Multiresolution analysis: Difference between revisions

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* ''Self-similarity'' in ''time'' demands that each subspace ''V_k'' is invariant under shifts by [[integer]] [[multiple (mathematics)|multiple]]s of ''2^k''. That is, for each <math>f\in V_k,\; m\in\Z</math> the function ''g'' defined as <math>g(x)=f(x-m2^{k})</math> also contained in <math>V_k</math>.

* ''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_Scaling (geometry)|scaling]] respectively [[Dilation (metric space)|dilation]] factor 2^''k-l''. I.e., for each <math>f\in V_k</math> there is a <math>g\in V_l</math> with <math>\forall x\in\R:\;g(x)=f(2^{k-l}x)</math>.

* 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]] ([[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_functionScaling function|scaling functions]]''' or '''[[father wavelets]]'''. In most cases one demands of those functions to be [[piecewise continuous]] with [[compact support]].

* ''Regularity'' demands that the model subspace ''V₀'' 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]].

* ''Completeness'' demands that those nested subspaces fill the whole space, i.e., their union should be [[dense set|dense]] in <math>L^2(\R)</math>, and that they are not too redundant, i.e., their intersection should only contain the zero element.

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Defining another function, known as '''mother wavelet''' or just '''the wavelet'''

:<math>\psi(x):=\sum_{k=-N}^N (-1)^k a_{1-k}\phi(2x-k),</math>

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>.{{cncitation needed|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{{cncitation needed|date=April 2013}}

:<math>L^2(\R)=\mbox{closure of }\bigoplus_{k\in\Z}W_k,</math>

thus the set

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* [[Wavelet]]

{{inlineno footnotes|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}}