Zeitschriftenartikel: „Hausdorff Distance“

1

Wu, Wei. „Quantized Gromov–Hausdorff distance“. Journal of Functional Analysis 238, Nr. 1 (September 2006): 58–98. http://dx.doi.org/10.1016/j.jfa.2005.02.017.

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2

Kraft, Daniel. „Computing the Hausdorff Distance of Two Sets from Their Distance Functions“. International Journal of Computational Geometry & Applications 30, Nr. 01 (März 2020): 19–49. http://dx.doi.org/10.1142/s0218195920500028.

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The Hausdorff distance is a measure of (dis-)similarity between two sets which is widely used in various applications. Most of the applied literature is devoted to the computation for sets consisting of a finite number of points. This has applications, for instance, in image processing. However, we would like to apply the Hausdorff distance to control and evaluate optimisation methods in level-set based shape optimisation. In this context, the involved sets are not finite point sets but characterised by level-set or signed distance functions. This paper discusses the computation of the Hausdorff distance between two such sets. We recall fundamental properties of the Hausdorff distance, including a characterisation in terms of distance functions. In numerical applications, this result gives at least an exact lower bound on the Hausdorff distance. We also derive an upper bound, and consequently a precise error estimate. By giving an example, we show that our error estimate cannot be further improved for a general situation. On the other hand, we also show that much better accuracy can be expected for non-pathological situations that are more likely to occur in practice. The resulting error estimate can be improved even further if one assumes that the grid is rotated randomly with respect to the involved sets.

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Ali, Mehboob, Zahid Hussain und Miin-Shen Yang. „Hausdorff Distance and Similarity Measures for Single-Valued Neutrosophic Sets with Application in Multi-Criteria Decision Making“. Electronics 12, Nr. 1 (31.12.2022): 201. http://dx.doi.org/10.3390/electronics12010201.

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Hausdorff distance is one of the important distance measures to study the degree of dissimilarity between two sets that had been used in various fields under fuzzy environments. Among those, the framework of single-valued neutrosophic sets (SVNSs) is the one that has more potential to explain uncertain, inconsistent and indeterminate information in a comprehensive way. And so, Hausdorff distance for SVNSs is important. Thus, we propose two novel schemes to calculate the Hausdorff distance and its corresponding similarity measures (SMs) for SVNSs. In doing so, we firstly develop the two forms of Hausdorff distance between SVNSs based on the definition of Hausdorff metric between two sets. We then use these new distance measures to construct several SMs for SVNSs. Some mathematical theorems regarding the proposed Hausdorff distances for SVNSs are also proven to strengthen its theoretical properties. In order to show the exact calculation behavior and distance measurement mechanism of our proposed methods in accordance with the decorum of Hausdorff metric, we utilize an intuitive numerical example that demonstrate the novelty and practicality of our proposed measures. Furthermore, we develop a multi-criteria decision making (MCDM) method under single-valued neutrosophic environment using the proposed SMs based on our defined Hausdorff distance measures, called as a single-valued neutrosophic MCDM (SVN-MCDM) method. In this connection, we employ our proposed SMs to compute the degree of similarity of each option with the ideal choice to identify the best alternative as well as to perform an overall ranking of the alternatives under study. We then apply our proposed SVN-MCDM scheme to solve two real world problems of MCDM under single-valued neutrosophic environment to show its effectiveness and application.

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BERINDE, VASILE, und MĂDĂLINA PĂCURAR. „"Why Pompeiu-Hausdorff metric instead of Hausdorff metric?"“. Creative Mathematics and Informatics 31, Nr. 1 (01.02.2022): 33–41. http://dx.doi.org/10.37193/cmi.2022.01.03.

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"The distance between two sets, commonly called Hausdorff metric, is a very important mathematical concept, with plenty of applications in almost all scientific research areas. We suggest in this paper an update of its name as Pompeiu-Hausdorff metric (distance). Based on historical evidence, this proposal follows the contemporary manner of appointing concepts in scientific writings, especially in mathematics."

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BERINDE, VASILE, und MADALINA PACURAR. „The role of the Pompeiu-Hausdorff metric in fixed point theory“. Creative Mathematics and Informatics 24, Nr. 2 (2015): 143–50. http://dx.doi.org/10.37193/cmi.2015.02.17.

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The main aim of this note is to highlight the role of the Pompeiu-Hausdorff metric in fixed point theory and, subsidiarily, to touch some issues related to the history of this fundamental concept in modern mathematics. This will allow us to conclude that what is nowadays almost generally called Hausdorff metric (distance) and very seldom Hausdorff-Pompeiu metric (distance) or Pompeiu-Hausdorff metric (distance), should be fairly and correctly named Pompeiu-Hausdorff metric (distance).

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BERINDE, VASILE, und MADALINA PACURAR. „The role of the Pompeiu-Hausdorff metric in fixed point theory“. Creative Mathematics and Informatics 22, Nr. 2 (2013): 143–50. http://dx.doi.org/10.37193/cmi.2013.02.13.

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The main aim of this note is to highlight the role of the Pompeiu-Hausdorff metric in fixed point theory and, subsidiarily, to touch some issues related to the history of this fundamental concept in modern mathematics. This will allow us to conclude that what is nowadays almost generally called Hausdorff metric (distance) and very seldom Hausdorff-Pompeiu metric (distance) or Pompeiu-Hausdorff metric (distance), should be fairly and correctly named Pompeiu-Hausdorff metric (distance).

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Bîrsan, Temistocle. „Convexity and Hausdorff-Pompeiu distance“. Mathematica Moravica 15, Nr. 1 (2011): 17–23. http://dx.doi.org/10.5937/matmor1101017b.

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8

Beer, Gerald, und Luzviminda Villar. „Borel measures and Hausdorff distance“. Transactions of the American Mathematical Society 307, Nr. 2 (01.02.1988): 763. http://dx.doi.org/10.1090/s0002-9947-1988-0940226-0.

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9

Kerr, David. „Matricial quantum Gromov–Hausdorff distance“. Journal of Functional Analysis 205, Nr. 1 (Dezember 2003): 132–67. http://dx.doi.org/10.1016/s0022-1236(03)00195-2.

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10

Sendov, Bl. „Hausdorff distance and image processing“. Russian Mathematical Surveys 59, Nr. 2 (30.04.2004): 319–28. http://dx.doi.org/10.1070/rm2004v059n02abeh000721.

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11

Bourgain, Jean. „Hausdorff dimension and distance sets“. Israel Journal of Mathematics 87, Nr. 1-3 (Februar 1994): 193–201. http://dx.doi.org/10.1007/bf02772994.

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12

Prus-Wiśniowski, Franciszek. „λ-Variation and Hausdorff Distance“. Mathematische Nachrichten 158, Nr. 1 (11.11.2006): 283–97. http://dx.doi.org/10.1002/mana.19921580120.

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13

Herron, David A., Abigail Richard und Marie A. Snipes. „Chordal Hausdorff Convergence and Quasihyperbolic Distance“. Analysis and Geometry in Metric Spaces 8, Nr. 1 (01.07.2020): 36–67. http://dx.doi.org/10.1515/agms-2020-0104.

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AbstractWe study Hausdorff convergence (and related topics) in the chordalization of a metric space to better understand pointed Gromov-Hausdorff convergence of quasihyperbolic distances (and other conformal distances).

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Bo, Guan, Liang Xu Liu, Jian Bo Fan und Jin Yang Chen. „An Efficient Trajectory Clustering Framework Based Relative Distance“. Applied Mechanics and Materials 241-244 (Dezember 2012): 3209–12. http://dx.doi.org/10.4028/www.scientific.net/amm.241-244.3209.

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along with more and more trajectory dataset being collected into application servers, the research in trajectory clustering has become increasingly important topic. This paper proposes a new mobile object trajectory Clustering algorithm (Trajectory Clustering based Improved Minimum Hausdorff Distance under Translation, TraClustMHD). In this framework, improved Minimum Hausdorff Distance under Translation is presented to measure the similarity between sub-segments. In additional, R-Tree is employed to improve the efficiency. The experimental results showed that this algorithm better than based on Hausdorff distance and based on line Hausdorff distance has good trajectory clustering performance.

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Cai, Wei, Wen Chen und Fajie Wang. „Three-dimensional Hausdorff derivative diffusion model for isotropic/anisotropic fractal porous media“. Thermal Science 22, Suppl. 1 (2018): 1–6. http://dx.doi.org/10.2298/tsci170630265c.

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The anomalous diffusion in fractal isotropic/anisotropic porous media is characterized by the Hausdorff derivative diffusion model with the varying fractal orders representing the fractal structures in different directions. This paper presents a comprehensive understanding of the Hausdorff derivative diffusion model on the basis of the physical interpretation, the Hausdorff fractal distance and the fundamental solution. The concept of the Hausdorff fractal distance is introduced, which converges to the classical Euclidean distance with the varying orders tending to 1. The fundamental solution of the 3-D Hausdorff fractal derivative diffusion equation is proposed on the basis of the Hausdorff fractal distance. With the help of the properties of the Hausdorff derivative, the Huasdorff diffusion model is also found to be a kind of time-space dependent convection-diffusion equation underlying the anomalous diffusion behavior.

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Ronse, Christian, Loic Mazo und Mohamed Tajine. „Correspondence between Topological and Discrete Connectivities in Hausdorff Discretization“. Mathematical Morphology - Theory and Applications 3, Nr. 1 (01.01.2019): 1–28. http://dx.doi.org/10.1515/mathm-2019-0001.

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Abstract We consider Hausdorff discretization from a metric space E to a discrete subspace D, which associates to a closed subset F of E any subset S of D minimizing the Hausdorff distance between F and S; this minimum distance, called the Hausdorff radius of F and written rH(F), is bounded by the resolution of D. We call a closed set F separated if it can be partitioned into two non-empty closed subsets F1 and F2 whose mutual distances have a strictly positive lower bound. Assuming some minimal topological properties of E and D (satisfied in ℝn and ℤn), we show that given a non-separated closed subset F of E, for any r > rH(F), every Hausdorff discretization of F is connected for the graph with edges linking pairs of points of D at distance at most 2r. When F is connected, this holds for r = rH(F), and its greatest Hausdorff discretization belongs to the partial connection generated by the traces on D of the balls of radius rH(F). However, when the closed set F is separated, the Hausdorff discretizations are disconnected whenever the resolution of D is small enough. In the particular case where E = ℝn and D = ℤn with norm-based distances, we generalize our previous results for n = 2. For a norm invariant under changes of signs of coordinates, the greatest Hausdorff discretization of a connected closed set is axially connected. For the so-called coordinate-homogeneous norms, which include the Lp norms, we give an adjacency graph for which all Hausdorff discretizations of a connected closed set are connected.

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Wang, Hui, Guo Jia Li, Jun Hui Pan und Fu Hua Shang. „An Improved Algorithm of Improved Computation Efficiency on LTS Hausdorff Distance“. Applied Mechanics and Materials 347-350 (August 2013): 3217–21. http://dx.doi.org/10.4028/www.scientific.net/amm.347-350.3217.

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The computation efficiency of traditional algorithm is not high, and there is more time consuming. This paper presents an effective method for improved hausdorff distance, depth correction of logging curves is based on improved Hausdorff distance. In this method. On the basis of existing LTS hausdorff distance, the contrast curve segment is divided into neighborhood in an area, the LTS hausdorff distance is calculated by using engineering approximate, and the improving methods of search path is put forward, which ensures that the improved algorithm is better than the original algorithm has high computing efficiency and accuracy in theory.

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van Kreveld, Marc, Tillmann Miltzow, Tim Ophelders, Willem Sonke und Jordi L. Vermeulen. „Between shapes, using the Hausdorff distance“. Computational Geometry 100 (Januar 2022): 101817. http://dx.doi.org/10.1016/j.comgeo.2021.101817.

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19

Kim, Kyeongtaek, und Ji Hun Kyung. „Character Matching Using a Hausdorff Distance“. Journal of Society of Korea Industrial and Systems Engineering 38, Nr. 2 (30.06.2015): 56–62. http://dx.doi.org/10.11627/jkise.2015.38.2.56.

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20

Huttenlocher, D. P., G. A. Klanderman und W. J. Rucklidge. „Comparing images using the Hausdorff distance“. IEEE Transactions on Pattern Analysis and Machine Intelligence 15, Nr. 9 (1993): 850–63. http://dx.doi.org/10.1109/34.232073.

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21

Kwon, Oh-Kyu. „Nonparametric hierarchical Hausdorff distance matching algorithm“. Optical Engineering 39, Nr. 7 (01.07.2000): 1917. http://dx.doi.org/10.1117/1.602576.

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22

Nutanong, Sarana, Edwin H. Jacox und Hanan Samet. „An incremental Hausdorff distance calculation algorithm“. Proceedings of the VLDB Endowment 4, Nr. 8 (Mai 2011): 506–17. http://dx.doi.org/10.14778/2002974.2002978.

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23

Van Rooij, Arnoud. „Fourier inversion and the Hausdorff distance“. Statistica Neerlandica 56, Nr. 2 (Mai 2002): 206–13. http://dx.doi.org/10.1111/1467-9574.00194.

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Rieffel, Marc A. „Vector Bundles and Gromov–Hausdorff Distance“. Journal of K-Theory 5, Nr. 1 (25.08.2009): 39–103. http://dx.doi.org/10.1017/is008008014jkt080.

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AbstractWe show how to make precise the vague idea that for compact metric spaces that are close together for Gromov–Hausdorff distance, suitable vector bundles on one metric space will have counterpart vector bundles on the other. Our approach employs the Lipschitz constants of projection-valued functions that determine vector bundles. We develop some computational techniques, and we illustrate our ideas with simple specific examples involving vector bundles on the circle, the two-torus, the two-sphere, and finite metric spaces. Our topic is motivated by statements concerning “monopole bundles” over matrix algebras in the literature of theoretical high-energy physics.

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Attouch, Hedy, Roberto Lucchetti und Roger J. B. Wets. „The topology of theρ-hausdorff distance“. Annali di Matematica Pura ed Applicata 160, Nr. 1 (Dezember 1991): 303–20. http://dx.doi.org/10.1007/bf01764131.

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Belogay, E., C. Cabrelli, U. Molter und R. Shonkwiler. „Calculating the Hausdorff distance between curves“. Information Processing Letters 64, Nr. 1 (Oktober 1997): 17–22. http://dx.doi.org/10.1016/s0020-0190(97)00140-3.

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27

Li, Hanfeng. „Order-unit quantum Gromov–Hausdorff distance“. Journal of Functional Analysis 231, Nr. 2 (Februar 2006): 312–60. http://dx.doi.org/10.1016/j.jfa.2005.03.016.

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28

Liu, Luo-fei. „Zero asymptotic Lipschitz distance and finite Gromov-Hausdorff distance“. Science in China Series A: Mathematics 50, Nr. 3 (März 2007): 345–50. http://dx.doi.org/10.1007/s11425-007-0009-4.

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29

Vegter, Gert, und Mathijs Wintraecken. „Refutation of a claim made by Fejes Tóth on the accuracy of surface meshes“. Studia Scientiarum Mathematicarum Hungarica 57, Nr. 2 (Juni 2020): 193–99. http://dx.doi.org/10.1556/012.2020.57.2.1454.

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AbstractFejes Tóth [3] studied approximations of smooth surfaces in three-space by piecewise flat triangular meshes with a given number of vertices on the surface that are optimal with respect to Hausdorff distance. He proves that this Hausdorff distance decreases inversely proportional with the number of vertices of the approximating mesh if the surface is convex. He also claims that this Hausdorff distance is inversely proportional to the square of the number of vertices for a specific non-convex surface, namely a one-sheeted hyperboloid of revolution bounded by two congruent circles. We refute this claim, and show that the asymptotic behavior of the Hausdorff distance is linear, that is the same as for convex surfaces.

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WANG, FAJIE, WEN CHEN, CHUANZENG ZHANG und QINGSONG HUA. „KANSA METHOD BASED ON THE HAUSDORFF FRACTAL DISTANCE FOR HAUSDORFF DERIVATIVE POISSON EQUATIONS“. Fractals 26, Nr. 04 (August 2018): 1850084. http://dx.doi.org/10.1142/s0218348x18500846.

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This study proposes the radial basis function (RBF) based on the Hausdorff fractal distance and then applies it to develop the Kansa method for the solution of the Hausdorff derivative Poisson equations. The Kansa method is a meshless global technique promising for high-dimensional irregular domain problems. It is, however, noted that the shape parameter of the RBFs can have a significant influence on the accuracy and robustness of the numerical solution. Based on the leave-one-out cross-validation algorithm proposed by Rippa, this study presents a new technique to choose the optimal shape parameter of the RBFs with the Hausdorff fractal distance. Numerical experiments show that the Kansa method based on the Hausdorff fractal distance is highly accurate and computationally efficient for the Hausdorff derivative Poisson equations.

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Ishiki, Yoshito. „Branching Geodesics of the Gromov-Hausdorff Distance“. Analysis and Geometry in Metric Spaces 10, Nr. 1 (01.01.2022): 109–28. http://dx.doi.org/10.1515/agms-2022-0136.

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Abstract In this paper, we first evaluate topological distributions of the sets of all doubling spaces, all uniformly disconnected spaces, and all uniformly perfect spaces in the space of all isometry classes of compact metric spaces equipped with the Gromov–Hausdorff distance.We then construct branching geodesics of the Gromov–Hausdorff distance continuously parameterized by the Hilbert cube, passing through or avoiding sets of all spaces satisfying some of the three properties shown above, and passing through the sets of all infinite-dimensional spaces and the set of all Cantor metric spaces. Our construction implies that for every pair of compact metric spaces, there exists a topological embedding of the Hilbert cube into the Gromov– Hausdorff space whose image contains the pair. From our results, we observe that the sets explained above are geodesic spaces and infinite-dimensional.

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BRANDENBURG, FRANZ J., ANDREAS GLEIßNER und ANDREAS HOFMEIER. „COMPARING AND AGGREGATING PARTIAL ORDERS WITH KENDALL TAU DISTANCES“. Discrete Mathematics, Algorithms and Applications 05, Nr. 02 (Juni 2013): 1360003. http://dx.doi.org/10.1142/s1793830913600033.

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Comparing and ranking information is an important topic in social and information sciences, and in particular on the web. Its objective is to measure the difference of the preferences of voters on a set of candidates and to compute a consensus ranking. Commonly, each voter provides a total order or a bucket order of all candidates, where bucket orders allow ties. In this work we consider the generalization of total and bucket orders to partial orders and compare them by the nearest neighbor and the Hausdorff Kendall tau distances. For total and bucket orders these distances can be computed in [Formula: see text] time. We show that the computation of the nearest neighbor Kendall tau distance is NP-hard, 2-approximable and fixed-parameter tractable for a total and a partial order. The computation of the Hausdorff Kendall tau distance for a total and a partial order is shown to be coNP-hard. The rank aggregation problem is known to be NP-complete for total and bucket orders, even for four voters and solvable in [Formula: see text] time for two voters. We show that it is NP-complete for two partial orders and the nearest neighbor Kendall tau distance. For the Hausdorff Kendall tau distance it is in [Formula: see text], but not in NP or coNP unless NP = coNP, even for four voters.

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HUANG, Hua, Kai YAN und Chun QI. „Adaptive Hausdorff Distance Based on Similarity Weighting“. Acta Automatica Sinica 35, Nr. 7 (10.08.2009): 882–87. http://dx.doi.org/10.3724/sp.j.1004.2009.00882.

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Ji, Yibo, und Alexey A. Tuzhilin. „Gromov–Hausdorff distance between interval and circle“. Topology and its Applications 307 (Februar 2022): 107938. http://dx.doi.org/10.1016/j.topol.2021.107938.

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35

Falconer, K. J. „On the Hausdorff dimensions of distance sets“. Mathematika 32, Nr. 2 (Dezember 1985): 206–12. http://dx.doi.org/10.1112/s0025579300010998.

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Qiu, Derong. „Geometry of non-Archimedean Gromov-Hausdorff distance“. P-Adic Numbers, Ultrametric Analysis, and Applications 1, Nr. 4 (15.11.2009): 317–37. http://dx.doi.org/10.1134/s2070046609040050.

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Liu, Hui, Zhichun Zhang und Dong Wei. „A Hausdorff Distance Based Image Registration Algorithm“. International Journal of Signal Processing, Image Processing and Pattern Recognition 8, Nr. 1 (31.01.2015): 125–34. http://dx.doi.org/10.14257/ijsip.2015.8.1.13.

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Noldus, Johan. „A Lorentzian Gromov–Hausdorff notion of distance“. Classical and Quantum Gravity 21, Nr. 4 (07.01.2004): 839–50. http://dx.doi.org/10.1088/0264-9381/21/4/007.

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Donoso-Aguirre, F., J. P. Bustos-Salas, M. Torres-Torriti und A. Guesalaga. „Mobile robot localization using the Hausdorff distance“. Robotica 26, Nr. 2 (März 2008): 129–41. http://dx.doi.org/10.1017/s0263574707003657.

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SUMMARYThis paper presents a novel method for localization of mobile robots in structured environments. The estimation of the position and orientation of the robot relies on the minimisation of the partial Hausdorff distance between ladar range measurements and a floor plan image of the building. The approach is employed in combination with an extended Kalman filter to obtain accurate estimates of the robot's position, heading and velocity. Good estimates of these variables were obtained during tests performed using a differential drive robot, thus demonstrating that the approach provides an accurate, reliable and computationally feasible alternative for indoor robot localization and autonomous navigation.

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Gao, Yongsheng, und Maylor K. H. Leung. „Line segment Hausdorff distance on face matching“. Pattern Recognition 35, Nr. 2 (Februar 2002): 361–71. http://dx.doi.org/10.1016/s0031-3203(01)00049-8.

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41

Zhao, Chunjiang, Wenkang Shi und Yong Deng. „A new Hausdorff distance for image matching“. Pattern Recognition Letters 26, Nr. 5 (April 2005): 581–86. http://dx.doi.org/10.1016/j.patrec.2004.09.022.

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Chaudhuri, B. „A modified Hausdorff distance between fuzzy sets“. Information Sciences 118, Nr. 1-4 (September 1999): 159–71. http://dx.doi.org/10.1016/s0020-0255(99)00037-7.

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P, Vivek E., und N. Sudha. „Robust Hausdorff distance measure for face recognition“. Pattern Recognition 40, Nr. 2 (Februar 2007): 431–42. http://dx.doi.org/10.1016/j.patcog.2006.04.019.

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Bartoň, Michael, Iddo Hanniel, Gershon Elber und Myung-Soo Kim. „Precise Hausdorff distance computation between polygonal meshes“. Computer Aided Geometric Design 27, Nr. 8 (November 2010): 580–91. http://dx.doi.org/10.1016/j.cagd.2010.04.004.

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Rieffel, Marc A. „Gromov-Hausdorff distance for quantum metric spaces“. Memoirs of the American Mathematical Society 168, Nr. 796 (2004): 0. http://dx.doi.org/10.1090/memo/0796.

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46

ALT, HELMUT, und LUDMILA SCHARF. „COMPUTING THE HAUSDORFF DISTANCE BETWEEN CURVED OBJECTS“. International Journal of Computational Geometry & Applications 18, Nr. 04 (August 2008): 307–20. http://dx.doi.org/10.1142/s0218195908002647.

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The Hausdorff distance between two sets of curves is a measure for the similarity of these objects and therefore an interesting feature in shape recognition. If the curves are algebraic computing the Hausdorff distance involves computing the intersection points of the Voronoi edges of the one set with the curves in the other. Since computing the Voronoi diagram of curves is quite difficult we characterize those points algebraically and compute them using the computer algebra system SYNAPS. This paper describes in detail which points have to be considered, by what algebraic equations they are characterized, and how they actually are computed.

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Herron, David A. „Gromov–Hausdorff distance for pointed metric spaces“. Journal of Analysis 24, Nr. 1 (Juni 2016): 1–38. http://dx.doi.org/10.1007/s41478-016-0001-x.

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Banič, Iztok, und Andrej Taranenko. „Measuring Closeness of Graphs—The Hausdorff Distance“. Bulletin of the Malaysian Mathematical Sciences Society 40, Nr. 1 (02.11.2015): 75–95. http://dx.doi.org/10.1007/s40840-015-0259-1.

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49

Irpino, Antonio, und Valentino Tontodonato. „Clustering reduced interval data using Hausdorff distance“. Computational Statistics 21, Nr. 2 (Juni 2006): 271–88. http://dx.doi.org/10.1007/s00180-006-0263-x.

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Choi, Sung Woo, und Hans-Peter Seidel. „Hyperbolic Hausdorff Distance for Medial Axis Transform“. Graphical Models 63, Nr. 5 (September 2001): 369–84. http://dx.doi.org/10.1006/gmod.2001.0556.

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