0.1 Multiresolution analysis  (Page 2/2)

The main interest of MRA lies in the fact that, whenever a collection of closed subspaces satisfies the conditions in [link] , there exists an orthonormal wavelet basis, noted $\{{\psi }_{j,k},k\in \mathrm{Z}\mathrm{Z}\}$ of $W_j.$ Hence, we can say in very general terms that the projection of a function $f$ onto $V_{j+1}$ can be decomposed as

where ${\mathcal{P}}_j$ is the projection operator onto $V_j,$ and ${\mathcal{Q}}_j$ is the projection operator onto $W_j.$ Before proceeding further, let us state clearly what the term “orthogonal wavelet ” involves.

Recapitulation: orthogonal wavelet

We introduce above the class of orthogonal wavelets. Let us define this precisely.

An orthogonal wavelet basis is associated with an orthogonal multiresolution analysis that can be defined as follows. We talk about orthogonal MRA when the wavelet spaces $W_j$ are defined as the orthogonal complement of $V_j$ in $V_{j+1}.$ Consequently, the spaces $W_j,$ with $j\in \mathrm{Z}\mathrm{Z}$ are all mutually orthogonal, the projections ${\mathcal{P}}_j$ and ${\mathcal{Q}}_j$ are orthogonal, so that the expansion

is orthogonal. Moreover, as ${\mathcal{Q}}_j$ is orthogonal, the projection onto $W_j$ can explicitely be written as:

with

A sufficient condition for a MRA to be orthogonal is:

or $<\psi ,\varphi (.-l)>=0$ for $l\in \mathrm{Z}\mathrm{Z},$ since the other conditions simply follow from scaling.

An orthogonal wavelet is a function $\psi$ such that the collection of functions $\left\{\psi \right(x-l\left)\right|l\in \mathrm{Z}\mathrm{Z}\}$ is an orthonormal basis of $W_0.$ This is the case if $<\psi ,\psi (.-l)>={\delta }_{l,0}.$

In the following, we consider only orthogonal wavelets. We now outline how to construct a wavelet function $\psi \left(x\right)$ starting from $\varphi \left(x\right),$ and thereafter we show what this construction gives with the Haar function.

How to construct ψ(χ) starting from ρ(χ)?

Suppose we have an orthonormal basis (ONB) $\{{\varphi }_{j,k},k\in \mathrm{Z}\mathrm{Z}\}$ for $V_j$ and we want to construct ${\psi }_{jk}$ such that

• ${\psi }_{jk},k\in \mathrm{Z}\mathrm{Z}$ form an ONB for $W_j$
• $V_j\perp W_j,\text{i.e.}<{\varphi }_{jk},{\psi }_{j{k}^{\text{'}}}>=0\forall k,k^{\text{'}}$
• $W_j\perp W_{j^{\text{'}}}\text{for}j\ne j^{\text{'}}.$

It is natural to use conditions given by the MRA aspect to obtain this. More specifically, the following relationships are used to characterize $\psi :$

1. Since $\varphi \in V_0\subset V_1,$ and the ${\varphi }_{1,k}$ are an ONB in $V_1,$ we have:
   $$\varphi \left(x\right)=\sum _k{h}_k{\varphi }_{1,k},h_k=<\varphi ,{\varphi }_{1,k}>,\sum _{k\in \mathrm{Z}\mathrm{Z}}{\left|h_k\right|}^2=1.$$
   (refinement equation)
2. ${\delta }_{k,0}=\int \varphi \left(x\right)\overline{\varphi (x-k)}dx$
   (orthonormality of $\varphi (.-k)$ )
3. Let us now characterize $W_0:f\in W_0$ is equivalent to $f\in V_1$ and $f\perp V_0.$ Since $f\in V_1,$ we have:
   $$f=\sum _n{f}_n{\varphi }_{1,n},\text{with}{f}_n=<f,{\varphi }_{1,n}>.$$
   The constraint $f\perp V_0$ is implied by $f\perp {\varphi }_{0,k}$ for all $k.$
4. Taking the general form of $f\in W_0,$ we can deduce a candidate for our wavelet. We then need to verify that the ${\psi }_{0,k}$ are indeed an ONB of $W_0.$

In fact, in our setting, the conditions given above can be re-written in the Fourier domain, where the manipulations become easier (for details,see [link] , chapter 5). Let us now state the result of these manipulations.

One possibility for the construction of the wavelet $\psi$ is to take

(with convergence of this serie is $L_2$ sense).

Example(continued)

Let us see what the recipe of [link] gives for the Haar multiresolution analysis. In that case, $\varphi \left(x\right)=1$ for $0\le x\le 1,0$ otherwise. Hence:

Consequently,

Homogeneous and inhomogeneous representation of

Inhomogeneous representation

If we consider a first (coarse) approximation of $f\in V_0,$ and then “refine” this approximation with detail spaces $W_j,$ the decomposition of $f$ can be written as:

where ${\alpha }_k=<f,{\varphi }_{0,k}>$ and ${\beta }_{j,k}=<f,{\psi }_{j,k}>.$ In this case, we talk about inhomogeneous representation of $f.$

Homogeneous representation

If we use the fact that

we can decompose $f$ as a linear combination of functions ${\psi }_{j,k}$ only:

We then talk about homogeneous representation of $f$ .

Properties of the homogeneous representation

• Each coefficient ${\beta }_{j,k}$ in [link] depends only locally on $f$ because
  $${\beta }_{j,k}=\int f\left(x\right){\psi }_{j,k}\left(x\right)dx,$$
  and the wavelet ${\psi }_{j,k}\left(x\right)$ has (essentially) bounded support.
• ${\beta }_{j,k}$ gives information on scale $2^{-j},$ near position $2^{-j}k$
• a discontinuity in $f$ only affects a small proportion of coefficients– a fixed number at each frequency level.