0.1 Multiresolution analysis

The scaling function and the subspaces

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

1. $...\subset V_{-1}\subset V_0\subset V_1...$
2. The spaces $V_j$ satisfy
   $$\bigcup _{j\in \mathrm{Z}\mathrm{Z}}{V}_j\text{is}\text{dense}\text{in}{L}_2\left(\mathrm{I}\mathrm{R}\right)\text{and}\bigcap _{j\in \mathrm{Z}\mathrm{Z}}{V}_j=\left\{0\right\}$$
3. 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 $V_0.$
4. If $f\in V_0,f(.-k)\in V_0,k\in \mathrm{Z}\mathrm{Z}.$ That is, $V_0$ (and hence all the $V_j$ ) is invariant under translation.
5. There exists $\varphi \in V_0$ such that $\{{\varphi }_{0,n};n\in \mathrm{Z}\mathrm{Z}\}$ 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

1. $\varphi \left(x\right)\in V_0$
2. $\left\{\varphi \right(.-k),k\in \mathrm{Z}\mathrm{Z}\}$ is an orthonormal basis for $V_0$
3. $\varphi \left(2^{j}x\right)\in V_j$ ,

we obtain that, for fixed $j$ , $\{{\varphi }_{j,k}\left(x\right)=2^{j/2}\varphi (2^{j}x-k),k\in \mathrm{Z}\mathrm{Z}\}$ 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 $L_2\left(\mathrm{I}\mathrm{R}\right),$ we are guaranteed that any given function of $L_2$ can be approximated arbitrarily close by such projections, i.e.:

for all $f$ in $L_2.$ Note that the orthogonal projection of $f$ onto $V_j$ can be written as:

where ${\alpha }_k=<f,{\varphi }_{j,k}>.$

Example

The simplest example of a scaling function is given by the Haar function:

Hence we have that

and

The function $\varphi$ generates, by translation and scaling, a multiresolution analysis for the spaces $V_j$ defined by:

The wavelet function and the detail spaces wj

The detail space wj

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 $V_{j+1}$ :

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: