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<H1><A NAME="SECTION002020000000000000000">
The continuous wavelet transform</A>
</H1>

<P>
The Morlet-Grossmann definition of the continuous wavelet
transform [<A
 HREF="node370.html#grossmann">17</A>] for a 1<I>D</I> signal 
<!-- MATH: $f(x)\in L^2(R)$ -->
<IMG
 WIDTH="132" HEIGHT="48" ALIGN="MIDDLE" BORDER="0"
 SRC="img581.gif"
 ALT="$f(x)\in L^2(R)$">
is:
<BR>
<DIV ALIGN="CENTER"><A NAME="eqn_wave">&#160;</A>
<!-- MATH: \begin{eqnarray}
W(a,b)=\frac{1}{\sqrt a}\int_{-\infty}^{+\infty}f(x) \psi^*(\frac{x-b}{a}) dx
\end{eqnarray} -->

<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
 WIDTH="360" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="img582.gif"
 ALT="$\displaystyle W(a,b)=\frac{1}{\sqrt a}\int_{-\infty}^{+\infty}f(x) \psi^*(\frac{x-b}{a}) dx$"></TD>
<TD>&nbsp;</TD>
<TD>&nbsp;</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.1)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
where <I>z</I><SUP>*</SUP> denotes the complex conjugate of <I>z</I>, <IMG
 WIDTH="59" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="img583.gif"
 ALT="$\psi^*(x)$">
is the
analyzing wavelet, <I>a</I> (&gt;0) is the scale parameter and <I>b</I> is
the position parameter.  The transform is characterized by the
following three properties: 
<DL COMPACT>
<DT>1.
<DD>it is a linear transformation,
<DT>2.
<DD>it is covariant under translations:
<BR>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{eqnarray}
f(x) \longrightarrow f(x-u) \qquad W(a,b)\longrightarrow W(a,b-u)
\end{eqnarray} -->

<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
 WIDTH="450" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="img584.gif"
 ALT="$\displaystyle f(x) \longrightarrow f(x-u) \qquad W(a,b)\longrightarrow W(a,b-u)$"></TD>
<TD>&nbsp;</TD>
<TD>&nbsp;</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.2)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P> <DT>3.
<DD>it is covariant under dilations:  
<BR>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{eqnarray}
f(x) \longrightarrow f(sx) \qquad W(a,b)\longrightarrow s^{-\frac{1}
{2}}W(sa,sb)
\end{eqnarray} -->

<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
 WIDTH="438" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="img585.gif"
 ALT="$\displaystyle f(x) \longrightarrow f(sx) \qquad W(a,b)\longrightarrow s^{-\frac{1}
{2}}W(sa,sb)$"></TD>
<TD>&nbsp;</TD>
<TD>&nbsp;</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.3)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P></DL>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.

<P>
In Fourier space, we have:
<BR>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{eqnarray}
\hat W(a,\nu)=\sqrt a \hat f(\nu)\hat{\psi}^*(a\nu)
\end{eqnarray} -->

<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
 WIDTH="246" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="img586.gif"
 ALT="$\displaystyle \hat W(a,\nu)=\sqrt a \hat f(\nu)\hat{\psi}^*(a\nu)$"></TD>
<TD>&nbsp;</TD>
<TD>&nbsp;</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.4)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
When the scale <I>a</I> varies, the filter 
<!-- MATH: $\hat{\psi}^*(a\nu)$ -->
<IMG
 WIDTH="70" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="img587.gif"
 ALT="$\hat{\psi}^*(a\nu)$">
is only reduced or
dilated while keeping the same pattern. 

<P>
Now consider a function <I>W</I>(<I>a</I>,<I>b</I>) which is the wavelet transform of a
given function <I>f</I>(<I>x</I>). It has been shown
[#grossmann<#14252<A
 HREF="node370.html#>"></A>,#holschn<#14253<A
 HREF="node370.html#>"></A>] that <I>f</I>(<I>x</I>) can be restored using the
formula:
<BR>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{eqnarray}
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}
\end{eqnarray} -->

<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
 WIDTH="467" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="img588.gif"
 ALT="$\displaystyle f(x)=\frac{1}{C_{\chi}} \int_0^{+\infty}\int_{-\infty}^{+\infty} \frac{1}{\sqrt
a}W(a,b)\chi(\frac{x-b}{a})\frac{da.db}{a^2}$"></TD>
<TD>&nbsp;</TD>
<TD>&nbsp;</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.5)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
where:
<BR>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{eqnarray}
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
\end{eqnarray} -->

<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
 WIDTH="440" HEIGHT="83" ALIGN="MIDDLE" BORDER="0"
 SRC="img589.gif"
 ALT="$\displaystyle C_{\chi}=\int_0^{+\infty} \frac{\hat{\psi}^*(\nu)\hat{\chi}(\nu)}{\nu}d\nu
=\int_{-\infty}^0 \frac{\hat{\psi}^*(\nu)\hat{\chi}(\nu)}{\nu}d\nu$"></TD>
<TD>&nbsp;</TD>
<TD>&nbsp;</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.6)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
Generally 
<!-- MATH: $\chi(x)=\psi(x)$ -->
<IMG
 WIDTH="123" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="img590.gif"
 ALT="$\chi(x)=\psi(x)$">,
but  other choices can enhance certain features 
for some applications.

<P>
The reconstruction is only available if <IMG
 WIDTH="32" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img591.gif"
 ALT="$C_{\chi}$">
is defined (admissibility
condition). In the case of 
<!-- MATH: $\chi(x)=\psi(x)$ -->
<IMG
 WIDTH="123" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="img592.gif"
 ALT="$\chi(x)=\psi(x)$">,
this condition implies
 
<!-- MATH: $\hat \psi(0)=0$ -->
<IMG
 WIDTH="88" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="img593.gif"
 ALT="$\hat \psi(0)=0$">,
<I>i.e.</I> the mean of the wavelet function is 0. 

<P>
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
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