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<H2><A NAME="SECTION00534000000000000000">
Image Restoration</A>
</H2>
Both the imaging system and observing conditions will cause a
degradation of the resolution of the image.  In principle it is
possible to reduce this smearing effect by deconvolving the frame with
the point spread function.  The degree to which this can be done
depends on the actual sampling of the image.  Basically, it is not
possible to retrieve information on features with frequencies higher
than the Nyquist frequency (see Equation&nbsp;<A HREF="node12.html#eq:nyquist">2.2</A>).  Several
different techniques are used depending on the data (see Wells 1980
for a general discussion of the methods).

<P>
Fourier transforms are often used since convolutions in the frequency
space become multiplications (see Section&nbsp;<A HREF="node22.html#trans">2.3.3</A>).  Combining
Equation&nbsp;<A HREF="node12.html#eq:image">2.1</A> and Equation&nbsp;<A HREF="node22.html#eq:fourier-conv">2.26</A> the
original image 
<!-- MATH: $I = {\cal O_F(I)}$ -->
<IMG
 WIDTH="109" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="img110.gif"
 ALT="$I = {\cal O_F(I)}$">
is obtained by division
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
{\cal I}(u,v) = {\cal O_F} [ F(j,k) \otimes P(j,k) + N(j,k)] / {\cal P}(u,v)
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:fourier-deconv">&#160;</A><IMG
 WIDTH="467" HEIGHT="40"
 SRC="img111.gif"
 ALT="\begin{displaymath}{\cal I}(u,v) = {\cal O_F} [ F(j,k) \otimes P(j,k) + N(j,k)] / {\cal P}(u,v)
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(2.27)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
if the transformed PSF <IMG
 WIDTH="23" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="img112.gif"
 ALT="$\cal P$">
is non-zero and the noise <I>N</I> is
insignificant.  For data with low or medium signal-to-noise ratio
(i.e. <I>S</I>/<I>N</I> &lt; 100), as for most optical observations, this technique
introduces artifacts in the deconvolved image.  These effects can be
reduced by filtering the transforms with Wiener filters (Helstrom
1967, Horner 1970).

<P>
Another group of image restoration algorithms use iterative methods to
obtain a solution which is consistent with the data.
The maximum entropy method uses either the entropy :
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
H1 = - \sum_j I_j \log( I_j) \;\; {\em or} \;\; H2 = \sum_j \log( I_j )
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:entropy">&#160;</A><IMG
 WIDTH="398" HEIGHT="64"
 SRC="img113.gif"
 ALT="\begin{displaymath}H1 = - \sum_j I_j \log( I_j) \;\; {\em or} \;\; H2 = \sum_j \log( I_j )
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(2.28)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
in the optimizing procedure (Frieden 1972).
It tends to enhance sharp features but a solution may depend on the 
initial guess and therefore not be unique.

<P>
A different scheme was introduced by Lucy (1974) who uses a correction
term based on the ratio between the image and the guess.  A first
guess <IMG
 WIDTH="32" HEIGHT="25" ALIGN="BOTTOM" BORDER="0"
 SRC="img114.gif"
 ALT="$\Psi^0$">
must be specified (e.g. a constant) to start the
iteration.  The first step in the iteration performs a convolution
with the PSF :
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
\varphi_{i,j}^r = \Psi_{i,j}^r \otimes P_{i,j}.
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:lucy-deconv-1">&#160;</A><IMG
 WIDTH="163" HEIGHT="43"
 SRC="img115.gif"
 ALT="\begin{displaymath}\varphi_{i,j}^r = \Psi_{i,j}^r \otimes P_{i,j}.
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(2.29)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
The second step computes a correction factor based on this frame and 
the original image 
<!-- MATH: $\tilde{\varphi}$ -->
<IMG
 WIDTH="20" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="img116.gif"
 ALT="$\tilde{\varphi}$">
:
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
\Psi_{i,j}^{r+1} = \Psi_{i,j}^r \left(
\frac{\tilde{\varphi}_{i,j}}{\varphi_{i,j}^r} 
                        \otimes P_{i,j} \right) .
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:lucy-deconv-2">&#160;</A><IMG
 WIDTH="254" HEIGHT="69"
 SRC="img117.gif"
 ALT="\begin{displaymath}\Psi_{i,j}^{r+1} = \Psi_{i,j}^r \left(
\frac{\tilde{\varphi}_{i,j}}{\varphi_{i,j}^r}
\otimes P_{i,j} \right) .
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(2.30)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
The procedure repeats these two steps until the corrections are
sufficiently small.  After Equation&nbsp;<A HREF="node23.html#eq:lucy-deconv-2">2.30</A> is computed
the iteration continues with Equation&nbsp;<A HREF="node23.html#eq:lucy-deconv-1">2.29</A>.  This
scheme reaches a stable solution very quickly (i.e. 3-5 steps) and is
little affected by noise.  This makes it very useful for low
signal-to-noise data.  A photographic picture of a galaxy is used to
illustrate this technique (see Figure&nbsp;<A HREF="node23.html#fig:lucy-deconv">2.11</A>).  A fit
to the stellar image was used to define the PSF.
<BR>
<DIV ALIGN="CENTER"><A NAME="fig:lucy-deconv">&#160;</A><A NAME="815">&#160;</A>
<TABLE WIDTH="50%">
<CAPTION><STRONG>Figure 2.11:</STRONG>
Deconvolution a photographic image with the Lucy method:
           (A) original and (B) restored image after 3 iterations.</CAPTION>
<TR><TD><IMG
 WIDTH="892" HEIGHT="445"
 SRC="img118.gif"
 ALT="\begin{figure}\psfig{figure=fig11_deconvolution.eps,clip=} \end{figure}"></TD></TR>
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</DIV>
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
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