21 research outputs found

    On the Optimality of Functionals over Triangulations of Delaunay Sets

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    In this short paper, we consider the functional density on sets of uniformly bounded triangulations with fixed sets of vertices. We prove that if a functional attains its minimum on the Delaunay triangulation, for every finite set in the plane, then for infinite sets the density of this functional attains its minimum also on the Delaunay triangulations

    Optimality of Delaunay Triangulations

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    In this paper, we begin by defining and examining the properties of a Voronoi diagram and extend it to its dual, the Delaunay triangulations. We explore the algorithms that construct such structures. Furthermore, we define several optimal functionals and criterions on the set of all triangulations of points in Rd that achieve their minimum on the Delaunay triangulation. We found a new result and proved that Delaunay triangulation has lexicographically the least circumradii sequence. We discuss the CircumRadii-Area (CRA) conjecture that the circumradii raised to the power of alpha times the area of the triangulation holds true for all α ≥ 1. We took it upon ourselves to prove that CRA conjecture is true for α =1, FRV quadrilaterals, and TRV quadrilaterals. Lastly, we demonstrate counterexamples for alpha\u3c1

    Well-Centered Triangulation

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    Meshes composed of well-centered simplices have nice orthogonal dual meshes (the dual Voronoi diagram). This is useful for certain numerical algorithms that prefer such primal-dual mesh pairs. We prove that well-centered meshes also have optimality properties and relationships to Delaunay and minmax angle triangulations. We present an iterative algorithm that seeks to transform a given triangulation in two or three dimensions into a well-centered one by minimizing a cost function and moving the interior vertices while keeping the mesh connectivity and boundary vertices fixed. The cost function is a direct result of a new characterization of well-centeredness in arbitrary dimensions that we present. Ours is the first optimization-based heuristic for well-centeredness, and the first one that applies in both two and three dimensions. We show the results of applying our algorithm to small and large two-dimensional meshes, some with a complex boundary, and obtain a well-centered tetrahedralization of the cube. We also show numerical evidence that our algorithm preserves gradation and that it improves the maximum and minimum angles of acute triangulations created by the best known previous method.Comment: Content has been added to experimental results section. Significant edits in introduction and in summary of current and previous results. Minor edits elsewher

    Hybrid cell-centred/vertex model for multicellular systems

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    This thesis presents a hybrid vertex/cell-centred approach to mechanically simulate planar cellular monolayers undergoing cell reorganisation. Cell centres are represented by a triangular nodal network, while the cell boundaries are formed by an associated vertex network. The two networks are coupled through a kinematic constraint which we allow to relax progressively. Cell-cell connectivity changes due to cell reorganisation or remodelling events, are accentuated. These situations are handled by using a variable resting length and applying an Equilibrium-Preserving Mapping (EPM) on the new connectivity, which computes a new set of resting lengths that preserve nodal and vertex equilibrium. As a by-product, the proposed technique enables to recover fully vertex or fully cell-centred models in a seamless manner by modifying a numerical parameter of the model. The properties of the model are illustrated by simulating monolayers subjected to imposed extension and during a wound healing process. The evolution of forces and the EPM are analysed during the remodelling events.Esta tesis presenta un modelo híbrido para la simulación mecánica de monocapas celulares. Este modelo combina métodos de vértices y centrados en la célula, y está orientado al análisis de deformaciones con reorganización celular. Los núcleos vienen representados por nodos que forman una malla triangular, mientras que las contornos (membranas y córtex) forman una malla poligonal de vértices. Las dos mallas se acoplan a través de una restricción cinemática que puede ser relajada de forma controlada. El estudio hace especial hincapié en los cambios de conectividad, tanto debidos a la reorganización celular como el remodelado del citoesqueleto. Estas situaciones se abordan a través de una longitud de referencia variable y aplicando un Mapeo con Conservación de Equilibrio (EPM) que minimiza el error en el equilibrio nodal y en los vértices. La técnica resultante puede ser adaptada progresivamente a través de un parámetro, dando lugar a un modelo exclusivamente de vértices o a uno de centros. Sus propiedades se ilustran en simulaciones de monocapas sujetas a una extensión impuesta y durante el proceso de cicatrizado de heridas. La evolución de las fuerzas y los efectos del EPM durante el remodelado se analizan en estos ejemplos

    Higher signature Delaunay decompositions

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    A Delaunay decomposition is a cell decomposition in R^d for which each cell is inscribed in a Euclidean ball which is empty of all other vertices. This article introduces a generalization of the Delaunay decomposition in which the Euclidean balls in the empty ball condition are replaced by other families of regions bounded by certain quadratic hypersurfaces. This generalized notion is adaptable to geometric contexts in which the natural space from which the point set is sampled is not Euclidean, but rather some other flat semi-Riemannian geometry, possibly with degenerate directions. We prove the existence and uniqueness of the decomposition and discuss some of its basic properties. In the case of dimension d = 2, we study the extent to which some of the well-known optimality properties of the Euclidean Delaunay triangulation generalize to the higher signature setting. In particular, we describe a higher signature generalization of a well-known description of Delaunay decompositions in terms of the intersection angles between the circumscribed circles.Comment: 25 pages, 6 figure

    Topology-preserving watermarking of vector graphics

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    Watermarking techniques for vector graphics dislocate vertices in order to embed imperceptible, yet detectable, statistical features into the input data. The embedding process may result in a change of the topology of the input data, e.g., by introducing self-intersections, which is undesirable or even disastrous for many applications. In this paper we present a watermarking framework for two-dimensional vector graphics that employs conventional watermarking techniques but still provides the guarantee that the topology of the input data is preserved. The geometric part of this framework computes so-called maximum perturbation regions (MPR) of vertices. We propose two efficient algorithms to compute MPRs based on Voronoi diagrams and constrained triangulations. Furthermore, we present two algorithms to conditionally correct the watermarked data in order to increase the watermark embedding capacity and still guarantee topological correctness. While we focus on the watermarking of input formed by straight-line segments, one of our approaches can also be extended to circular arcs. We conclude the paper by demonstrating and analyzing the applicability of our framework in conjunction with two well-known watermarking techniques
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