0.2 Multiresolution analysis and wavelets

Definition of subspacesAnd of scaling functions

A natural way to introduce wavelets is through the multiresolution analysis. Given a function $f\in L_2\left(\mathbb{R}\right)$ , a multiresolution of $L_2\left(\mathbb{R}\right)$ will provide us with a sequence of spaces $V_j,V_{j+1},...$ such that the projections of $f$ onto these spaces give finer and finer approximations (as $j\to \infty$ ) of the function $f$ .

1. $...\subset V_{-1}\subset V_0\subset V_1\subset ...$
2. The spaces $V_j$ satisfy
   $$\bigcup _{j\in \mathbb{Z}}{V}_j\text{is}\text{dense}\text{in}{L}_2\left(\mathbb{R}\right)\text{and}\bigcap _{j\in \mathbb{Z}}{V}_j=\left\{0\right\}.$$
3. If $f\left(x\right)\in V_0,f\left(2^{j}x\right)\in V_j,$ i.e. the spaces $V_j$ are scaled versions of the central space $V_0.$
4. If $f\in V_0,f(.-k)\in V_0,k\in \mathbb{Z}$ , that is, $V_0$ (and hence all the $V_j$ ) is invariant under translation.
5. There exists $\varphi \in V_0$ such that $\left\{\varphi \right(x-k);k\in \mathbb{Z}\}$ is a Riesz basis in $V_0.$

We will call `level' of a MRA one of the subspaces $V_j$ . From [link] , it follows that, for fixed $j$ , the set $\{{\varphi }_{jk}\left(x\right)=2^{j/2}\varphi (2^{j}x-k);k\in \mathbb{Z}\}$ of scaled and translated versions of $\varphi$ is a Riesz 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 two-scale equation or refinement equation . It is a fundamental equation in MRA since it tells us how to go from a fine level $V_1$ to a coarser level $V_0$ . The function $\varphi$ is called the scaling function .

As said before, the spaces $V_j$ will be used to approximate general functions. This will be done by defining appropriate projections onto these spaces. Since the union of all the $V_j$ is dense in $L_2\left(\mathbb{R}\right),$ we are guaranteed that any given function of $L_2\left(\mathbb{R}\right)$ can be approximated arbitrarily close by such projections. As an example, define the space $V_j$ as

Then the scaling function $\varphi \left(x\right)=1_{[0,1)}\left(x\right)$ , called the Haar scaling function, generates by translation and dilatation a MRA for the sequence of spaces $\{V_j,j\in \mathbb{Z}\}$ defined in [link] , see [link] , [link] .

The detail space and the wavelet function

Rather than considering all the nested spaces $V_j,$ it would be more efficient to code only the information needed to go from $V_j$ to $V_{j+1}.$ Hence we consider the space $W_j$ which complements $V_j$ in $V_{j+1}$ :

The space $W_j$ is not necessarily orthogonal to $V_j$ , but it always 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 equation [link] , we have for any $j_0\in \mathbb{Z}$ , the decomposition

With the notational convention that $W_{j_0-1}:=V_{j_0}$ , we call the sequence

${\left\{W_j\right\}}_{j\ge j_0-1}$ a multiscale decomposition ( MSD ).

We call $\psi$ a wavelet function whenever the set $\left\{\psi \right(x-k);k\in \mathbb{Z}\}$ is a Riesz basis of $W_0$ . Since $W_0\subset V_1$ , there also exist a refinement equation for $\psi$ , similarly to [link] :

The collection of wavelet functions $\{{\psi }_{jk}=2^{j/2}\psi (2^{j}x-k);k\in \mathbb{Z},j\in \mathbb{Z}\}$ is then a Riesz basis for $L_2\left(\mathbb{R}\right)$ . One of the main features of the wavelet functions is that they possess a certain number of vanishing moments.

We now mention two interesting cases of wavelet bases.

Orthogonal bases

In an orthogonal multiresolution analysis , the spaces $W_j$ are defined as the orthogonal complement of $V_j$ in $V_{j+1}$ . The following theorem tells us one of the main advantages of such a MRA.