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<H3><A NAME="SECTION002044200000000000000"> </A>
<A NAME="sec_pyr_dir"> </A>
<BR>
Pyramidal Algorithm with one Wavelet
</H3>
To modify the previous algorithm in order to have an isotropic
wavelet transform, we compute the difference signal by:
<BR>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{eqnarray}
w_{j+1}(k) = c_j(k) - \tilde c_j (k)
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="232" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
SRC="img688.gif"
ALT="$\displaystyle w_{j+1}(k) = c_j(k) - \tilde c_j (k)$"></TD>
<TD> </TD>
<TD> </TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.43)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
but
<!-- MATH: $\tilde c_j$ -->
<IMG
WIDTH="23" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
SRC="img689.gif"
ALT="$\tilde c_j$">
is computed without reducing the number of samples:
<BR>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{eqnarray}
\tilde{c}_j(k) = \sum_l h(k-l) c_j(k)
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="237" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
SRC="img690.gif"
ALT="$\displaystyle \tilde{c}_j(k) = \sum_l h(k-l) c_j(k)$"></TD>
<TD> </TD>
<TD> </TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.44)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
and <I>c</I><SUB><I>j</I>+1</SUB> is obtained by:
<BR>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{eqnarray}
c_{j+1}(k) = \sum_l h(l-2k) c_j(l)
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="264" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
SRC="img691.gif"
ALT="$\displaystyle c_{j+1}(k) = \sum_l h(l-2k) c_j(l)$"></TD>
<TD> </TD>
<TD> </TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.45)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
The reconstruction method is the same as with the laplacian pyramid,
but the reconstruction is not exact. However, the exact reconstruction
can be performed by an iterative algorithm. If <I>P</I><SUB>0</SUB> represents
the wavelet coefficients pyramid, we look for an image such that the wavelet
transform of this image gives <I>P</I><SUB>0</SUB>. Van Cittert's iterative algorithm gives:
<BR>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{eqnarray}
P_{n+1} = P_0 + P_n - R(P_n)
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><I>P</I><SUB><I>n</I>+1</SUB> = <I>P</I><SUB>0</SUB> + <I>P</I><SUB><I>n</I></SUB> - <I>R</I>(<I>P</I><SUB><I>n</I></SUB>)</TD>
<TD> </TD>
<TD> </TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.46)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
where
<UL>
<LI><I>P</I><SUB>0</SUB> is the pyramid to be reconstructed
<LI><I>P</I><SUB><I>n</I></SUB> is the pyramid after n iterations
<LI><I>R</I> is an operator which consists in doing a reconstruction followed
by a wavelet transform.
</UL>The solution is obtained by reconstructing the pyramid <I>P</I><SUB><I>n</I></SUB>.
<P>
We need no more than 7 or 8 iterations to converge. Another way to
have a pyramidal wavelet transform with an isotropic wavelet is
to use a scaling function with a cut-off frequency.
<P>
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
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