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<H2><A NAME="SECTION002081000000000000000">
Introduction</A>
</H2>
Consider an image characterized by its intensity
distribution <I>I</I>(<I>x</I>,<I>y</I>), corresponding to the observation of an
object <I>O</I>(<I>x</I>,<I>y</I>) through an optical system. If the
imaging system is linear and shift-invariant, the relation between
the object and the image in the same coordinate frame is a
convolution:
<BR>
<DIV ALIGN="CENTER"><A NAME="eqn_1">&#160;</A>
<!-- MATH: \begin{eqnarray}
I(x,y)= O(x,y) * P(x,y) + N(x,y)
\end{eqnarray} -->

<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><I>I</I>(<I>x</I>,<I>y</I>)= <I>O</I>(<I>x</I>,<I>y</I>) * <I>P</I>(<I>x</I>,<I>y</I>) + <I>N</I>(<I>x</I>,<I>y</I>)</TD>
<TD>&nbsp;</TD>
<TD>&nbsp;</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.99)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P><I>P</I>(<I>x</I>,<I>y</I>) is the point spread function (PSF) of the imaging system, and
<I>N</I>(<I>x</I>,<I>y</I>) is an additive noise. In Fourier space we have:
<BR>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{eqnarray}
\hat I(u,v)= \hat O(u,v) \hat P(u,v) + \hat N(u,v)
\end{eqnarray} -->

<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
 WIDTH="330" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="img830.gif"
 ALT="$\displaystyle \hat I(u,v)= \hat O(u,v) \hat P(u,v) + \hat N(u,v)$"></TD>
<TD>&nbsp;</TD>
<TD>&nbsp;</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.100)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
We want to determine <I>O</I>(<I>x</I>,<I>y</I>) knowing <I>I</I>(<I>x</I>,<I>y</I>) and <I>P</I>(<I>x</I>,<I>y</I>). This
inverse problem has led to a large amount of work, the main difficulties 
being the existence of: (i) a cut-off frequency of the
PSF, and (ii) an intensity noise (see for example [<A
 HREF="node370.html#cornwell">6</A>]).

<P>
Equation <A HREF="node336.html#eqn_1">14.99</A> is always an ill-posed problem.
This means that there is not a unique least-squares solution of minimal norm

<!-- MATH: $\parallel I(x,y) - P(x,y) * O(x,y) \parallel^2$ -->
<IMG
 WIDTH="293" HEIGHT="48" ALIGN="MIDDLE" BORDER="0"
 SRC="img831.gif"
 ALT="$\parallel I(x,y) - P(x,y) * O(x,y) \parallel^2$">
and a regularization is
necessary.

<P>
The best restoration algorithms are generally iterative [<A
 HREF="node370.html#katsaggelos">24</A>]. 
Van Cittert [<A
 HREF="node370.html#cittert">41</A>] proposed the following iteration:
<BR>
<DIV ALIGN="CENTER"><A NAME="vanvan">&#160;</A>
<!-- MATH: \begin{eqnarray}
O^{(n+1)} (x,y) = O^{(n)} (x,y) + \alpha(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"><IMG
 WIDTH="569" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="img832.gif"
 ALT="$\displaystyle O^{(n+1)} (x,y) = O^{(n)} (x,y) + \alpha(I(x,y) - P(x,y)* O^{(n)} (x,y))$"></TD>
<TD>&nbsp;</TD>
<TD>&nbsp;</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.101)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
where <IMG
 WIDTH="19" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="img833.gif"
 ALT="$\alpha$">
is a converging parameter generally taken as 1.  In
this equation, the object distribution is modified by adding a term
proportional to the residual.  But this algorithm diverges when we
have noise [<A
 HREF="node370.html#frieden">12</A>].  Another iterative algorithm is provided by
the minimization of the norm 
<!-- MATH: $\parallel I(x,y) - P(x,y)* O(x,y)
\parallel^2$ -->
<IMG
 WIDTH="294" HEIGHT="48" ALIGN="MIDDLE" BORDER="0"
 SRC="img834.gif"
 ALT="$\parallel I(x,y) - P(x,y)* O(x,y)
\parallel^2$">[<A
 HREF="node370.html#Landweber">21</A>] and leads to:
<BR>
<DIV ALIGN="CENTER"><A NAME="eqn_carre">&#160;</A>
<!-- MATH: \begin{eqnarray}
O^{(n+1)} (x,y) = O^{(n)} (x,y) + \alpha P_s(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"><IMG
 WIDTH="656" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="img835.gif"
 ALT="$\displaystyle O^{(n+1)} (x,y) = O^{(n)} (x,y) + \alpha P_s(x,y) * [I(x,y) - P(x,y) *
O^{(n)} (x,y)]$"></TD>
<TD>&nbsp;</TD>
<TD>&nbsp;</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.102)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
where 
<!-- MATH: $P_s(x,y)=P(-x,-y)$ -->
<I>P</I><SUB><I>s</I></SUB>(<I>x</I>,<I>y</I>)=<I>P</I>(-<I>x</I>,-<I>y</I>). 

<P>
Tikhonov's regularization [<A
 HREF="node370.html#tikhonov">40</A>] consists of minimizing the term:
<BR>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{eqnarray}
\parallel I(x,y) - P(x,y)* O(x,y) \parallel^2 + \lambda \parallel H *
O\parallel^2
\end{eqnarray} -->

<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
 WIDTH="438" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="img836.gif"
 ALT="$\displaystyle \parallel I(x,y) - P(x,y)* O(x,y) \parallel^2 + \lambda \parallel H *
O\parallel^2$"></TD>
<TD>&nbsp;</TD>
<TD>&nbsp;</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.103)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
where <I>H</I> corresponds to a high-pass filter. 
This criterion contains two terms; 
the first one, 
<!-- MATH: $\parallel I(x,y) - P(x,y)* O(x,y) \parallel^2$ -->
<IMG
 WIDTH="293" HEIGHT="48" ALIGN="MIDDLE" BORDER="0"
 SRC="img837.gif"
 ALT="$\parallel I(x,y) - P(x,y)* O(x,y) \parallel^2$">,
expresses
 fidelity to the data <I>I</I>(<I>x</I>,<I>y</I>) and the second one, 
<!-- MATH: $\lambda \parallel H * O\parallel^2$ -->
<IMG
 WIDTH="126" HEIGHT="48" ALIGN="MIDDLE" BORDER="0"
 SRC="img838.gif"
 ALT="$\lambda \parallel H * O\parallel^2$">,
smoothness of the restored image. <IMG
 WIDTH="18" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
 SRC="img839.gif"
 ALT="$\lambda$">
is the
regularization parameter and represents the trade-off between
fidelity to the data and the restored image smoothness. Finding
the optimal value <IMG
 WIDTH="19" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
 SRC="img840.gif"
 ALT="$\lambda$">
necessitates use of numeric techniques such as
Cross-Validation [<A
 HREF="node370.html#golub">15</A>] [<A
 HREF="node370.html#galatsanos">14</A>]. 

<P>
This method works well, but it is relatively long
and produces smoothed images. This second point can be a real problem
when we seek compact structures as is the case in astronomical imaging. 
 
An iterative approach for computing maximum likelihood estimates may be used.
The Lucy method [#lucy<#15258<A
 HREF="node370.html#>"></A>,#katsaggelos<#15259<A
 HREF="node370.html#>"></A>,#adorf<#15260<A
 HREF="node370.html#>"></A>] uses such 
an iterative approach:
<BR>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{eqnarray}
O^{(n+1)} = O^{(n)} [ \frac{I}{I^{(n)}} \ast P^* ]
\end{eqnarray} -->

<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
 WIDTH="239" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="img841.gif"
 ALT="$\displaystyle O^{(n+1)} = O^{(n)} [ \frac{I}{I^{(n)}} \ast P^* ]$"></TD>
<TD>&nbsp;</TD>
<TD>&nbsp;</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.104)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
and
<BR>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{eqnarray}
I^{(n)} = P \ast O^{(n)}
\end{eqnarray} -->

<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
 WIDTH="150" HEIGHT="28" ALIGN="BOTTOM" BORDER="0"
 SRC="img842.gif"
 ALT="$\displaystyle I^{(n)} = P \ast O^{(n)}$"></TD>
<TD>&nbsp;</TD>
<TD>&nbsp;</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.105)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
where <I>P</I><SUP>*</SUP> is the conjugate of the PSF.

<P>
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<I>Petra Nass</I>
<BR><I>1999-06-15</I>
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