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<H2><A NAME="SECTION002084000000000000000">
Regularization from significant structures</A>
</H2>
If we use an iterative deconvolution algorithm, such as Van Cittert's or
Lucy's one, we define
<!-- MATH: $R^{(n)}(x,y)$ -->
<I>R</I><SUP>(<I>n</I>)</SUP>(<I>x</I>,<I>y</I>), the error at iteration <I>n</I>:
<BR>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{eqnarray}
R^{(n)}(x,y) = I(x,y) - P(x,y) * O^{(n)}(x,y)
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><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>)</TD>
<TD> </TD>
<TD> </TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.111)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
By using the <EM>à trous</EM> wavelet transform algorithm, <I>R</I><SUP>(<I>n</I>)</SUP>
can be defined by the sum of its <I>n</I><SUB><I>p</I></SUB> wavelet planes and the last smooth
plane (see equation <A HREF="node317.html#eqn_rec">14.33</A>).
<BR>
<DIV ALIGN="CENTER"><A NAME="resid"> </A>
<!-- MATH: \begin{eqnarray}
R^{(n)}(x,y) = c_{n_p}(x,y) + \sum_{j=1}^{n_p} w_j(x,y)
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="342" HEIGHT="85" ALIGN="MIDDLE" BORDER="0"
SRC="img856.gif"
ALT="$\displaystyle R^{(n)}(x,y) = c_{n_p}(x,y) + \sum_{j=1}^{n_p} w_j(x,y)$"></TD>
<TD> </TD>
<TD> </TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.112)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
The wavelet coefficients provide a mechanism to extract from the residuals
at each iteration only the significant structures. A large part of
these residuals are generally statistically non significant.
The significant residual is:
<BR>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{eqnarray}
\bar{R}^{(n)}(x,y) = c_{n_p}(x,y) + \sum_{j=1}^{n_p} \alpha(w_j(x,y), N_j) \
w_i(x,y)
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="488" HEIGHT="85" ALIGN="MIDDLE" BORDER="0"
SRC="img857.gif"
ALT="$\displaystyle \bar{R}^{(n)}(x,y) = c_{n_p}(x,y) + \sum_{j=1}^{n_p} \alpha(w_j(x,y), N_j) \
w_i(x,y)$"></TD>
<TD> </TD>
<TD> </TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.113)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
<I>N</I><SUB><I>j</I></SUB> is the standard deviation of the noise at scale <I>j</I>, and <IMG
WIDTH="20" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
SRC="img858.gif"
ALT="$\alpha$">
is
a function which is defined by:
<BR>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{eqnarray}
\alpha(a, \sigma) = \left\{
\begin{array}{ll}
1 & \mbox{if } \mid a \mid \geq k\sigma \\
0 & \mbox{if } \mid a \mid < k\sigma
\end{array}
\right.
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="279" HEIGHT="82" ALIGN="MIDDLE" BORDER="0"
SRC="img859.gif"
ALT="$\displaystyle \alpha(a, \sigma) = \left\{
\begin{array}{ll}
1 & \mbox{if } \mid a \mid \geq k\sigma \\
0 & \mbox{if } \mid a \mid < k\sigma
\end{array}\right.$"></TD>
<TD> </TD>
<TD> </TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.114)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
The standard deviation of the noise <I>N</I><SUB><I>j</I></SUB> is estimated from the
standard deviation of the noise in the image. This
is done from the study of noise variation in the wavelet space,
with the hypothesis of a white Gaussian noise.
<P>
We now show how the iterative deconvolution algorithms can be
modified in order to take into account only the significant
structure at each scale.
<P>
<BR><HR>
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<LI><A NAME="tex2html5672"
HREF="node340.html">Regularization of Van Cittert's algorithm</A>
<LI><A NAME="tex2html5673"
HREF="node341.html">Regularization of the one-step gradient method</A>
<LI><A NAME="tex2html5674"
HREF="node342.html">Regularization of Lucy's algorithm</A>
<LI><A NAME="tex2html5675"
HREF="node343.html">Convergence</A>
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
</ADDRESS>
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