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Up: The Wavelet Transform
Previous: Introduction
The Morlet-Grossmann definition of the continuous wavelet
transform [17] for a 1D signal
is:
![$\displaystyle W(a,b)=\frac{1}{\sqrt a}\int_{-\infty}^{+\infty}f(x) \psi^*(\frac{x-b}{a}) dx$](img582.gif) |
|
|
(14.1) |
where z* denotes the complex conjugate of z,
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:
![$\displaystyle f(x) \longrightarrow f(x-u) \qquad W(a,b)\longrightarrow W(a,b-u)$](img584.gif) |
|
|
(14.2) |
- 3.
- it is covariant under dilations:
![$\displaystyle f(x) \longrightarrow f(sx) \qquad W(a,b)\longrightarrow s^{-\frac{1}
{2}}W(sa,sb)$](img585.gif) |
|
|
(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:
![$\displaystyle \hat W(a,\nu)=\sqrt a \hat f(\nu)\hat{\psi}^*(a\nu)$](img586.gif) |
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(14.4) |
When the scale a varies, the filter
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:
![$\displaystyle f(x)=\frac{1}{C_{\chi}} \int_0^{+\infty}\int_{-\infty}^{+\infty} \frac{1}{\sqrt
a}W(a,b)\chi(\frac{x-b}{a})\frac{da.db}{a^2}$](img588.gif) |
|
|
(14.5) |
where:
![$\displaystyle C_{\chi}=\int_0^{+\infty} \frac{\hat{\psi}^*(\nu)\hat{\chi}(\nu)}{\nu}d\nu
=\int_{-\infty}^0 \frac{\hat{\psi}^*(\nu)\hat{\chi}(\nu)}{\nu}d\nu$](img589.gif) |
|
|
(14.6) |
Generally
,
but other choices can enhance certain features
for some applications.
The reconstruction is only available if
is defined (admissibility
condition). In the case of
,
this condition implies
,
i.e. the mean of the wavelet function is 0.
Next: Examples of Wavelets
Up: The Wavelet Transform
Previous: Introduction
http://www.eso.org/midas/midas-support.html
1999-06-15