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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 Hutchinson metric ： ウィキペディア英語版 Hutchinson metric In mathematics, the Hutchinson metric is a function which measures "the discrepancy between two images for use in fractal image processing" and "can also be applied to describe the similarity between DNA sequences expressed as real or complex genomic signals."〔 (Efficient computation of the Hutchinson metric between digitized images ) abstract〕〔 (HUTCHINSON METRIC IN FRACTAL DNA ANALYSIS -- A NEURAL NETWORK APPROACH )〕 ==Formal definition== Consider only nonempty, compact, and finite metric spaces. For a space $X\; \backslash ,$, let $P(X)\; \backslash ,$ denote the space of Borel probability measures on $X\; \backslash ,$, with :$\backslash delta\; :\; X\; \backslash rightarrow\; P(X)\; \backslash ,$ the embedding associating to $x\; \backslash in\; X$ the point measure $\backslash delta\_x\; \backslash ,$. The support $|\backslash mu|\; \backslash ,$ of a measure in P(X) is the smallest closed subset of measure 1. If :$f\; :\; X\_1\; \backslash rightarrow\; X\_2\; \backslash ,$ is Borel measurable then the induced map :$f\_$ * : P(X_1) \rightarrow P(X_2) \, associates to $\backslash mu\; \backslash ,$ the measure $f\_$ *(\mu) \, defined by :$f\_$ *(\mu)(B)= \mu(f^(B)) \, for all $B\; \backslash ,$ Borel in $X\_2\; \backslash ,$. Then the Hutchinson metric is given by :$d(\backslash mu\_1,\backslash mu\_2)=\backslash sup\; \backslash left\; \backslash lbrace\; \backslash int\; u(x)\; \backslash ,\; \backslash mu\_1(dx)\; -\; \backslash int\; u(x)\; \backslash ,\; \backslash mu\_2(dx)\; \backslash right\; \backslash rbrace$ where the $\backslash sup$ is taken over all real-valued functions ''u'' with Lipschitz constant $\backslash le\; 1\; \backslash ,.$ Then $\backslash delta\; \backslash ,$ is an isometric embedding of $X\; \backslash ,$ into $P(X)\; \backslash ,$, and if :$f\; :\; X\_1\; \backslash rightarrow\; X\_2\; \backslash ,$ is Lipschitz then :$f\_$ * : P(X_1) \rightarrow P(X_2) \, is Lipschitz with the same Lipschitz constant.〔 (Invariant Measures for Set-Valued Dynamical Systems Walter Miller; Ethan Akin Transactions of the American Mathematical Society, Vol. 351, No. 3. (Mar., 1999), pp. 1203-1225 ) 〕 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「Hutchinson metric」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.