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Artykuły w czasopismach na temat „Discrete Voronoi Diagram”

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Data publikacji: 6 września 2023

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1

Wang, Hui. "Discrete Construction of Compoundly Weighted Voronoi Diagram". Applied Mechanics and Materials 467 (grudzień 2013): 545–48. http://dx.doi.org/10.4028/www.scientific.net/amm.467.545.

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Compoundly weighted Voronoi diagram is difficult to construct because the bisector is fairly complex. In traditional algorithm, production process is always extremely complex and it is more difficult to graphic display because of the complex definition of mathematic formula. In this paper, discrete algorithms are used to construct compoundly weighted Voronoi diagrams. The algorithm can get over all kinds of shortcomings that we have just mentioned. So it is more useful and effective than the traditional algorithm. The results show that the algorithm is both simple and useful, and it is of high potential value in practice.

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2

Kang, Shun. "A Generic Statistics-Based Tessellation Method of Voronoi Diagram". Journal of Systems Science and Information 3, nr 6 (25.12.2015): 568–76. http://dx.doi.org/10.1515/jssi-2015-0568.

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AbstractIn terms of distance function and spatial continuity in Voronoi diagram, a generic generating method of Voronoi diagram, named statistical Voronoi diagram, is proposed in this paper based upon statistics with mean vector and covariance matrix. Besides, in order to make good on the discreteness of spatial Voronoi cell, the cross Voronoi cell accomplished the discrete ranges in its continuous domain. In the light of Mahalanobis distance, not only ordinary Voronoi and weighted Voronoi are implemented, but also the theory of Voronoi diagram is improved further. Last but not least, through Gaussian distribution on spatial data, the validation and soundness of this method are proofed by empirical results.

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3

Chaidee, S., P. Pakawanwong, V. Suppakitpaisarn i P. Teerasawat. "INTERACTIVE LAND-USE OPTIMIZATION USING LAGUERRE VORONOI DIAGRAM WITH DYNAMIC GENERATING POINT ALLOCATION". ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XLII-2/W7 (14.09.2017): 1091–98. http://dx.doi.org/10.5194/isprs-archives-xlii-2-w7-1091-2017.

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In this work, we devise an efficient method for the land-use optimization problem based on Laguerre Voronoi diagram. Previous Voronoi diagram-based methods are more efficient and more suitable for interactive design than discrete optimization-based method, but, in many cases, their outputs do not satisfy area constraints. To cope with the problem, we propose a force-directed graph drawing algorithm, which automatically allocates generating points of Voronoi diagram to appropriate positions. Then, we construct a Laguerre Voronoi diagram based on these generating points, use linear programs to adjust each cell, and reconstruct the diagram based on the adjustment. We adopt the proposed method to the practical case study of Chiang Mai University’s allocated land for a mixed-use complex. For this case study, compared to other Voronoi diagram-based method, we decrease the land allocation error by 62.557&thinsp;%. Although our computation time is larger than the previous Voronoi-diagram-based method, it is still suitable for interactive design.

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4

Chen, Jie, Gang Yang i Meng Yang. "Computation of Compact Distributions of Discrete Elements". Algorithms 12, nr 2 (18.02.2019): 41. http://dx.doi.org/10.3390/a12020041.

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In our daily lives, many plane patterns can actually be regarded as a compact distribution of a number of elements with certain shapes, like the classic pattern mosaic. In order to synthesize this kind of pattern, the basic problem is, with given graphics elements with certain shapes, to distribute a large number of these elements within a plane region in a possibly random and compact way. It is not easy to achieve this because it not only involves complicated adjacency calculations, but also is closely related to the shape of the elements. This paper attempts to propose an approach that can effectively and quickly synthesize compact distributions of elements of a variety of shapes. The primary idea is that with the seed points and distribution region given as premise, the generation of the Centroidal Voronoi Tesselation (CVT) of this region by iterative relaxation and the CVT will partition the distribution area into small regions of Voronoi, with each region representing the space of an element, to achieve a compact distribution of all the elements. In the generation process of Voronoi diagram, we adopt various distance metrics to control the shape of the generated Voronoi regions, and finally achieve the compact element distributions of different shapes. Additionally, approaches are introduced to control the sizes and directions of the Voronoi regions to generate element distributions with size and direction variations during the Voronoi diagram generation process to enrich the effect of compact element distributions. Moreover, to increase the synthesis efficiency, the time-consuming Voronoi diagram generation process was converted into a graphical rendering process, thus increasing the speed of the synthesis process. This paper is an exploration of elements compact distribution and also carries application value in the fields like mosaic pattern synthesis.

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5

Oku, Toshinobu. "Characteristics of Discrete Voronoi Diagram with Ambiguous Boundaries". Journal of the City Planning Institute of Japan 42.3 (2007): 463–68. http://dx.doi.org/10.11361/journalcpij.42.3.463.

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6

OKU, TOSHINOBU. "Characteristics of Discrete Voronoi Diagram with Ambiguous Boundaries". Journal of the City Planning Institute of Japan 42 (2007): 78. http://dx.doi.org/10.11361/cpij1.42.0.78.0.

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7

Vozáb, Jan, i Jan Vorel. "Generation of LDPM structure formed by Voronoi cells". Acta Polytechnica CTU Proceedings 40 (24.07.2023): 111–16. http://dx.doi.org/10.14311/app.2023.40.0111.

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A preliminary study of an approach to internal structure generation used in lattice discrete particle models (LDPMs) [1]. The presented method used for particle generation and placement is intended to help realistically capture the internal structure of materials. First, a method for structure generation using LDPM is presented. Then, the method of particle generation using a Voronoi diagram [2] is described. The last part is the optimizations on the algorithm that use Apollonius circles to calculate the specific points of the Voronoi diagram.

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8

BEREG, SERGEY, KEVIN BUCHIN, MAIKE BUCHIN, MARINA GAVRILOVA i BINHAI ZHU. "VORONOI DIAGRAM OF POLYGONAL CHAINS UNDER THE DISCRETE FRÉCHET DISTANCE". International Journal of Computational Geometry & Applications 20, nr 04 (sierpień 2010): 471–84. http://dx.doi.org/10.1142/s0218195910003396.

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Polygonal chains are fundamental objects in many applications like pattern recognition and protein structure alignment. A well-known measure to characterize the similarity of two polygonal chains is the (continuous/discrete) Fréchet distance. In this paper, for the first time, we consider the Voronoi diagram of polygonal chains in d-dimension under the discrete Fréchet distance. Given a set [Formula: see text] of n polygonal chains in d-dimension, each with at most k vertices, we prove fundamental properties of such a Voronoi diagram [Formula: see text]. Our main results are summarized as follows. • The combinatorial complexity of [Formula: see text] is at most O(ndk+∊). • The combinatorial complexity of [Formula: see text] is at least Ω(ndk) for dimension d = 1, 2; and Ω(nd(k-1)+2) for dimension d > 2.

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9

Lu, Xiaomin, Haowen Yan, Wende Li, Xiaojun Li i Fang Wu. "An Algorithm based on the Weighted Network Voronoi Diagram for Point Cluster Simplification". ISPRS International Journal of Geo-Information 8, nr 3 (27.02.2019): 105. http://dx.doi.org/10.3390/ijgi8030105.

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Points on maps that stand for geographic objects such as settlements are generally connected by road networks. However, in the existing algorithms for point cluster simplification, points are usually viewed as discrete objects or their distances are considered in Euclidean spaces, and therefore the point cluster generalization results obtained by these algorithms are sometimes unreasonable. To take roads into consideration so that point clusters can be simplified in appropriate ways, the network Voronoi diagram is used and a new algorithm is proposed in this paper. First, the weighted network Voronoi diagram is constructed taking into account the weights of the points and the properties of the related road segments. Second, the network Voronoi polygons are generated and two factors (i.e., the area of the network Voronoi polygon and the total length of the dilated road segments in the polygon) are considered as the basis for point simplification. Last, a Cartesian coordinate system is built based on the two factors and the point clusters are simplified by means of the “concentric quadrants”. Our experiments show that the algorithm can effectively and correctly transmit types of information in the process of point cluster simplification, and the results are more reasonable than that generated by the ordinary Voronoi-based algorithm and the weighted Voronoi-based algorithm.

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10

Alt, Helmut, Otfried Cheong i Antoine Vigneron. "The Voronoi Diagram of Curved Objects". Discrete & Computational Geometry 34, nr 3 (4.08.2005): 439–53. http://dx.doi.org/10.1007/s00454-005-1192-0.

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11

Everett, Hazel, Daniel Lazard, Sylvain Lazard i Mohab Safey El Din. "The Voronoi Diagram of Three Lines". Discrete & Computational Geometry 42, nr 1 (28.04.2009): 94–130. http://dx.doi.org/10.1007/s00454-009-9173-3.

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12

Aronov, Boris, Steven Fortune i Gordon Wilfong. "The furthest-site geodesic voronoi diagram". Discrete & Computational Geometry 9, nr 3 (marzec 1993): 217–55. http://dx.doi.org/10.1007/bf02189321.

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13

BASCH, JULIEN, HARISH DEVARAJAN, PIOTR INDYK i LI ZHANG. "PROBABILISTIC ANALYSIS FOR DISCRETE ATTRIBUTES OF MOVING POINTS". International Journal of Computational Geometry & Applications 13, nr 01 (luty 2003): 5–22. http://dx.doi.org/10.1142/s0218195903001050.

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We perform a probabilistic study of discrete attributes of moving points. In our probabilistic model, an item is given an initial position and a velocity drawn independently at random from the same distribution. We study the expected number of changes that happen to the closest pair, the Voronoi diagram, and the convex hull of a set of such moving items.

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14

Maniatty, W. A., i B. K. Szymanski. "Fine-grain discrete Voronoi diagram algorithms in L1 and L∞ norms". Mathematical and Computer Modelling 26, nr 4 (sierpień 1997): 71–78. http://dx.doi.org/10.1016/s0895-7177(97)00145-3.

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15

Jia, Kang, Jun Hong, Yun Long Li i Zong Bin Li. "Research on Constructing an Approximate Topological Graph and its Path Planning". Applied Mechanics and Materials 44-47 (grudzień 2010): 596–604. http://dx.doi.org/10.4028/www.scientific.net/amm.44-47.596.

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This paper proposes a new method to construct an approximate Topological Graph based on the traditional constructing voronoi diagram by discrete grids, which combines the PRM (Probabilistic Roadmap Method) path planning thought. By means of the approximate topological graph and the hierarchical thought, the efficiency of path planning is improved. Meanwhile, this method can reduce the unnecessary collision detection and can be effectively used in the path planning of mechanical produce. In the end, this method is analyzed and evaluated, and meanwhile some improvements and simulations about this method are given.

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16

Meethongjan, Khitikun, Mohamad Dzulkifli, Amjad Rehman, Ayman Altameem i Tanzila Saba. "An Intelligent Fused Approach for Face Recognition". Journal of Intelligent Systems 22, nr 2 (1.06.2013): 197–212. http://dx.doi.org/10.1515/jisys-2013-0010.

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AbstractFace detection plays important roles in many applications such as human–computer interaction, security and surveillance, face recognition, etc. This article presents an intelligent enhanced fused approach for face recognition based on the Voronoi diagram (VD) and wavelet moment invariants. Discrete wavelet transform and moment invariants are used for feature extraction of the facial face. Finally, VD and the dual tessellation (Delaunay triangulation, DT) are used to locate and detect original face images. Face recognition results based on this new fusion are promising in the state of the art.

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17

Kalantari, Bahman. "Polynomial Root-Finding Methods Whose Basins of Attraction Approximate Voronoi Diagram". Discrete & Computational Geometry 46, nr 1 (18.02.2011): 187–203. http://dx.doi.org/10.1007/s00454-011-9330-3.

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18

Sudha, N., i K. Sridharan. "A High-Speed VLSI Design and ASIC Implementation for Constructing Euclidean Distance-Based Discrete Voronoi Diagram". IEEE Transactions on Robotics and Automation 20, nr 2 (kwiecień 2004): 352–58. http://dx.doi.org/10.1109/tra.2004.824638.

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19

Arseneva, Elena, i Evanthia Papadopoulou. "Randomized incremental construction for the Hausdorff Voronoi diagram revisited and extended". Journal of Combinatorial Optimization 37, nr 2 (20.09.2018): 579–600. http://dx.doi.org/10.1007/s10878-018-0347-x.

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20

Chaidee, Supanut, i Kokichi Sugihara. "Laguerre Voronoi Diagram as a Model for Generating the Tessellation Patterns on the Sphere". Graphs and Combinatorics 36, nr 2 (14.01.2019): 371–85. http://dx.doi.org/10.1007/s00373-019-02006-5.

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21

McAllister, M., D. Kirkpatrick i J. Snoeyink. "A compact piecewise-linear voronoi diagram for convex sites in the plane". Discrete & Computational Geometry 15, nr 1 (styczeń 1996): 73–105. http://dx.doi.org/10.1007/bf02716580.

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22

Aggarwal, Alok, Leonidas J. Guibas, James Saxe i Peter W. Shor. "A linear-time algorithm for computing the voronoi diagram of a convex polygon". Discrete & Computational Geometry 4, nr 6 (grudzień 1989): 591–604. http://dx.doi.org/10.1007/bf02187749.

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23

Yap, Chee K. "AnO(n logn) algorithm for the voronoi diagram of a set of simple curve segments". Discrete & Computational Geometry 2, nr 4 (grudzień 1987): 365–93. http://dx.doi.org/10.1007/bf02187890.

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24

Fischer, A., A. Smolin i G. Elber. "Mid-Surfaces of Profile-based Freeforms for Mold Design". Journal of Manufacturing Science and Engineering 121, nr 2 (1.05.1999): 202–7. http://dx.doi.org/10.1115/1.2831206.

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Mid-surfaces of complex thin objects are commonly used in CAD applications for the analysis of casting and injection molding. However, geometrical representation in CAD typically takes the form of a solid representation rather than a mid-surface; therefore, a process for extracting the mid-surface is essential. Contemporary methods for extracting mid-surfaces are based on numerical computations using offsetting techniques or Voronoi diagram processes where the data is discrete and piecewise linear. These algorithms usually have high computational complexity, and their accuracy is not guaranteed. Furthermore, the geometry and topology of the object are not always preserved. To overcome these problems, this paper proposes a new approach for extracting a mid-surface from a freeform thin object. The proposed method reduces the mid-surface problem into a parametrization problem that is based on a matching technique in which a nonlinear optimization function is defined and solved according to mid-surface criteria. Then, the resulting mid-surface is dictated by a reparametrization process. The algorithm is implemented for freeform ruled, swept, and rotational surfaces, that are commonly used in engineering products. Reducing the problem to the profile curves of these surfaces alleviates the computational complexity of the 3D case and restricts it to a 2D case. Error is controlled globally through an iterative refinement process that utilizes continuous symbolic computations on the parametric representation. The feasibility of the proposed method is demonstrated through several examples.

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25

Zhao, W., X. Tong, H. Xie, Y. Jin, S. Liu, D. Wu, X. Liu, L. Guo i Q. Zhou. "SIMULATION EXPERIMENT ON LANDING SITE SELECTION USING A SIMPLE GEOMETRIC APPROACH". ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XLII-3/W1 (25.07.2017): 213–18. http://dx.doi.org/10.5194/isprs-archives-xlii-3-w1-213-2017.

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Safe landing is an important part of the planetary exploration mission. Even fine scale terrain hazards (such as rocks, small craters, steep slopes, which would not be accurately detected from orbital reconnaissance) could also pose a serious risk on planetary lander or rover and scientific instruments on-board it. In this paper, a simple geometric approach on planetary landing hazard detection and safe landing site selection is proposed. In order to achieve full implementation of this algorithm, two easy-to-compute metrics are presented for extracting the terrain slope and roughness information. Unlike conventional methods which must do the robust plane fitting and elevation interpolation for DEM generation, in this work, hazards is identified through the processing directly on LiDAR point cloud. For safe landing site selection, a Generalized Voronoi Diagram is constructed. Based on the idea of maximum empty circle, the safest landing site can be determined. In this algorithm, hazards are treated as general polygons, without special simplification (e.g. regarding hazards as discrete circles or ellipses). So using the aforementioned method to process hazards is more conforming to the real planetary exploration scenario. For validating the approach mentioned above, a simulated planetary terrain model was constructed using volcanic ash with rocks in indoor environment. A commercial laser scanner mounted on a rail was used to scan the terrain surface at different hanging positions. The results demonstrate that fairly hazard detection capability and reasonable site selection was obtained compared with conventional method, yet less computational time and less memory usage was consumed. Hence, it is a feasible candidate approach for future precision landing selection on planetary surface.

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26

Boissonnat, Jean-Daniel, Mael Rouxel-Labbé i Mathijs H. M. J. Wintraecken. "Anisotropic Triangulations via Discrete Riemannian Voronoi Diagrams". SIAM Journal on Computing 48, nr 3 (styczeń 2019): 1046–97. http://dx.doi.org/10.1137/17m1152292.

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27

Kim, K., S. Lee i H. B. Jung. "Assessing Roundness Errors Using Discrete Voronoi Diagrams". International Journal of Advanced Manufacturing Technology 16, nr 8 (3.07.2000): 559–63. http://dx.doi.org/10.1007/s001700070045.

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28

Ustimenko, V. "On small world non-Sunada twins and cellular Voronoi diagrams". Algebra and Discrete Mathematics 30, nr 1 (2020): 118–42. http://dx.doi.org/10.12958/adm1343.

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Special infinite families of regular graphs of unbounded degree and of bounded diameter (small world graphs) are considered. Two families of small world graphs Gi and Hi form a family of non-Sunada twins if Gi and Hi are isospectral of bounded diameter but groups Aut(Gi) and Aut(Hi) are nonisomorphic. We say that a family of non-Sunada twins is unbalanced if each Gi is edge-transitive but each Hi is edge-intransitive. If all Gi and Hi are edge-transitive we have a balanced family of small world non-Sunada twins. We say that a family of non-Sunada twins is strongly unbalanced if each Gi is edge-transitive but each Hi is edge-intransitive. We use term edge disbalanced for the family of non-Sunada twins such that all graphs Gi and Hi are edge-intransitive. We present explicit constructions of the above defined families. Two new families of distance-regular—but not distance-transitive—graphs will be introduced.

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29

Canny, John, i Bruce Donald. "Simplified Voronoi diagrams". Discrete & Computational Geometry 3, nr 3 (wrzesień 1988): 219–36. http://dx.doi.org/10.1007/bf02187909.

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30

Allen, Sarah R., Luis Barba, John Iacono i Stefan Langerman. "Incremental Voronoi Diagrams". Discrete & Computational Geometry 58, nr 4 (4.10.2017): 822–48. http://dx.doi.org/10.1007/s00454-017-9943-2.

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31

Edelsbrunner, Herbert, i Raimund Seidel. "Voronoi diagrams and arrangements". Discrete & Computational Geometry 1, nr 1 (marzec 1986): 25–44. http://dx.doi.org/10.1007/bf02187681.

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32

Velić, Mirko, Dave May i Louis Moresi. "A Fast Robust Algorithm for Computing Discrete Voronoi Diagrams". Journal of Mathematical Modelling and Algorithms 8, nr 3 (16.12.2008): 343–55. http://dx.doi.org/10.1007/s10852-008-9097-6.

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33

Canas, Guillermo D., i Steven J. Gortler. "Orphan-Free Anisotropic Voronoi Diagrams". Discrete & Computational Geometry 46, nr 3 (4.08.2011): 526–41. http://dx.doi.org/10.1007/s00454-011-9372-6.

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34

Rosenthal, Paul, i Lars Linsen. "Enclosing Surfaces for Point Clusters Using 3D Discrete Voronoi Diagrams". Computer Graphics Forum 28, nr 3 (czerwiec 2009): 999–1006. http://dx.doi.org/10.1111/j.1467-8659.2009.01448.x.

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35

Sequeira, R. E., i F. J. Preteux. "Discrete Voronoi diagrams and the SKIZ operator: a dynamic algorithm". IEEE Transactions on Pattern Analysis and Machine Intelligence 19, nr 10 (1997): 1165–70. http://dx.doi.org/10.1109/34.625128.

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36

Sud, Avneesh, Naga Govindaraju, Russell Gayle, Ilknur Kabul i Dinesh Manocha. "Fast proximity computation among deformable models using discrete Voronoi diagrams". ACM Transactions on Graphics 25, nr 3 (lipiec 2006): 1144–53. http://dx.doi.org/10.1145/1141911.1142006.

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37

Aurenhammer, F. "A relationship between Gale transforms and Voronoi diagrams". Discrete Applied Mathematics 28, nr 2 (sierpień 1990): 83–91. http://dx.doi.org/10.1016/0166-218x(90)90108-o.

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38

Mehlhorn, K., St Meiser i C. Ó'Dúnlaing. "On the construction of abstract voronoi diagrams". Discrete & Computational Geometry 6, nr 2 (czerwiec 1991): 211–24. http://dx.doi.org/10.1007/bf02574686.

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39

Aurenhammer, Franz. "A new duality result concerning voronoi diagrams". Discrete & Computational Geometry 5, nr 3 (czerwiec 1990): 243–54. http://dx.doi.org/10.1007/bf02187788.

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40

Dwyer, R. A. "Voronoi diagrams of random lines and flats". Discrete & Computational Geometry 17, nr 2 (marzec 1997): 123–36. http://dx.doi.org/10.1007/bf02770869.

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41

Yin, Quanjun, Long Qin, Xiaocheng Liu i Yabing Zha. "Incremental Construction of Generalized Voronoi Diagrams on Pointerless Quadtrees". Mathematical Problems in Engineering 2014 (2014): 1–14. http://dx.doi.org/10.1155/2014/456739.

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In robotics, Generalized Voronoi Diagrams (GVDs) are widely used by mobile robots to represent the spatial topologies of their surrounding area. In this paper we consider the problem of constructing GVDs on discrete environments. Several algorithms that solve this problem exist in the literature, notably the Brushfire algorithm and its improved versions which possess local repair mechanism. However, when the area to be processed is very large or is of high resolution, the size of the metric matrices used by these algorithms to compute GVDs can be prohibitive. To address this issue, we propose an improvement on the current algorithms, using pointerless quadtrees in place of metric matrices to compute and maintain GVDs. Beyond the construction and reconstruction of a GVD, our algorithm further provides a method to approximate roadmaps in multiple granularities from the quadtree based GVD. Simulation tests in representative scenarios demonstrate that, compared with the current algorithms, our algorithm generally makes an order of magnitude improvement regarding memory cost when the area is larger than210×210. We also demonstrate the usefulness of the approximated roadmaps for coarse-to-fine pathfinding tasks.

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42

Valette, S., J. M. Chassery i R. Prost. "Generic Remeshing of 3D Triangular Meshes with Metric-Dependent Discrete Voronoi Diagrams". IEEE Transactions on Visualization and Computer Graphics 14, nr 2 (marzec 2008): 369–81. http://dx.doi.org/10.1109/tvcg.2007.70430.

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43

Dwyer, Rex A. "Higher-dimensional voronoi diagrams in linear expected time". Discrete & Computational Geometry 6, nr 3 (wrzesień 1991): 343–67. http://dx.doi.org/10.1007/bf02574694.

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44

Koltun, Vladlen, i Micha Sharir. "Polyhedral Voronoi Diagrams of Polyhedra in Three Dimensions". Discrete and Computational Geometry 31, nr 1 (1.01.2004): 83–124. http://dx.doi.org/10.1007/s00454-003-2950-5.

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45

Barequet, G., M. T. Dickerson i M. T. Goodrich. "Voronoi diagrams for convex polygon-offset distance functions". Discrete & Computational Geometry 25, nr 2 (marzec 2001): 271–91. http://dx.doi.org/10.1007/s004540010081.

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46

Gafarova, Yu A. "Numerical implementation of finite-element method of control volume using irregular mesh". Proceedings of the Mavlyutov Institute of Mechanics 7 (2010): 98–108. http://dx.doi.org/10.21662/uim2010.1.008.

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To solve problems with complex geometry it is considered the possibility of application of irregular mesh and the use of various numerical methods using them. Discrete analogues of the Beltrami-Mitchell equations are obtained by the control volume method using the rectangular grid and the finite element method of control volume using the Delaunay triangulation. The efficiency of using the Delaunay triangulation, Voronoi diagrams and the finite element method of control volume in a test case is demonstrated.

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Lê, Ngoc-Minh. "On voronoi diagrams in theL p-metric in ℝD". Discrete & Computational Geometry 16, nr 2 (luty 1996): 177–96. http://dx.doi.org/10.1007/bf02716806.

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Dwyer, R. A. "Voronoi Diagrams and Convex Hulls of Random Moving Points". Discrete & Computational Geometry 23, nr 3 (marzec 2000): 343–65. http://dx.doi.org/10.1007/pl00009505.

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Kim, K., S. Lee, Y. Choi i J. Jeong. "Determining a Pair of Concentric Spheres for Assessing Sphericity Errors Using Discrete Voronoi Diagrams". International Journal of Advanced Manufacturing Technology 16, nr 10 (21.08.2000): 728–32. http://dx.doi.org/10.1007/s001700070025.

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Kühn, Ulrich. "A randomized parallel algorithm for Voronoi diagrams based on symmetric convex distance functions". Discrete Applied Mathematics 109, nr 1-2 (kwiecień 2001): 177–96. http://dx.doi.org/10.1016/s0166-218x(00)00235-3.

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