 8, contrairement aux approches classiques qui implémentent l’approche ”levelsets” par différences finies, nous décomposons la fonction implicite, dont le niveau 0 représentel’interface en mouvement, sur une base de fonctions de base radiale à support compact. Nousmontrons l’intérêt d’une telle modélisation dans le cadre de modèles numériques, ainsi que sur desimages médicales de différentes modalités.

Introduction

Segmentation is the separation of structures of interest from the background. Segmentationis an essential analysis function in image processing. The goal of the segmentation is topartition an image into regions, that are also sometimes called classes or subsets. These par-

titions should be homogeneous with respect to one or more characteristics or features of interest.

In image processing, segmentation is an initial stage for many further processings, such astracking, detection, or extracting relevant informations to be quantified. For example, in medicalimaging, image segmentation is used for analyzing anatomical structures and tissue types, for an-alyzing how behaves an organ of interest, such as the heart. The applications in medical imaginginclude quantification of arterial or renal stenosis (see Fig. 6.24), multiple sclerosis lesion quantifi-cation, tumor dection (see Fig. 6.25), measuring tumor volume and its reponse to therapy, studyingbrain development, detection of microcalcification on mamograms, etc... Segmentation can alsobe used as an initial step for visualization and compression.

Fig. 6.24. Contrast Enhanced Magnetic Resonance Angiography image of a severe (∼ 80%) renal stenosis.This image is courtesy of Leonardo Flórez Valencia and Maciej Orkisz.

Since the beginning of medical imaging, segmentation has been performed by physicians. Thisstage is known to be time consuming, thus automatic segmentation is a challenging problem.

A wide variety of segmentation techniques have been proposed in the literature, see [Fu andMui, 1981, Haralick and Shapiro, 1985, Pal and Pal, 1993, Zouagui, 2004]. Unfortunately, thisproblem can not be solved by one method for any case, so this problem is generally solved byspecific methods for one application or/and for one type of images. Some segmentation methodclassifications have been proposed [Fu and Mui, 1981, Haralick and Shapiro, 1985, Pal and Pal,1993,Zouagui, 2004]. Three kinds of methods appear in all classifications: thresholding methods,

111

112

(a) (b)

Fig. 6.25. Illustration of a lung tumor to be segmented in transversal (a) and sagital slides (b). These imagesare courtesy of Massachusetts General Hospital in Boston.

region based methods and contour based methods.

Here we decide to work with contour based methods, and more precisely with active contoursmethods, where the stress is put on interfaces which separate different regions. Active contoursmake interfaces evolve in order to separate properly every region.

Chapter 7

Active Contours

Contents7.1 Active Contours Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1137.2 Level Set Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.1 Active Contours Principle

Image segmentation, like many other problems from image processing (e.g. tracking, classifi-cation), can be cast in optimization theory framework where an energy has to be minimized. Theinitialization step is usually not trivial and many methods have been proposed to reach a minimum.In this context, active contours have been proposed to solve these problems using both variationalapproaches orand partial differential equation (PDE).

(a) step 0 (b) step 40 (c) step 90 (d) step 150

Fig. 7.1. Example of segmentation using active contour.

Fig. 7.1 illustrates active contours principle used for segmentation purpose. In this simpleexample, the object to be segmented (white region) has two holes and blurred contours. Froma particular initialization, here including the whole object, the interface automatically evolves inorder to properly detect the shape of the object of interest.

Let us consider a family of evolving curves defined parametricaly by Γ(p, t) : [0, 1]× [0, t]→R2 where p parameterizes the geometry and t parameterizes the family of evolving curves. Thegeneral evolution of the contour is generally given by the following PDE:

113

114 CHAPTER 7. ACTIVE CONTOURS

∂Γ(p, t)∂t

= 〈V,n〉 (7.1)

Γ(t = 0) = Γ0 (7.2)

where n is the unit normal of the interface Γ and V is the velocity vector1.

There are two approaches in active contours framework to represent the evolving interface Γ:

Lagrangian representation: the interface Γ is explicitly represented. The evolution of the inter-face is performed by modifying, according to Eq. 7.1-7.2, the location of some characteristicpoints, e.g. vertices of a discretized interface, control points such as in a b-splines parametricrepresentation of the continuous interface [Precioso et al., 2005]. The most popular methodusing such representation is snake [Kass et al., 1988].

As we saw in part III, explicit representations, i.e. mesh and parametric interface, topolog-ical modifications during the evolution are difficult to perform. Thus, the initial interfaceshould have the same topological properties as the solution. On the other hand, snakes aredifficult to implement for higher dimensions than 2, and important problems about inter-face sampling come out for large deformations. For these reasons, the initialization is reallyimportant, and authors generally consider an initialization close to the solution.

Implicit representation: this method is generally referred as the level set method. The interfaceis represented as the zero set of an implicit function. The main advantage of this methodis the independence of the interface sampling, the possibility to change the topology dur-ing the interface evolution, and the simplicity to describe the evolution for any dimension.However the main inconvenient of such methods is that it is almost impossible to constrainthe topology of the solution.

An interesting comparison between these two methods can be found in [Delingette and Mon-tagnat, 2001], [Montagnat et al., 2001].

7.2 Level Set Formulation

The level set-based segmentation consists in capturing the shape to be recovered by propa-gating an interface which evolves according to the solution of a PDE, which could be directlyspecified by the user (with using some edge detection related velocity term) or could derive froman energy functional.

In the case where the PDE comes from an energy functional, this energy criterion is designedin such a way that its minimum corresponds to the solution of the given problem. The energyfunctional is then minimized using variational calculus techniques [Tsai and Osher, 2003, Aubertet al., 2003] and gradient descent method to get a PDE governing the motion of the interface.

In the level set formalism, the interface Γ inRd is represented as the zero level set of a Lipschitzcontinuous function f of dimension d+ 1, satisfying:

f(p, t) > 0, for p ∈ Ωin(t), (7.3)

f(p, t) < 0, for p ∈ Ωout(t), (7.4)

f(p, t) = 0, for p ∈ ∂Ωin(t) = Γ(t) . (7.5)1Generally authors only write a velocity in the normal direction, i.e. Vn. It is obvious that these two expressions

are strictly equivalent.

7.2. LEVEL SET FORMULATION 115

where Ω is an open region in Rd+1, Ωin is a region in Ω bounded by Γ. Ωout is defined asΩout = Ω \ Ωin (see Fig. 7.2).

(a)

Ωout

Ω

Ωin

in

in

Ω

(b)

Fig. 7.2. Level set function for a two dimensional domain Ω represented as an elevation map S , i.e. z =f(p) (a). The interface Γ corresponds to the zero set of the level set function, i.e. the intersection ofS and the horizontal plane z = 0. Corresponding regions Ωin and Ωout are represented on (b).

For brevity sake, in the following we consider the classical problem of segmenting one ob-ject (possibly having several non connected components) from the background2. This problem istypically handled by the evolution of one level set whose steady state partitions the image intotwo regions delimiting the boundaries of the object to be segmented. In this framework, a gen-eral expression of the energy functional which are generally used for driving the level set can beformulated as [Zhao et al., 1996, Jehan-Besson et al., 2003]:

J(f) = Jb(f) + Jr(f) (7.6)

where

• the first term Jb(f) is an energy criterion attached to the interface Γ (often referred to ascontour term). Jehan [Jehan-Besson et al., 2003] expresses this term as follows

Jb(f) =∫

Γkb(p)dA, (7.7)

where kb is a border descriptor and dA is an area element. Following the same idea, wepropose here an alternative general expression for the contour term:

Jb(Γ) =∫

Ωh (p, f (p)) · δ (f (p)) ‖∇f (p)‖ dp (7.8)

where h is a border descriptor function (it depends on the zero level set of f ), δ is theunivariate dirac function.

• The second Jr(Γ) (often referred to as region term) is an energy criterion related to theinside and outside regions delimited by the interface Γ. Jehan [Jehan-Besson et al., 2003]gives the following expression for this term:

Jr(Γ) =∫

Ωin

kin (p,Ωin) dp +∫

Ωout

kout (p,Ωout) dp, (7.9)

2The interested reader will find in [Zhao et al., 1996,Chan and Vese, 2001] approaches which extend this method toseveral regions.


where kin and kout are region descriptors for region inside and outside the delimited by Γ,respectively. This criterion can also be rewritten as follows:

Jr(f) = µ1 ·∫

Ωk (p, f) · u (f) dp + µ2 ·

∫

Ωg (p, f) · (1− u (f)) dp, (7.10)

where µ1 and µ2 are some positive hyper-parameters, u is the univariate Heaviside function,and k and g are the region descriptor functions for region inside and outside the delimitedinterface Γ, respectively.

Using variational calculus [Tsai and Osher, 2003, Chan and Vese, 2001], the minimization ofEq. 7.6 leads to the following general evolution equation:

∂f(p, t)∂t

+ V (p, t) · δε (f(p, t)) = 0 (7.11)

where δε is a regularized version of the univariate Dirac function [Chan and Vese, 2001] given as

δε(x) =1

πε ·(

1 +(xε

)2) (7.12)

where ε is a real positive constant and V is a velocity function which is derived from the variationalscheme.

Depending on the specific application, V can be given by the user, or can derive from an en-ergy functional to be minimized. V can be a function of the position p, of geometrical propertiesof the interface and of images properties reflecting the object to be segmented (can be related toan edge detection preprocessing of the image).

Some authors modified the evolution equation (Eq. 7.11) by substituing δε by ‖∇f‖. Thisoperation does not affect the steady state solution and remove stiffness near the zero level set[Zhao et al., 1996]. Moreover, the equation becomes independent of the scaling of the level setfunction used and the problem becomes morphological. For more details refer to [Tsai and Osher,2003, Marquina and Osher, 2000, Alvarez et al., 1993]. The evolution equation is then:

∂f(p, t)∂t

+ V (p, t) · ‖∇f(p, t)‖ = 0 (7.13)

Some authors also prefer using a propagation using a velocity vector field (Eq. 7.14) [Mark-stein, 1964, Williams, 1985]. Such propagation is generally referred as advection propagation.

∂f(p, t)∂t

+ 〈V(p, t),∇f(p, t)〉 = 0 (7.14)

However it is important to notice that this expression is slightly equivalent to the previous one,indeed tangential velocity components vanish when plugged into Eq. 7.14. For examples, in twodimensions with V = Vn · n + Vt · t, the previous level set equation becomes

∂f(p, t)∂t

+ 〈Vn · n + Vt · t,∇f(p, t)〉 = 0 (7.15)

is equivalent to

∂f(p, t)∂t

+ Vn · ‖∇f(p, t)‖ = 0 (7.16)


since 〈t,∇f〉 = 0. Furthermore, since 〈n,∇f〉 = ‖∇f‖, we can rewrite Eq. 7.14 as Eq. 7.13 byconsidering that V(p, t) is the velocity along the normal direction.

These evolution equations (Eq. 7.11 - 7.14) are commonly implemented using finite differencemethods. These numerical schemes have been developed to obtain accurate and unique solutions,and involve upwind differencing, essentially non oscillatory schemes borrowed from the numeri-cal solutions of conservation laws and Hamilton Jacobi equations.

In traditional level set methods [Caselles et al., 1997, Malladi et al., 1995, Osher and Fed-kiw, 2002], the level set function can develop flat or steep regions leading to difficulties in bothnumerical approximation of the derivatives and speed of convergence. In order to overcome thisdifficulty, the following scheme is generally used:

• initialize the level set as the distance function (relative to the interface). This choice is gen-erally motivated by the interesting numerical properties of such function (e.g. ‖∇f(p)‖ =1, ∀ p ∈ Ω);

• reshape the level set function periodically in order to ressurect this distance function prop-erty.

Indeed, Mulder et al. showed in [Mulder et al., 1992] that initializing f to a distance func-tion and keeping its corresponding properties during the evolution process leads to more accuratenumerical solutions than initializing f to a Heaviside function.

The most straightforward way of implementing the re-intialization operation is to extract theinterface f = 0 and then explicitely compute the distance function from it. However, this methodis generally slow and time consuming. In this perspective, a now widely accepted method hasbeen proposed [Sussman and Fatemi, 1999, Peng et al., 1999] in order to re-initialize the level setfunction by solving the following PDE:

∂f(p, t)∂t

= sign (f0(p, t)) · (1− ‖∇f(p, t)‖) (7.17)

where f0 is the function to be re-intialized and sign (·) is the sign function.However, if f0 is not smooth or much steeper on one side of the interface than the other [Osher

and Fedkiw, 2002], the zero level set of the resulting function f can be moved away from that ofthe original function. For this reason, Li et al. recently proposed in [Li et al., 2005] to add a newenergy term to the general criterion (7.6):

Jreg(f) =∫

Ω

12

(‖∇f(p, t)‖ − 1)2 dp (7.18)

This expression corresponds to a regularization term that penalizes the deviation of the levelset function from a signed distance function. This method has the main advantage to keep the levelset function as a signed distance function without the need of the re-initialization.

Finally, it has to be noted that in order to reduce the computational cost, the level set methodis generally not computed on the whole image domain, but in a narrow band around the interface.Unfortunately this implementation aspect makes the method more sensitive to the initialization.

The basic properties of the level sets are illustrated in Fig. 7.3, which shows the evolution ofthe embedding function f and its zero level set Γ, segmenting a synthetic image. Topology changesare handled naturally, since the splitting or merging of the zero level set is handled naturally by


the evolution of the embedding function. In this particular example, f is initialized as a signeddistance function and is periodically reinitialized. The evolution described in Eq. 7.13 is appliedon the whole image domain i.e. on the fixed rectangular coordinate system of the image I (nooptimization is made such as the narrow-banding).


(a) (b)

(c) (d)

(e) (f)

(g) (h)

Fig. 7.3. Evolution of the implicit function f (b,d,f,h), along with the evolution of its zero level set (red) whilesegmenting a synthetic image (a,c,e,g). The black curve on (a,c,e,g) corresponds to the projection ofthe zero level on the synthetic image.

Chapter 8

Our Contribution

Contents8.1 Implicit Function Decomposition in the Level Set Framework . . . . . . . . 122

8.2 Level Sets with CSRBF Collocation . . . . . . . . . . . . . . . . . . . . . . 122

8.2.1 Unified representation . . . . . . . . . . . . . . . . . . . . . . . . . . 122

8.2.2 Solving Level Set With RBF Collocation . . . . . . . . . . . . . . . . 123

8.2.3 Compactly Supported Radial Basis Functions . . . . . . . . . . . . . . 124

8.2.4 RBF Centers Distribution and Velocity Sampling . . . . . . . . . . . . 124

8.2.5 Resolution of the ODEs . . . . . . . . . . . . . . . . . . . . . . . . . 125

8.2.6 Bounded Implicit Function . . . . . . . . . . . . . . . . . . . . . . . . 126

8.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

8.3.1 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 128

8.3.2 Experimental Medical Data . . . . . . . . . . . . . . . . . . . . . . . 132

8.3.3 Computation Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

As we saw in the previous chapter, level set methods are implemented using finite differencemethods. These numerical schemes have been developed to obtain accurate and unique solution,and involve upwind differencing, essentially non oscillatory schemes borrowed from the numeri-cal solutions of conservation laws and Hamilton Jacobi equations. Such an approach is howevertime consuming. In order to speed up the process, some authors consider the level set equationlocally, i.e. on a narrow band that embeds the zero level of the implicit function. The general levelset evolution equation is then applied on a limited set of points (corresponding to pixels) whosenumber varies with the evolution of the narrow band.

We propose here to expand the PDE solution at (collocation) points where the solution is con-sidered space and time separable with some time invariant basis. Thus, in this chapter, we proposean alternative technique to the usual finite difference framework, which consists in changing thelevel set PDE into coefficients ordinary differential equation. Finally, we propose one collocationmethod using a compactly supported radial basis functions.

121

122 CHAPTER 8. OUR CONTRIBUTION : LEVEL SET WITH CSRBF

8.1 Implicit Function Decomposition in the Level Set Framework

The most common approach to express a searched function f is to assume that it is a linearcombination of some basis functions ui, such that the space spanned by these basis functions is anHaar space.

By assuming that space and time are separable, i.e. the time dependence of f is only due tothe expansion coefficients. In such case, this naturally leads to the following expansion:

f(p, t) =N−1∑

i=0

ci(t) · ui(p) (8.1)

By substituing Eq. 8.1 into Eq. 7.11 yields

N−1∑

i=0

ui(p) · dcidt

+ V (p, t) · δε(

N−1∑

i=0

ci(t) · ui(p)

)= 0 (8.2)

or into Eq. 7.13

N−1∑

i=0

ui(p) · dcidt

+ V (p, t) ·∥∥∥∥∥

N−1∑

i=0

ci(t) ·∇ui(p)

∥∥∥∥∥ = 0 (8.3)

or into Eq. 7.14

N−1∑

i=0

ui(p) · dcidt

+

⟨V(p, t),

N−1∑

i=0

ci(t) ·∇ui(p)

⟩= 0 (8.4)

These equations can thus be applied to every (collocation) point p of the domain Ω. The levelset, a PDE based evolution formulation, is now cast into an Ordinary Differential Equations (ODE).

Here, we decide to consider an implicit function f expressed in terms of radial basis functions,and the evolution equation provided by Eq. 7.11.

8.2 Solving The Level-Sets Evolution Equation using CSRBF collo-cation

In order to define the most general framework for level set evolution and radial basis functions[Gelas et al., 2007a], we first propose an unified representation for the evaluation of the implicitfunction f and to compute the coefficients α, like in the case of the interpolation. Then, we presenthow to solve the level sets evolution using RBF collocation.

8.2.1 Unified representation

From Eq. 2.4 and Eq. 2.7, a unified representation for the function f may be obtained forSPD- or CPD-RBF. The evaluation of f for any point p is expressed as the product of the one-linevector Φ(p) and a column vector µ:

f(p) = Φ(p) · µ, (8.5)

and the associated system linking data s and the expansion coefficients is noted as

8.2. LEVEL SETS WITH CSRBF COLLOCATION 123

H · µ = q (8.6)

In the positive definite case, we have:

Φ(p) =[φ (‖p− p0‖) , . . . ,φ

(∥∥p− pN−1

∥∥)],

µ = α,

H = A,

q = s.

In the conditionally positive definite case, we have:

Φ(p) =[φ (‖p− p0‖) , . . . ,φ

(∥∥p− pN−1

∥∥), g0(p), . . . , gQ−1(p)

],

µ =[

απ

],

H =[A GT

G 0

],

q =[

s0

].

8.2.2 Solving Level Set With RBF Collocation

In such case, this naturally leads to the following decomposition:

f(p, t) = Φ(p) · µ(t) (8.7)

and to the following evolution equations by substituing Eq. 8.7 into Eq. 7.11 yields

Φ(p) · dµ(t)dt

+ V (p, t) · δε (Φ(p) · µ(t)) = 0, (8.8)

or into Eq. 7.13 yields

Φ(p) · dµ(t)dt

+ V (p, t) · ‖∇Φ(p) · µ(t)‖ = 0, (8.9)

or into Eq. 7.14 yields

Φ(p) · dµ(t)dt

+ 〈V(p, t),∇Φ(p) · µ(t)〉 = 0. (8.10)

where depending on the type of RBF used

∇Φ(p)

[∇φ (‖p− p0‖) , . . . ,∇φ(∥∥p− pN−1

∥∥)], for SPD RBF, (8.11)[∇φ (‖p− p0‖) , . . . ,∇φ

(∥∥p− pN−1

∥∥),

∇g0(p), . . . ,∇gQ−1(p)] , for CPD RBF. (8.12)

Equations (8.8)-(8.10) apply to every point p of the domain Ω. In order to solve for thecoefficients µ, these equations have to be sampled at N distinct locations, traditionally calledcollocation points. In the framework of RBF formulation, the collocation points are chosen to be


the RBF centers, i.e. the pi, which is generally referred as unsymmetric collocation. This approachmay thus be summarized through the following unique equation:

H · dµ(t)dt

= B(µ(t), t), (8.13)

where H is the interpolation matrix defined in the previous section, and B(µ) is a column vectorrelated to the level set formalism used in (8.8), i.e.

B(µ(t), t) =

V (pi, t) · δε (Φ (pi) · µ (t)) , (8.14)

V (pi, t) · ‖∇Φ(pi) · µ(t))‖, (8.15)

〈V(pi, t),∇Φ(pi) · µ(t)〉 , respectively. (8.16)

As mentionned before, from Eq. 8.13, the level set evolution is now cast into a Ordinary Dif-ferential Equations (ODE).

In the next section we give implementation issues for solving these ODEs with a low compu-tational complexity and obtaining an efficient evolution of the level set.

8.2.3 Compactly Supported Radial Basis Functions

From Eq. 8.13, the evolution of the level set relies on matrix H , and the implicit functionevaluation on vector Φ(p), which in turn depends on the choice of the underlying RBF. In orderto obtain a sparse interpolation matrix, and the possibility of a fast evaluation of the reconstructedfunction f , we decide to use compactly supported radial basis functions (see section 2.6), with akd-tree data structure for centers in order to perform efficient evaluation of the implicit function(see section 2.6.2.1).

The use of CSRBF implies the choice of the support size σ that we decide to connect to the filldistance1 hP,Ω (Eq. 3.7), through the following expression

σ = ρ · hP,Ω (8.17)

where ρ > 1.

8.2.4 RBF Centers Distribution and Velocity Sampling

As mentioned before, the RBF-based methods do not require any underlying mesh, or grid,i.e. we can use any centers spatial distribution adapted to the target application. When no a prioriinformation about the shape to be segmented is available, the RBF centers may thus be simplylocated on a regular grid (see Fig. 8.1). Conversely, if some knowledge about this shape is athand, an interesting perspective consists in incorporating this information through an adapted cen-ters distribution. In most examples given in the next section, no a priori known shape is assumedand regular grids are therefore used for RBF centers, while the interest of using an adapted, nonregular, center placement is illustrated on a particular case.

Once the centers have been placed, the collocation approach implies the evaluation of theapplication dependent velocity term V (pi) in Eq. (8.14)-(8.16) at each RBF center pi. This isstraightforward if the velocity is defined for any point in the computational domain Ω, as it is the

1The fill distance can be interpreted as the radius of the largest ball which is completely contained in Ω and whichdoes not contain any data site pi.

8.2. LEVEL SETS WITH CSRBF COLLOCATION 125

(a) (b)

Fig. 8.1. RBF Center located on regular grid, (a) rectangular one, (b) triangular one.

case when the velocity term relies on region-based features [Chan and Vese, 2001,Zhu and Yuille,1996]. This may be an issue when the targeted segmentation application yields velocity terms thatare defined only on the interface, i.e. depend on boundary-based features. As for conventionallevel set implementation, this issue may nevertheless be easily addressed using the velocity exten-sion methods developed for conventional level set implementation [Tsai and Osher, 2003].

Conventional RBF collocation methods [Kansa, 1990a, Kansa, 1990b, Fasshauer, 2006] usepoint wise sampling of V (pi) at RBF centers, which is suitable when the underlying data are givenwith a very good accuracy as in the case of theoretical PDE resolution [Kansa, 1990a, Kansa,1990b]. Experimental images may however be corrupted by noise, making this point wise sam-pling inappropriate, since the evolution of the interface would thus be driven by unreliable velocityterms. In such a case, the velocity is computed as the weighted average of the neighbooring veloc-ities.

8.2.5 Resolution of the ODEs

The resolution of the ODEs obtained through collocation requires the definition of an initialimplicit function f0, i.e. the initial RBF expansion coefficients α0. In practice, the initializationmay be provided as an interface, available either from a priori knowledge about the object to besegmented or from user interaction. In such case, f0 may be easily built as an implicit functionwhose zero set interpolates or approximates the given interface (see section 5-6). In the case wherethe initialization is given itself as an implicit function (see [Huang et al., 2006, Tsai et al., 2003]),Eq. 2.5 may be directly applied to obtain f0.

Many numerical schemes may be then applied to solve the ODEs in Eq. 8.13, Euler’s andRunge-Kutta being the most traditional. Our current implementation uses a simple first orderforward Euler’s method. As shown in the results section, this scheme yields accurate segmentationresults and provides a fast evolution of the interface. Applying Euler’s method to Eq. 8.13 yields:

αn+1 = αn − τ ·H−1 ·Bn(αn), (8.18)

where τ is the step size.

A straightforward approach for implementing the evolution equation (Eq. 8.18) would consistin computing H−1 as an initial stage and then performing the propagation by computing the prod-uct H−1 ·Bn(αn) at each iteration. This approach is inefficient since H−1 is not sparse and thus


requires an important space in memory.

Fortunately, Wendland’s CSRBF are positive definite functions, and, as a consequence, theassociated matrix H is also positive definite. Since H is moreover symmetric, we may use aLDLT or a Cholesky decomposition. In the current implementation, we use a sparse Choleskydecomposition from MUMPS library [Amestoy et al., 2006], i.e. H = L · LT where L is a lowertriangular matrix. The Cholesky decomposition (i.e. the matrix L) is computed as an initial stage,and for each iteration we solve the following triangular systems:

L · un = Bn(αn)

LT · vn = un.

The RBF coefficients are finally given as:

αn+1 = αn − τ · vn (8.19)

8.2.6 Bounded Implicit Function

As mentioned in section 7.2, periodically reshaping the level as the interface signed distancefunction is a common strategy used for avoiding developing of steep regions in the implicit func-tion near the zero level. This scheme increases the computational cost and reduces the topologicalflexibility of the method since it prevents the level set from creating new zero level componentsfar away from the initial interface.

A similar problem appears in our collocation method: the implicit interface reaches a stablesolution, however RBF coefficients can go on increasing. This may be shown as follows:

Proposition 8.1. In Eq. 8.18 ‖α‖1 can increase slowly at each step, depending on τ .

Proof. By computing ‖αn+1‖1, it follows

‖αn+1‖1 6 ‖αn‖1 + τ · ‖H−1‖ · ‖Bn(αn)‖1 (8.20)

Depending on the choice of the used approach (Eq. 7.11 - 7.14), the higher bound of the l1-normof is

‖Bn(αn)‖1 6

maxpi∈P

V (pi) ·maxx∈R

δε(x) (8.21)

maxpi∈P

V (pi) ·maxp∈Ω

‖∇f(p)‖2 (8.22)

maxpi∈P

‖V(pi)‖ ·maxp∈Ω

‖∇f(p)‖2 (8.23)

where maxx∈R

δε(x) = 1πε and ‖∇f(p)‖ 6 K ′ · ‖α‖1, where K ′ = max

r∈[0,∞[|φ′(r)|.

Thus, it follows that:‖αn+1‖ 6 ‖αn‖+ γ, (8.24)

where γ = τπ·ε · ‖H−1‖ ·max

pi∈PV (pi) > 0. By choosing a low τ value, at least a slow increasing of

‖αn‖1 can be obtained.

8.3. RESULTS 127

We propose here to take advantage of the linearity of the RBF expansion (Eq. 2.4) in order toavoid the steep gradient problem and to preserve the topological flexibility at a low computationalcost. In order to bound the implicit function f (hence the norm of gradient ‖∇f‖), while preserv-ing the global shape of the implicit function and the implicit interface, we bound the expansioncoefficients.

As a starting point, we note that the multiplication of an implicit function f by a non-null coef-ficient k does not change its associated interface2. Since f is represented through the coefficientsα of a RBF expansion, k · f simply corresponds to k ·α.

For SPD-RBF, especially Wendland’s ones (see Table 2.3), this provides us an easy way tobound the values of the implicit function. Indeed, due to the linearity of the SPD-RBF expansion,we may link this constraint to the L1-norm of α. Formally we have:

|f(p)| =

∣∣∣∣∣N−1∑

i=0

αi · φ (‖p− pi‖)∣∣∣∣∣ 6

N−1∑

i=0

|αi| · |φ (‖p− pi‖)| 6 K · ‖α‖1,

‖∇f(p)‖2 =

∥∥∥∥∥N−1∑

i=0

αi ·∇φ (‖p− pi‖)∥∥∥∥∥

2

6N−1∑

i=0

|αi| · ‖∇φ (‖p− pi‖)‖2 6 K ′ · ‖α‖1

where K = maxr∈[0,∞[

|φ(r)| = φ(0), and K ′ = maxr∈[0,∞[

|φ′(r)|, and N is the number of RBF

functions.In our implementation, for d = 2, 3 we use C2 Wendland’s function (see Table 2.3 and Fig.

2.3(b)), where K and K ′ are equal to 1 and 13564 , respectively.

In order to bound the implicit function (hence its gradient norm), we apply a normalization onRBF coefficients which bounds ‖αn‖1. The evolution equation becomes

L · un = Bn(αn) (8.25)

LT · vn = un (8.26)

αn+1 = αn − τ · vn (8.27)

αn+1 =β

‖αn+1‖1 ·αn+1 (8.28)

where β is a positive constant. It is important to notice that the computational cost of this normal-ization is in O(N).

In our implementation, β is set to σ, so the values of f are in the interval [−σ,+σ].

8.3 Results

In this section we evaluate the proposed approach using simulated and medical images. In allexperiments we use the general evolution equation given in Eq. 7.11 and consequently apply theRBF collocation framework given in Eq. 8.8, Eq. 8.14 and Eq. 8.25 through Eq. 8.28.

Such an approach implies the choice of the support ε of the regularized Dirac measure δε(x)defined in Eq. 7.12. The fact that f is bounded through Eq. 8.28 allows an easy selection of ε.

2We avoid k = 0 since the zero set of f would be the whole domain Ω.


As noted by Chan [Chan and Vese, 2001], this parameter has indeed to be large enough so theevolution equation acts on all level curves and yields a global minimizer. Since f is now boundedin the interval [−σ,+σ] from Eq. 8.25, we thus simply set ε to 2σ in all experiments.

The use of CSRBF collocation implies a choice of the RBF support size σ. As mentioned insection 8.2.3, this parameter is related to the fill distance hP,Ω as σ = ρ · hP,Ω. Unless otherwisementionned we set ρ = 4 in our experiments, in order to ensure continuity of the reconstructedimplicit function. A larger support would provide a smoother implicit function at the cost of ahigher computational cost.

In Eq. 8.18, we use a forward Euler approach for time discretization, which implies the choiceof the time step value τ . Experimentaly, we observe that for τ values varying in the range [0.1, 10],the result of the segmentation remains unchanged. This result shows the robustness of the proposedmethod in the choice of the time step. Thus, we arbitrarily fix a value of τ to one for each experi-ment.

Finally, we provide for each experiment the cpu time corresponding to each step of the pro-posed algorithm. Table 8.1 and Table 8.2 summarize the obtained results. The calculations wereperformed on a 3.6 GHz Pentium IV with 2 Go of RAM.

8.3.1 Numerical Simulations

The segmentation examples given in this part are based on the Chan-Vese functional [Chan andVese, 2001], which aims at partitioning the image into regions with piecewise constant intensity.This approach corresponds to a particular case of the Mumford-Shah functional [Mumford andShah, 1989], known as the minimal partition problem. This functional is given as:

JMum(C1(f), C2(f), f) = λ1 ·∫

Ωδ (f) ‖∇f‖ dp + λ2 ·

∫

Ω(I(p)− C1(f))2 · u (f) dp

+ λ3 ·∫

Ω(I(p)− C2(f))2 · (1− u (f)) dp

where λ1, λ2 and λ3 are hyperparameters, C1(f) and C2(f) are constant calculated below and Iis the image to be segmented.

Due to the intrinsic smoothness of the RBF formulation, the smoothness term of the functionalis not used (we set λ1 = 0). The two remaining hyperparameters (λ2 and λ3) are set to one. C1(f)and C2(f) are computed at each iteration using the following expressions:

C1(f) =

∫Ω I(p) · u (f(p, t)) dp∫

Ω u (f(p, t)) dp(8.29)

C2(f) =

∫Ω I(p) · (1− u (f(p, t))) dp∫

Ω (1− u (f(p, t))) dp(8.30)

In this framework, the velocity term used in our examples is then given as

V (p, t) = − (I(p)− C1(f))2 + (I(p)− C2(f))2 (8.31)

8.3. RESULTS 129

(a) Without noise (b) SNR = 30 dB (c) SNR = 20 dB

(d) Result without noise (e) result SNR = 30 dB (f) result SNR = 20 dB

Fig. 8.2. Segmentation of an image containing a shape with two holes and blurred contours. The method isapplied for various additive noise levels. 100× 100 RBF have been used in the experiment. Top row:original images and initial interfaces. Bottom row: corresponding segmentation results.


8.3.1.1 Simulation 1

Fig. 8.2 illustrates the application of the method to an image containing a shape with two holesand blurred contours. The method has been applied for various additive noise levels, as shown onFig. 8.2(a) through Fig. 8.2(c). For this simulation no a priori knowledge about the shape tosegment is considered and the RBF centers are thus positioned on a regular rectangular grid with100× 100 nodes.

Fig. 8.2(d) shows the result obtained from the image without noise. The model properly detectsthe shape and automatically handles the required topology changes. Fig. 8.2(e) and Fig. 8.2(f)shows that in the presence of additional noise (corresponding to a SNR value of 30 dB and 20 dB,respectively) the model still provides a correct segmentation. However, we note that the detectedinterface is not as smooth as the original shape. As previously mentionned, the smoothness of theinterface may be adjusted by selecting an appropriate support size σ. This is demonstrated in Fig.8.3, which shows the results obtained with larger supports in the case of a 20 dB SNR.

(a) (b) (c)

Fig. 8.3. Segmentation of the image with a 20 dB SNR noise level given in Fig. 8.2(c) for increasing CSRBFsupport size. (a) σ = 4 · hP,Ω, (b) σ = 8 · hP,Ω. (c) σ = 16 · hP,Ω.

(a) without normalization (b) with normalization

Fig. 8.4. Representation of the implicit function f after 100 iterations, (a) without normalization, (b) withnormalization.

The influence of the normalization procedure described in section 8.2.6 is studied from the

8.3. RESULTS 131

results corresponding to Fig. 8.2(b). Fig. 8.4 shows the level set after 100 iterations when nor-malisation is not applied and illustrates how steep gradients results. As previously mentioned, thiscould be accounted for by the conventional reinitialisation of the level set to a signed distancefunction. Fig. 8.4(a) presents the obtained level set when normalisation is applied: the values ofthe implicit function f are bounded in the range [−σ,+σ], thus preventing high gradients to showup.

The influence of the normalisation may be further investigated by studying the variation of theRBF coefficient versus the iteration step number. We measure here this variation by computingthe L∞-norm of the difference of the RBF coefficients between consecutive iterations. Withoutnormalisation Fig. 8.5(a) shows that the variation of the RBF coefficients reaches a constant,non-zero value after 15 iterations, which corresponds to the convergence of the energy criterionJMum (see Fig. 8.5(b)). This illustrates the predicted, and well known, phenomenon of the section8.2.6: the implicit function keeps on evolving, even though the interface has reached the desiredsolution, i.e. convergence may be evaluated only locally. Fig. 8.5(c) gives the variation of theRBF coefficients when normalization is applied. In this case, the variation tends to stabilize tozero. This indicates that the level set globally converges to a stable solution. This observationsuggests a simple way to check whether the implicit interface reached the solution, by lookingat the variation of RBF coefficients. This interesting result remains however to be theoreticallydemonstrated in a general setting.

0 5 10 15 20 25

0.8

1

1.2

1.4

1.6

Number of steps

Var

iati

on

of

RB

F c

oef

fici

ent

(a)

0 5 10 15 20 252.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8x 10

6

Number of steps

En

erg

y fu

nct

ion

al J

Mu

m

(b)

0 5 10 15 20 250

1

2

3

4

5x 10

−3

Number of steps

Var

iatio

n of

RB

F c

oeffi

cien

t

(c)

0 5 10 15 20 252.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8x 10

6

Number of steps

En

erg

y fu

nct

ion

J M

um

(d)

Fig. 8.5. Variation of RBF Coefficients versus the iteration step number without normalization (a), withnormalization (c). Energy JMum versus the iteration step number without normalization (b), withnormalization (d).


8.3.1.2 Simulation 2

The second simulation addresses the influence of the number of CSRBF centers on the results,using an image having high local curvature values (see Fig. 8.6).

(a) 20× 20 RBF (b) 40× 40 RBF (c) 150× 150 RBF

Fig. 8.6. Segmentation of a shape containing high curvatures for increasing number of CSRBFs. The CS-RBF centers are located on a regular rectangular grid.

Segmentation is first performed using a regular rectangular grid applied on the whole domainwith varying number of centers (400, 1.600 and 22.500). The results presented on Fig. 8.6 showhow the accuracy of the obtained segmentation increases with the number of centers involved.

As previously mentioned, the spatial distribution of the RBF centers may be arbitrary (i.e. theyneed not to lie on a regular grid) and this feature may be used to incorporate a priori knowledge onshape. This is illustrated in Fig. 8.7, where the RBF centers are distributed according to the shapeto be recovered. In this simple example, the density of RBF centers is increased near the boundaryto be detected. Fig. 8.7(a), 8.7(b) and Fig. 8.7(d), 8.7(e) show the obtained grid for 5.000 and10.000 centers, respectively. As in the previous example, the segmented results presented in Fig.8.7(c) and Fig. 8.7(f) show that the number of centers directly influences the accuracy of the result.However comparison of Fig. 8.6(c) and Fig. 8.7(f) indicates that the adapted centers distributionyields qualitatively the same accuracy using less centers. Such a strategy could obviously beimproved by designing more general shape-based centers distribution.

8.3.2 Experimental Medical Data

8.3.2.1 Calcaneus bone in 3D

The proposed segmentation approach has been applied to 3D CT images of calcaneus bone.The data have been obtained using synchrotron radiation, with a voxel size 80 µm3. Due to itscomplex topology, calcaneus bone is an attractive example for testing a level set approach. As inthe simulation section, Chan-Vese functional [Chan and Vese, 2001] and the associated velocityterm (Eq. 8.31) was used. Because of the complexity of the shape to be recovered, the RBF centerswere distributed on a regular rectangular grid. Fig. 8.8 provides a 3D visualization of the obtainedresults as well as two corresponding image slices. These results show the ability of the model tohandle complex topology.

An attractive feature of our approach lies in the fact that it provides a continuous representationof the implicit function embedding the interface. As mentioned in section 4.1.2, this feature pro-vides analytical access to the differential properties of the interface, such as Gaussian and Meancurvatures. This is illustrated in Fig. 8.9, which presents the distribution of these quantities overthe surface of the segmented calcaneus bone.

8.3. RESULTS 133

(a) 5.000 (b) 5.000 (c) 5.000

(d) 10.000 (e) 10.000 (f) 10.000

Fig. 8.7. Segmentation of a shape containing high curvatures using irregular CSRBF centers placement.Top row: results for 5.000 RBF (a) Centers distribution along with the interface (b) Close-up ofthe centers distribution (c) Resulting segmentation. Bottom row: results for 10.000 RBF (d) Centersdistribution along with the interface (e) Close-up of the centers distribution (f) Resulting segmentation.

(a) (b) (c)

Fig. 8.8. Segmentation of 3D CT images of calcaneus bone. (a) 3D rendering of the resulting segmentation,(b) and (c) Two slices through the original data volume.


(a) (b)

Fig. 8.9. Distribution of the Gaussian (a) and mean (b) curvature over the surface of the segmented calca-neus bone.

8.3.2.2 Ultrasound images

In this section we apply our approach to echocardiographic ultrasound images. Segmentationof echocardiographic images is a difficult task, due to the specificities of the ultrasound acquisitionwhich yields speckle and blurred boundaries related to diffuse scattering and attenuation.

In this context, we use the framework initially described by [Zhu and Yuille, 1996]3 which re-lies on evolving the interface in such a way that the resulting segmentation maximizes a posterioriprobability of distribution of the image intensity in the inside and outside region. The velocity termmay then be derived by following the approach we described in [Bernard et al., 2006a, Bernardet al., 2006b], which shows that the statistics of the ultrasound signal corresponding to blood andmyocardial regions may be described by Generalized Gaussian distributions. The velocity term isthen given as

V (p, t) = − log

(m∏

i=1

(P (Ii/ξin, ϑin)P (Ii/ξout, ϑout)

)1/m)

(8.32)

where P corresponds to the Generalized Gaussian distribution, ξ and ϑ are the two GeneralizedGaussian parameters computed inside and outside the moving interface at each iteration, m is thesize of a circular window centered at p and Ii is the intensity of the treated image at pixel locationi.

For this test, the RBF centers are distributed on a regular rectangular grid with 600 nodes. Fig.8.10 provides the result obtained from a parasternal long axis view with a narrow angle focusedon the inferolateral wall (see 8.11). This type of image is frequently used in clinical routine inorder to accurately locate and track a region of the myocardium during the cardiac cycle. Froman initialisation located inside the left ventricle, our model yields proper segmentation of all theblood/tissue interfaces in the image. This result shows the ability of the approach to deal withnoisy images using a statistical region-based segmentation.

3Note that in [Herbulot et al., 2006] the velocity is related to the entropy.

8.3. RESULTS 135

(a) initialization (b) during evolution (c) final result

Fig. 8.10. Segmentation of echocardiographic data corresponding to a parasternal long axis view of theheart. (a) Original image along with the initial interface, (b) intermediate state of the interfaceduring evolution of the level set, (c) final segmentation result.

Fig. 8.11. Segmentation of an echocardiographic image acquired in vivo for a parasternal long axis orien-tation.


8.3.3 Computation Times

We present in Table 8.1 and Table 8.2 the computation times associated to the above de-scribed experiments. Table 8.1 shows that the initialization stage, i.e. computing the matrix Hand Cholesky decomposition is very fast. Table 8.2 shows that the computation times associatedto the Euler steps are reasonable, on the order of a few seconds for 2D images and of a few minutesfor the 3D calcaneaous image. The total time needed to obtain the segmentation is much larger.This time is however not linked to the proposed RBF collocation method, since it mainly resultsfrom the computation of the parameters associated to the velocity terms (i.e. C1(f) and C2(f) forthe Chan-Vese method [Chan and Vese, 2001] and ξ and ϑ for the Generalized Gaussian for theZhu-Yuille method [Zhu and Yuille, 1996]). This computation overhead would thus be the samefor a conventional level set implementation.

TABLE 8.1 TIME IN SECONDS FOR INITIALIZATION STAGE AND FOR VARIOUS IMAGES.

Name Image SizeNumber of

RBFβ Building of H

Choleskydecomposition

Simulation 1 311× 311 100× 100 σ 0.63 0.31Simulation 2 446× 446 150× 150 σ 1.47 1.18

U.S. 249× 40 60× 10 σ 0.03 0.01Calcaneous 1003 373 σ 12.93 23.14

TABLE 8.2 TIME IN SECONDS FOR MAIN PROCESS FOR VARIOUS IMAGES.

Name Image SizeNumber of

RBFβ

Computation ofαn+1 (Euler

steps)

Parametersestimation

Total

Simulation 1 311× 311 100× 100 σ 14.23 31.08 45.31Simulation 2 446× 446 150× 150 σ 23.51 52.71 76.22

U.S. 249× 40 60× 10 σ 2.26 20.64 22.90Calcaneous 1003 373 σ 401.15 7604.32 8005.47

Conclusion

Bilan

Dans cette partie, nous avons proposé un formalisme pour résoudre l’équation de propagationdes ensembles de niveaux. Contrairement aux méthodes classiques, qui résolvent ce problème pardifférences finies, nous décomposons la fonction implicite sur une base de fonctions, permettantde transformer l’EDP, qui régit le mouvement de l’interface, en une simple équation différentiellesur les coefficients. Pour concrétiser cette proposition, nous avons décomposé la fonction implicitesur une base de fonctions de base radiale à support compact. Nous proposons aussi de contraindrela fonction implicite pour éviter toute phase de réinitialisation pendant son évolution. Ainsi, lasegmentation obtenue est plus ”flexible” que les méthodes utilisant des ”bandes étroites”, car notreméthode permet l’apparition de nouvelles composantes (niveaux 0) pendant l’évolution, et parconséquent elle est donc moins sensible aux conditions initiales. La flexibilité et les performancesde l’approche proposée sont illustrées sur des simulations, et sur des images médicales.

Conclusion

In this part, we proposed a compactly supported RBF collocation approach for segmentationusing level sets. In contrast to the conventional finite difference method with narrow band imple-mentations, this numerical scheme yields overall control of the level set over the computationaldomain and provides a parametric continuous solution, in the sense that the solution is representedby a set of parameters. Due to this overall control and using compactly supported RBF represen-tation, the level set may be easily constrained. This feature avoids periodical reinitialisation of thelevel set in the course of its propagation. In this way, the obtained segmentation is topologicallymore flexible, since it may develop new zero level components and is thus less dependent upon theinitialisation. The flexibility and performances of the proposed approach have been prooved fromsimulations as well as medical image segmentations.

137

Part V

Conclusion And Perspectives

139

Bilan et Perspectives

Bilan

Dans cette thèse, nous avons étudié l’utilisation des fonctions de base radiale à supportcompact (CSRBF) pour des applications en reconstruction de signaux multidimension-nels, reconstruction de surfaces, et en segmentation d’images.

Dans la partie II, nous nous sommes intéressés à la reconstruction de signaux multidimension-nels à partir d’échantillons irréguliers par des méthodes dites ”meshless”, ne nécessitant pas demaillages sous jacents. Dans un premier temps, nous avons fait un bref tour d’horizon de ces mé-thodes, avant de s’attarder plus en détails sur les méthodes utilisant les fonction de bases radiales.

Nous avons proposé une nouvelle méthode multirésolution d’approximation, qui approximeles échantillons à différents niveaux de résolution, ou de détails. Pour cela, nous effectuons toutd’abord une classification hiérarchique utilisant une analyse en composante principale, puis nousutilisons, à un niveau de détail donné, les centres de chaque classe comme centres de CSRBF. Enne conservant que les centres où l’erreur locale, par rapport au niveau précédent, est la plus grande,nous ajoutons de l’information seulement là où elle est requise. Enfin les coefficients des CSRBFsont calculés en minimisant l’erreur quadratique entre deux niveaux successifs.

Nous avons également présenté deux méthodes d’approximation adaptative, dans lesquelles lesupport de chaque CSRBF est adapté localement à la répartition des centres. Dans le premier algo-rithme, nous travaillons avec un budget fixé de CSRBF et calculons le support de chaque CSRBFà partir d’un ”diagramme de Voronoï borné”. L’erreur quadratique est minimisée par un moindre-carré discret sur les échantillons. Dans le second algorithme, au lieu de travailler à budget fixé deCSRBF, nous travaillons avec une erreur fixée. Pour cela, nous utilisons une méthode multirésolu-tion où l’on ajoute des CSRBF uniquement sur les régions où elles sont nécessaires pour atteindrecet objectif.

Dans la partie III, nous nous sommes intéressés à la reconstruction de surfaces implicites àpartir de nuages de points. Nous avons commencé par décrire les propriétés des surfaces impliciteset par présenter quelques méthodes de la littérature. Parmi toutes ces méthodes, nous nous sommesplus particulièrement intéressés à celles qui utilisent les fonctions de base radiale. Les méthodestraditionnelles ne peuvent pas traiter à un coût calculatoire raisonnable d’importants nuages depoints. D’autres méthodes développent des surfaces externes indésirables, ou encore ne peuventtraiter des nuages de points irréguliers.

Par conséquent, nous avons présenté deux nouvelles approches pour résoudre ces problèmes.La première approche ”composite” divise le problème global de reconstruction en problèmes lo-caux. Cette division de l’espace, et des données, est réalisée par le biais d’une décomposition bi-naire de l’espace de type kd-tree. Pour chaque partition, nous calculons une approximation localeen minimisant une fonctionnelle composée d’un terme d’erreur et d’un terme régularisant. Enfinla solution globale est définie comme la somme pondérée de chacune de ces solutions locales,

141

142

en suivant le principe de partition de l’unité. Cette méthode permet un traitement de ”grands”nuages de points à un faible coût calculatoire. Dans la seconde approche, nous effectuons une ap-proximation adaptative avec un budget donné de CSRBF. Le support de chaque CSRBF s’adaptelocalement à la répartition des centres, à l’information géométrique et topologique qu’ils contien-nent, en calculant, au préalable, l’axe médian correspondant. Enfin les coefficients des CSRBFsont calculés en minimisant une fonctionnelle régularisante.

Dans la partie IV, nous nous sommes intéressés à la segmentation d’images en utilisant leformalisme des ”ensembles de niveaux”, où la propagation de l’interface est régie par une équa-tion aux dérivées partielles (EDP). Contrairement aux méthodes classiques, résolvant l’équationau dérivées partielles par différences finies, nous proposons une approche tout à fait différenteen décomposant la fonction implicite sur une base de fonctions. Une telle représentation per-met la définition d’une interface continue et paramétrique (la fonction implicite étant représentéepar un jeu de paramètres). Nous avons développé cette approche pour le cas particulier des CS-RBF, et contraint l’évolution de la fonction implicite, permettant ainsi d’éviter une réinitialisationpériodique pendant la propagation de l’interface. La méthode de segmentation proposée permetune grande flexibilité des changements topologiques, et, de par son principe, est beaucoup moinsdépendante des conditions initiales.

Perspectives

Les résultats des approches, que nous avons proposées, pour la reconstruction de signaux mul-tidimensionnels, pour la reconstruction de surface et le formalisme que nous avons développépour les ensembles de niveaux en segmentaion d’images, offrent de nombreuses perspectives derecherche, pour différentes applications. Ici nous présentons quelques une de ces voies futures derecherche qui nous apparaissent les plus intéressantes.

Dans la partie II, nous avons testé nos méthodes en 1 et 2D. Dans un premier temps, il nousfaudrait valider notre méthode sur des exemples en plus grande dimension. Il serait aussi très in-téressant d’utiliser nos méthodes pour la reconstruction IRM, où la reconstruction se passe dans ledomaine de Fourier (espace des k) avec des points d’acquisition irréguliers.

Dans la partie III, nous avons présenté des méthodes de reconstruction de surface qui peuvents’avérer très utiles imagerie médicale. En effet, ces méthodes permettent une création rapide demodèles, en sélectionnant directement des points sur des images 3D, pour la segmentation, pourla validation d’algorithmes de segmentation et d’estimation de mouvements. Ces méthodes pour-raient aussi être utilisées pour définir des volumes d’intérêt sur des images 3D. Dans cette thèse,nous nous sommes intéressés au problème de reconstruction avec un budget fixé de CSRBF, maispour certaines applications, il serait intéressant de traiter le problème dual, à savoir la reconstruc-tion de surface à erreur fixée.

Le formalisme que nous avons présenté dans la partie IV offre de nombreuses perspectivespour modéliser l’évolution de l’interface en segmentation d’images. En effet la propagation del’interface est maintenant représentée par un jeu de paramètres, et nous pouvons imaginer de con-traindre ceux-ci dans une séquence d’image pour le suivi d’objets, d’organes d’intérêt, tout au longde la séquence. Nous pensons aussi qu’il serait très intéressant d’étudier de nouvelles stratégiespour la répartition des centres sur l’image, pour l’initialisation de l’interface, afin de décrire uneapproche utilisant les informations issus de modèles.

Conclusion and Perspectives

Conclusion

In this thesis, we study the use of Compactly Supported Radial Basis Functions (CSRBF) forapplication in mutlidimensional signal reconstruction, surface reconstruction and image seg-mentation.

In Part II, we focused on the reconstruction of multidimensional signals from irregular sampleswith meshless methods. We started by presenting various techniques, before dealing with radialbasis functions based methods.

We presented a new multiresolution approximation method, which performs approximation atdifferent level of resolution, or details. Samples are first clustered following a binary space par-titioning tree, by using a principal component analysis based classification, and secondly CSRBFcenters for one level of resolution are defined as class centers. By selecting centers where thelargest local error is at the previous level, we only add information wherever it is required. Thenthe mean square is minimized for the computation of CSRBF coefficients.

We also presented an adaptive approximation method, where the support size of each CSRBFis locally adapted to the center distribution. In the first proposed algorithm, we work with a fixednumber of CSRBF and compute the support size by first computing the bounded Voronoï diagramof the center set. The mean square error is then minimized by considering a discrete least-squaresapproximation. In the second proposed algorithm, instead of dealing with the number of CSRBF,we deal with the fitting accuracy. To this end, we use an adaptive multiresolution method, whereCSRBF are added wherever it is required regarding to the prescribed error.

In Part III, we focused on the reconstruction of implicit surfaces from given point sets. Westarted by presenting facts about implicit surfaces and some previous works. Among all the im-plicit surface reconstruction techniques, we decide to study RBF based methods. We have seenthat common methods can not deal with large point sets due to high computational cost, someother produce some undesired extra off-surfaces or can not process irregular point sets.

Consequently, we presented two new methods that overcome these problems. The first methodis a composite approach which divides the global reconstruction problem into several local ones.This division is done by performing an adaptive overlapping domain and data decomposition basedon a kd-tree decomposition. The local approximation is performed by minimizing a quadratic func-tional which includes a data closeness term and regularization one. The global solution is finallyobtained by blending all local solutions by using partition of unity principle. This method allowsconsidering any unorganized point set with a low computational cost. In the second method, weuse an adaptive approximating method, for a fixed budget of CSRBF. The support size of eachCSRBF is locally adapted to the geometry and ”topology” of the centers, by first extracting anapproximation of the medial axis. Coefficients are then calculated by minimizing a regularizingfunctional.

143

144

In Part IV, we focus on solving the problem of image segmentation with using the level setformalism. While conventional methods use finite difference with narrow band implementations,we propose to decompose the resulting implicit function into basis functions defined over thecomputational domain. Such representation provides a parametric continuous definition of theresulting implicit function, in the sense that the solution is represented by a set of coefficients. Weimplement the proposed method for CSRBF, and constrain the evolution which avoids periodicalreinitialisation of the level set during its propagation. The obtained segmentation is topologicallymore flexible, since it may develop new zero level components and is thus less dependent upon theinitialisation.

Perspectives

The results of our approaches for multidimensional signal and surface reconstruction and theformalism we developed for level set evolution give rise to various future research directions, tovarious applications, and we present here some directions that seem to be particularly valuable.

In Part II, we demonstrate the effectiveness of our proposed method for 1 and 2D case. First,we should validate our proposed methods for higher dimensions to ensure their effectiveness. Itwould also be of interest to validate use our proposed methods for MRI Reconstruction, where thereconstruction takes place in the Fourier domain with irregular (k-space) acquisition points.

In Part III, we presented some methods for surface reconstruction that could be useful in med-ical imaging. Indeed, these methods allow to create quickly models for segmentation by selectingsome points on 3D images. Thus, it would be really interesting to use these methods to createheart atlas. Moreover, in the same way, we believe that these methods can be also used to drawor select volume of interest on 3D images. In this thesis, we focus on surface reconstruction witha fixed budget of CSRBF, however in some applications it would be required to consider the dualproblem, i.e. reconstruction with a fixed accuracy.

The formalism presented in Part IV offers an interesting modelling of the evolving interface.Indeed the evolving interface is now represented by a set of parameters; we can easily imagineto constraint them along an image sequence in order to track objects, or organs, of interest into asequence.

We also think that it could be really interesting to investigate new strategies for center locationin the image and about the initialization interface in order to produce a model based level setapproach.

Appendix

145

Appendix A

From Variational Problem to RadialBasis Functions

As we saw in section 2.3, that the general solution to the variational problem is the functionwhich minimizes the following functional:

H[f ] =N−1∑

i=0

(f(pi)− si)2 + λ (R[f ]− r0) (A.1)

where the smoothness term is given by:

R[f ] =∫

Rd

|f(ω)|2Φ(ω)

dω (A.2)

Here, we just present demonstration from [Girosi et al., 1993], but for more mathematicaldetails we ask readers to refer to the literature [Wendland, 2005b].

Depending on the choice of Φ, the functionalR[f ] can have a non-empty null space, and there-fore there is a certain class of functions that are ”invisible” to it. Authors express the functionalA.1 in terms of Fourier transform of f :

f(p) = C

∫

Rd

f(ω) eip·ωdω (A.3)

obtaining:

H[f ] =N−1∑

i=0

(si − C

∫

Rd

f(ω) ei·pi·ωdω)2

+ λ

∫

Rd

|f(ω)|2Φ(ω)

dω (A.4)

Since we consider real function, the Fourier transform satisfies the even constraints, i.e. f(−ω)= ¯f(ω). Though the functional can be rewritten as follows:

H[f ] =N−1∑

i=0

(si − C

∫

Rd

f(ω) ei·pi·ωdω)2

+ λ

∫

Rd

¯f(−ω) f(ω)

Φ(ω)dω (A.5)

In order to find the minimum of this functional, they consider functional derivatives with re-spect to f :

∂H[f ]

∂f(u)= 0 ∀u ∈ Rd (A.6)

147

148 APPENDIX A. FROM VARIATIONAL PROBLEM TO RBF

They proceed to compute the functional derivatives of the first and the second term of H[f ].For the data closeness term, we have:

∂

∂f(u)

N−1∑

i=0

(si − C

∫

Rd

f(ω) eip·ωdω)2

= 2N−1∑

i=0

(si − f(pi))∫

Rd

∂f(ω)

∂f(u)ei·pi·ωdω

= 2N−1∑

i=0

(si − f(pi))∫

Rd

δ (ω − u) ei·pi·ωdω

= 2N−1∑

i=0

(si − f(pi)) ei·pi·u

For the smoothness term, we have:

∂

∂f(u)

∫

Rd

f(−ω) f(ω)Φ(ω)

dω = 2∫

Rd

f(−ω)Φ(ω)

∂f(ω)

∂f(u)dω

= 2∫

Rd

f(−ω)Φ(ω)

δ(ω − u)dω

= 2f(−u)Φ(u)

The equation A.6 can be re written:

N−1∑

i=0

(si − f(pi)) · ei·pi·u + λ · f(−u)Φ(u)

By changing u by −u and multiplying by Φ(u) on both sides of the previous equation:

f(u) = Φ(−u) ·N−1∑

i=0

si − f(pi)λ

· ei·pi·u

They define coefficients:

αi =si − f(pi)

λi = 0, . . . , N − 1

and assume that the kernel function is real (its Fourier transform is even), and take the FourierTransform of the last equation, obtaining:

f(p) =N−1∑

i=0

αi · δ(pi − p) ∗ Φ(p) =N−1∑

i=0

αi · Φ(p− pi)

where ∗ denotes the convolution.They had defined as equivalent all the functions differing by a term that lies in the null space

of R[f ], and therefore they obtain the most general solution of the minimization problem:

149

f(p) =N−1∑

i=0

αi · Φ(pi − p) + g(p) (A.7)

where g(p) is a term that lies in the null space of R[f ].

150 APPENDIX A. FROM VARIATIONAL PROBLEM TO RBF

Appendix B

kd-tree

We present the detailed procedure to build the kd-tree in the Algorithm 5. By following thisalgorithm, kd-tree can be constructed inO(dN logN) time and requiresO(dN) space in memory.With such data structure, range query search can be computed in O(logN) (see algorithm 6).

Algorithm 5 Build kd-treeInput: Point set X = x0, . . . ,xp, and box BOutput: Root of a kd-tree.

if X contains at most M points thenReturn a leaf storing indices of these points.

elseDetermine splitting dimension and value.Split the current box B into two new ones B1 and B2.Split the current point set X into two new ones X1 and X2.Create two new nodes T1 and T2 with corresponding box and point set.Apply this algorithm to the two new nodes T1 and T2.Create a splitting node T , store the splitting dimension, splitting value, and the two children

T1 and T2.Return T .

end if

151

152 APPENDIX B. KD-TREE

Algorithm 6 Range query in kd-tree data structureInput: Query regionR ⊆ Ω and root node T of kd-tree.Output: All points contained inR.

if T is a leaf thenReport all points stored in T that are inR.

elseLet denote T1 and T2 the left and the right child of T , and let B1 and B2 their associated

boxes.for i=1,2 do

if B1 is fully contained inR thenReport all points in the tree rooted at Ti.

elseifR intersects B then

Apply this algorithm toR and Ti.end if

end ifend for

end if

Appendix C

3D Partition of Unity

C.1 General formulation

The partition of unity of blend M local solutions fi with weight function ω, by the followingrelationship:

f =

M−1∑i=0

ωi · fi

M−1∑i=0

ωi

The gradient ∇f of the corresponding implicit function f is given by

∇f =

M−1∑i=0

ωi ·(

M−1∑j=0

(fj ·∇ωj + ωj ·∇fj)

)−

M−1∑i=0

∇ωi ·(

M−1∑j=0

ωj · fj

)

(M−1∑j=0

ωj

)2

The hessian matrix Hessf of the corresponding implicit function f is given by

Hessf =

M−1∑i=0

(2∇fi ·∇ωT

i + ωi · Hessfi + fi · Hessωi

)

M−1∑j=0

ωj(p)

−2

M−1∑i=0

∇ωi ·(

M−1∑j=0

(ωj ·∇fT

j + fj ·∇ωTj

))+

M−1∑i=0

Hessωi ·(

M−1∑j=0

ωj · fj

)

(M−1∑j=0

ωj(p)

)2

+

2M−1∑i=0

∇ωi ·∇ωTi ·

(M−1∑j=0

ωj · fj

)

(M−1∑j=0

ωj(p)

)3

153

154 APPENDIX C. 3D PARTITION OF UNITY

C.2 Weight Function

The weight function are generally given as a decay function ψ and a distance function di. Herewe give some useful expression for the computation of the quantity given in the previous section:

ωi = ψ(di)

∇ωi = ∇di · ψ′(di)

Hessωi = Hessdi · ψ′(di) + ∇di ·∇dTi · ψ′′(di)

C.2.1 Spherical Weight Function

In the case of a spherical weight function, i.e. radial distance function di we have:

di(p) =‖p− ci‖

σ

∇di(p) =p− ci

σ · ‖p− ci‖

Hessdi =‖ci − p‖ · I −∇di(p) ·∇Tdi(p)

σ · ‖ci − p‖3

C.2.2 Cubical Weight Function

In the case of a cubical weight function for a cube spanned between a = [xa, ya, za]T andb = [xb, yb, zb]T , as we have presented in this document, we have:

di(x, y, z) = 1− 64∏

r∈x,y,z

(r − ra) (r − rb)(ra − rb)2

∂di

∂v(x, y, z) = −64

[(r − ra) + (r − rb)]∏r∈x,y,z

(ra − rb)2 ·∏

r∈x,y,z−v(r − ra)(r − rb), v ∈ x, y, z

∂2di

∂v2(x, y, z) = − 128∏

r∈x,y,z(ra − rb)2 ·

∏

r∈x,y,z−v(r − ra)(r − rb)

∂2di

∂u∂v(x, y, z) =

64∏r∈x,y,z

(ra − rb)2 · (r − ra) (r − rb)

· [(u− ua) + (u− ub)] [(v − va) + (v − vb)]

where u 6= v ∈ x, y, z and r 6= u, v.

Appendix D

Convergence proof elements for ournew Level Set approach

Here, we provide some convergence proof elements about the convergence of our proposedLevel Set approach in the case Eq. 7.11.

A global convergence of the level set is obtained if the following proposition is satisfied:∥∥αn+1 − αn

∥∥1

6∥∥αn − αn−1

∥∥1

(D.1)

To prove Eq. D.1, we pursue the following steps.

Lemma D.1. ∥∥αn+1 − αn∥∥

16

∥∥αn+1 −αn∥∥

1(D.2)

Proof. The normalization operator used in Eq. 8.28 is a projection operator onto the convex set Cof l1-norm β-bounded vectors.

• Clearly C is a closed convex set, i.e.

u,v ∈ C ⇒ µ · u + (1− µ) · v ∈ C for 0 6 µ 6 1. (D.3)

To prove Eq. D.3 we consider u,v ∈ C, then ‖u‖1 6 β, ‖v‖1 6 β. It follows

‖µ · u + (1− µ) · v‖1 6 µ‖u‖1 + (1− µ) · ‖v‖1 6 β (D.4)

It is easy to prove the closure by considering a given sequence un ⊂ C such that

limn→∞ ‖un − u‖1 = 0 =⇒ u ∈ C (D.5)

• Now consider the operatorP (w) =

β

‖w‖1 ·w, if ‖w‖1 > β (D.6)

P (w) = w, else

This operator is a projection operator onto the closed set C (POCS). In order to verify thisconjecture it is sufficient to show that for an arbitrary w /∈ C its projection onto C is itsclosest point of C

155

156 APPENDIX D. CONVERGENCE PROOF ELEMENTS

‖w − P (w)‖1 = infv∈C‖w − v‖1 (D.7)

This condition can be easily verified as follows.

Clearly ‖w − v‖1 is minimum if v is collinear to w, i.e. v = ‖v‖1‖w‖1 w. Then

infv∈C‖w − v‖1 = inf

v∈C‖w‖1 ·

∣∣∣∣1−‖v‖1‖w‖1

∣∣∣∣

= ‖w‖1 ·∣∣∣∣1−

β

‖w‖1

∣∣∣∣

According to Eq. D.6 the left side of Eq. D.7 can be rewritted as

‖w − P (w)‖1 = ‖w‖1 ·∣∣∣∣1−

β

‖w‖1

∣∣∣∣ (D.8)

That proves the equality of both sides of Eq. D.7

A POCS is a non expansive operator

‖P (w1)− P (w2)‖1 6 ‖w1 −w2‖1 , for w1,w2 /∈ C (D.9)

That proves the lemma 1.

We recall Eq. 8.28 αn+1 = αn − τH−1Bn(αn). By combining two consecutive equationswe can write

∥∥αn+1 −αn∥∥

1=

∥∥αn − αn−1 − τ ·H−1 · [Bn (αn)−Bn−1(αn−1

)]∥∥1

(D.10)

In the general case the terms Bn(αn) and Bn−1(αn−1) are data dependent, and cancel oneanother. More precisely, according to Eq. 8.14 with the choice ε = 2σ = β (see section 8.2.6 and8.3) the term Bn(αn) weakly depends on αn:

Bn(αn) ≈ 1πεV n (D.11)

where V n is the velocity function at iteration n.

Using the hypothesis of a consistent choice of the velocity, V n becomes stationary and conse-quently V n − V n−1 vanishes.

Then∥∥αn+1 −αn

∥∥1

=∥∥αn − αn−1

∥∥1

and according to lemma 1, it demonstrates the con-vergence condition Eq. D.1.

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FOLIO ADMINISTRATIFTHÈSE SOUTENUE DEVANT L’INSTITUT NATIONAL

DES SCIENCES APPLIQUÉES DE LYON

NOM : GELAS DATE DE SOUTENANCE : 27 Novembre 2006PRÉNOM : Arnaud

TITRE :Compactly Supported Radial Basis Functions: Multidimensional Reconstruction and Applica-tions

NATURE : Doctorat NUMÉRO D’ORDRE : 2006-ISAL-0095FORMATION DOCTORALE : Sciences de l’Information, des Dispositifs et des SystèmesFILIÈRE : Instrumentation, Système, Signal & Image

COTE B.I.U. LYON : T 50/210/19 / et bis CLASSE :

RÉSUMÉ : Cette thèse traite l’application des fonctions de base radiale, à support com-pact (CSRBF), pour la reconstruction multidimensionnelle de signaux et de surfaces à partird’échantillons, ainsi que pour la segmentation des images. En particulier, cette approche per-met de considérer l’échantillonnage irrégulier.Dans chacune de ces applications nous proposons de nouvelles méthodes. Pour la reconstruc-tion multidimensionnelle de signaux nous proposons une approximation multirésolution baséesur une classification hiérarchique des échantillons, puis une approximation adaptative aveccalcul local du support de chaque CSRBF. Pour la reconstruction de surfaces implicites, nousproposons une méthode composite qui associe la partition de l’unité et les CSRBF, puis uneapproximation adaptative où le support de chaque CSRBF est déterminé à partir d’une approxi-mation de l’axe médian. Enfin, nous proposons un formalisme de collocation pour la résolutionde l’équation de propagation des ensembles de niveaux en segmentation, par représentation dela fonction implicite sur une base de fonctions. En particulier, nous illustrons cette méthodeavec les CSRBF en segmentation d’images médicales.

MOTS-CLÉS : Reconstruction, Approximation, Interpolation, Régularisation, Multirésolu-tion, Partition de l’unité, Surface implicite, Axe médian, Segmentation d’image, Ensemble deniveaux, Collocation, Fonctions de base radiale

LABORATOIRE DE RECHERCHES :Centre de Recherche et d’Applications en Traitement de l’Image et du Signal (CREATIS), UMRCNRS 5515, U630 Inserm

DIRECTEUR DE THÈSE : Rémy PROST

PRÉSIDENT DU JURY : Jean-Marc CHASSERY

COMPOSITION DU JURY : Michel BARLAUD (rapporteur), Jean-Marc CHASSERY (prési-dent), Olivier DEVILLERS (rapporteur), Denis FRIBOULET, Takashi KANAI, YutakaOHTAKE, Rémy PROST (directeur).

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