<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <!--Converted with LaTeX2HTML 98.1p1 release (March 2nd, 1998) originally by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds * revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan * with significant contributions from: Jens Lippmann, Marek Rouchal, Martin Wilck and others --> <HTML> <HEAD> <TITLE>Pyramidal Algorithm with one Wavelet</TITLE> <META NAME="description" CONTENT="Pyramidal Algorithm with one Wavelet"> <META NAME="keywords" CONTENT="vol2"> <META NAME="resource-type" CONTENT="document"> <META NAME="distribution" CONTENT="global"> <META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1"> <LINK REL="STYLESHEET" HREF="vol2.css"> <LINK REL="previous" HREF="node319.html"> <LINK REL="up" HREF="node318.html"> <LINK REL="next" HREF="node321.html"> </HEAD> <BODY > <!--Navigation Panel--> <A NAME="tex2html5453" HREF="node321.html"> <IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="icons.gif/next_motif.gif"></A> <A NAME="tex2html5450" HREF="node318.html"> <IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="icons.gif/up_motif.gif"></A> <A NAME="tex2html5446" HREF="node319.html"> <IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="icons.gif/previous_motif.gif"></A> <A NAME="tex2html5452" HREF="node1.html"> <IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="icons.gif/contents_motif.gif"></A> <BR> <B> Next:</B> <A NAME="tex2html5454" HREF="node321.html">Multiresolution with scaling functions</A> <B> Up:</B> <A NAME="tex2html5451" HREF="node318.html">Pyramidal Algorithm</A> <B> Previous:</B> <A NAME="tex2html5447" HREF="node319.html">The Laplacian Pyramid</A> <BR> <BR> <!--End of Navigation Panel--> <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> <HR> <!--Navigation Panel--> <A NAME="tex2html5453" HREF="node321.html"> <IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="icons.gif/next_motif.gif"></A> <A NAME="tex2html5450" HREF="node318.html"> <IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="icons.gif/up_motif.gif"></A> <A NAME="tex2html5446" HREF="node319.html"> <IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="icons.gif/previous_motif.gif"></A> <A NAME="tex2html5452" HREF="node1.html"> <IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="icons.gif/contents_motif.gif"></A> <BR> <B> Next:</B> <A NAME="tex2html5454" HREF="node321.html">Multiresolution with scaling functions</A> <B> Up:</B> <A NAME="tex2html5451" HREF="node318.html">Pyramidal Algorithm</A> <B> Previous:</B> <A NAME="tex2html5447" HREF="node319.html">The Laplacian Pyramid</A> <!--End of Navigation Panel--> <ADDRESS> <I>Petra Nass</I> <BR><I>1999-06-15</I> </ADDRESS> </BODY> </HTML>