There are two ways to introduce wavelets: one is through the continuous wavelet transform, and the other is through multiresolution analysis (MRA), which is the presentation adopted here. Here we start by defining multiresolution analysis and thereafter we give one example of such MRA.
Definition
( Multiresolution analysis)
A multiresolution analysis of
${L}_{2}\left(\mathrm{I}\phantom{\rule{-0.166667em}{0ex}}\mathrm{R}\right)$ is defined as a sequence of closed subspaces
${V}_{j}\subset {L}_{2}\left(\mathrm{I}\phantom{\rule{-0.166667em}{0ex}}\mathrm{R}\right),j\in \mathrm{Z}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\mathrm{Z}$ with the following properties:
If
$f\left(x\right)\in {V}_{0},f\left({2}^{j}x\right)\in {V}_{j}.$ This property means that all the spaces
${V}_{j}$ are scaled versions of the central space
${V}_{0}.$
If
$f\in {V}_{0},f(.-k)\in {V}_{0},k\in \mathrm{Z}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\mathrm{Z}.$ That is,
${V}_{0}$ (and hence all the
${V}_{j}$ ) is invariant under translation.
There exists
$\varphi \in {V}_{0}$ such that
$\{{\varphi}_{0,n};n\in \mathrm{Z}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\mathrm{Z}\}$ is an orthonormal basis in
${V}_{0}.$
Condition 5 in
[link] seems to be quite contrived, but it can be relaxed (i.e.,instead of taking orthonormal basis, we can take Riesz basis). We will use the following terminology: a
level of a multiresolution analysis is one of the
${V}_{j}$ subspaces and one level is
coarser (respectively
finer ) with respect to another whenever the index of the corresponding subspace is smaller (respectively bigger).
Consequence of the definition
Let us make a couple of simple observations concerning this definition. Combining the facts that
$\varphi \left(x\right)\in {V}_{0}$
$\left\{\varphi \right(.-k),k\in \mathrm{Z}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\mathrm{Z}\}$ is an orthonormal basis for
${V}_{0}$
$\varphi \left({2}^{j}x\right)\in {V}_{j}$ ,
we obtain that, for fixed
$j$ ,
$\{{\varphi}_{j,k}\left(x\right)={2}^{j/2}\varphi ({2}^{j}x-k),k\in \mathrm{Z}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\mathrm{Z}\}$ is an orthonormal basis for
${V}_{j}$ .
Since
$\varphi \in {V}_{0}\subset {V}_{1},$ we can express
$\varphi $ as a linear combination of
$\left\{{\varphi}_{1,k}\right\}:$
[link] is called the
refinement equation , or the
two scales difference equation. The function
$\varphi \left(x\right)$ is called the
scaling function .
Under very general condition,
$\varphi $ is uniquely defined by its refinement equation and the normalisation
The spaces
${V}_{j}$ will be used to approximate general functions (see an example below). This will be done by defining appropriate projections onto these spaces. Since the union of all the
${V}_{j}$ is dense in
${L}_{2}\left(\mathrm{I}\phantom{\rule{-0.166667em}{0ex}}\mathrm{R}\right),$ we are guaranteed that any given function of
${L}_{2}$ can be approximated arbitrarily close by such projections, i.e.:
Rather than considering all our nested spaces
${V}_{j},$ we would like to code only the information needed to go from
${V}_{j}$ to
${V}_{j+1}.$ Hence we define by
${W}_{j}$ the space complementing
${V}_{j}$ in
${V}_{j+1}$ :
$${V}_{j+1}={V}_{j}\oplus {W}_{j}$$
This space
${W}_{j}$ answers our question: it contains the “detail” information needed to go from an approximation at resolution
$j$ to an approximation at resolution
$j+1.$ Consequently, by using recursively the
[link] , we have:
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Source:
OpenStax, Multiresolution analysis, filterbank implementation, and function approximation using wavelets. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10568/1.2
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