<!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>Introduction</TITLE> <META NAME="description" CONTENT="Introduction"> <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="next" HREF="node316.html"> <LINK REL="previous" HREF="node314.html"> <LINK REL="up" HREF="node314.html"> <LINK REL="next" HREF="node316.html"> </HEAD> <BODY > <!--Navigation Panel--> <A NAME="tex2html5398" HREF="node316.html"> <IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="icons.gif/next_motif.gif"></A> <A NAME="tex2html5395" HREF="node314.html"> <IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="icons.gif/up_motif.gif"></A> <A NAME="tex2html5389" HREF="node314.html"> <IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="icons.gif/previous_motif.gif"></A> <A NAME="tex2html5397" 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="tex2html5399" HREF="node316.html">Multiresolution Analysis</A> <B> Up:</B> <A NAME="tex2html5396" HREF="node314.html">The discrete wavelet transform</A> <B> Previous:</B> <A NAME="tex2html5390" HREF="node314.html">The discrete wavelet transform</A> <BR> <BR> <!--End of Navigation Panel--> <H2><A NAME="SECTION002041000000000000000"> Introduction</A> </H2> For processing classical images the sampling is made in accordance with Shannon's [<A HREF="node370.html#shannon">32</A>] well-known theorem. The discrete wavelet transform (DWT) can be derived from this theorem if we process a signal which has a cut-off frequency. For such images the frequency band is always limited by the size of the camera aperture. <P> A digital analysis is provided by the discretisation of formula <A HREF="node310.html#eqn_wave">14.1</A>, with some simple considerations on the modification of the wavelet pattern by dilation. Usually the wavelet function <IMG WIDTH="59" HEIGHT="44" ALIGN="MIDDLE" BORDER="0" SRC="img601.gif" ALT="$\psi^*(x)$"> has no cut-off frequency and it is necessary to suppress the values outside the frequency band in order to avoid aliasing effects. We can work in Fourier space, computing the transform scale by scale. The number of elements for a scale can be reduced, if the frequency bandwidth is also reduced. This is possible only for a wavelet which also has a cut-off frequency. The decomposition proposed by Littlewood and Paley [<A HREF="node370.html#littlewood">22</A>] provides a very nice illustration of the reduction of elements scale by scale. This decomposition is based on an iterative dichotomy of the frequency band. The associated wavelet is well localized in Fourier space where it allows a reasonable analysis to be made although not in the original space. The search for a discrete transform which is well localized in both spaces leads to multiresolution analysis. <P> <HR> <!--Navigation Panel--> <A NAME="tex2html5398" HREF="node316.html"> <IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="icons.gif/next_motif.gif"></A> <A NAME="tex2html5395" HREF="node314.html"> <IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="icons.gif/up_motif.gif"></A> <A NAME="tex2html5389" HREF="node314.html"> <IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="icons.gif/previous_motif.gif"></A> <A NAME="tex2html5397" 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="tex2html5399" HREF="node316.html">Multiresolution Analysis</A> <B> Up:</B> <A NAME="tex2html5396" HREF="node314.html">The discrete wavelet transform</A> <B> Previous:</B> <A NAME="tex2html5390" HREF="node314.html">The discrete wavelet transform</A> <!--End of Navigation Panel--> <ADDRESS> <I>Petra Nass</I> <BR><I>1999-06-15</I> </ADDRESS> </BODY> </HTML>
