On Taxicab Distance Mean Functions and their Geometric Applications: Methods, Implementations and Examples

• Autorzy: Vincze Csaba , Nagy Abris

Identyfikatory

DOI

10.3233/FI-222156

• Języki publikacji: EN

Abstrakty

EN

A distance mean function measures the average distance of points from the elements of a given set of points (focal set) in the space. The level sets of a distance mean function are called generalized conics. In case of infinite focal points the average distance is typically given by integration over the focal set. The paper contains a survey on the applications of taxicab distance mean functions and generalized conics' theory in geometric tomography: bisection of the focal set and reconstruction problems by coordinate X-rays. The theoretical results are illustrated by implementations in Maple, methods and examples as well.

Słowa kluczowe

EN

distance mean functions; generalized conics; taxicab distance; parallel x-rays

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2022
• Tom: Vol. 189, nr 2
• Strony: 145--169
• Opis fizyczny: Bibliogr. 21 poz., ys.

Twórcy

autor

Vincze Csaba

• nstitute of Mathematics University of Debrecen P.O.Box 400, H-4002 Debrecen, Hungary

autor

Nagy Abris

• nstitute of Mathematics University of Debrecen Debrecen, Hungary

Bibliografia

• [1] Vincze C, Kov´acs Z, Csorv´assy Z. On the generalization of Erd˝os-Vincze’s theorem about the approximation of closed convex plane curves by polyellipses. Annales Mathematicae et Informaticae, 2018. 49:181-197. doi:10.33039/ami.2018.11.002.
• [2] Vincze C, Nagy A. On the theory of generalized conics with applications in geometric tomography. J. of Approx. Theory, 2012. 164:371-390. doi:10.1016/j.jat.2011.11.004.
• [3] Vincze C, Nagy A. On the average taxicab distance function and its applications. Acta Appl. Math., 2019. 161:201-220. doi:10.1007/s10440-018-0210-1.
• [4] Barczy M, Nagy A, Nosz´aly C, Vincze C. A Robbins-Monro type algorithm for computing global minimizer of generalized conic functions. Optimization, 2015. 64(9):1999-2020. doi:10.1080/02331934.2014.919499.
• [5] Garey MR, Johnson DS, Preparata FP, Tarjan RE. Triangulating a simple polygon. Information Processing Letters, 1978. 7(4):175-179. doi:10.1016/0020-0190(78)90062-5.
• [6] Lee D, Preparata FP. Location of a point in a planar subdivision and its applications. SIAM Journal on Computing, 1977. 6(3):594-606. doi:10.1137/0206043.
• [7] Meisters GH. Polygons have ears. The American Mathematical Monthly, 1975. 82(6):648-651. doi:
• 10.2307/2319703.
• [8] Gardner RJ. Geometric Tomography. Cambridge University Press, New York, 2006.
• [9] Vincze C, Nagy A. Reconstruction of hv-convex sets by their coordinate X-ray functions. Journal of Mathematical Imaging and Vision, 2014. 49(3):569 - 582. doi:10.1007/s10851-013-0487-7.
• [10] Vincze C, Nagy A. Generalized conic functions of hv-convex planar sets: continuity properties and X-rays. Aequationes Mathematicae, 2015. 89:1015 - 1030. doi:10.1007/s00010-014-0322-2.
• [11] Vincze C, Nagy A. An algorithm for the reconstruction of hv-convex planar bodies by finitely many and noisy measurements of their coordinate X-rays. Fundamenta Informaticae, 2015. 141(2-3):169 - 189. doi:10.3233/FI-2015-1270.
• [12] Bianchi G, Burchard A, Gronchi P, Volcic A. Convergence in Shape of Steiner Symmetrization. Indiana University Math. Journal, 2012. 61(4):1695-1709. doi:10.1512/iumj.2012.61.5087.
• [13] Gardner RJ, Kiderlen M. A solution to Hammer’s X-ray reconstruction problem. Advances in Mathematics, 2007. 214(1):323-343. doi:10.1016/j.aim.2007.02.005.
• [14] Li D, Sun X. Nonlinear Integer Programming. Springer, New York, 2006.
• [15] Vincze C. On the taxicab distance sum function and its applications in discrete tomography. Periodica Mathematica Hungarica, 2019. 79:177 - 190. doi:10.1007/s10998-018-00278-7.
• [16] Ryser HJ. Combinatorial properties of matrices of zeros and ones. Canadian Journal of Mathematics, 1957. 9:371-377. doi:10.4153/CJM-1957-044-3.
• [17] Ryser HJ. Matrices of zeros and ones. Bulletin of the American Mathematical Society, 1960. 66(6):442-464. doi:10.1090/S0002-9904-1960-10494-6.
• [18] Gale D. A theorem on flows in networks. Pacific Journal of Mathematics, 1957. 7(2):1973-1982. doi: 10.2140/pjm.1957.7.1073.
• [19] Anstee RP. The network flow approach for matrices with given row and column sums. Discrete Mathematics, 1983. 44(2):125-138. doi:10.1016/0012-365X(83)90053-5.
• [20] Batenburg KJ. A network flow algorithm for reconstructing binary images from discrete X-rays. Journal of Mathematical Imaging and Vision, 2007. 27:175-191. doi:10.1007/s10851-006-9798-2.
• [21] Mirsky L. Combinatorial theorems and integral matrices. Journal of Combinatorial Theory, 1968. 5(1):30-44. doi:10.1016/S0021-9800(68)80026-2.