<!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>