Monday, February 9, 2009

The continuous wavelet transform

The Morlet-Grossmann definition of the continuous wavelet transform [17] for a 1D signal $f(x)\in L^2(R)$ is:

$\displaystyle W(a,b)=\frac{1}{\sqrt a}\int_{-\infty}^{+\infty}f(x) \psi^*(\frac{x-b}{a}) dx$ (14.1)

where z* denotes the complex conjugate of z, $\psi^*(x)$ is the analyzing wavelet, a (>0) is the scale parameter and b is the position parameter. The transform is characterized by the following three properties:
1.
it is a linear transformation,
2.
it is covariant under translations:
$\displaystyle f(x) \longrightarrow f(x-u) \qquad W(a,b)\longrightarrow W(a,b-u)$ (14.2)

3.
it is covariant under dilations:
$\displaystyle f(x) \longrightarrow f(sx) \qquad W(a,b)\longrightarrow s^{-\frac{1} {2}}W(sa,sb)$ (14.3)

The last property makes the wavelet transform very suitable for analyzing hierarchical structures. It is like a mathematical microscope with properties that do not depend on the magnification.

In Fourier space, we have:

$\displaystyle \hat W(a,\nu)=\sqrt a \hat f(\nu)\hat{\psi}^*(a\nu)$ (14.4)

When the scale a varies, the filter $\hat{\psi}^*(a\nu)$ is only reduced or dilated while keeping the same pattern.

Now consider a function W(a,b) which is the wavelet transform of a given function f(x). It has been shown [#grossmann<#14252,#holschn<#14253] that f(x) can be restored using the formula:

$\displaystyle f(x)=\frac{1}{C_{\chi}} \int_0^{+\infty}\int_{-\infty}^{+\infty} \frac{1}{\sqrt a}W(a,b)\chi(\frac{x-b}{a})\frac{da.db}{a^2}$ (14.5)

where:
$\displaystyle C_{\chi}=\int_0^{+\infty} \frac{\hat{\psi}^*(\nu)\hat{\chi}(\nu)}{\nu}d\nu =\int_{-\infty}^0 \frac{\hat{\psi}^*(\nu)\hat{\chi}(\nu)}{\nu}d\nu$ (14.6)

Generally $\chi(x)=\psi(x)$, but other choices can enhance certain features for some applications.

The reconstruction is only available if $C_{\chi}$ is defined (admissibility condition). In the case of $\chi(x)=\psi(x)$, this condition implies $\hat \psi(0)=0$, i.e. the mean of the wavelet function is 0.

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