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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 fast wavelet transform ： ウィキペディア英語版 fast wavelet transform The Fast Wavelet Transform is a mathematical algorithm designed to turn a waveform or signal in the time domain into a sequence of coefficients based on an orthogonal basis of small finite waves, or wavelets. The transform can be easily extended to multidimensional signals, such as images, where the time domain is replaced with the space domain. It has as theoretical foundation the device of a finitely generated, orthogonal multiresolution analysis (MRA). In the terms given there, one selects a sampling scale ''J'' with sampling rate of 2J per unit interval, and projects the given signal ''f'' onto the space $V\_J$; in theory by computing the scalar products :$s^\_n:=2^J\; \backslash langle\; f(t),\backslash phi(2^J\; t-n)\; \backslash rangle,$ where $\backslash phi$ is the scaling function of the chosen wavelet transform; in practice by any suitable sampling procedure under the condition that the signal is highly oversampled, so :$P\_J()(x):=\backslash sum\_\; s^\_n\backslash ,\backslash phi(2^Jx-n)$ is the orthogonal projection or at least some good approximation of the original signal in $V\_J$. The MRA is characterised by its scaling sequence :$a=(a\_,\backslash dots,a\_0,\backslash dots,a\_N)$ or, as Z-transform, $a(z)=\backslash sum\_^Na\_nz^$ and its wavelet sequence :$b=(b\_,\backslash dots,b\_0,\backslash dots,b\_N)$ or $b(z)=\backslash sum\_^Nb\_nz^$ (some coefficients might be zero). Those allow to compute the wavelet coefficients $d^\_n$, at least some range ''k=M,...,J-1'', without having to approximate the integrals in the corresponding scalar products. Instead, one can directly, with the help of convolution and decimation operators, compute those coefficients from the first approximation $s^$. == Forward DWT == One computes recursively, starting with the coefficient sequence $s^$ and counting down from ''k=J-1'' to some ''M:$$ s^_n:=\frac12 \sum_^N a_m s^_ or $$ s^(z):=(\downarrow 2)(a^ *(z)\cdot s^(z)) and :$$ d^_n:=\frac12 \sum_^N b_m s^_ or $$ d^(z):=(\downarrow 2)(b^ *(z)\cdot s^(z)) , for ''k=J-1,J-2,...,M'' and all $n\backslash in\backslash Z$. In the Z-transform notation: : * The downsampling operator $(\backslash downarrow\; 2)$ reduces an infinite sequence, given by its Z-transform, which is simply a Laurent series, to the sequence of the coefficients with even indices, $(\backslash downarrow\; 2)(c(z))=\backslash sum\_c\_z^$. : * The starred Laurent-polynomial $a^$ *(z) denotes the adjoint filter, it has ''time-reversed'' adjoint coefficients, $a^$ *(z)=\sum_^N a_^ *z^. (The adjoint of a real number being the number itself, of a complex number its conjugate, of a real matrix the transposed matrix, of a complex matrix its hermitian adjoint). : * Multiplication is polynomial multiplication, which is equivalent to the convolution of the coefficient sequences. It follows that :$P\_k()(x):=\backslash sum\_\; s^\_n\backslash ,\backslash phi(2^kx-n)$ is the orthogonal projection of the original signal ''f'' or at least of the first approximation $P\_J()(x)$ onto the subspace $V\_k$, that is, with sampling rate of 2k per unit interval. The difference to the first approximation is given by :$P\_J()(x)=P\_k()(x)+D\_k()(x)+\backslash dots+D\_()(x)$, where the difference or detail signals are computed from the detail coefficients as :$D\_k()(x):=\backslash sum\_\; d^\_n\backslash ,\backslash psi(2^kx-n)$, with $\backslash psi$ denoting the ''mother wavelet'' of the wavelet transform. 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「fast wavelet transform」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.