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<H2><A NAME="SECTION002083000000000000000"> </A>
<A NAME="direct_dec"> </A>
<BR>
Tikhonov's regularization and multiresolution analysis
</H2>
If <I>w</I><SUB><I>j</I></SUB><SUP>(<I>I</I>)</SUP> are the wavelet coefficients of
the image <I>I</I> at the scale j, we have:
<BR>
<DIV ALIGN="CENTER"><A NAME="eq_cpphi"> </A>
<!-- MATH: \begin{eqnarray}
\hat{w}_j^{(I)}(u,v) & = & \hat{g}(2^{j-1}u, 2^{j-1}v)
\prod_{i=j-2}^{i=0}\hat{h}(2^{i}u, 2^{i}v) \hat{I}(u,v)
\nonumber \\& = & {\hat{\psi}(2^{j}u, 2^{j}v) \over
\hat{\phi}(u,v)} \hat{P}(u,v) \hat{O}(u,v) \\& = &
\hat{w}_j^{(P)} \hat{O}(u,v) \nonumber
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="95" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
SRC="img843.gif"
ALT="$\displaystyle \hat{w}_j^{(I)}(u,v)$"></TD>
<TD ALIGN="CENTER" NOWRAP>=</TD>
<TD ALIGN="LEFT" NOWRAP><IMG
WIDTH="366" HEIGHT="87" ALIGN="MIDDLE" BORDER="0"
SRC="img844.gif"
ALT="$\displaystyle \hat{g}(2^{j-1}u, 2^{j-1}v)
\prod_{i=j-2}^{i=0}\hat{h}(2^{i}u, 2^{i}v) \hat{I}(u,v)$"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
</TD></TR>
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"> </TD>
<TD ALIGN="CENTER" NOWRAP>=</TD>
<TD ALIGN="LEFT" NOWRAP><IMG
WIDTH="250" HEIGHT="83" ALIGN="MIDDLE" BORDER="0"
SRC="img845.gif"
ALT="$\displaystyle {\hat{\psi}(2^{j}u, 2^{j}v) \over
\hat{\phi}(u,v)} \hat{P}(u,v) \hat{O}(u,v)$"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.106)</TD></TR>
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"> </TD>
<TD ALIGN="CENTER" NOWRAP>=</TD>
<TD ALIGN="LEFT" NOWRAP><IMG
WIDTH="117" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
SRC="img846.gif"
ALT="$\displaystyle \hat{w}_j^{(P)} \hat{O}(u,v)$"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
where
<!-- MATH: $w_{j}^{(P)}$ -->
<I>w</I><SUB><I>j</I></SUB><SUP>(<I>P</I>)</SUP> are the wavelet coefficients of the PSF at the scale <I>j</I>.
The wavelet coefficients of the image <I>I</I> are the product of convolution
of object <I>O</I> by the wavelet coefficients of the PSF.
<P>
To deconvolve the image, we have to minimize for each scale j:
<BR>
<DIV ALIGN="CENTER"><A NAME="eqn_min1"> </A>
<!-- MATH: \begin{eqnarray}
\parallel {\hat \psi(2^ju, 2^jv)\over \hat\phi(u, v)} \hat P(u,v) \hat O(u,v) - \hat w_j^{(I)}(u,v)\parallel^2
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="410" HEIGHT="83" ALIGN="MIDDLE" BORDER="0"
SRC="img847.gif"
ALT="$\displaystyle \parallel {\hat \psi(2^ju, 2^jv)\over \hat\phi(u, v)} \hat P(u,v) \hat O(u,v) - \hat w_j^{(I)}(u,v)\parallel^2$"></TD>
<TD> </TD>
<TD> </TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.107)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
and for the plane at the lower resolution:
<BR>
<DIV ALIGN="CENTER"><A NAME="eqn_min2"> </A>
<!-- MATH: \begin{eqnarray}
\parallel {\hat \phi(2^{n-1}u, 2^{n-1}v)\over \hat\phi(u, v)} \hat P(u,v) \hat O(u,v) - \hat c_{n-1}^{(I)}(u,v)\parallel^2
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="461" HEIGHT="83" ALIGN="MIDDLE" BORDER="0"
SRC="img848.gif"
ALT="$\displaystyle \parallel {\hat \phi(2^{n-1}u, 2^{n-1}v)\over \hat\phi(u, v)} \hat P(u,v) \hat O(u,v) - \hat c_{n-1}^{(I)}(u,v)\parallel^2$"></TD>
<TD> </TD>
<TD> </TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.108)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P><I>n</I> being the number of planes of the wavelet transform ((<I>n</I>-1) wavelet
coefficient planes and one plane for the image at the lower resolution).
The problem has not generally a unique solution, and we need to do
a regularization [<A
HREF="node370.html#tikhonov">40</A>]. At each scale, we add the term:
<BR>
<DIV ALIGN="CENTER"><A NAME="eqn_min3"> </A>
<!-- MATH: \begin{eqnarray}
\gamma_j \parallel w_j^{(O)} \parallel^2 \mbox{ min }
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="175" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
SRC="img849.gif"
ALT="$\displaystyle \gamma_j \parallel w_j^{(O)} \parallel^2 \mbox{ min }$"></TD>
<TD> </TD>
<TD> </TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.109)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
This is a smoothness constraint. We want to have the minimum information
in the restored object. From equations <A HREF="node338.html#eqn_min1">14.107</A>, <A HREF="node338.html#eqn_min2">14.108</A>,
<A HREF="node338.html#eqn_min3">14.109</A>, we find:
<BR>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{eqnarray}
\hat D(u,v) \hat O(u,v) = \hat N(u,v)
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="243" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
SRC="img850.gif"
ALT="$\displaystyle \hat D(u,v) \hat O(u,v) = \hat N(u,v)$"></TD>
<TD> </TD>
<TD> </TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.110)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
with:
<BR><P></P>
<DIV ALIGN="CENTER">
<IMG
WIDTH="712" HEIGHT="64"
SRC="img851.gif"
ALT="\begin{eqnarray*}\hat D(u,v) = \sum_j \mid \hat\psi(2^ju, 2^jv) \mid^2 (\mid\hat...
... \gamma_j) + \mid \hat \phi(2^{n-1}u,2^{n-1}v)\hat P(u,v) \mid^2
\end{eqnarray*}">
</DIV><P></P>
<BR CLEAR="ALL">
and:
<BR><P></P>
<DIV ALIGN="CENTER">
<IMG
WIDTH="742" HEIGHT="64"
SRC="img852.gif"
ALT="\begin{eqnarray*}\hat N(u,v) = \hat\phi(u, v) [ \sum_j \hat P^*(u,v)\hat\psi^*(2...
... \hat P^*(u,v) \hat\phi^*(2^{n-1}u,2^{n-1}v) \hat c_{n-1}^{(I)}]
\end{eqnarray*}">
</DIV><P></P>
<BR CLEAR="ALL">
if the equation is well constrained, the object can be computed by a
simple division of <IMG
WIDTH="25" HEIGHT="27" ALIGN="BOTTOM" BORDER="0"
SRC="img853.gif"
ALT="$\hat N$">
by <IMG
WIDTH="25" HEIGHT="27" ALIGN="BOTTOM" BORDER="0"
SRC="img854.gif"
ALT="$\hat D$">.
An iterative algorithm
can be used to do this inversion if we want to add other constraints such as
positivity. We have in fact a multiresolution Tikhonov's regularization.
This method has the advantage to furnish a solution quickly, but
optimal regularization parameters <IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img855.gif"
ALT="$\gamma_j$">
cannot be found directly,
and several tests are generally necessary before finding an acceptable
solution. Hovewer, the method can be interesting if we need to deconvolve
a big number of images with the same noise characteristics. In this case,
parameters have to be determined only the first time. In a general way,
we prefer to use one of the following iterative algorithms.
<P>
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
</ADDRESS>
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