Geometric modeling and optimization over regular domains for graphics and visual computing

The effective construction of parametric representation of complicated geometric objects can facilitate many design, analysis, and simulation tasks in Computer-Aided Design (CAD), Computer-Aided Manufacturing (CAM), and Computer-Aided Engineering (CAE). Given a 3D shape, the procedure of finding such a parametric representation upon a canonical domain is called geometric parameterization. Regular geometric regions, such as polycubes and spheres, are desirable domains for parameterization. Parametric representations defined upon regular geometric domains have many desirable mathematical properties and can facilitate or simplify various surface/solid modeling and processing computation. This dissertation studies the construction of parameterization on regular geometric domains and explores their applications in shape modeling and computer-aided design. Specifically, we studies (1) the surface parameterization on the spherical domain for closed genus-zero surfaces; (2) the surface parameterization on the polycube domain for general closed surfaces; and (3) the volumetric parameterization for 3D-manifolds embedded in 3D Euclidean space. We propose novel computational models to solve these geometric problems. Our computational models reduce to nonlinear optimizations with various geometric constraints. Hence, we also need to explore effective optimization algorithms. The main contributions of this dissertation are three-folded. (1) We developed an effective progressive spherical parameterization algorithm, with an efficient nonlinear optimization scheme subject to the spherical constraint. Compared with the state-of-the-art spherical mapping algorithms, our algorithm demonstrates the advantages of great efficiency, lower distortion, and guaranteed bijectiveness, and we show its applications in spherical harmonic decomposition and shape analysis. (2) We propose a first topology-preserving polycube domain optimization algorithm that simultaneously optimizes polycube domain together with the parameterization to balance the mapping distortion and domain simplicity. We develop effective nonlinear geometric optimization algorithms dealing with variables with and without derivatives. This polycube parameterization algorithm can benefit the regular quadrilateral mesh generation and cross-surface parameterization. (3) We develop a novel quaternion-based optimization framework for 3D frame field construction and volumetric parameterization computation. We demonstrate our constructed 3D frame field has better smoothness, compared with state-of-the-art algorithms, and is effective in guiding low-distortion volumetric parameterization and high-quality hexahedral mesh generation

3D mesh metamorphosis from spherical parameterization for conceptual design

Engineering product design is an information intensive decision-making process that consists of several phases including design specification definition, design concepts generation, detailed design and analysis, and manufacturing. Usually, generating geometry models for visualization is a big challenge for early stage conceptual design. Complexity of existing computer aided design packages constrains participation of people with various backgrounds in the design process. In addition, many design processes do not take advantage of the rich amount of legacy information available for new concepts creation. The research presented here explores the use of advanced graphical techniques to quickly and efficiently merge legacy information with new design concepts to rapidly create new conceptual product designs. 3D mesh metamorphosis framework 3DMeshMorpher was created to construct new models by navigating in a shape-space of registered design models. The framework is composed of: i) a fast spherical parameterization method to map a geometric model (genus-0) onto a unit sphere; ii) a geometric feature identification and picking technique based on 3D skeleton extraction; and iii) a LOD controllable 3D remeshing scheme with spherical mesh subdivision based on the developedspherical parameterization. This efficient software framework enables designers to create numerous geometric concepts in real time with a simple graphical user interface. The spherical parameterization method is focused on closed genus-zero meshes. It is based upon barycentric coordinates with convex boundary. Unlike most existing similar approaches which deal with each vertex in the mesh equally, the method developed in this research focuses primarily on resolving overlapping areas, which helps speed the parameterization process. The algorithm starts by normalizing the source mesh onto a unit sphere and followed by some initial relaxation via Gauss-Seidel iterations. Due to its emphasis on solving only challenging overlapping regions, this parameterization process is much faster than existing spherical mapping methods. To ensure the correspondence of features from different models, we introduce a skeleton based feature identification and picking method for features alignment. Unlike traditional methods that align single point for each feature, this method can provide alignments for complete feature areas. This could help users to create more reasonable intermediate morphing results with preserved topological features. This skeleton featuring framework could potentially be extended to automatic features alignment for geometries with similar topologies. The skeleton extracted could also be applied for other applications such as skeleton-based animations. The 3D remeshing algorithm with spherical mesh subdivision is developed to generate a common connectivity for different mesh models. This method is derived from the concept of spherical mesh subdivision. The local recursive subdivision can be set to match the desired LOD (level of details) for source spherical mesh. Such LOD is controllable and this allows various outputs with different resolutions. Such recursive subdivision then follows by a triangular correction process which ensures valid triangulations for the remeshing. And the final mesh merging and reconstruction process produces the remeshing model with desired LOD specified from user. Usually the final merged model contains all the geometric details from each model with reasonable amount of vertices, unlike other existing methods that result in big amount of vertices in the merged model. Such multi-resolution outputs with controllable LOD could also be applied in various other computer graphics applications such as computer games

Numerical and variational aspects of mesh parameterization and editing

A surface parameterization is a smooth one-to-one mapping between the surface and a parametric domain. Typically, surfaces with disk topology are mapped onto the plane and genus-zero surfaces onto the sphere. As any attempt to flatten a non-trivial surface onto the plane will inevitably induce a certain amount of distortion, the main concern of research on this topic is to minimize parametric distortion. This thesis aims at presenting a balanced blend of mathematical rigor and engineering intuition to address the challenges raised by the mesh parameterization problem. We study the numerical aspects of mesh parameterization in the light of parallel developments in both mathematics and engineering. Furthermore, we introduce the concept of quasi-harmonic maps for reducing distortion in the fixed boundary case and extend it to both the free boundary and the spherical case. Thinking of parameterization in a more general sense as the construction of one or several scalar fields on a surface, we explore the potential of this construction for mesh deformation and surface matching. We propose an \u27;on-surface parameterization\u27; for guiding the deformation process and performing surface matching. A direct harmonic interpolation in the quaternion domain is also shown to give promising results for deformation transfer.Eine Flächenparameterisierung ist eine globale bijektive Abbildung zwischen der Fläche und einem zugehörigen parametrischen Gebiet. Gewöhnlich werden Flächen mit scheibenförmiger Topologie auf eine Kreisscheibe und Flächen mit Genus Null auf eine Sphäre abgebildet. Das Hauptinteresse der Forschung an diesem Thema ist die Minimierung der parametrischen Verzerrung, die unweigerlich bei jedem Versuch, eine nicht triviale Fläche über einer Ebene zu parameterisieren, erzeugt wird. Diese Arbeit strebt zur Behandlung des Parametrisierungsproblems eine ausgeglichene Mischung zwischen mathematischer Präzision und ingenieurwissenschaftlicher Intuition an. Wir behandeln dabei die numerischen Aspekte des Parameterisierungsproblems im Hinblick auf die aktuellen parallelen Entwicklungen in der Mathematik und den Ingenieurwissenschaften. Weiterhin führen wir das Konzept der quasi-harmonischen Abbildungen ein, um die Verzerrung bei gegebenen Randbedingungen zu verringern. Anschließend verallgemeinern wir dieses Konzept auf den sphärischen Fall und auf den Fall mit freien Randbedingungen. Durch allgemeinere Betrachtung der Parameterisierung als Konstruktion eines oder mehrerer skalarer Felder auf einer Fläche ergibt sich ein neuer Ansatz zur Netzdeformation und der Erzeugung von Flächenkorrespondenzen. Wir stellen eine \u27;on-surface parameterization\u27; vor, welche den Deformationsprozess leitet und Flächenkorrespondenzen erstellt. Darüber hinaus zeigt eine direkte harmonische Interpolation in der Domäne der Quaternionen auch vielversprechende Resultate für die Übertragung von Deformationen

Métamorphose de maillage 3D

Cette thèse de doctorat aborde spécifiquement le problème de la métamorphose entre différents maillages 3D, qui peut assurer un niveau élevé de qualité pour la séquence de transition, qui devrait être aussi lisse et progressive que possible, cohérente par rapport à la géométrie et la topologie, et visuellement agréable. Les différentes étapes impliquées dans le processus de transformation sont développées dans cette thèse. Nos premières contributions concernent deux approches différentes des paramétrisations: un algorithme de mappage barycentrique basé sur la préservation des rapports de longueur et une technique de paramétrisation sphérique, exploitant la courbure Gaussien. L'évaluation expérimentale, effectuées sur des modèles 3D de formes variées, démontré une amélioration considérable en termes de distorsion maillage pour les deux méthodes. Afin d aligner les caractéristiques des deux modèles d'entrée, nous avons considéré une technique de déformation basée sur la fonction radial CTPS C2a approprié pour déformer le mappage dans le domaine paramétrique et maintenir un mappage valide a travers le processus de mouvement. La dernière contribution consiste d une une nouvelle méthode qui construit un pseudo metamaillage qui évite l'exécution et le suivi des intersections d arêtes comme rencontrées dans l'état-of-the-art. En outre, notre méthode permet de réduire de manière drastique le nombre de sommets normalement nécessaires dans une structure supermesh. Le cadre générale de métamorphose a été intégré dans une application prototype de morphing qui permet à l'utilisateur d'opérer de façon interactive avec des modèles 3D et de contrôler chaque étape du processusThis Ph.D. thesis specifically deals with the issue of metamorphosis of 3D objects represented as 3D triangular meshes. The objective is to elaborate a complete 3D mesh morphing methodology which ensures high quality transition sequences, smooth and gradual, consistent with respect to both geometry and topology, and visually pleasant. Our first contributions concern the two different approaches of parameterization: a new barycentric mapping algorithm based on the preservation of the mesh length ratios, and a spherical parameterization technique, exploiting a Gaussian curvature criterion. The experimental evaluation, carried out on 3D models of various shapes, demonstrated a considerably improvement in terms of mesh distortion for both methods. In order to align the features of the two input models, we have considered a warping technique based on the CTPS C2a radial basis function suitable to deform the models embeddings in the parametric domain maintaining a valid mapping through the entire movement process. We show how this technique has to be adapted in order to warp meshes specified in the parametric domains. A final contribution consists of a novel algorithm for constructing a pseudo-metamesh that avoids the complex process of edge intersections encountered in the state-of-the-art. The obtained mesh structure is characterized by a small number of vertices and it is able to approximate both the source and target shapes. The entire mesh morphing framework has been integrated in an interactive application that allows the user to control and visualize all the stages of the morphing processEVRY-INT (912282302) / SudocSudocFranceF

In Vivo Human Right Ventricle Shape and Kinematic Analysis with and without Pulmonary Hypertension

Pulmonary hypertension (PH) is a severe cardio-pulmonary illness which has been commonly observed to induce substantial and ultimately deleterious changes to the human right ventricle (RV) shape and function. As such, the functional state of the RV is thought to be a major determinant of symptoms and survival rates for PH. However, there has been little success to-date to identify clinically obtainable metrics of RV shape and deformation as a means to detect the onset and progression of PH. This difficulty is largely the result of the absence of a proven approach that is generally applicable for consistent and reliable quantitative analysis of anatomical shapes, particularly the RV, between patients and over time. Therefore, a computational framework which can quantitatively analyze RV shape and deformation could be a key to assist in clinically detecting the onset and progression of PH. Statistical shape analysis techniques were developed, implemented, and assessed to analyze variations in human RV endocardial surface (RVES) shapes and kinematics from noninvasive clinical medical imaging data with respect to a spectrum of hemodynamic states. A computational framework for the quantitative analysis and statistical decomposition of sets of 3D genus-0 shapes that combines a modified harmonic mapping approach directly with proper orthogonal decomposition (DM-POD) is presented. The DM-POD approach is shown to be a robust technique for recovering inherent shape-related features through the analysis of sets of artificially generated shapes. The DM-POD approach is then applied to obtain kinematic features of the human RV based on the relative change in shape of the endocardial surface using cardiac computed tomography images. In addition, the kinematic features of the RVES obtained by the DM-POD approach are shown to be consistent and associated with intrinsically physiological components of the heart, and thus may potentially provide a more accurate means for classifying the progressive change in RV function caused by PH, in comparison to traditional clinical hemodynamic and volume-based metrics. Statistical shape analysis for the human RV is further evaluated through analysis of alternate components of the DM-POD approach, as well as through comparison of the DM-POD workflow with an alternate spherical harmonic function-based workflow (SPHARM), with respect to the aspects of surface representation, alignment, and decomposition. Additionally, different ways of utilizing the available imaging data with respect to the classification potential are investigated by considering analysis results when applying both the various DM-POD and SPHARM approaches with several different combinations of the phases captured throughout a single cardiac cycle for the patient set. Lastly, a novel statistical decomposition technique known as independent component analysis (ICA) was incorporated into the statistical shape analysis framework (i.e., DM-POD) to produce an alternative workflow (DM-ICA). Both the DM-POD and DM-ICA approaches are applied to analyze sets of artificially generated data and the human RVES datasets, and the respective results are compared. The DM-POD and DM-ICA workflows are shown to produce consistent, but substantially different results due to the various principles and views of each of the two statistical decomposition algorithms (i.e., POD and ICA). Most importantly, the results from the DM-POD and DM-ICA workflows appear to relate to RV function in unique ways, with respect to both traditional clinical metrics and each other, and have the potential to provide new metrics for better understanding of the human RV and its relationship to PH

Images géométriques de genre arbitraire dans le domaine sphérique

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal

Curvilinear Spherical Parameterization

We present an efficient approach for solving the spherical parameterization problem. The essence of the approach is to look for a solution in the curvilinear coordinate system, without requiring the additional spherical constraints usually needed in cartesian formulations. This setup allows us to take full advantage of some existing techniques originally developed for planar parameterization. Our results substantiate the efficiency of the method and confirm its robustness. Meshes of non-trivial geometry with tens of thousands of triangles are processed in a few seconds, always yielding bijective maps. This computational achievement bridges a so far wide gap in performance between spherical and planar parameterization