A BAYESIAN APPROACH TO SHAPE RECONSTRUCTION OF A COMPACT OBJECT FROM A FEW NUMBER OF PROJECTIONS
{\sc Ali Mohammad-Djafari}
{\em Laboratoire des Signaux et Syst\`emes (CNRS-ESE-UPS)}
{\em \'Ecole Sup\'erieure d'\'Electricit\'e,}
{\em Plateau de Moulon, 91192 Gif-sur-Yvette, France.}
} \def\eabs{
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\title{A BAYESIAN APPROACH TO SHAPE RECONSTRUCTION OF A COMPACT 
OBJECT FROM A FEW NUMBER OF PROJECTIONS}  
\author{
{\sc Ali Mohammad-Djafari} \\ 
{\em Laboratoire des Signaux et Syst\`emes (CNRS-ESE-UPS)} \\  
{\em \'Ecole Sup\'erieure d'\'Electricit\'e,} \\ 
{\em Plateau de Moulon, 91192 Gif-sur-Yvette, France.} \\ 
{\em E-mail: djafari@lss.supelec.fr}
}
%\pagestyle{empty}                %without page numbering
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\markboth{A. Mohammad--Djafari}
{A BAYESIAN APPROACH TO SHAPE RECONSTRUCTION} 

\input{latexdef.tex}
\input{trdef.tex}
\input{spgraph.tex}

\def\Hij{H_{ij}}
\def\zjm{z_{j-1}}
\def\zjp{z_{j+1}}
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\begin{document}
\maketitle
\thispagestyle{empty}

\begin{abstract} 
Image reconstruction in X ray tomography consists in determining an object  
from its projections. In many applications such as non destructive testing, 
we look for an image who has a constant value inside a region (default) 
and another constant value outside that region (homogeneous region 
surrounding the default). 
The image reconstruction problem becomes then the determination 
of the shape of that region. 
In this work we model the object (the default region) as a polygonal disc 
and propose a new method for the estimation of the coordinates of its 
vertices directly from a very limited number of its projections.  
\end{abstract} 

\keywords{Computed Imaging, Tomography, Shape reconstruction, 
Non destructive testing, Regularization, Bayesian estimation, 
Deformable contours}

\section{Introduction}
Gammagraphy is a well known technique in non destructive testing (NDT) 
and non destructive evaluation (NDE) applications. 
Tomographic image reconstruction in these applications is more recent 
and consists of determining an object from its projections. 
The relation between the object $f(x,y)$ and its projections $p(r,\phi)$ 
is frequently modeled by the Radon transform: 
\beq \label{eq.1}
p(r,\phi)=\intd f(x,y) \delta(r-x\cosf-y\sinf) \d{x}\d{y}
\eeq
In many image reconstruction applications, especially in NDT and NDE, 
we look for an image $f(x,y)$ who has a constant value $c_1$ inside a 
region (default region D) and another constant value $c_2$ outside that 
region (homogeneous surrounding safe region), e.g. metal \& air. 
The image reconstruction problem becomes then the determination of the 
shape of the default region. 

In this communication, without loss of generality, we assume that $c_1=1$ 
and $c_2=0$: 
\beq \label{eq.2}
f(x,y)=\left\{\barr{ll} 
1 & \hbox{if}\quad (x,y)\in D, \\ 
0 & \hbox{elsewhere} \earr \right.,
\eeq
where $D$ represents the default region. 

There has been many works in image reconstruction and computed tomography 
dealing with this problem.  
To emphasis the originality and the place of this work, we give here a 
summary of the different approaches for this problem: 

\medskip\noindent$\bullet$ 
In the first approach, one starts by discretizing the equation (\ref{eq.1}) 
to obtain:
\beq \label{eq.3}
\pb=\Hb\fb+\nb
\eeq
where, 
$\fb$ is the discretized values of the object $f(x,y)$ (the pixel values of 
the image), 
$\pb$ is values of the projection data $p(r,\phi)$, 
$\nb$ is a vector to represent the modeling and measurement errors (noise) and 
$\Hb$ the discretized Radon operator. 
Then the solution is defined as the argument which minimizes the regularization 
criterion 
\beq \label{eq.4}
J(\fb)=||\pb - \Hb \fb||^2 + \lambda \Omega(\fb), 
\eeq
where $\lambda$ is the regularization parameter. \\  
$\Omega(\fb)$ has to be chosen appropriately to reflect the fact that $\fb$ 
must represent a binary image. This is the classical approach of general 
image reconstruction problem. 
In fact, one can also interpret $J(\fb)$ as the maximum a posteriori (MAP) 
criterion in the Bayesian estimation framework where 
$Q(\fb)=||\pb - \Hb \fb||^2$ represents the 
likelihood term and $\expf{-\lambda \Omega(\fb)}$ the prior 
probability law.

This approach has been used with success in many applications 
(e.g. \cite{Herman87,Kak87,Geman87,Demoment89})  
but the cost of its calculation is huge due to the great dimension of $\fb$. 
Many works have been done on choosing appropriate regularization functionals or 
equivalently appropriate prior probability laws for $\fb$ to enforce some special 
properties of the image such as smoothness, positivity or piecewise 
smoothness 
\cite{Brette94a,Djafari96a,Bouman93,Bedini94,Nikolova94,Nikolova95d}.  
Among these, one can mention mainly two types of functions for $\Omega(\fb)$: 

\noindent{Entropic laws:}
\[
\Omega(\fb)=\sum_{j=1}^N \phi(f_j), \hbox{~~with~~} 
\phi(x)=\left\{x^p, -x\log x, \log x, \cdots\right\}
\]
Homogeneous Markovian laws:
\[
\sum_{j=1}^N \sum_{i\in{\cal N}_j} \phi(f_j,f_i), 
\hbox{~with~} 
\phi(x,y)=\left\{|x-y|^p, -|x-y|\log\frac{x}{y}, \log\cosh|x-y|, \cdots\right\} 
\]
See for example \cite{Djafari96a} for the entropic laws, 
\cite{Bouman93,Brette94a} for scale invariant markovian laws and 
\cite{Bedini94,Nikolova94,Nikolova95d} for other specific choices.

\medskip\noindent$\bullet$ 
In the second approach, one starts by giving a parametric model for the object 
and then tries to estimate these parameters using least squares (LS) 
or maximum likelihood (ML) methods. 
In general, in this approach one chooses a parametric model such as 
superposition of circular or elliptical discs to be able to relate   
analytically the projections to these parameters. 
For example, for a superposition of elliptical discs we have:
\beq \label{eq.5}
f(x,y)=\sum_{k=1}^K d_k f_k(x,y)
\eeq
with
\beq \label{eq.6}
f_k(x,y)=\left\{\barr{ll} 
1 & \hbox{if}\quad (x-\alpha_k)^2+(y-\beta_k)^2<g_k^2(\theta), \\ 
0 & \hbox{elsewhere} \earr \right.,
\eeq
where 
\beq \label{eq.7}
g_k(\theta)=\sqrt{a_k^2 \cos^2\theta+b_k^2 \sin^2\theta}.
\eeq
and where $\thetab=\{d_k, \alpha_k, \beta_k, a_k, b_k, k=1,\cdots,K\}$ 
is a vector of parameters defining the parametric model of the image 
(density values, coordinates of the centers and the two diameters of 
the ellipses). It is then easy to calculate analytically the projections 
and the relation between the data and the unknown parameters becomes:
\beq \label{eq.8}
p(r,\phi)=h(r,\phi;\thetab)+n(r,\phi)
\eeq
where
\beq \label{eq.9}
h(r,\phi;\thetab)=\sum_{k=1}^K d_k p_k(r,\phi)
\hbox{~~with~~}
p_k(r,\phi)=\left\{
\barr{ll} 
 \frac{2 a_k b_k}{g_k^2(\phi)} \sqrt{g_k^2(\phi)-r^2)} 
   & \hbox{if}\quad r<g_k(\phi), \\ 
 0 & \hbox{elsewhere} 
\earr\right. 
\eeq
The LS or the ML estimate when the noise is assumed to be Gaussian 
is then given by:
\beq \label{eq.10}
\wh{\thetab}=\argmins{\thetab}{\|p(r,\phi)-h(r,\phi;\thetab)\|^2}
\eeq
This approach has also been used with success 
in image reconstruction \cite{Shepp82,Devaney91,Rossi84,Prince90,Prince91}. 
But, the range of applicability of these methods is 
limited to the cases where the parametric models are actually appropriate.  

\medskip\noindent$\bullet$ 
In the third approach which is more appropriate to our problem of shape 
reconstruction, one starts by modeling directly the contour of the object 
by a function, say $g(\theta)$ such as:
\beq \label{eq.11}
D=\left\{(x,y): \rho^2(\theta)=x^2+y^2 <g^2(\theta) \right\}.
\eeq
The next step is then to relate the projections $p(r,\phi)$ to $g(\theta)$ 
which, in this case is:
\beq \label{eq.12}
p(r,\phi) = \izdpi\int_0^{g(\theta)} \delta(r-\rho\cos(\phi-\theta))
\rho\d{\rho}\d{\theta}.
\eeq
and finally to discretize this relation to obtain: 
\beq \label{eq.13}
\pb=\hb(\gb)+\nb
\eeq
where $\gb$ represents the discretized values of $g(\theta)$ defining the 
contour of the object and $\hb(\gb)$ represents the discretized version of 
the nonlinear operator (\ref{eq.12}) relating projection data $\pb$ and $\gb$. 
Then, one defines the solution as the argument which minimizes 
\beq \label{eq.14}
J(\gb)=||\pb - \hb(\gb)||^2 + \lambda \Omega(\gb), 
\eeq
where $\Omega(\gb)$ has to be chosen appropriately to reflect some regularity 
property of the object's contour. 

In this case also one can consider $J(\gb)$ as the MAP criterion with 
$Q(\gb)=||\pb - \hb(\gb)||^2$ as the likelihood term and $\Omega(\gb)$ as 
the prior one. \\ 
This approach has been used in image restoration \cite{Friedland92}, 
but it seems to be new in image reconstruction applications and 
the proposed method in this work is in this category. 
The originality of our work is to model the contour of the object by a 
piecewise linear function which means that the object is modeled as a 
polygonal disc whose vertices are estimated directly from the 
projection data. 

Now before going further in details, let compare this last approach 
with the first one, by noting the following:

\medskip\noindent$\bullet$ 
In (\ref{eq.3}) and (\ref{eq.4}), $\fb$ represents the pixel values of the 
image (a very great dimensional vector depending on the image dimensions), 
but in (\ref{eq.13}) and (\ref{eq.14}), $\gb$ represents the discretized 
values of $g(\theta)$ defining the contour of the object. 
The dimension of this vector is moderate and independent of the image 
dimensions. 

\medskip\noindent$\bullet$ 
In (\ref{eq.3}) and (\ref{eq.4}), $\Hb\fb$ is a linear function of $\fb$ 
and so $Q(\fb)$ is a quadratic function of it, but in (\ref{eq.13}) 
and (\ref{eq.14}), $\hb(\gb)$ is not a linear function of $\gb$ and so 
$Q(\gb)$ will not be a quadratic function of it. \\ 
We will discuss more the consequences of these remarks in the next section. 

\section{Proposed method}
In this paper we propose to model the contour of the object (default region) 
as a periodic piecewise linear function or equivalently to model the shape 
of the object as a polygonal disc with a great number $N$ of vertices to be 
able to approximate any shape. 
Then we propose to estimate directly the coordinates 
$\{(x_j,y_j), j=1,\cdots,N\}$ 
of the vertices of this polygonal disc from the projection data 
(see Fig.~\ref{fig1}). 

\bfig[htb] \label{fig1}
 \hfil \input{tomox5.pic}
 \caption{Proposed shape reconstruction modeling.}
\efig

\medskip 
The idea of modeling the shape of the object as a polygonal disc is not 
new and some works have been done in image reconstruction applications, 
but, in general in these works, a hypothesis of convexity of the polygonal 
disc has been used which is very restrictive in real applications. 
In our work we do not make this hypothesis and also we choose $N$ 
appropriately great to to be able to approximate any shape. 

As we deal with inverse problems, the solution is then defined as the 
argument which minimizes the following criterion 
\beq \label{eq.15}
J(\zb)=||\pb - \hb(\zb)||^2 + \lambda \Omega(\zb), 
\eeq
where $\zb=\xb+i\yb$ is a complex vector whose real and imaginary 
parts represent the $x$ and the $y$ coordinates of the polygon vertices, 
$\hb(\zb)$ represents the direct operator which calculates the projections 
for any given $\zb$ and $\Omega(\zb)$ is chosen to be a function which 
reflects the regularity of the object contour. 
In this work we used the following: 
\beq \label{eq.17}
\Omega(\zb)=\sum_{j=1}^N |z_{j-1}-2 z_{j}+z_{j+1}|^2.
\eeq
Note that 
$|z_{j-1}-2 z_{j}+z_{j+1}|^2$ is just the Euclidian distance between 
the point $z_{j}$ and the line segment passing through $z_{j-1}$ and 
$z_{j+1}$ and so this choice favors a shape whose local curvature is limited. 
We can also give a probabilistic interpretation to this choice. In 
fact we can consider $z_{j}$ as random variables with the following 
Markovian law:
\beq
p(z_{j}|\zb)=p(z_{j}|z_{j-1},z_{j+1})\propto
\expf{-\frac{1}{2\sigma^{2}}|z_{j-1}-2 z_{j}+z_{j+1}|^2}
\eeq 
Other functions are possible and are studied in this work.  

In both cases, the criterion $J(\zb)$ is multimodal essentially due to the 
fact that $\hb(\zb)$ is a nonlinear function of $\zb$. 
Calculating the optimal solution corresponding to the global minimum of 
(\ref{eq.15}) needs then carefully designed algorithms. For this we propose 
the following strategies: 

\medskip\noindent$\bullet$ 
The first is to use a global optimization technique such as simulated 
annealing (SA). This technique has given satisfactory result as it can be 
seen from the simulations in the next section. 
However, this algorithm needs a great number of iterations and some skills 
for choosing the first temperature and cooling schedule, but the overall 
calculations is not very important due to the fact that we do not need to 
calculate the gradient of the criterion (\ref{eq.15}). 

\medskip\noindent$\bullet$ 
The second is to find an initial solution in the attractive region of the global 
optimum and to use a local descent type algorithm to find the solution. 

The main problem here is how to find this initial solution. For this, we used a 
moment based method proposed by Milanfar, Karl \& Wilsky 
\cite{Milanfar93,Milanfar94} 
which is accurate enough to obtain an initial solution which is not very far 
from the optimum. 
The basic idea of this method is to relate the moments of the projections to 
the moments of a class of polygonal discs obtained by an affine transformation 
of a centered regular polygonal disc, and so to estimate a polygonal disc 
whose vertices are on an ellipse and whose moments up to the second order 
matches those of the projections.

However, there is no theoretical proof that this initial solution will be 
in the attractive region of the global optimum. 
In the simulation results section we will show some results comparing the 
performances of these two methods as well as a comparison with some other 
classical methods.

\section{Simulation results}
To measure the performances of the proposed method and keeping the objective 
of using this method for NDT applications where the number of projections 
are very limited, we simulated a case where the object is a polygonal disc 
with $N=40$ vertices (hand-made) and calculated its projections for only 
5 directions: 
\[
\phi=\{-45,-22.5,0,+22.5,+45 \hbox{~degrees}\}
\]
\bfig[hbt] \label{fig2}
 \includegraphics[width=9cm,height=9cm]{fig1}
 \caption{Original image and noisy projections.}
\efig

\medskip 
Then, we added some noise (white, Gaussian and centered) on them to simulate 
the measurement errors. The S/N ratio was chosen 20dB. Finally, from these data 
we estimated the solution by either of the two proposed methods. 
Figures~\ref{fig3} and \ref{fig4} show these results. 

In Fig.~\ref{fig3}, we give the reconstruction results obtained by simulated 
annealing (SA) algorithm and in Fig.~\ref{fig4} those obtained by a 
moment-based initialization and a local descent-based 
optimization algorithm. Note that, the SA is independent of initialization, 
however, in these figures we show the results obtained by the proposed method. 

\bfig[htb] \label{fig3}
 \btab{cc} 
  \includegraphics[width=6cm,height=6cm]{fig2} & 
  \includegraphics[width=6cm,height=6cm]{fig2b}
 \etab 
 \caption[Reconstruction using simulated annealing.]{
 Reconstruction using simulated annealing.\\ 
 a) Original, Initialization and Reconstructed objects \\
 b) Evolution of the criterion :  
 $J=J_1+\lambda J_2$ where $J1=Q(\zb)$ and $J_2=\Omega(\zb)$ 
 }
\efig

\medskip 
\bfig[htb] \label{fig4}
 \btab{cc} 
 \includegraphics[width=6cm,height=6cm]{fig3} & 
 \includegraphics[width=6cm,height=6cm]{fig3b}
 \etab 
 \caption[Reconstruction using a local minimizer.]{
 Reconstruction using a moment-based initialization and a local minimizer.\\ 
 a) Original, Initialization and Reconstructed objects \\
 b) Evolution of the criterion $J=J_1+\lambda J_2$ during the iterations.
 }  
\efig

\medskip 
In Fig.~\ref{fig5} we show a comparison between the results obtained by the 
proposed method and those obtained either by a classical backprojection 
method or by some other methods in the first approach using (\ref{eq.3}) 
and (\ref{eq.4}) with different regularization functionals $\Omega(\fb)$ 
among those in (\ref{eq.5}). 
Also, for the purpose of curiosity we show the result of a binary 
segmented image obtained by thresholding these last images. 

%\newpage
\bfig[htb] \label{fig5}
 \hspace*{-5mm}
 \btab{ccc}
 \includegraphics[width=4cm,height=4cm]{orig1} & 
 \includegraphics[width=4cm,height=4cm]{back1}     &
 \includegraphics[width=4cm,height=4cm]{gm}        
 \\ 
 a & c & e  \\ 
 \includegraphics[width=4cm,height=4cm]{geomet} & 
 \includegraphics[width=4cm,height=4cm]{tback}  & 
 \includegraphics[width=4cm,height=4cm]{tgm}    
 \\       
 b & d & f \\ 
 \etab
 \caption[A comparison with backprojection and some other classical methods]
 {
 A comparison with backprojection and some other classical methods  \\ 
 a) Original,  b) Proposed method,  \\
 c) Backprojection, d) Binary threshold of c),  \\
 e) Gaussian Markov Random Field (GMRF) modeling  
   and the MAP estimation reconstruction, 
 f) Binary threshold of e).
 }
\efig

\section{Conclusions}
A new method for tomographic image reconstruction of a compact object from 
its limited angle projections is proposed. 
The basic idea of the proposed method is to model the object as a polygonal 
disc whose vertices coordinates are estimated directly from the projections 
using the Bayesian MAP estimation framework or equivalently by optimizing 
a regularized criterion. 

This criterion is not unimodal. To optimize it two methods are examined: 
a global optimization method based on simulated annealing and a local 
gradient-based method with a good initialization obtained using a moment 
based method. 
The first one seems to give entire satisfaction and better results. 
The final destination of the proposed method is for non destructive 
testing (NDT) and evaluation (NDE) image reconstruction 
applications including X-rays, ultrasound or Eddy currents 
\cite{Gautier95b,Premel95,Nikolova96}. 

{\small 
\input{revuedef.tex}
\input{month-bi.tex}
\bibliographystyle{maxent}
\bibliography{gpibase,gpipubli,amd}
}
\edoc



revuedef.tex
% fichier : revuedef.tex
% Auteur: AMD, GLB
% Mise a jour: 06 septembre 1992, 10:00
%
% Modifie et completee par	L.Lemitre le 05 10 94
% 				J.-F. Gio le 15-02-95
%				H. C. le 27-06-95
%				J. I. le 02-08-95
%
% Ce fichier contient des alias sur les noms de revues
% pour faciliter la saisie de fiches biblio.
%
\def\AA{Astrononmy and Astrophysics}
\def\AAP{Advances in Applied Probability}			%Ajout FC
\def\ABE{Annals of Biomedical Engineering}
\def\AT{Annales des T\'el\'ecommunications}
\def\AMS{Annals of Mathematical Statistics}
\def\AISM{Annals of Institute of Statistical Mathematics}
\def\AO{Applied Optics}
\def\AP{The Annals of Probability}
\def\AST{The Annals of Statistics}
\def\BMK{Biometrika}
\def\CPAM{Communications on Pure and Applied Mathematics}
\def\EMK{Econometrica}
\def\CRAS{Compte-rendus de l'acad\'emie des sciences}
\def\CVGIP{Computer Vision and Graphics and Image Processing}
\def\GJRAS{Geophysical Journal of the Royal Astrononomical Society}
\def\GSC{Geoscience}
\def\GPH{Geophysics}
\def\GRETSI#1{Actes du #1$^{\mbox{e}}$ Colloque GRETSI} 
%\de%\GRETSI89{Actes du 12 Colloque GRETSI} 
%\de%\GRETSI91{Actes du 13 Colloque GRETSI} 
%\de%\GRETSI93{Actes du 14 Colloque GRETSI} 
%\de%\GRETSI95{Actes du 15 Colloque GRETSI} 
\def\CGIP{Computer Graphics and Image Processing}
\def\ICASSP{Proceedings of IEEE ICASSP}
\def\ICEMBS{Proceedings of IEEE EMBS}
\def\ICIP{Proceedings of the International Conference on Image Processing}
\def\ieeP{Proceedings of the IEE}
\def\ieeeAC{IEEE Transactions on Automatic and Control}
\def\ieeeAES{IEEE Transactions on Aerospace and Electronic Systems}
\def\ieeeAP{IEEE Transactions on Antennas and Propagation}
\def\ieeeASSP{IEEE Transactions on Acoustics Speech and Signal Processing}
\def\ieeeBME{IEEE Transactions on Biomedical Engineering}
\def\ieeeCS{IEEE Transactions on Circuits and Systems}
\def\ieeeCT{IEEE Transactions on Circuit Theory}
\def\ieeeC{IEEE Transactions on Communications}
\def\ieeeGE{IEEE Transactions on Geoscience and Remote Sensing}
\def\ieeeGEE{IEEE Transactions on Geosciences Electronics}
\def\ieeeIP{IEEE Transactions on Image Processing}
\def\ieeeIT{IEEE Transactions on Information Theory}
\def\ieeeMI{IEEE Transactions on Medical Imaging}
\def\ieeeMTT{IEEE Transactions on Microwave Theory and Technology}
\def\ieeeM{IEEE Transactions on Magnetics}
\def\ieeeNS{IEEE Transactions on Nuclear Sciences}
\def\ieeePAMI{IEEE Transactions on Pattern Analysis and Machine Intelligence}
\def\ieeeP{Proceedings of the IEEE}
\def\ieeeRS{IEEE Transactions on Radio Science}
\def\ieeeSMC{IEEE Transactions on Systems, Man and Cybernetics}
\def\ieeeSP{IEEE Transactions on Signal Processing}
\def\ieeeSSC{IEEE Transactions on Systems Science and Cybernetics}
\def\ieeeSU{IEEE Transactions on Sonics and Ultrasonics}
\def\ieeeUFFC{IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control}
\def\IJC{International Journal of Control}
\def\IJCV{International Journal of Computer Vision}
\def\IJIST{International Journal of Imaging Systems and Technology}
\def\IP{Inverse Problems}
\def\ISR{International Statistical Review}
\def\IUSS{Proceedings of International Ultrasonics Symposium}
\def\JAPH{Journal of Applied Physics}
\def\JAP{Journal of Applied Probability}
\def\JAS{Journal of Applied Statistics}
\def\JASA{Journal of Acoustical Society America}
\def\JASAS{Journal of American Statistical Association}
\def\JBME{Journal of Biomedical Engineering}			% Ajout JFG
\def\JCAM{Journal of Computational and Applied Mathematics}	% Ajout HC
\def\JCAT{Journal of Computer Assisted Tomography}
\def\JEWA{Journal of Electromagnetic Waves and Applications}	% Ajout HC
\def\JMO{Journal of Modern Optics}
\def\JNDE{Journal of Nondestructive Evaluation}		        % Ajout HC
\def\JMP{Journal of Mathematical Physics}
\def\JOSA{Journal of Optical Society America}
\def\JP{Journal de Physique}
\def\JRSSA{Journal of the Royal Statistical Society A}
\def\JRSSB{Journal of the Royal Statistical Society B}
\def\JRSSC{Journal of the Royal Statistical Society C}
\def\JSPI{Journal of Statistical Planning and Inference}        %Ajout MF
\def\JTSA{Journal of Time Series Analysis}                      %Ajout MF
\def\JVCIR{Journal of Visual Communication and Image Representation} 
	\def\MMAS{???} % Trouve dans gpi base 
\def\MNAS{Mathematical Methods in Applied Science}
\def\MNRAS{Monthly Notes of Royal Astronomical Society}
\def\MP{Mathematical Programming}
	\def\NSIP{NSIP}  % Trouve dans gpi base 
\def\OC{Optics Communication}
\def\PRA{Physical Review A}
\def\PRB{Physical Review B}
\def\PRC{Physical Review C}
\def\PRD{Physical Review D}				% Ajout stef
\def\PRL{Physical Review Letters}			% Ajout HC	
\def\RGSP{Review of Geophysics and Space Physics}	% Ajout HC			% Ajout HC	
\def\RS{Radio Science}					% Ajout HC	
\def\SP{Signal Processing}
\def\siamAM{SIAM Journal of Applied Mathematics}
\def\siamCO{SIAM Journal of Control and Optimization}
\def\siamJO{SIAM Journal of Optimization}		% Ajout HC pour JFB
\def\siamMA{SIAM Journal of Mathematical Analysis}
\def\siamNA{SIAM Journal of Numerical Analysis}
\def\siamO{SIAM Journal of Optimization}
\def\siamR{SIAM Review}
\def\SSR{Stochastics and Stochastics Reports}           % Ajout MF
\def\TPA{Theory of Probability and its Applications}	% Ajout FC
\def\TMK{Technometrics}
\def\TS{Traitement du Signal}
\def\UMB{Ultrasound in Medecine and Biology}
\def\US{Ultrasonics}
\def\USI{Ultrasonic Imaging}

spgraph.tex
% SpGpraph Version 0.0  < April 6, 1988 >
%        Written by Conrad Kwok
%
% The macros are in public domain.
% You may distribute or modify it in any ways you like.
%
% You need an dvi-to-printer program that supports the \special
% commands used by the macros. The commands are subset of the commands
% used by tpic program. For more information, read the READ.ME file
%
% Usage:
%    For LaTeX
%  Include the file after the \documentstyle command by saying
%  \include spgraph
%
%    For TeX
% Include the file after including the file latexpicobjs.tex
%  \include latexpicobj
%  \include spgraph
% in the beginning of the TeX file.
%
% The macros are for use with LaTeX picture environment
% By including the macro file, you can draw
%   1) \line in any slope
%   2) \circle with any radius
%   3) \ellipse with any x-axis and y-axis length
%   4) Circular \arc by giving starting and ending angle (in radius)
%
% Furthermore you may draw lines in any thickness.
%
%
\makeatletter

% Create a new counter to store the line thickness in thousandth of an inch
\newcount\@gphlinewidth
\@gphlinewidth\@wholewidth \divide\@gphlinewidth 4736

% Redefine \thinlines, \thicklines and define \Thickline
% See also latex.tex
\def\thinlines{\let\@linefnt\tenln \let\@circlefnt\tencirc
    \@wholewidth\fontdimen8\tenln \@halfwidth .5\@wholewidth
    \@gphlinewidth\@wholewidth \divide\@gphlinewidth 4736\relax}
\def\thicklines{\let\@linefnt\tenlnw \let\@circlefnt\tencircw
    \@wholewidth\fontdimen8\tenlnw \@halfwidth .5\@wholewidth
    \@gphlinewidth\@wholewidth \divide\@gphlinewidth 4736
    \advance\@gphlinewidth\@ne   % Make the output looks better
    \relax}
\def\Thicklines{\let\@linefnt\tenlnw \let\@circlefnt\tencircw
    \@wholewidth\fontdimen8\tenlnw \@wholewidth 1.5\@wholewidth
    \@halfwidth .5\@wholewidth
    \@gphlinewidth\@wholewidth \divide\@gphlinewidth 4736\relax}
%
% Generate the \special command for drawing arc
\def\@circlespecial#1#2#3#4{%
       \special{pn \the\@gphlinewidth}%
       \special{ar 0 0 #1 #2 #3 #4}
}
%
% Command for drawing an arc. Elliptical arc command can be generated
% but all iptex program I saw so far does not support that.
\def\@arc#1#2#3#4{%
% convert TeX dimension to length in terms thousandth of an inch
 \@tempdima #1\unitlength
 \@tempdimb #2\unitlength
        \@tempcnta\@tempdima \advance\@tempcnta 4736 \divide\@tempcnta 9473
 \@tempcntb\@tempdimb \advance\@tempcntb 4736 \divide\@tempcntb 9473
 \setbox\@tempboxa\hbox{%
     \@circlespecial{\the\@tempcnta}{\the\@tempcntb}{#3}{#4}}%
        \wd\@tempboxa\z@ \box\@tempboxa
}
%
% Command for drawing Circle and Ellipse in terms of \@arc
% replace original \@circle
\def\@circle#1{\@arc{#1}{#1}{0}{6.2832}}
\def\ellipse#1#2{\@arc{#1}{#2}{0}{6.2832}}
\def\arc#1#2#3{\@arc{#1}{#1}{#2}{#3}}
%
% Generate \special commands for drawing line
\def\@linespecial#1#2{%
       \special{pn \the\@gphlinewidth}%
       \special{pa 0 0}%
       \special{pa #1 #2}%
       \special{fp}
}
%
% Replace original \@sline
% For description, see latex.tex
\def\@sline{%
 \ifnum\@xarg< 0
   \@negargtrue \@xarg -\@xarg \@tempdima -\@linelen
 \else
   \@negargfalse \@tempdima\@linelen
 \fi
% truncation is used in arithmetic
 \@tempcnta\@linelen \divide\@tempcnta 4736
        \@yyarg -\@yarg \multiply\@yyarg \@tempcnta \divide\@yyarg\@xarg
  \if@negarg
     \@tempcnta -\@tempcnta
 \fi
 \setbox\@linechar\hbox{\@linespecial{\the\@tempcnta}{\the\@yyarg}}%
 \wd\@linechar\@tempdima
 \@clnht\@linelen
        \multiply\@clnht\@yarg
        \divide\@clnht\@xarg
 \ifnum\@yarg< 0
   \@clnht -\@clnht
   \ht\@linechar\z@ \dp\@linechar\@clnht
 \else
   \ht\@linechar\@clnht \dp\@linechar\z@
 \fi
 \box\@linechar
% Codes below (till end of the command) is only necessary 
% when used with \@svector
 \if@negarg
   \@yyarg -\@yarg
 \else
   \@yyarg \@yarg
 \fi
 \setbox\@linechar\hbox{\@linefnt\@getlinechar(\@xarg,\@yyarg)}%
 \ifnum\@yarg> 0
   \let\@upordown\raise
   \advance\@clnht -\ht\@linechar
 \else
   \let\@upordown\lower
 \fi
 \if@negarg \kern\wd\@linechar \fi
}
%
\makeatother


trdef.tex
%TRDEF.LATEX
%LaTeX document
%Author: Ali Mohammad-djafari
%Date  : 22 Augoust 1991.

%BOLD MATH
\def\bm#1{\mbox{\boldmath $#1$}}

%Numbers
\def\zerob{{\bm 0}}
\def\oneb{{\bm 1}}

%Small letters
\def\ab{{\bm a}}
\def\bb{{\bm b}}
\def\cb{{\bm c}}
\def\db{{\bm d}}
\def\eb{{\bm e}}
\def\fb{{\bm f}}
\def\gb{{\bm g}}
\def\hb{{\bm h}}
\def\ib{{\bm i}}
\def\jb{{\bm j}}
\def\kb{{\bm k}}
\def\lb{{\bm l}}
\def\mb{{\bm m}}
\def\nb{{\bm n}}
\def\ob{{\bm o}}
\def\pb{{\bm p}}
\def\qb{{\bm q}}
\def\rb{{\bm r}}
\def\sb{{\bm s}}
\def\tb{{\bm t}}
\def\ub{{\bm u}}
\def\vb{{\bm v}}
\def\wb{{\bm w}}
\def\xb{{\bm x}}
\def\yb{{\bm y}}
\def\zb{{\bm z}}

%Capital letters
\def\Ab{{\bm A}}
\def\Bb{{\bm B}}
\def\Cb{{\bm C}}
\def\Db{{\bm D}}
\def\Eb{{\bm E}}
\def\Fb{{\bm F}}
\def\Gb{{\bm G}}
\def\Hb{{\bm H}}
\def\Ib{{\bm I}}
\def\Jb{{\bm J}}
\def\Kb{{\bm K}}
\def\Lb{{\bm L}}
\def\Mb{{\bm M}}
\def\Nb{{\bm N}}
\def\Ob{{\bm O}}
\def\Pb{{\bm P}}
\def\Qb{{\bm Q}}
\def\Rb{{\bm R}}
\def\Sb{{\bm S}}
\def\Tb{{\bm T}}
\def\Ub{{\bm U}}
\def\Vb{{\bm V}}
\def\Wb{{\bm W}}
\def\Xb{{\bm X}}
\def\Yb{{\bm Y}}
\def\Zb{{\bm Z}}

%Small Greek letters
\def\alphab{\bm{\alpha}}
\def\betab{\bm{\beta}}
\def\deltab{\bm{\delta}}
\def\epsilonb{\bm{\epsilon}}
\def\gammab{\bm{\gamma}}
\def\omegab{\bm{\omega}}
\def\thetab{\bm{\theta}}
\def\xib{\bm{\xi}}
\def\lambdab{\bm{\lambda}}
\def\phib{\bm{\phi}}
\def\mub{\bm{\mu}}
\def\psib{\bm{\psi}}
\def\chib{\bm{\chi}}
\def\sigmab{\bm{\sigma}}

%Capital Greek letters
\def\Deltab{\bm{\Delta}}
\def\Lambdab{\bm{\Lambda}}
\def\Phib{\bm{\Phi}}
\def\Psib{\bm{\Psi}}
\def\Sigmab{\bm{\Sigma}}

%Caligraphic capital letters
\def\Ac{{\cal A}}
\def\Bc{{\cal B}}
\def\Cc{{\cal C}}
\def\Dc{{\cal D}}
\def\Ec{{\cal E}}
\def\Fc{{\cal F}}
\def\Gc{{\cal G}}
\def\Hc{{\cal H}}
\def\Ic{{\cal I}}
\def\Jc{{\cal J}}
\def\Kc{{\cal K}}
\def\Lc{{\cal L}}
\def\Mc{{\cal M}}
\def\Nc{{\cal N}}
\def\Oc{{\cal O}}
\def\Pc{{\cal P}}
\def\Qc{{\cal Q}}
\def\Rc{{\cal R}}
\def\Sc{{\cal S}}
\def\Tc{{\cal T}}
\def\Uc{{\cal U}}
\def\Vc{{\cal V}}
\def\Wc{{\cal W}}
\def\Xc{{\cal X}}
\def\Yc{{\cal Y}}
\def\Zc{{\cal Z}}

%widetilde and widehat
\def\wt#1{\widetilde{#1}}
\def\wh#1{\widehat{#1}}
%

%Arrows
\def\ra{\rightarrow}
\def\la{\leftarrow}
\def\da{\downarrow}
\def\ua{\uparrow}

\def\Ra{\Rightarrow}
\def\La{\Leftarrow}
\def\Da{\Downarrow}
\def\Ua{\Uparrow}

\def\lra{\longrightarrow}
\def\lla{\longleftarrow}
\def\Lra{\Longrightarrow}
\def\Lla{\longleftarrow}

\def\lrarr{\leftrightarrow}
\def\Lrarr{\Leftrightarrow}
\def\udarr{\updownarrow}
\def\Uparr{\Updownarrow}

%
\def\d#1{\,\mbox{d}#1}
\def\dxdy{\d{x} \d{y}}
\def\dwxdwy{\d{\omega_x} \d{\omega_y}}
\def\dxdydz{\d{x} \d{y} \d{z}}

%DEFINITIONS
\def\disp#1{{\displaystyle #1}}
\def\diag#1{\mbox{diag}\left\{#1\right\}}

\def\Prob#1{\mbox{Pr}\left\{#1\right\}}
\def\var#1{\mbox{Var}\left\{#1\right\}}
\def\cov#1{\mbox{Cov}\left\{#1\right\}}
\def\corr#1{\mbox{Corr}\left\{#1\right\}}
\def\trace#1{\mbox{Tr}\left\{#1\right\}}
\def\rang#1{\mbox{rang}\left\{#1\right\}}
\def\det#1{\mbox{d\'et}\left\{#1\right\}}

\def\cosf{\cos \phi}
\def\sinf{\sin \phi}
\def\cost{\cos \theta}
\def\sint{\sin \theta}

\def\sgn{\mbox{sgn}}
\def\sinc{\mbox{sinc}}
\def\rect{\mbox{rect}}
\def\sincf#1{\mbox{sinc}\left(#1\right)}
\def\rectf#1{\mbox{rect}\left(#1\right)}
\def\trif#1{\mbox{tri}\left(#1\right)}
%
%VECTEURS
\def\xvec#1#2#3{\left\{#1_#2,\ldots,#1_#3\right\}}

\def\vx{\left[x_1,\ldots, x_n\right]^t}
\def\vz{\left[z_1,\ldots, z_n\right]^t}
\def\vw{\left[\omega_1,\ldots, \omega_n\right]^t}
\def\vxi{\left[\xi_1,\ldots, \xi_n\right]^t}
%
%INTEGRALES
\def\iii{\int_{-\infty}^{+\infty}}
\def\izi{\int_{0}^{\infty}}
\def\izpi{\int_{0}^{\pi}}
\def\izdpi{\int_{0}^{2\pi}}
\def\intd{\int\kern-.8em\int}
\def\intt{\int\kern-.8em\int\kern-.8em\int}
\def\intg{\int\kern-1.1em\int}
%
\def\sumd{\mathop{\sum\sum}}

\def\sumi{\sum_{i=1}^{M}}
\def\sumj{\sum_{i=1}^{N}}
\def\sumk{\sum_{k=1}^{K}}
\def\sumn{\sum_{n=1}^{N}}
\def\summ{\sum_{m=1}^{M}}

%
%TRANSFORMS
\def\TA#1{{\cal A}\left\{ {#1} \right\}}
\def\TH#1{{\cal H}\left\{ {#1} \right\}}
\def\TP#1{{\cal P}\left\{ {#1} \right\}}
\def\TR#1{{\cal R}\left\{ {#1} \right\}}
\def\TRa#1{{\cal R}^{\dag}\left\{ {#1} \right\}}
\def\BR#1{{\cal B}\left\{ {#1} \right\}}
\def\TF#1{{\cal F}\left\{ {#1} \right\}}
\def\TFI#1{{\cal F}^{-1}\left\{ {#1} \right\}}
\def\TFn#1#2{{\cal F}_{#1}\left\{ {#2} \right\}}
\def\TFnI#1#2{{\cal F}_{#1}^{-1}\left\{ {#2} \right\}}
%
\def\Im#1{{\cal I}\mbox{m}\left(#1\right)}
\def\Ker#1{{\cal K}\mbox{er}\left(#1\right)}
%
\def\Imag#1{\mbox{Im}\left(#1\right)}
\def\Re#1{\mbox{Re}\left(#1\right)}
%
\def\expf#1{\exp\left[ {#1} \right]}

%derivee
\def\dfdx#1#2{{\mbox{d} {#1}\over{\mbox{d} {#2}}}}
\def\dfdxd#1#2{{\mbox{d}^2 {#1}\over{\mbox{d} {#2}^2}}}
\def\dfdxt#1#2{{\mbox{d}^3 {#1}\over{\mbox{d} {#2}^3}}}
\def\dfdxn#1#2{{\mbox{d}^n {#1}\over{\mbox{d} {#2}^n}}}
\def\dfdxk#1#2{{\mbox{d}^k {#1}\over{\mbox{d} {#2}^k}}}

%derivee partielle
\def\dpdx#1#2{{{\partial {#1}\over \partial {#2}}}}
\def\dpdxd#1#2{{{\partial^2 {#1}}\over{\partial {#2}^2}}}
\def\dpdxdy#1#2#3{{{\partial ^2 {#1}}\over{\partial {#2} \partial {#3}}}}

\def\arg{\mbox{arg}}
\def\argmins#1#2{\mbox{arg}\min_{#1}\left\{{#2}\right\}}
\def\argmaxs#1#2{\mbox{arg}\max_{#1}\left\{{#2}\right\}}
\def\argmin#1#2{\mathop{\mbox{arg}\min}_{#1}\left\{{#2}\right\}}
\def\argmax#1#2{\mathop{\mbox{arg}\max}_{#1}\left\{{#2}\right\}}

\def\esp#1{\mbox{E}\left\{ #1 \right\}}
\def\espx#1#2{\mbox{E}_{#1}\left\{ #2 \right\}}

\def\wth#1{\widehat{\widetilde{\phantom{#1}}}\!\!\!\! #1}

%OTHERS
\def\lrf{L_{r,\phi}}
\def\fw{\widehat{f}(\omegab)}
\def\fthwxi{\wth{f}(\Omega,\xib)}
\def\fthwfi{\wth{f}(\Omega,\phi)}
\def\ftrfi{\widetilde{f}(r,\phi)}

\def\fwxwy{\widehat{f}(\omega_x, \omega_y)}
\def\wxpwy{(\omega_x \, x + \omega_y \, y)}

\def\wtx{\omegab^t \cdot \xb}
\def\ejwtx{\exp\left[j \omegab^t \cdot \xb\right]}
\def\xitx{\xib^t \cdot \xb}

%
\def\ftrxi{\widetilde{f}(r,\xib)}
\def\ent{-\int p(x) \, \ln p(x) \d{x}}

%YOURS

\def\mean#1{\left< #1 \right>}
\def\slnhn{\sum_{n=1}^N \lambda_n h_n(\rb)}
\def\slngn{\sum_{n=1}^N \lambda_n g_n(\rb)}
\def\smngn{\sum_{n=1}^N \mu_n g_n(\rb)}
\def\slmhm{\sum_{m=1}^N \lambda_m h_m(\rb)}
\def\vlambda{\bm{\lambda} = [\lambda_1,\ldots,\lambda_n]}

\def\apriori{{\em a priori} }
\def\aposteriori{{\em a posteriori} }

\def\titre#1{\bcc{\Large\bf #1}\ecc}

% My definitions
\def\AMD{Ali Mohammad--Djafari}
\def\LSSa{Laboratoire des Signaux et Syst\`emes 
(CNRS--ESE--UPS) \\ 
\'Ecole Sup\'erieure d'\'Electricit\'e \\ 
Plateau de Moulon, 91192 Gif sur Yvette Cedex, France.}

\def\ME{maximum entropy}
\def\pdf{probability distribution function}
\def\lm{Lagrange multipliers}
%%
\def\fix#1{\phi _#1(x)}
\def\fin{\fix n}
\def\fik{\fix k}
\def\fiz{\fix 0}
\def\sfinz{\sum_{n=0}^N \lambda_n \, \fin}
\def\sfinu{\sum_{n=1}^N \lambda_n \, \fin}
\def\bl{\bm{\lambda}}
\def\bd{\bm{\delta}}
\def\blz{\bl ^0}
\def\gnl{G _n(\bl)}
\def\gnlz{G _n(\blz)}
\def\un{n=1,\dots, N}
\def\nn{n=0,\dots, N}

\def\finn{\fin , \nn}
\def\esfinz{\exp\,\left[ -\sfinz \right] }
\def\esfinu{\exp\,\left[ -\sfinu \right] }
\def\esxm{\exp\,\left[ -\sum_{m=0}^N \lambda_m \, x^m \right] }
\def\efin{\esp \fin }
\def\zl{Z(\bl)}
\def\finxi{\phi _n(x_i)}
\def\snfinxi{\sum_{n=1}^N \lambda_n \finxi}
\def\esnfinxi{\exp \left[ - \snfinxi \right]}
\def\smfinxi{\sum_{i=1}^M \finxi}

\def\ejnw{\exp \left( -j n \omega_0 x \right) }
\def\eejnw{\mbox{E} \left\lbrace \ejnw \right\rbrace}

\def\signed#1{{\unskip\nobreak\hfil\penalty50\hskip2em\mbox{}
\nobreak\hfil\tt#1\parfillskip=0pt \finalhyphendemerits=0 \par}}

\def\uncatcodespecials{\def\do##1{\catcode`##1=12 }\dospecials}
\def\listing#1{\par\begingroup\setupverbatim\input#1 \endgroup}
\newcount\lineno
\def\setupverbatim{\tt \lineno=0
 \obeylines \uncatcodespecials \obeyspaces
 \everypar{\advance\lineno by1 \llap{\sevenrm\the\lineno\ \ }}}
{\obeyspaces\global\let =\ }

\def\defined{\stackrel{\mbox{def}}{=}}
\def\str{\stackrel}

\def\ER{\mbox{I\kern-.25em R}}
\def\EC{\mbox{C\kern-.8em C}}
\def\EZ{\mbox{Z\kern-.55em Z}}
\def\EN{\mbox{N\kern-.8em N}}

\def\singles{
 \abovedisplayskip 12pt plus 3pt minus 9pt
 \belowdisplayskip 12pt plus 3pt minus 9pt
 \abovedisplayshortskip 0pt plus 3pt
 \belowdisplayshortskip 7pt plus 3pt minus 4pt
 \baselineskip 14.4pt
 \lineskip 1pt
 \lineskiplimit 0pt}
\def\oneandhalf{
 \abovedisplayskip 18pt plus 3pt minus 9pt
 \belowdisplayskip 18pt plus 3pt minus 9pt
 \abovedisplayshortskip 0pt plus 3pt
 \belowdisplayshortskip 9.333pt plus 3pt
 \baselineskip 20pt
 \lineskip 2pt
 \lineskiplimit 1pt}

\def\double{
 \abovedisplayskip 24pt plus 3pt minus 9pt
 \belowdisplayskip 24pt plus 3pt minus 9pt
 \abovedisplayshortskip 0pt plus 3pt
 \belowdisplayshortskip 12pt plus 3pt
 \baselineskip 27pt
 \lineskip 3pt
 \lineskiplimit 2pt}

\def\dadb{\d{\alpha}\d{\beta}}

\def\ffbox#1{\fbox{\mbox{\vbox{#1}}}}

\def\rot{\mbox{rot}}
\def\case#1#2#3#4{
    \left\{
           \begin{array}{ll}
            {\displaystyle #1} & {\displaystyle #2} \cr 
            {\displaystyle #3} & {\displaystyle #4}
           \end{array}
    \right. }

\def\beqnarr#1&#2&#3\\#4&#5&#6\eeqnarr{
    \left\{
           \begin{array}{lcl}
            {\displaystyle #1} & #2 & {\displaystyle #3} \\ 
            {\displaystyle #4} & #5 & {\displaystyle #6} 
           \end{array}
    \right. }

\def\pyx{p(\yb|\xb)}
\def\pxy{p(\xb|\yb)}

\newtheorem{definition}{D\'efinition}
\newtheorem{theorem}{Th\'eor\`eme}

\def\ie{{\em i.e.}}
\def\unsdpi{\left(\frac{1}{2\pi}\right)}
\def\unspi{\left(\frac{1}{\pi}\right)}
one.bbl
\begin{thebibliography}{10}

\bibitem{Herman87}
G.~Herman, H.~Tuy, H.~Langenberg, and P.~Sabatier, {\em Basic Methods of
  Tomography and Inverse Problems}.
\newblock Adams Hilger, 1987.

\bibitem{Kak87}
A.~Kak and M.~Slaney, {\em Principles of Computerized Tomographic Imaging}.
\newblock New York, NY: IEEE Press, 1987.

\bibitem{Geman87}
S.~Geman and D.~McClure, ``Statistical methods for tomographic image
  reconstruction,'' in {\em Proc. of the 46-th Session of the ISI, Bulletin of
  the ISI}, vol.~52, pp.~22--26, 1987.

\bibitem{Demoment89}
G.~Demoment, ``Image reconstruction and restoration : Overview of common
  estimation structure and problems,'' {\em ieeeASSP}, vol.~ASSP-37,
  pp.~2024--2036, Dec. 1989.

\bibitem{Brette94a}
S.~Brette, J.~Idier, and A.~Mohammad-Djafari, ``Scale invariant
  \uppercase{M}arkov models for linear inverse problems,'' in {\em Proc. of the
  Section on \uppercase{B}ayesian Statistical Sciences}, (Alicante, Spain),
  pp.~266--270, American Statistical Association, 1994.

\bibitem{Djafari96a}
A.~Mohammad-Djafari and J.~Idier, {\em A scale invariant \uppercase{B}ayesian
  method to solve linear inverse problems}, pp.~121--134.
\newblock Maximum entropy and \uppercase{B}ayesian methods, Santa Barbara,
  U.S.A.: Kluwer Academic Publ., g. heidbreder~ed., 1996.

\bibitem{Bouman93}
C.~Bouman and K.~Sauer, ``A generalized \uppercase{G}aussian image model for
  edge-preserving {\sc map} estimation,'' {\em \ieeeIP}, vol.~IP-2,
  pp.~296--310, \Jul 1993.

\bibitem{Bedini94}
L.~Bedini, I.~Gerace, and A.~Tonazzini, ``A deterministic algorithm for
  reconstructing images with interacting discontinuities,'' {\em \CVGIP},
  vol.~56, pp.~109--123, \Mar 1994.
\newblock AMD.

\bibitem{Nikolova94}
M.~Nikolova, A.~Mohammad-Djafari, and J.~Idier, ``Inversion of large-support
  ill-conditioned linear operators using a \uppercase{M}arkov model with a line
  process,'' in {\em ICASSP}, vol.~V, (Adelaide, Australia), pp.~357--360,
  1994.

\bibitem{Nikolova95d}
M.~Nikolova, J.~Idier, and A.~Mohammad-Djafari, ``Inversion of large-support
  ill-posed linear operators using a piecewise \uppercase{G}aussian {\sc
  mrf},'' tech. rep., {\sc gpi--lss}, submitted to {\ieeeIP}, Gif-sur-Yvette,
  France, 1995.

\bibitem{Shepp82}
L.~A. Shepp and Y.~Vardi, ``Maximum likelihood reconstruction for emission
  tomography,'' {\em \ieeeMI}, vol.~MI-1, pp.~113--122, 1982.

\bibitem{Devaney91}
A.~J. Devaney and G.~A. Tsihrintzis, ``Maximum likelihood estimation of object
  location in diffraction tomography,'' {\em \ieeeSP}, vol.~SP-39,
  pp.~672--681, Mar. 1991.

\bibitem{Rossi84}
D.~J. Rossi and A.~S. Wilsky, ``Reconstruction from projections based on
  detection and estimation of objects,'' {\em ieeeASSP}, vol.~ASSP-32, no.~4,
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\end{thebibliography}
