Surface parameterization over regular domains

Surface parameterization has been widely studied and it has been playing a critical role in many geometric processing tasks in graphics, computer-aided design, visualization, vision, physical simulation and etc. Regular domains, such as polycubes, are favored due to their structural regularity and geometric simplicity. This thesis focuses on studying the surface parameterization over regular domains, i.e. polycubes, and develops effective computation algorithms. Firstly, the motivation for surface parameterization and polycube mapping is introduced. Secondly, we briefly review existing surface parameterization techniques, especially for extensively studied parameterization algorithms for topological disk surfaces and parameterizations over regular domains for closed surfaces. Then we propose a polycube parameterization algorithm for closed surfaces with general topology. We develop an efficient optimization framework to minimize the angle and area distortion of the mapping. Its applications on surface meshing, inter-shape morphing and volumetric polycube mapping are also discussed

Dihedral angle-based maps of tetrahedral meshes

International audienceWe present a geometric representation of a tetrahedral mesh that is solely based on dihedral angles. We first show that the shape of a tetrahedral mesh is completely defined by its dihedral angles. This proof leads to a set of angular constraints that must be satisfied for an immersion to exist in R 3. This formulation lets us easily specify conditions to avoid inverted tetrahedra and multiply-covered vertices, thus leading to locally injective maps. We then present a constrained optimization method that modifies input angles when they do not satisfy constraints. Additionally, we develop a fast spectral reconstruction method to robustly recover positions from dihedral angles. We demonstrate the applicability of our representation with examples of volume parameterization, shape interpolation, mesh optimization, connectivity shapes, and mesh compression

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

User-appropriate viewer for high resolution interactive engagement with 3D digital cultural artefacts.

The core mission of museums and cultural institutions is the preservation, study and presentation of cultural heritage content. In this technological age, the creation of digital datasets and archives has been widely adopted as one way of seeking to achieve some or all of these goals. However, there are many challenges with the use of these data, and in particular the large numbers of 3D digital artefacts that have been produced using methods such as non- contact laser scanning. As public expectation for more open access to information and innovative digital media increases, there are many issues that need to be rapidly addressed. The novel nature of 3D datasets and their visualisation presenting unique issues that impede use and dissemination. Key questions include the legal issues associated with 3D datasets created from cultural artefacts; the complex needs of users who are interacting with them; a lack of knowledge to texture and assess the visual quality of the datasets; and how the visual quality of the presented dataset relates to the perceptual experience of the user. This engineering doctorate, based on an industrial partnership with the National Museums of Liverpool and Conservation Technologies, investigates these questions and offers new ways of working with 3D cultural heritage datasets. The research outcomes in the thesis provide an improved understanding of the complexity of intellectual property law in relation to 3D cultural heritage datasets and how this impacts dissemination of these types of data. It also provides tools and techniques that can be used to understand the needs of a user when interacting with 3D cultural content. Additionally, the results demonstrate the importance of the relationship between texture and polygonal resolution and how this can affect the perceived visual experience of a visitor. It finds that there is an acceptable cost to texture and polygonal resolution to offer the best perceptual experience with 3D digital cultural heritage. The results also demonstrate that a non-textured mesh may be as highly received as a high resolution textured mesh. The research presented provides methodologies and guidelines to improve upon the dissemination and visualisation of 3D cultural content; enhancing and communicating the significance of their 3D collections to their physical and virtual visitors. Future opportunities and challenges for disseminating and visualising 3D cultural content are also discussed

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version