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A '''multiresolution analysis (MRA)''' or '''multiscale approximation (MSA)''' is the design method of most of the practically relevant [[discrete wavelet transform]]s (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 equation]]s (the ''[[ironing method]]'') and the [[pyramid (image processing)|pyramid method]]s of [[image processing]] as introduced in 1981/83 by Peter J. Burt, Edward H. Adelson and James Crowley.
== overwhelmed win over the soul of the house ==


== Definition ==
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A ''multiresolution analysis'' of the [[Lp space|space]] <math>L^2(\mathbb{R})</math> consists of a [[sequence]] of nested [[linear subspace|subspaces]]
== 'Come on ==


::<math>\{0\}\dots\subset V_0\subset V_1\subset\dots\subset V_n\subset V_{n+1}\subset\dots\subset L^2(\R)</math>
Tsing Yi with the girls in the juvenile behind.<br><br>'Come on, Ling old, say good-bye.'<br><br>Xiao Yan raised his head, looked before the mountain, he could feel the breath of a large number of forceful, where fast cohesion, and, in the [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-2.html カシオ腕時計 g-shock] other two directions Mountains, also came bursts of spatial fluctuations, it [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-15.html カシオ ソーラー 腕時計] seems Yan Lei II strong family is dispatched<br><br>voice down, Xiao Yan is no longer hesitated, holding the hand Kaoru children, stature flash, is swept [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-1.html カシオ スタンダード 腕時計] into the sky.<br><br>'Xiao Yan Master, I wish triumphant [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html 腕時計 メンズ casio] return!'<br><br>looked back duo Xiao Yan Ling shadow is smile, muttered to himself.<br><br>when Xiao Yan horizon when the two appear in the first mountain, where the air is already shadows scattered, many of the ancient tribe of the strong launch into the sky, forming a neat formation in the sky, the vast surging between breath, like a remote one, that like power and influence, even into the territory of the soul of Emperor [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html casio 腕時計 g-shock] Xiao Yan, are
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that satisfies certain self-similarity relations in time/space and scale/frequency, as well as [[completeness]] and regularity relations.
== and under this acid itch ==


* ''Self-similarity'' in ''time'' demands that each subspace ''V<sub>k</sub>'' is invariant under shifts by [[integer]] [[multiple (mathematics)|multiple]]s of ''2<sup>-k</sup>''. That is, for each <math>f\in V_k,\; m\in\mathbb Z</math> there is a <math>g\in V_k</math> with <math>\forall x\in\mathbb R:\;f(x)=g(x+m2^{-k})</math>.
Gu fine blood into the body, Xiao Yan is covered with the skin immediately harnessed into a bright Jinmang, [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-1.html casio 時計] the vast energy in the body as [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-0.html カシオ 腕時計 チタン] the storm swept in general open, had exhausted the meridians, almost in an instant, that is once again being gurgle vindictive, quickly filled.<br><br>Moreover, in this energy crisis sweeping the room, Xiao Yan also feel the body's bones, muscles and even the meridians, etc., are among faint heard a strange acid itch, and under this acid itch, as if bones, muscles, etc., are slowly becoming more and more tough and [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html 時計 カシオ] strong<br><br>'This is the essence and blood cologne energy [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-8.html カシオ gショック 腕時計] inherent in it, really amazing.'<br><br>feel the change in the body, Xiao Yan could not help but come to mind is some amazing promise of 'color', this ability, almost all can, and some higher-order Dan 'medicine' compared to this really worthy [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-11.html カシオ腕時計 g-shock] cologne Void Top-like existence of Warcraft community, just fine blood force, is that it has
* ''Self-similarity'' in ''scale'' demands that all subspaces <math>V_k\subset V_l,\; k<l,</math> are time-scaled versions of each other, with [[Scaling_(geometry)|scaling]] respectively [[Dilation (metric space)|dilation]] factor ''2<sup>l-k</sup>''. I.e., for each <math>f\in V_k</math> there is a <math>g\in V_l</math> with <math>\forall x\in\mathbb R:\;g(x)=f(2^{l-k}x)</math>. If f has limited [[support (mathematics)|support]], then as the support of g gets smaller, the resolution of the ''l''-th subspace is higher than the resolution of the ''k''-th subspace.
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* ''Regularity'' demands that the model subspace ''V<sub>0</sub>'' be generated as the [[linear hull]] ([[algebraic closure|algebraically]] or even [[topologically closed]]) of the integer shifts of one or a finite number of generating functions <math>\phi</math> or <math>\phi_1,\dots,\phi_r</math>. Those integer shifts should at least form a frame for the subspace <math>V_0\subset L^2(\R)</math>, which imposes certain conditions on the decay at infinity. The generating functions are also known as '''[[Wavelet#Scaling_function|scaling functions]]''' or '''[[father wavelets]]'''. In most cases one demands of those functions to be [[piecewise continuous]] with [[compact support]].
== 'But you want me to pan the door completely closed hand ==


* ''Completeness'' demands that those nested subspaces fill the whole space, i.e., their union should be [[dense set|dense]] in <math>L^2(\mathbb{R})</math>, and that they are not too redundant, i.e., their intersection should only contain the zero element.
Means, still useless. '<br><br>'is not it?' Han busy mouth slightly, her eyes are as cold as ice.<br><br>'But you want me to pan the door completely closed hand, non-stick Dan' medicine 'sell this one, is also OK! long as you promise me a proposal.' Xiao Yan suddenly took on the face a little [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-10.html casio 腕時計 データバンク] sneer, looking across [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-0.html カシオ 腕時計 バンド] the Korea idle.<br><br>four hundred and ninetieth chapters conflict (under)<br><br>four hundred and ninetieth chapters conflict<br><br>'proposal? What proposal?'<br><br>hear the words of Xiao Yan, Han busy startled for a while, it seems that it is unexpected, after a long while, just some awareness of road.<br><br>'Since everyone was mixing with' medicine 'division, if genuine Ming dry like [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-8.html カシオ gショック 腕時計] ordinary people, then, is somewhat inconsistent status so [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-2.html casio 腕時計 デジタル] long idle if Korea have the ability, we large available refining' medicine 'approach a division showdown , [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html 腕時計 メンズ casio] if I lose, the future nonstick pan door slightest Dan '
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== Important conclusions ==
== we are here.' ==
In the case of one continuous (or at least with bounded variation) compactly supported scaling function with orthogonal shifts, one may make a number of deductions. The proof of existence of this class of functions is due to [[Ingrid Daubechies]].


Assuming the scaling function has compact support, then <math>V_0\subset V_1</math> implies that there is a finite sequence of coefficients <math>a_k=2 \langle\phi(x),\phi(2x-k)\rangle</math> for <math>|k|\leq N</math>, and  <math>a_k=0</math> for <math>|k|>N</math>, such that
'Do not look, we are here.'<br><br>in five young people turned blue grabbing, faint laughter, suddenly sounded on [http://alleganycountyfair.org/sitemap.xml http://alleganycountyfair.org/sitemap.xml] the branches, [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html 時計 casio] the former hastened the rise, but it is Qiaode surrounding open space in the trunk, do not know when there's three men and two woman five young people, but it is already surrounded them in it.<br><br>blue youth are to be five freshmen, a pedestrian suddenly appeared a bit scared, a little bit [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-1.html カシオ 時計 メンズ] back, but God, these freshmen are not the same as the one with the mice to escape it? How these guys [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-6.html 電波腕時計 カシオ] even dare to openly appear in front of them?<br><br>'yo, pretty girl, you, will start to be available tap, or leave a bad impression, that in the future may have no chance, oh.' After a while, the young blue five [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-9.html カシオ 掛け時計] people finally recovered , staring across a few people on the trunk, the last big bright eyes stopped at the two girls who suddenly joking laughter, loud
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:<math>\phi(x)=\sum_{k=-N}^N a_k\phi(2x-k).</math>
<ul>
 
 
Defining another function, known as '''mother wavelet''' or just '''the wavelet'''
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:<math>\psi(x):=\sum_{k=-N}^N (-1)^k a_{1-k}\phi(2x-k).</math>
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One can show that the space <math>W_0\subset V_1</math>, which is defined as the (closed) linear hull of the mother wavelet's integer shifts, is the orthogonal complement to <math>V_0</math> inside <math>V_1</math>.{{cn|date=April 2013}} Or put differently, <math>V_1</math> is the [[orthogonal direct sum|orthogonal sum]] (denoted by <math>\oplus</math>) of <math>W_0</math> and <math>V_0</math>. By self-similarity, there are scaled versions <math>W_k</math> of <math>W_0</math> and by completeness one has{{cn|date=April 2013}}
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:<math>L^2(\mathbb R)=\mbox{closure of }\bigoplus_{k\in\Z}W_k,</math>
</ul>
 
thus the set
 
:<math>\{\psi_{k,n}(x)=\sqrt2^k\psi(2^kx-n):\;k,n\in\Z\}</math>
 
is a countable complete [[orthonormal wavelet]] basis in <math>L^2(\R)</math>.
 
==See also==
* [[Multiscale modeling]]
* [[Scale space]]
* [[Wavelet]]
 
{{inline|date=April 2013}}
 
==References==
* {{cite book|first=Charles K.|last=Chui|title=An Introduction to Wavelets|year=1992|publisher=Academic Press|location=San Diego|isbn=0-585-47090-1}}
 
* {{cite book|author1-link=Ali Akansu|first1=A.N.|last1=Akansu|first2=R.A.|last2=Haddad|title=Multiresolution signal decomposition: transforms, subbands, and wavelets|publisher=Academic Press|year=1992|isbn=978-0-12-047141-6}}
 
* {{cite book|author1-link=C. Sidney Burrus|first1=C.S.|last1=Burrus|first2=R.A.|last2=Gopinath|first3=H.|last3=Guo|title=Introduction to Wavelets and Wavelet Transforms: A Primer|publisher=Prentice-Hall|year=1997|isbn=0-13-489600-9}}
 
* {{cite book|first=S.G.|last=Mallat|url=http://www.cmap.polytechnique.fr/~mallat/book.html|title=A Wavelet Tour of Signal Processing|publisher=Academic Press|year=1999|isbn=0-12-466606-X}}
 
== External links ==
*[http://www.docstoc.com/docs/160022503/A-Concise-Introduction-to-Wavelets A Concise Introduction to Wavelets (with  coverage of multiresolution analysis)] by René Puchinger.
 
[[Category:Wavelets]]
[[Category:Time–frequency analysis]]

Latest revision as of 02:15, 1 October 2014

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Kind of near-blind reverence, like pious believers like.

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'Come on

Tsing Yi with the girls in the juvenile behind.

'Come on, Ling old, say good-bye.'

Xiao Yan raised his head, looked before the mountain, he could feel the breath of a large number of forceful, where fast cohesion, and, in the カシオ腕時計 g-shock other two directions Mountains, also came bursts of spatial fluctuations, it カシオ ソーラー 腕時計 seems Yan Lei II strong family is dispatched

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'Xiao Yan Master, I wish triumphant 腕時計 メンズ casio return!'

looked back duo Xiao Yan Ling shadow is smile, muttered to himself.

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Means, still useless. '

'is not it?' Han busy mouth slightly, her eyes are as cold as ice.

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