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<H3><A NAME="SECTION002084100000000000000"> </A>
<A NAME="cittert_dec"> </A>
<BR>
Regularization of Van Cittert's algorithm
</H3>
Van Cittert's iteration is:
<BR>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{eqnarray}
O^{(n+1)} (x,y) = O^{(n)} (x,y) + \alpha{R}^{(n)}(x,y)
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="374" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
SRC="img860.gif"
ALT="$\displaystyle O^{(n+1)} (x,y) = O^{(n)} (x,y) + \alpha{R}^{(n)}(x,y)$"></TD>
<TD> </TD>
<TD> </TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.115)</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) + \alpha {\bar{R}}^{(n)}(x,y)
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="373" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
SRC="img861.gif"
ALT="$\displaystyle O^{(n+1)} (x,y) = O^{(n)} (x,y) + \alpha {\bar{R}}^{(n)}(x,y)$"></TD>
<TD> </TD>
<TD> </TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.116)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
The basic idea of our method consists of detecting, at each scale,
structures of a given size in the residual
<!-- MATH: $R^{(n)}(x,y)$ -->
<I>R</I><SUP>(<I>n</I>)</SUP>(<I>x</I>,<I>y</I>) and putting
them in the restored image
<!-- MATH: $O^{(n)}(x,y)$ -->
<I>O</I><SUP>(<I>n</I>)</SUP>(<I>x</I>,<I>y</I>). The process finishes when
no more structures are detected. Then, we have separated the image
<I>I</I>(<I>x</I>,<I>y</I>) into two images
<!-- MATH: $\tilde O(x,y)$ -->
<IMG
WIDTH="74" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
SRC="img862.gif"
ALT="$\tilde O(x,y)$">
and <I>R</I>(<I>x</I>,<I>y</I>). <IMG
WIDTH="22" HEIGHT="27" ALIGN="BOTTOM" BORDER="0"
SRC="img863.gif"
ALT="$\tilde O$">
is
the restored image, which does not contain any noise, and <I>R</I>(<I>x</I>,<I>y</I>) is
the final residual which does not contain any structure. <I>R</I> is our
estimation of the noise <I>N</I>(<I>x</I>,<I>y</I>).
<P>
<BR><HR>
<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
</ADDRESS>
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