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<H3><A NAME="SECTION002084200000000000000"> </A>
<A NAME="grad_dec"> </A>
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Regularization of the one-step gradient method
</H3>
The one-step gradient iteration is:
<BR>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{eqnarray}
O^{(n+1)} (x,y) = O^{(n)} (x,y) + P(-x,-y) * {R}^{(n)}(x,y)
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><I>O</I><SUP>(<I>n</I>+1)</SUP> (<I>x</I>,<I>y</I>) = <I>O</I><SUP>(<I>n</I>)</SUP> (<I>x</I>,<I>y</I>) + <I>P</I>(-<I>x</I>,-<I>y</I>) * <I>R</I><SUP>(<I>n</I>)</SUP>(<I>x</I>,<I>y</I>)</TD>
<TD> </TD>
<TD> </TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.117)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
with
<!-- MATH: ${R}^{(n)}(x,y) = I(x,y) - P(x,y) * O^{(n)} (x,y)$ -->
<I>R</I><SUP>(<I>n</I>)</SUP>(<I>x</I>,<I>y</I>) = <I>I</I>(<I>x</I>,<I>y</I>) - <I>P</I>(<I>x</I>,<I>y</I>) * <I>O</I><SUP>(<I>n</I>)</SUP> (<I>x</I>,<I>y</I>).
The regularization by the significant structures leads to:
<BR>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{eqnarray}
O^{(n+1)} (x,y) = O^{(n)} (x,y) + P(-x,-y) * {\bar{R}}^{(n)}(x,y)
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="482" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
SRC="img864.gif"
ALT="$\displaystyle O^{(n+1)} (x,y) = O^{(n)} (x,y) + P(-x,-y) * {\bar{R}}^{(n)}(x,y)$"></TD>
<TD> </TD>
<TD> </TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.118)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
</ADDRESS>
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