0.2 Multiresolution analysis and wavelets

An immediate consequence of [link] is that $\{{\psi }_{jk},k\in \mathbb{Z}\}$ constitutes an orthogonal basis for the orthogonal complement $W_j$ of $V_j$ in $V_{j+1}$ . In this section, let ${\mathcal{P}}_j$ (resp. ${\mathcal{Q}}_j$ ) be the orthogonal projection operator onto $V_j$ (resp. $W_j$ ). The orthogonal expansion

tells us that a first, coarse approximation of $f$ in $V_{j_0}$ is further refined with the projection of $f$ onto the detail spaces $W_j$ .

[link] shows two examples of orthogonal wavelet functions. The first is the Haar wavelet, associated to the Haar scaling function defined in "Definition of subspaces V j and of scaling functions" .

The Haar wavelet has only one vanishing moment and consequently is optimal only to represent functions having a low degree of regularity, like, for example, $\beta -$ Hölder functions with $0<\beta <1$ .

Daubechies constructed in [link] , [link] compactly supported wavelets which have more than one vanishing moment. Compactly supported wavelets are desirable from a numerical point of view, while having more than one vanishing moment allows to reconstruct exactly polynomials of higher order. These wavelets cannot, in general, be written in a closed analytic form. However, their graph can be computed with arbitrarily high precision using a subdivision scheme algorithm. [link] (b) represents the Daubechies Least Asymmetric wavelet with $N=4$ vanishing moments.

This figure also illustrates the reason behind the name `wavelet': since wavelets are functions with a certain number of vanishing moments, they have the shape of a `little wave' or `wavelet'.

Biorthogonal bases

Having an orthogonal MRA puts strong constraints on the construction of a wavelet basis. For example, the Haar wavelet is the only real-valued function which is compactly supported and symmetric.However, if we relax orthogonality for biorthogonality , then it becomes possible to have real-valued wavelet bases of fixed but arbitrary high order (see Definition 1 from Approximation of Functions ) which are symmetric and compactly supported [link] . In a biorthogonal setting, a dual scaling function $\tilde{\varphi }$ and a dual wavelet function $\tilde{\psi }$ exist. They generate a dual MRA with subspaces ${\tilde{V}}_j$ and complement spaces ${\tilde{W}}_j$ such that

In other words,

Moreover, the dual functions also have to satisfy

where ${\delta }_{k,0}$ is the Kronecker symbol. By construction, the dual scaling and wavelet functions satisfy a refinement equation, similarly to the equations [link] and [link] .

In this work, we use the following convention: the dual MSD will be used to decompose a function (or a signal), while the original, or primal MSD reconstructs the function. This yields the following representation of a function $f\in L_2\left(\mathbb{R}\right)$

[link] shows an example of a biorthogonal wavelet basis built by Cohen, Daubechies and Feauveau in [link] , (called CDF-wavelets hereafter).