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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 Multiresolution analysis ： ウィキペディア英語版 Multiresolution analysis A multiresolution analysis (MRA) or multiscale approximation (MSA) is the design method of most of the practically relevant discrete wavelet transforms (DWT) and the justification for the algorithm of the fast wavelet transform (FWT). It was introduced in this context in 1988/89 by Stephane Mallat and Yves Meyer and has predecessors in the microlocal analysis in the theory of differential equations (the ''ironing method'') and the pyramid methods of image processing as introduced in 1981/83 by Peter J. Burt, Edward H. Adelson and James Crowley. == Definition == A ''multiresolution analysis'' of the Lebesgue space $L^2(\backslash mathbb)$ consists of a sequence of nested subspaces ::$\backslash \backslash dots\backslash subset\; V\_1\backslash subset\; V\_0\backslash subset\; V\_\backslash subset\backslash dots\backslash subset\; V\_\backslash subset\; V\_\backslash subset\backslash dots\backslash subset\; L^2(\backslash R)$ that satisfies certain self-similarity relations in time/space and scale/frequency, as well as completeness and regularity relations. * ''Self-similarity'' in ''time'' demands that each subspace ''Vk'' is invariant under shifts by integer multiples of ''2k''. That is, for each $f\backslash in\; V\_k,\backslash ;\; m\backslash in\backslash Z$ the function ''g'' defined as $g(x)=f(x-m2^)$ also contained in $V\_k$. * ''Self-similarity'' in ''scale'' demands that all subspaces $V\_k\backslash subset\; V\_l,\backslash ;\; k>l,$ are time-scaled versions of each other, with scaling respectively dilation factor 2''k-l''. I.e., for each $f\backslash in\; V\_k$ there is a $g\backslash in\; V\_l$ with $\backslash forall\; x\backslash in\backslash R:\backslash ;g(x)=f(2^x)$. * In the sequence of subspaces, for ''k''>''l'' the space resolution 2''l'' of the ''l''-th subspace is higher than the resolution 2''k'' of the ''k''-th subspace. * ''Regularity'' demands that the model subspace ''V0'' be generated as the linear hull (algebraically or even topologically closed) of the integer shifts of one or a finite number of generating functions $\backslash phi$ or $\backslash phi\_1,\backslash dots,\backslash phi\_r$. Those integer shifts should at least form a frame for the subspace $V\_0\backslash subset\; L^2(\backslash R)$, which imposes certain conditions on the decay at infinity. The generating functions are also known as scaling functions or father wavelets. In most cases one demands of those functions to be piecewise continuous with compact support. * ''Completeness'' demands that those nested subspaces fill the whole space, i.e., their union should be dense in $L^2(\backslash R)$, and that they are not too redundant, i.e., their intersection should only contain the zero element. 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「Multiresolution analysis」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.