<!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>Regularization of Lucy's algorithm</TITLE>
<META NAME="description" CONTENT="Regularization of Lucy's algorithm">
<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="node343.html">
<LINK REL="previous" HREF="node341.html">
<LINK REL="up" HREF="node339.html">
<LINK REL="next" HREF="node343.html">
</HEAD>
<BODY >
<!--Navigation Panel-->
<A NAME="tex2html5707"
HREF="node343.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
SRC="icons.gif/next_motif.gif"></A>
<A NAME="tex2html5704"
HREF="node339.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
SRC="icons.gif/up_motif.gif"></A>
<A NAME="tex2html5698"
HREF="node341.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
SRC="icons.gif/previous_motif.gif"></A>
<A NAME="tex2html5706"
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="tex2html5708"
HREF="node343.html">Convergence</A>
<B> Up:</B> <A NAME="tex2html5705"
HREF="node339.html">Regularization from significant structures</A>
<B> Previous:</B> <A NAME="tex2html5699"
HREF="node341.html">Regularization of the one-step</A>
<BR>
<BR>
<!--End of Navigation Panel-->
<H3><A NAME="SECTION002084300000000000000"> </A>
<A NAME="lucy_dec"> </A>
<BR>
Regularization of Lucy's algorithm
</H3>
Now, define
<!-- MATH: $I^{(n)}(x,y) = P(x,y) * O^{(n)} (x,y)$ -->
<I>I</I><SUP>(<I>n</I>)</SUP>(<I>x</I>,<I>y</I>) = <I>P</I>(<I>x</I>,<I>y</I>) * <I>O</I><SUP>(<I>n</I>)</SUP> (<I>x</I>,<I>y</I>). Then
<!-- MATH: $R^{(n)}(x,y) = I(x,y) - I^{(n)}(x,y)$ -->
<I>R</I><SUP>(<I>n</I>)</SUP>(<I>x</I>,<I>y</I>) = <I>I</I>(<I>x</I>,<I>y</I>) - <I>I</I><SUP>(<I>n</I>)</SUP>(<I>x</I>,<I>y</I>), and
hence
<!-- MATH: $I(x,y) = I^{(n)}(x,y) + R^{(n)}(x,y)$ -->
<I>I</I>(<I>x</I>,<I>y</I>) = <I>I</I><SUP>(<I>n</I>)</SUP>(<I>x</I>,<I>y</I>) + <I>R</I><SUP>(<I>n</I>)</SUP>(<I>x</I>,<I>y</I>).
Lucy's equation is:
<BR>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{eqnarray}
O^{(n+1)}(x,y) = O^{(n)}(x,y) [ \frac{I^{(n)}(x,y) +
R^{(n)}(x,y)}{I^{(n)}(x,y)} * P(-x,-y) ]
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="584" HEIGHT="80" ALIGN="MIDDLE" BORDER="0"
SRC="img865.gif"
ALT="$\displaystyle O^{(n+1)}(x,y) = O^{(n)}(x,y) [ \frac{I^{(n)}(x,y) +
R^{(n)}(x,y)}{I^{(n)}(x,y)} * P(-x,-y) ]$"></TD>
<TD> </TD>
<TD> </TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.119)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
and the regularization leads [<A
HREF="node370.html#starck5">39</A>] to:
<BR>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{eqnarray}
O^{(n+1)}(x,y) = O^{(n)}(x,y) [ \frac{I^{(n)}(x,y) +
{\bar{R}}^{(n)}(x,y)}{I^{(n)}(x,y)} * P(-x,-y) ]
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="585" HEIGHT="80" ALIGN="MIDDLE" BORDER="0"
SRC="img866.gif"
ALT="$\displaystyle O^{(n+1)}(x,y) = O^{(n)}(x,y) [ \frac{I^{(n)}(x,y) +
{\bar{R}}^{(n)}(x,y)}{I^{(n)}(x,y)} * P(-x,-y) ]$"></TD>
<TD> </TD>
<TD> </TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.120)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
<BR><HR>
<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
</ADDRESS>
</BODY>
</HTML>
