Monday, February 9, 2009

Examples of Wavelets

Morlet's Wavelet

The wavelet defined by Morlet is [16]:
$\displaystyle \hat{g}(\omega) = e^{ -2 \pi^2(\nu - \nu_0)^2}$ (14.7)

it is a complex wavelet which can be decomposed in two parts, one for the real part, and the other for the imaginary part.
\begin{eqnarray*}g_r(x) & = & \frac{1}{\sqrt{2 \pi}} e^{-\frac{x^2}{2}} \cos(2\p... ... = & \frac{1}{\sqrt{2 \pi}} e^{-\frac{x^2}{2}} \sin(2\pi\nu_0 x) \end{eqnarray*}

where $\nu_0$ is a constant. The admissibility condition is verified only if 0.8$" align="middle" border="0" height="39" width="82">. Figure 14.1 shows these two functions.
Figure 14.1: Morlet's wavelet: real part at left and imaginary part at right.
\begin{figure} \centerline{ \hbox{\psfig{figure=fig_morlet.ps,bbllx=1cm,bblly=13.5cm,bburx=20.5cm,bbury=27cm,height=5cm,width=15cm,clip=} }} \end{figure}

Mexican Hat

The Mexican hat defined by Murenzi [30] is:
$\displaystyle g(x) = (1 - x^2) e^{-\frac{1}{2}x^2}$ (14.8)

it is the second derivative of a Gaussian (see figure 14.2).
Figure 14.2: Mexican Hat
\begin{figure} \centerline{ \hbox{\psfig{figure=fig_mexicain.ps,bbllx=1cm,bblly=13.5cm,bburx=10.5cm,bbury=27cm,height=5cm,width=10cm,clip=} }} \end{figure}
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