Multicriteria-optimized triangulations

Kolingerová, Ivana; Ferko, Andrej

doi:10.1007/s003710100125

Cited by 11 publications

(6 citation statements)

References 38 publications

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“…Evolutionary algorithms for remeshing Evolutionary algorithms are powerful for computing approximate solutions to all types of problems. They have been successfully applied to many remeshing problems [YLH18,LXY ∗ 16,CW99,HH03,QWG97,KF01]. A sequence of edge operations are optimized by differential evolution to simplify the input mesh to a Delaunay mesh containing a specified number of vertices [YLH18].…”

Section: Related Workmentioning

confidence: 99%

Constrained Remeshing Using Evolutionary Vertex Optimization

Zhang

Wang

Guo

et al. 2022

Computer Graphics Forum

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We propose a simple yet effective method to perform surface remeshing with hard constraints, such as bounding approximation errors and ensuring Delaunay conditions. The remeshing is formulated as a constrained optimization problem, where the variables contain the mesh connectivity and the mesh geometry. To solve it effectively, we adopt traditional local operations, including edge split, edge collapse, edge flip, and vertex relocation, to update the variables. Central to our method is an evolutionary vertex optimization algorithm, which is derivative‐free and robust. The feasibility and practicability of our method are demonstrated in two applications, including error‐bounded Delaunay mesh simplification and error‐bounded angle improvement with a given number of vertices, over many models. Compared to state‐of‐the‐art methods, our method achieves higher remeshing quality.

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Section: Related Workmentioning

confidence: 99%

Constrained Remeshing Using Evolutionary Vertex Optimization

Zhang

Wang

Guo

et al. 2022

Computer Graphics Forum

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“…6 simulation data for narrow lattice stripes are compared to the exact results. All measurements are averaged over 5 independent runs, for large lattices an energy cut-off (5) with the empirical high = 0.75 is used. While the accuracy of single Wang-Landau simulations is limited and the density of states does not change after reaching a certain modification factor (saturation of error) [38,39], for almost all considered system sizes the relative error of the simulation data is below 10 −3 , so a systematic error can be neglected.…”

Section: Energy Cut-offs and Initial Estimatesmentioning

confidence: 99%

“…-Triangulations of spaces are relevant for a broad range of physical phenomena. They serve as discretisation of all kinds of surfaces, hypersurfaces and volumes [1], yielding applications of computational geometry in physics, material science, medical image processing or even in computer graphics and visualisation [2][3][4][5][6]. Many physical systems can be described by random surface models [7] -based on random triangulations.…”

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confidence: 99%

Entropy of unimodular lattice triangulations

Knauf¹,

Krüger²,

Mecke³

2015

EPL

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Triangulations are important objects of study in combinatorics, finite element simulations and quantum gravity, where its entropy is crucial for many physical properties. Due to their inherent complex topological structure even the number of possible triangulations is unknown for large systems. We present a novel algorithm for an approximate enumeration which is based on calculations of the density of states using the Wang-Landau flat histogram sampling. For triangulations on two-dimensional integer lattices we achive excellent agreement with known exact numbers of small triangulations as well as an improvement of analytical calculated asymptotics. The entropy density is C = 2.196(3) consistent with rigorous upper and lower bounds. The presented numerical scheme can easily be applied to other counting and optimization problems.Introduction. -Triangulations of spaces are relevant for a broad range of physical phenomena. They serve as discretisation of all kinds of surfaces, hypersurfaces and volumes [1], yielding applications of computational geometry in physics, material science, medical image processing or even in computer graphics and visualisation [2][3][4][5][6]. Many physical systems can be described by random surface models [7] -based on random triangulations. For instance, biological membranes and vesicles can be modelled using triangulated surfaces with curvature-dependent Hamiltonians [8][9][10][11][12][13].Triangulations are also used as a random graph model for real world networks: Random Apollonian networks [14][15][16], which are the dual graphs of classical Apollonian packed granular matter and therewith triangulations, show both small-world and scale-free behaviour, as many real world networks. The properties of triangulations of closed surfaces with arbitrary genus are of much interest, since each graph can be embedded into a closed surface with high enough genus [17,18].The (Causal) Dynamical Triangulation approach even tries to describe quantum gravity from scratch with an ensemble of random space-time triangulations as their central entity [19]. Based on a discrete version of general relativity, where spacetime is approximated by triangles or higher-dimensional analogues, the curvatures become determined purly by the topological structure of the underlying triangulation, e.g. the number of triangles. The resulting action of the theory can be used to extract a

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“…Triangulations are one of the key topics for computer graphics. Therefore, it is recommended to present not only the traditional Delaunay triangulation and greedy triangulations, but also triangulations useful for special applications, such as constrained triangulations [Ang97], [Slo93], data dependent triangulations [DLR90] or multicriteria-optimised triangulations [KF01] and to point out their special properties. For Delaunay triangulation (as well as for Voronoi diagrams), algorithms based on nearly all algorithmic strategies have been developed and parallel algorithms of different categories exist, this topic can be used to show all common and general features of the particular algorithmic strategies, so that the students are able to use them for problems which they may face in the future.…”

Section: Planar Triangulationsmentioning

confidence: 99%

Computational Geometry Education for Computer Graphics Students

Kolingerová

2008

Computer Graphics Forum

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The paper surveys main features of computational geometry and presents the argument that a course oriented to applied computational geometry should be a part of the computer graphics curriculum, as it teaches effective algorithmic methods and helps to develop abstract thinking. Possible contents of the course and forms suitable and interesting for computer graphics students are discussed. The students' feedback on such a course has been mostly positive.Computational geometry is the study of algorithms and data structures for geometrically formulated problems. Typical tasks solved by computational geometry are convex hulls, point location, intersections, visibility, triangulations and other partitions in 2D and 3D, motion planning. These problems are inspired by applied sciences, such as robotics, databases, image and pattern recognition, cartography, GIS, biology, civil engineering and many others; see a more detailed list in [App96]. However, most of the problems in the scope of computational geometry are inspired by computer graphics. With some simplification, computational geometry can be understood as fundamental research for computer graphics.The main advantages of this scientific discipline are its exactness in language as well as in algorithms, the systematic use of complexity evaluation, beauty of its algorithms and † supported by Ministry of Education, Youth and Sports of the Czech Republic -project No. LC 06008 efficiency of the proposed solutions. Like mathematics, computational geometry not only teaches tools but also develops abstract reasoning.Computational geometry has also its drawbacks, such as its high abstraction, which causes difficulties to more practically oriented engineers. If an engineer unused to the high formalism of computational geometry tries to read papers from this area, they may not get too much of it because of 'somebody else's language', an absence of illustrations, sometimes also an absence of implementation results. Computational geometry sometimes also shows an unhealthy distance from practical life (it usually does not care whether the solved problem is academic or not). Due to this, computational geometry has a low influence on applied research and solutions to practical problems. Asymptotically optimised algorithms which are the focus of computational geometry may not necessarily work well in practice -if the question of expected complexity and expected type of input data are not properly considered in the algorithm proposal. The algorithms can depend on complicated data structures or on theoretically sound but never implemented algorithms.Special cases are often not discussed or the published algorithm has been proved but not implemented.Most of these negative features are gradually disappearing with the growing maturity of the computational geometry discipline. Experts in computational geometry have worked hard to cure the problems. Computational geometry has already grown to the stage where it is able to serve applied sciences as a marvellous tool, see [deB97], [Dob92]....

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Multicriteria-optimized triangulations

Cited by 11 publications

References 38 publications

Constrained Remeshing Using Evolutionary Vertex Optimization

Constrained Remeshing Using Evolutionary Vertex Optimization

Entropy of unimodular lattice triangulations

Computational Geometry Education for Computer Graphics Students

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