Multiresolution analysis: Difference between revisions

Revision as of 18:58, 12 March 2014

Revision as of 08:47, 7 April 2014

Line 5:

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

::<math>\{0\}\dots\subset V_1\subset V_0\subset V_1V_{-1}\subset\dots\subset V_nV_{-n}\subset V_{-n+1}\subset\dots\subset L^2(\R)</math>

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 ''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\mathbb Z</math> therethe isfunction a <math>''g\in'' V_k</math>defined withas <math>\forall x\in\mathbb R:\;fg(x)=gf(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_(geometry)|scaling]] respectively [[Dilation (metric space)|dilation]] factor ''2^l-''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\mathbb R:\;g(x)=f(2^{k-l-k}x)</math>.

* If f has limited [[support (mathematics)|support]], then asIn the supportsequence of gsubspaces, getsfor smaller,''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_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(\mathbb{R})</math>, and that they are not too redundant, i.e., their intersection should only contain the zero element.

== Important conclusions ==

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_1V_{-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

:<math>\phi(x)=\sum_{k=-N}^N a_k\phi(2x-k).</math>

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>

Oneone can show that the space <math>W_0\subset V_1V_{-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_1V_{-1}</math>.{{cn|date=April 2013}} Or put differently, <math>V_1V_{-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}}

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

:<math>L^2(\mathbb R)=\mbox{closure of }\bigoplus_{k\in\Z}W_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>.{{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}}

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

thus the set

:<math>\{\psi_{k,n}(x)=\sqrt2^{-k}\psi(2^kx{-k}x-n):\;k,n\in\Z\}</math>

:<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>.