Les courbes algébriques trigonométriques à hodographe pythagorien pour résoudre des problèmes d'interpolation deux et trois-dimensionnels et leur utilisation pour visualiser les informations dentaires dans des volumes tomographiques 3D

Algebraic-trigonometric Pythagorean hodograph curves for solving planar and spatial interpolation problems and their use for visualizing dental information within 3D tomographic volumes

• Auteur / Autrice : Cindy González
• Direction : Gudrun Albrecht, Marco Paluszny
• Type : Thèse de doctorat
• Discipline(s) : Mathématiques. Mathématiques appliquées
• Date : Soutenance le 25/01/2018
• Etablissement(s) : Valenciennes
• Ecole(s) doctorale(s) : École doctorale Sciences pour l'ingénieur (Lille)
• Partenaire(s) de recherche : Laboratoire : Laboratoire de mathématiques et leurs applications de Valenciennes (2006-2021)
• Partenaire(s) de recherche : Communauté d'Universités et Etablissements (ComUE) : Communauté d'universités et d'établissements Lille Nord de France (2009-2013)
• Jury : Président / Présidente : Felix Ali Mehmeti
• Jury : Examinateurs / Examinatrices : Gudrun Albrecht, Marco Paluszny, Jorge Estrada-Sarlabous, Géraldine Morin, Franca Caliò
• Jury : Rapporteurs / Rapporteuses : Jorge Estrada-Sarlabous, Géraldine Morin

Mots clés

Mots clés contrôlés

Imagerie tridimensionnelle

Hodographes

Splines, Théorie des

Courbes algébriques

Mots clés libres

Courbe algébrique trigonométrique à hodographe pythagorien

Courbequintique à hodographe pythagorien

Courbe B-Spline cubique

Facette développable

Volume odontologique

Algebraic-Trigonometric Pythagorean-Hodograph curve

Pythagorean-Hodograph quintic curve

Cubic B-Spline curve

Developable surface

Computed tomography

Résumé

Les problèmes d'interpolation ont été largement étudiés dans la Conception Géométrique Assistée par Ordinateur. Ces problèmes consistent en la construction de courbes et de surfaces qui passent exactement par un ensemble de données. Dans ce cadre, l'objectif principal de cette thèse est de présenter des méthodes d'interpolation de données 2D et 3D au moyen de courbes Algébriques Trigonométriques à Hodographe Pythagorien (ATPH). Celles-ci sont utilisables pour la conception de modèles géométriques dans de nombreuses applications. En particulier, nous nous intéressons à la modélisation géométrique d'objets odontologiques. À cette fin, nous utilisons les courbes spatiales ATPH pour la construction de surfaces développables dans des volumes odontologiques. Initialement, nous considérons la construction de courbes planes ATPH avec continuité C² qui interpolent une séquence ordonnée de points. Nous employons deux méthodes pour résoudre ce problème et trouver la « bonne » solution. Nous étendons les courbes ATPH planes à l'espace tridimensionnel. Cette caractérisation 3D est utilisée pour résoudre le problème d'interpolation Hermite de premier ordre. Nous utilisons ces splines ATPH spatiales C¹ continues pour guider des facettes développables, qui sont déployées à l'intérieur de volumes tomodensitométriques odontologiques, afin de visualiser des informations d'intérêt pour le professionnel de santé. Cette information peut être utile dans l'évaluation clinique, diagnostic et/ou plan de traitement.

Interpolation problems have been widely studied in Computer Aided Geometric Design (CAGD). They consist in the construction of curves and surfaces that pass exactly through a given data set, such as point clouds, tangents, curvatures, lines/planes, etc. In general, these curves and surfaces are represented in a parametrized form. This representation is independent of the coordinate system, it adapts itself well to geometric transformations and the differential geometric properties of curves and surfaces are invariant under reparametrization. In this context, the main goal of this thesis is to present 2D and 3D data interpolation schemes by means of Algebraic-Trigonometric Pythagorean-Hodograph (ATPH) curves. The latter are parametric curves defined in a mixed algebraic-trigonometric space, whose hodograph satisfies a Pythagorean condition. This representation allows to analytically calculate the curve's arc-length as well as the rational-trigonometric parametrization of the offsets curves. These properties are usable for the design of geometric models in many applications including manufacturing, architectural design, shipbuilding, computer graphics, and many more. In particular, we are interested in the geometric modeling of odontological objects. To this end, we use the spatial ATPH curves for the construction of developable patches within 3D odontological volumes. This may be a useful tool for extracting information of interest along dental structures. We give an overview of how some similar interpolating problems have been addressed by the scientific community. Then in chapter 2, we consider the construction of planar C2 ATPH spline curves that interpolate an ordered sequence of points. This problem has many solutions, its number depends on the number of interpolating points. Therefore, we employ two methods to find them. Firstly, we calculate all solutions by a homotopy method. However, it is empirically observed that only one solution does not have any self-intersections. Hence, the Newton-Raphson iteration method is used to directly compute this \good" solution. Note that C2 ATPH spline curves depend on several free parameters, which allow to obtain a diversity of interpolants. Thanks to these shape parameters, the ATPH curves prove to be more exible and versatile than their polynomial counterpart, the well known Pythagorean-Hodograph (PH) quintic curves and polynomial curves in general. These parameters are optimally chosen through a minimization process of fairness measures. We design ATPH curves that closely agree with well-known trigonometric curves by adjusting the shape parameters. We extend the planar ATPH curves to the case of spatial ATPH curves in chapter 3. This characterization is given in terms of quaternions, because this allows to properly analyze their properties and simplify the calculations. We employ the spatial ATPH curves to solve the first-order Hermite interpolation problem. The obtained ATPH interpolants depend on three free angular values. As in the planar case, we optimally choose these parameters by the minimization of integral shape measures. This process is also used to calculate the C1 interpolating ATPH curves that closely approximate well-known 3D parametric curves. To illustrate this performance, we present the process for some kind of helices. In chapter 4 we then use these C1 ATPH splines for guiding developable surface patches, which are deployed within odontological computed tomography (CT) volumes, in order to visualize information of interest for the medical professional. Particularly, we construct piecewise conical surfaces along smooth ATPH curves to display information related to the anatomical structure of human jawbones. This information may be useful in clinical assessment, diagnosis and/or treatment plan. Finally, the obtained results are analyzed and conclusions are drawn in chapter 5.