<!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>Hierarchical adaptive filtering</TITLE>
<META NAME="description" CONTENT="Hierarchical adaptive filtering">
<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="node332.html">
<LINK REL="up" HREF="node328.html">
<LINK REL="next" HREF="node334.html">
</HEAD>
<BODY >
<!--Navigation Panel-->
<A NAME="tex2html5598"
HREF="node334.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
SRC="icons.gif/next_motif.gif"></A>
<A NAME="tex2html5595"
HREF="node328.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
SRC="icons.gif/up_motif.gif"></A>
<A NAME="tex2html5591"
HREF="node332.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
SRC="icons.gif/previous_motif.gif"></A>
<A NAME="tex2html5597"
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="tex2html5599"
HREF="node334.html">Comparison using a multiresolution</A>
<B> Up:</B> <A NAME="tex2html5596"
HREF="node328.html">Noise reduction from the</A>
<B> Previous:</B> <A NAME="tex2html5592"
HREF="node332.html">Adaptive filtering from the</A>
<BR>
<BR>
<!--End of Navigation Panel-->
<H2><A NAME="SECTION002065000000000000000"> </A>
<A NAME="sec_filt_2"> </A>
<BR>
Hierarchical adaptive filtering
</H2>
In the previous algorithm we do not use the hierarchy of structures.
We have explored many approaches for introducing a non-linear
hierarchical law in the adaptive filtering and we found that the best
way was to link the threshold to the wavelet coefficient of the
previous plane <I>w</I><SUB><I>h</I></SUB>. We get:
<BR><P></P>
<DIV ALIGN="CENTER">
<IMG
WIDTH="317" HEIGHT="73"
SRC="img818.gif"
ALT="\begin{eqnarray*}W_i(x) = & w_i(x) & \mbox{ if } \mid w_i(x) \mid \geq L \\
W_i(x) = & 0 & \mbox{ if } \mid w_i(x) \mid < L
\end{eqnarray*}">
</DIV><P></P>
<BR CLEAR="ALL">and <I>L</I> is a threshold estimated by:
<BR><P></P>
<DIV ALIGN="CENTER">
<IMG
WIDTH="408" HEIGHT="105"
SRC="img819.gif"
ALT="\begin{eqnarray*}\mbox{ if } \mid w_i(x) \mid \geq kB_i & \mbox{ then } L = & kB...
... < kB_i & \mbox{ then } L = & kB_i t(\mid\frac{w_h}{S_h}\mid)\\
\end{eqnarray*}">
</DIV><P></P>
<BR CLEAR="ALL">where <I>S</I><SUB><I>h</I></SUB> is the standard deviation of <I>w</I><SUB><I>h</I></SUB>. The function <I>t</I>(<I>a</I>)must return a value between 0 and 1. A possible function for <I>t</I> is:
<UL>
<LI><I>t</I>(<I>a</I>) = 0 if <IMG
WIDTH="59" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img820.gif"
ALT="$a \geq k$">
<LI>
<!-- MATH: $t(a) = 1 - \frac{1}{k} a$ -->
<IMG
WIDTH="134" HEIGHT="49" ALIGN="MIDDLE" BORDER="0"
SRC="img821.gif"
ALT="$t(a) = 1 - \frac{1}{k} a$">
if <I>a</I> < <I>k</I>
</UL>
<P>
<BR><HR>
<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
</ADDRESS>
</BODY>
</HTML>
