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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Abstract cell complex ： ウィキペディア英語版 Abstract cell complex In mathematics, an abstract cell complex is an abstract set with Alexandrov topology in which a non-negative integer number called dimension is assigned to each point. The complex is called “abstract” since its points called “cells” are not subsets of a Hausdorff space as it is the case in Euclidean and CW complex. Abstract cell complexes play an important role in image analysis and computer graphics. ==History== The idea of abstract cell complexes (also named abstract cellular complexes) relates to J. Listing (1862) 〔Listing J.: "Der Census räumlicher Complexe". ''Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen'', v. 10, Göttingen, 1862, 97–182.〕 und E. Steinitz (1908).〔Steinitz E.: "Beiträge zur Analysis". ''Sitzungsbericht Berliner Mathematischen Gesellschaft'', Band. 7, 1908, 29–49.〕 Also A.W Tucker (1933),〔Tucker A.W.: "An abstract approach to manifolds", Annals Mathematics, v. 34, 1933, 191-243.〕 K. Reidemeister (1938),〔Reidemeister K.: "Topologie der Polyeder und kombinatorische Topologie der Komplexe". Akademische Verlagsgesellschaft Geest & Portig, Leipzig, 1938 (second edition 1953)〕 P.S. Aleksandrov (1956) 〔Aleksandrov P.S.: Combinatorial Topology, Graylock Press, Rochester, 1956,〕 as well as R. Klette und A. Rosenfeld (2004) 〔Klette R. und Rosenfeld. A.: "Digital Geometry", Elsevier, 2004.〕 have described abstract cell complexes. E. Steinitz has defined an abstract cell complex as $C=(E,B,dim)$ where ''E'' is an abstract set, ''B'' is an asymmetric, irreflexive and transitive binary relation called the bounding relation among the elements of ''E'' and ''dim'' is a function assigning a non-negative integer to each element of ''E'' in such a way that if $B(a,\; b)$, then V. Kovalevsky (1989) 〔Kovalevsky, V.: "Finite Topology as Applied to Image Analysis", ''Computer Vision, Graphics and Image Processing'', v. 45, No. 2, 1989, 141–161.〕 described abstract cell complexes for 3D and higher dimensions. He also suggested numerous applications to image analysis. In his book (2008) 〔http://www.geometry.kovalevsky.de.〕 he has suggested an axiomatic theory of locally finite topological spaces which are generalization of abstract cell complexes. The book contains among others new definitions of topological balls and spheres independent of metric, a new definition of combinatorial manifolds and many algorithms useful for image analysis. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Abstract cell complex」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.