Optimal Voronoi Tessellations with Hessian-based Anisotropy

International audienceThis paper presents a variational method to generate cell complexes with local anisotropy conforming to the Hessian of any given convex function and for any given local mesh density. Our formulation builds upon approximation theory to offer an anisotropic extension of Centroidal Voronoi Tessellations which can be seen as a dual form of Optimal Delaunay Triangulation. We thus refer to the resulting anisotropic polytopal meshes as Optimal Voronoi Tessel-lations. Our approach sharply contrasts with previous anisotropic versions of Voronoi diagrams as it employs first-type Bregman diagrams , a generalization of power diagrams where sites are augmented with not only a scalar-valued weight but also a vector-valued shift. As such, our OVT meshes contain only convex cells with straight edges, and admit an embedded dual triangulation that is combinatorially-regular. We show the effectiveness of our technique using off-the-shelf computational geometry libraries

VoroCrust: Voronoi Meshing Without Clipping

Polyhedral meshes are increasingly becoming an attractive option with particular advantages over traditional meshes for certain applications. What has been missing is a robust polyhedral meshing algorithm that can handle broad classes of domains exhibiting arbitrarily curved boundaries and sharp features. In addition, the power of primal-dual mesh pairs, exemplified by Voronoi-Delaunay meshes, has been recognized as an important ingredient in numerous formulations. The VoroCrust algorithm is the first provably-correct algorithm for conforming polyhedral Voronoi meshing for non-convex and non-manifold domains with guarantees on the quality of both surface and volume elements. A robust refinement process estimates a suitable sizing field that enables the careful placement of Voronoi seeds across the surface circumventing the need for clipping and avoiding its many drawbacks. The algorithm has the flexibility of filling the interior by either structured or random samples, while preserving all sharp features in the output mesh. We demonstrate the capabilities of the algorithm on a variety of models and compare against state-of-the-art polyhedral meshing methods based on clipped Voronoi cells establishing the clear advantage of VoroCrust output.Comment: 18 pages (including appendix), 18 figures. Version without compressed images available on https://www.dropbox.com/s/qc6sot1gaujundy/VoroCrust.pdf. Supplemental materials available on https://www.dropbox.com/s/6p72h1e2ivw6kj3/VoroCrust_supplemental_materials.pd

Explicit Topology Optimization of Conforming Voronoi Foams

Topology optimization is able to maximally leverage the high DOFs and mechanical potentiality of porous foams but faces three fundamental challenges: conforming to free-form outer shapes, maintaining geometric connectivity between adjacent cells, and achieving high simulation accuracy. To resolve the issues, borrowing the concept from Voronoi tessellation, we propose to use the site (or seed) positions and radii of the beams as the DOFs for open-cell foam design. Such DOFs cover extensive design space and have clear geometrical meaning, which makes it easy to provide explicit controls (e.g. granularity). During the gradient-based optimization, the foam topology can change freely, and some seeds may even be pushed out of the shape, which greatly alleviates the challenges of prescribing a fixed underlying grid. The mechanical property of our foam is computed from its highly heterogeneous density field counterpart discretized on a background mesh, with a much improved accuracy via a new material-aware numerical coarsening method. We also explore the differentiability of the open-cell Voronoi foams w.r.t. its seed locations, and propose a local finite difference method to estimate the derivatives efficiently. We do not only show the improved foam performance of our Voronoi foam in comparison with classical topology optimization approaches, but also demonstrate its advantages in various settings, especially when the target volume fraction is extremely low

Computing a high-dimensional euclidean embedding from an arbitrary smooth riemannian metric

International audienceThis article presents a new method to compute a self-intersection free high-dimensional Euclidean embedding (SIFHDE) for surfaces and volumes equipped with an arbitrary Riemannian metric. It is already known that given a high-dimensional (high-d) embedding, one can easily compute an anisotropic Voronoi diagram by back-mapping it to 3D space. We show here how to solve the inverse problem, i.e., given an input metric, compute a smooth intersection-free high-d embedding of the input such that the pullback metric of the embedding matches the input metric. Our numerical solution mechanism matches the deformation gradient of the 3D → higher-d mapping with the given Riemannian metric. We demonstrate applications of the method, by being used to construct anisotropic Restricted Voronoi Diagram (RVD) and anisotropic meshing, that are otherwise extremely difficult to compute. In the SIFHDE-space constructed by our algorithm, difficult 3D anisotropic computations are replaced with simple Euclidean computations, resulting in an isotropic RVD and its dual mesh on this high-d embedding. The results are compared with the state-ofthe-art in anisotropic surface and volume meshings using several examples and evaluation metrics

Visualization of implicit geographic information through map-like graphics

Viele von Web 2.0-Benutzern gesammelte Daten sind ortsbezogen, wobei der Ort meist nur eine Information unter vielen ist und dem Ortsbezug häufig keine besondere Bedeutung zugesprochen wird. Jedoch werden auch mehr und mehr geographische Informationen von Laien gesammelt und im Netz veröffentlicht. Trotz der technischen Möglichkeiten, die im Web-2.0 geboten werden, ist es für Nutzer ohne entsprechendes Expertenwissen meist nicht möglich, gut lesbare und ansprechende Karten zu erzeugen. Dieses Problem besteht, da der Nutzer den darzustellenden Inhalt bestimmt, ohne dass überprüft wird, ob die daraus resultierende Karte den kartographischen Ansprüchen genügt. Des Weiteren hat der Nutzer keinerlei Möglichkeiten, die Standardkartenbilder zu bearbeiten, um beispielsweise für seine Thematik irrelevante Objekte auszublenden oder durch Verdrängung relevante Objekte freizustellen. Besonders bei der Darstellung von POIs treten Überlappungen zwischen Signaturen häufig auf. Trotz des Bedarfs existiert aus verschiedenen Gründen keine etablierte Methode zur Verdrängung von Punktdaten. Daher liegt der Schwerpunkt dieser Arbeit auf der Entwicklung von Verfahren zur Verdrängung von Punktsignaturen. Als Hilfsstrukturen werden dazu Voronoi-Diagramme benutzt und als nutzergenerierte Information werden Sentiments visualisiert. Für den Entwurf der Visualisierungen werden relevante kartographische Bedingungen berücksichtigt und durch zugehörige Qualitätsmaße bewertet. Für die Darstellung von Sentiments werden neben der Verwendung von Punktsignaturen zwei weitere Darstellungsarten erstellt: Anpassung gegebener Signaturen und die Darstellung von Sentiments als Kontinua. Es werden Verdrängungsverfahren für Punktsignaturen entworfen und implementiert. Zur Bestimmung der Verschiebungsrichtung werden zwei verschieden Heuristiken vorgeschlagen und untersucht. Des Weiteren wird eine Möglichkeit zur Steigerung der Effizienz durch Aufteilung der Punktmenge aufgezeigt. Die Bewertung der entworfenen Punktsignaturen erfolgt durch eine Umfrage. Anschließend wird das realisierte Verfahren für gleich große Kreissignaturen in drei Aspekten evaluiert: Grad der Reduzierung der Iterationsschritte durch Zerlegung der Punktmenge, erreichte Verminderung der Überlappungsfläche und Veränderung der relativen Lage der Punkte.A lot of user generated information accumulated in the web is related to a place, with the location usually being just one piece of information among many, which gets no special attention. However, more and more geographic information is collected by laymen and published on the web. Despite the technical possibilities that are offered in Web 2.0, it is usually not possible for users without expert knowledge to produce legible and appealing maps. This problem exists because the user determines the content to be displayed without checking whether the resulting map meets the cartographic requirements. Furthermore, the user has no possibilities to edit the standard map images, for example, hide for his subject irrelevant objects or reducing overlap of relevant objects by displacement. Especially when displaying POIs, overlaps between point symbols often occur. Despite the need, there is no established method for displacing point data for various reasons. Therefore, the focus of this work is the development of methods for the displacement of point signatures. Voronoi diagrams are used as auxiliary structures and sentiment is visualized as user-generated information. For the design of the visualizations relevant cartographic requirements are taken into account and evaluated by quality measures. For the depiction of sentiments, in addition to the use of point symbols, two further types of visualizations are created: adaptation of given map symbols and the representation of sentiments as continua. Displacement techniques for point symbols are designed and implemented. To determine the direction of displacement two different heuristics are proposed and examined. Furthermore, a way to increase the efficiency by dividing the point set is shown. The evaluation of the designed point symbols is done by a survey. Subsequently, the realized method for circular symbols with equal size is evaluated in three aspects: degree of reduction of the iteration steps by decomposition of the point set, achieved reduction of the overlap area and change of the relative position of the points

Optimal Voronoi tessellations with Hessian-based anisotropy

This paper presents a variational method to generate cell complexes with local anisotropy conforming to the Hessian of any given convex function and for any given local mesh density. Our formulation builds upon approximation theory to offer an anisotropic extension of Centroidal Voronoi Tessellations which can be seen as a dual form of Optimal Delaunay Triangulation. We thus refer to the resulting anisotropic polytopal meshes as Optimal Voronoi Tessellations. Our approach sharply contrasts with previous anisotropic versions of Voronoi diagrams as it employs first-type Bregman diagrams, a generalization of power diagrams where sites are augmented with not only a scalar-valued weight but also a vector-valued shift. As such, our OVT meshes contain only convex cells with straight edges, and admit an embedded dual triangulation that is combinatorially-regular. We show the effectiveness of our technique using off-the-shelf computational geometry libraries