Daubechies wavelets

A wavelet is a function $\psi \in L ^ { 2 } ( \mathbf{R} )$ that yields a basis in $L ^ { 2 } ( \mathbf{R} )$ by means of translations and dyadic dilations of itself, i.e.,

\begin{equation*} f ( x ) = \sum _ { j = - \infty } ^ { \infty } \sum _ { k = - \infty } ^ { \infty } a _ { j , k } \psi ( 2 ^ { j } x - k ), \end{equation*}

for all $f \in L ^ { 2 } ( \mathbf{R} )$ (cf. also Wavelet analysis). Such a decomposition is called the discrete wavelet transform.

In 1988, the Belgian mathematician I. Daubechies constructed [a2] a class of wavelet functions $\psi _ { N }$, $N \in \mathbf{N} \backslash \{ 0 \}$, that satisfy some special properties. First of all, the collection $\psi _ { N } ( x - k )$, $k \in \mathbf{Z}$, is an orthonormal system for fixed $N \in \mathbf{N} \backslash \{ 0 \}$. Furthermore, each wavelet $\psi _ { N }$ is compactly supported (cf. also Function of compact support). Moreover, $\operatorname { supp } ( \psi _ { N } ) = [ 0,2 N - 1 ]$. The index number $N$ is also related to the number of vanishing moments, i.e.,

\begin{equation*} \int _ { - \infty } ^ { \infty } x ^ { k } \psi _ { N } ( x ) d x = 0,0 \leq k \leq N. \end{equation*}

A last important property of the Daubechies wavelets is that their regularity increases linearly with their support width. In fact,

\begin{equation*} \exists \lambda > 0 \forall N \in \mathbf{N} , N > 2 : \psi _ { N } \in C ^ { \lambda N }. \end{equation*}

For large $N$ one has $\lambda \approx 0.2$.

The Daubechies wavelets are neither symmetric nor anti-symmetric around any axis, except for $\Psi_1$, which is in fact the Haar wavelet [a3]. Satisfying symmetry conditions cannot go together with all other properties of the Daubechies wavelets.

The Daubechies wavelets can also be used for the continuous wavelet transform, i.e.

\begin{equation*} W _ { \psi } [ f ] ( a , b ) = \frac { 1 } { \sqrt { a } } \int _ { - \infty } ^ { \infty } f ( x ) \psi \overline{\left( \frac { x - b } { a } \right)} d x, \end{equation*}

for $f \in L ^ { 2 } ( \mathbf{R} )$, $a \in \mathbf{R} ^ { + }$ and $b \in \mathbf{R}$. The parameters $a$ and $b$ denote scale and translation/position of the transform. A stable reconstruction formula exists for the continuous wavelet transform if and only if the following admissibility condition holds:

\begin{equation*} 0 < C _ { \psi } = 2 \pi \int _ { 0 } ^ { \infty } \frac { \left| \widehat { \psi } ( a \omega ) \right| ^ { 2 } } { a } d a < \infty , \end{equation*}

where $\widehat { \psi }$ denotes the Fourier transform of $\psi$. The reconstruction formula reads:

\begin{equation*} f ( x ) = \frac { 1 } { C _ { \psi } } \int _ { 0 } ^ { \infty } \int _ { - \infty } ^ { \infty } W _ { \psi } [ f ] ( a , b ) \psi ( \frac { x - b } { a } ) d b \frac { d a } { a \sqrt { a } }. \end{equation*}

This result holds weakly in $L ^ { 2 } ( \mathbf{R} )$. For $f \in L ^ { 1 } ( \mathbf{R} ) \cap L ^ { 2 } ( \mathbf{R} )$ and $\hat { f } \in L ^ { 1 } ( \mathbf{R} )$, this results also holds pointwise.

All Daubechies wavelets satisfy the admissibility condition and thus guarantee a stable reconstruction.

References

[a1]	I. Daubechies, "Ten lectures on wavelets" , SIAM (1992)
[a2]	I. Daubechies, "Orthonormal bases of compactly supported wavelets" Commun. Pure Appl. Math. , 41 (1988) pp. 909–996
[a3]	A. Haar, "Zur theorie der orthogonalen Funktionensysteme" Math. Ann. , 69 (1910) pp. 331–371