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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 Legendre wavelet ： ウィキペディア英語版 Legendre wavelet In functional analysis, compactly supported wavelets derived from Legendre polynomials are termed Legendre wavelets or spherical harmonic wavelets.〔Lira et al〕 Legendre functions have widespread applications in which spherical coordinate system is appropriate.〔Gradshetyn and Ryzhik〕〔Colomer and Colomer〕〔Ramm and Zaslavsky〕 As with many wavelets there is no nice analytical formula for describing these harmonic spherical wavelets. The low-pass filter associated to Legendre multiresolution analysis is a finite impulse response (FIR) filter. Wavelets associated to FIR filters are commonly preferred in most applications.〔 An extra appealing feature is that the Legendre filters are ''linear phase'' FIR (i.e. multiresolution analysis associated with linear phase filters). These wavelets have been implemented on MATLAB (wavelet toolbox). Although being compactly supported wavelet, legdN are not orthogonal (but for ''N'' = 1).〔Herley and Vetterli〕 == Legendre multiresolution filters == Associated Legendre polynomials are the colatitudinal part of the spherical harmonics which are common to all separations of Laplace's equation in spherical polar coordinates.〔 The radial part of the solution varies from one potential to another, but the harmonics are always the same and are a consequence of spherical symmetry. Spherical harmonics $P\_n(z)$ are solutions of the Legendre $2^$-order differential equation, ''n'' integer: : $(1-z^2)\; \backslash frac\; -\; 2z\; \backslash frac\; +\; n(n+1)y=0.$ $P\_n(\; \backslash cos\; )$ polynomials can be used to define the smoothing filter $H(\; \backslash omega)$ of a multiresolution analysis (MRA).〔Mallat〕 Since the appropriate boundary conditions for an MRA are $|H(0)|=1$ and $|H(\; \backslash pi)|=0$, the smoothing filter of an MRA can be defined so that the magnitude of the low-pass $|H(\; \backslash omega)|$ can be associated to Legendre polynomials according to: $\backslash nu\; =\; 2\; n+1$. : $|H\_(\backslash omega)|=|\; \backslash frac\; )\}\}\; |$ Illustrative examples of filter transfer functions for a Legendre MRA are shown in figure 1, for $\backslash nu$=1,3 and 5. A low-pass behaviour is exhibited for the filter ''H'', as expected. The number of zeroes within is equal to the degree of the Legendre polynomial. Therefore, the roll-off of side-lobes with frequency is easily controlled by the parameter $\backslash nu$. The low-pass filter transfer function is given by : $H\_\; (\backslash omega)=-e^\}\; P\_(\; \backslash cos\; (\backslash frac\; ))$ The transfer function of the high-pass analysing filter $G\_\; (\backslash omega)$ is chosen according to Quadrature mirror filter condition,〔〔Vetterli and Herley〕 yielding: : $H\_\; (\backslash omega)=-e^\; \}\; P\_(\; \backslash sin\; (\backslash frac\; ))$ Indeed, $|G\_(0)|=0$ and $|G\_(\; \backslash pi)|=1$, as expected. 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「Legendre wavelet」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.