NEW ADVANCES IN BAYESIAN CALCULATION FOR LINEAR AND NONLINEAR INVERSE PROBLEMS
{\sc A. Mohammad-Djafari, H. Carfantan and M. Nikolova}
{\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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latexdef.tex
%LATEXDEF.LATEX
%LaTeX document
%Author: Ali Mohammad-djafari
%Date  : 22 Avril 1991.
%
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\def\edoc{\end{document}}

\def\babs{\begin{abstract}}
\def\eabs{\end{abstract}}

\def\bbib{\begin{thebibliography}{999}}
\def\ebib{\end{thebibliography}}

\def\beq{\begin{equation}}
\def\eeq{\end{equation}}

\def\beqn{\begin{eqnarray}}
\def\eeqn{\end{eqnarray}}

\def\beqnn{\begin{eqnarray*}}
\def\eeqnn{\end{eqnarray*}}

\def\barr{\begin{array}}
\def\earr{\end{array}}

\def\bqu{\begin{quote}}
\def\equ{\end{quote}}

\def\bqun{\begin{quotation}}
\def\equn{\end{quotation}}

\def\bit{\begin{itemize}}
\def\eit{\end{itemize}}

\def\ben{\begin{enumerate}}
\def\een{\end{enumerate}}

\def\bpic{\begin{picture}}
\def\epic{\end{picture}}

\def\bfig{\protect \begin{figure}}
\def\efig{\protect \end{figure}}

\def\bcc{\begin{center}}
\def\ecc{\end{center}}

\def\brr{\begin{flushright}}
\def\err{\end{flushright}}

\def\bll{\begin{flushleft}}
\def\ell{\end{flushleft}}

\def\btab{\begin{tabular}}
\def\etab{\end{tabular}}

\def\ul{\underline}
\def\ol{\overline}


month-bi.tex
%--------------------- month-fr.tex ---------------------------
% Fichier de d\'efinition pour les mois, en bilingue
%
\def\jan{janvier }
\def\feb{f\'evrier }
\def\mar{mars }
\def\apr{avril }
\def\may{mai }
\def\jun{juin }
\def\jul{juillet }
\def\aug{ao\^ut }
\def\sep{septembre }
\def\oct{octobre }
\def\nov{novembre }
\def\dec{d\'ecembre }
%
\def\Jan{January }	
\def\Feb{February }
\def\Mar{March }
\def\Apr{April }
\def\May{May }
\def\Jun{June }
\def\Jul{July }
\def\Aug{August }
\def\Sep{September }
\def\Oct{October }
\def\Nov{November }
\def\Dec{December }
% plus
\def\sub{soumis \`a }
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}

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}}

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two.tex
\documentclass{book}
\usepackage{maxent,xspace}
\usepackage{graphicx,epic,eepic}
\graphicspath{{Eps/}}

\title{NEW ADVANCES IN BAYESIAN CALCULATION FOR LINEAR AND 
NONLINEAR INVERSE PROBLEMS}
\author{
{\sc A. Mohammad-Djafari, H. Carfantan and M. Nikolova} \\
{\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
\pagestyle{myheadings}            %with page numbering
\markboth{A. Mohammad--Djafari et al.}
{NEW ADVANCES IN BAYESIAN CALCULATION\ldots} 

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

\begin{document}
\maketitle
\thispagestyle{empty}

\begin{abstract} 
The Bayesian approach has proved to be a coherent approach to handle 
ill posed Inverse problems. 
However, the Bayesian calculations need either an optimization or an 
integral calculation. The maximum a posteriori (MAP) estimation requires 
the minimization of a compound criterion which, in general, has two parts: 
a data fitting part and a prior part. \\ 
In many situations the criterion to be minimized becomes multimodal. 
The cost of the Simulated Annealing (SA) based techniques is in general 
huge for inverse problems. Recently a deterministic optimization 
technique, based on Graduated Non Convexity (GNC), have been proposed 
to overcome this difficulty. \\ 
The objective of this paper is to show two specific implementations of 
this technique for the following situations: \\ 
-- Linear inverse problems where the solution is modeled as a piecewise 
continuous function. The non convexity of the criterion is then due to the 
special choice of the prior;  \\ 
-- A nonlinear inverse problem which arises in inverse scattering 
where the non convexity of the criterion is due to the likelihood 
part.
\end{abstract} 

\keywords{Inverse problems, Regularization, Bayesian calculation, 
Global optimization, Graduated Non Convexity}

\section{Introduction}
We consider the case of general inverse problems:
\beq
\yb=\Ab(\xb)+\bb,
\eeq
where $\xb$ is the vector of unknown variables, $\yb$ is the data, 
$\Ab$ is a linear or non linear operator and $\bb$ represents the 
errors which are assumed, hereafter, additive, zero-mean, white and Gaussian.  

The Bayesian approach has proved to be a coherent approach to handle 
these problems. 
However, the Bayesian calculations need either an optimization or an 
integral calculation. The maximum a posteriori (MAP) estimation needs 
the minimization of a compound criterion:
\beq
J(\xb)=-\log p(\xb|\yb)=Q(\xb)+\lambda\Omega(\xb)
\eeq
which, in general, has two parts: a data fitting part: 
\beq
Q(\xb)=-\log p(\yb|\xb)=\|\yb-\Ab(\xb)\|^{2}
\eeq
and a prior part:
\beq
\Omega(\xb)=-\log p(\xb)=\sum_{r}\sum_{s\in{\cal N}_{r}} \phi(x_{r}-x_{s}). 
\eeq
This last expression is due to general Markov modeling where 
$r$ and $s$ are two site indexes and ${\cal N}_{r}$ means neighbor 
sites of $r$. 

In many situations the criterion $J(\xb)$ becomes multimodal. 
We consider here two cases: 
The first is the case of general linear inverse problems with markovian priors 
with non convex energies, and the second is, 
the case of non-linear inverse problem. 

In both cases we need a global optimization technique to 
determine the solution. In the first case the non convexity is due to 
the second termed and in the second case the non convexity is more due 
to the first termed. 

The cost of the Simulated Annealing (SA) based techniques is 
mainly dependent to the neighborhood size of the posterior marginal 
probability distribution $p(x_{j}|\xb,\yb)$ which is directly related 
to the neighborhood size of the prior marginal probability distribution 
$p(x_{j}|\xb)$ and the support of the operator $\Ab$. When $\Ab$ is a 
local operator with very small support, i.e.; when the data element $y_{i}$ 
depends only to a few number of the unknown variables $x_{j}$, then 
SA can be implemented efficiently \cite{Geman84,Younes88,Jeng90}. 
But, unfortunately, this is not the 
case of many inverse problems, where the support of the operator is not 
small. The cost of SA is then, in general, huge for these problems. 

\medskip 
Recently a deterministic relaxation algorithm, 
inspired by the Graduated Non Convexity (GNC) principle, 
has been proposed by Blake and Zisserman in~\cite{Blake87,Blake89} 
for the optimization of the multimodal MAP criteria. 
They have shown its efficiency in practical applications 
for noise cancellation and segmentation.  
This algorithm has been extended to the general linear 
ill-posed inverse problem by Nikolva $et~al.$~in~\cite{Nikolova94}.  

\medskip 
The object of this presentation is to show two specific implementations 
of this technique for two specific cases of the two 
aforementioned situations, i.e.; \\ 
-- The linear inverse problems where the solution is modeled as a 
piecewise continuous function using a compound markov modeling 
(for example the intensity and the line process in image reconstruction),  
where the non convexity of the criterion is due to the 
markovian priors; and \\ 
-- A special non-linear inverse problem which arises in 
inverse scattering and diffraction tomography imaging applications 
where the non convexity of the criterion is due to the 
likelihood part. 

\medskip 
The paper is organized as follows: The next section presents the main 
idea of the GNC principle. Sections 3 and 4 will consider the 
two aforementioned specific cases and, finally, some simulation 
results will illustrate the performances of the proposed method in two 
special applications. 

\section{Graduated Non Convexity scheme}

The principle of this algorithm is very simple. 
It consists of approximating the non convex criterion $J(\xb)$ with a 
sequence of continuously derivable criteria $J_{c_{k}}(\xb)$ such 
that: \\ 
$\bullet$ the first one $J_{c_{0}}(\xb)$ be convex; \\ 
$\bullet$ the final one (the limit) $J_{c_{k}}(\xb)$ converges to 
$J(\xb)$:
\[
\lim_{k\mapsto\infty} J_{c_{k}}(\xb)=J(\xb), \quad \forall \xb, 
\]
where $c_{k}>0$ are increasing relaxation parameters; and then \\ 
$\bullet$ for each $k$, a relaxed solution is calculated by 
minimizing locally, initialized by the previous solution, as follows:
\beq
\wh{\xb}_{c_{k}}=\argmins{\xb\in V(\wh{\xb}_{c_{k-1}})}{J_{c_{k}}(\xb)}
\eeq
Fig.~\ref{pgnc} illustrates such a scheme. The hope is that the sequence 
$\wh{\xb}_{c_{k}}$ converges to the global minimizer of $J(\xb)$. 
Note that there is no theoretical ground for this hope, 
however, in many practical applications it seems to be realist. 

\begin{figure}[htb] \label{pgnc}
 \includegraphics[width=9cm,height=9cm]{pgnc} 
 \caption{GNC scheme.}
\end{figure}

\bigskip 
\section{Linear inverse problems with a piecewise Gaussian prior}
Let first consider a very simple noise filtering problem:
\beq
\yb=\xb+\bb,
\eeq
where we know that $\xb$ represents the samples of a piecewise 
continuous function $x(t)$. 
Blake and Zisserman in~\cite{Blake87} proposed to estimate $\xb$ by 
searching the global minimum of the following criterion:
\beq
J(\xb)=\|\yb-\xb\|^{2}+\Omega(\xb)
\eeq
with
\beq
\Omega(\xb)=\sum_{j}\phi(t_{j}), \quad t_{j}=x_{j}-x_{j-1}
\eeq
and
\beq \label{quadt}
\phi(t)=\left\{\barr{ll}
(\lambda t)^{2}                & \hbox{if}\quad |t|<T\\ 
\alpha=(\lambda T)^{2} & \hbox{if}\quad |t|>T 
\earr\right.
\eeq

\begin{figure}[htb] \label{quadt1}
 \bcc
  \setlength{\unitlength}{0.92pt}
  \input{quadt.pic}
 \ecc
 \caption{Truncated quadratic function.}
\end{figure}

\bigskip 
Note that, for $\phi(t)=t^{2}$ the criterion $J(\xb)$ can be 
considered either as the MAP criterion with Gaussian prior or as the 
Tikhonov regularization one. 
The choice (\ref{quadt}) for  $\phi(t)$ is done to preserve the 
discontinuities in $x(t)$.  
With this choice, obviously, $J(\xb)$ is multimodal. 
The GNC idea was then to construct:
\beq
J_{c}(\xb)=\|\yb-\xb\|^{2}+\Omega_{c}(\xb)
\eeq
with
\beq
\Omega_{c}(\xb)=\sum_{j}\phi_{c}(t_{j}),
\eeq
and
\beq
\phi_{c}(t)=\left\{\barr{ll}
(\lambda t)^{2} & \hbox{if}\quad |t|<q_{c}\\ 
\alpha-\frac{1}{2} c (|t|-r_{c})^{2} & \hbox{if}\quad q_{c}>|t|>r_{c} \\ 
\alpha=(\lambda T)^{2} & \hbox{if}\quad |t|>r_{c} 
\earr\right., \quad 
\left\{\barr{l}
q_{c}=T(1+2\lambda^{2}/c)^{-1/2}\\ 
r_{c}=T(1+2\lambda^{2}/c)^{1/2}\earr\right.
\eeq
and find a $c_{0}$ such that $J_{c}$ for $c<c_{0}$ be convex and then 
for a given sequence of relaxations parameters $\{c_{0},c_{1},\ldots,c_{k}\}$ do: 
\beq
\wh{\xb}_{c_{k}}=\argmins{\xb\in V(\wh{\xb}_{c_{k-1}})}{J_{c_{k}}(\xb)}
\eeq

\begin{figure}[htb] \label{quadt2}
 \bcc
  \setlength{\unitlength}{0.92pt}
  \input{quadtr.pic} 
 \ecc
 \caption{Truncated quadratic function and it's relaxation scheme.}
\end{figure}

\medskip 
Blake and Zisserman in~\cite{Blake87} showed the existence of $c_{0}$ 
such that $J_{c_{0}}(\xb)$ be convex 
and global convergence of this algorithm. However, for the case of 
inverse problems $\yb=\Ab\xb+\bb$ where $\Ab$ is singular or 
ill-conditioned, the existence of $c_{0}$ is no more insured. 
Nikolova $et~al.$~\cite{Nikolova94,Nikolova94b} extended this work by proposing 
a doubly relaxed criterion:
\beq
J_{c}(\xb)=\|\yb-\xb\|^{2}+\Omega_{a,c}(\xb)
\eeq
with
\beq
\Omega_{a,c}(\xb)=\sum_{j}\left[\phi_{c}(t_{j})+a t_{j}^{2}\right]
\eeq
and the following double relaxation scheme:
\beqn 
\hbox{for fixed~~} c=c_{0} 
 && \hbox{and for~~} a=a_{0},\ldots,0 \hbox{~~do:} \nonumber \\ 
 && \wh{\xb}_{a_{k}}=\argmins{\xb\in V(\wh{\xb}_{a_{k-1}})}{J_{a_{k},c_{0}}(\xb)} \\ 
\hbox{for fixed~~} a=0 
 && \hbox{and for~~} c=c_{0},\ldots,\infty \hbox{~~do:} \nonumber \\ 
 && \wh{\xb}_{c_{k}}=\argmins{\xb\in V(\wh{\xb}_{c_{k-1}})}{J_{c_{k},0}(\xb)}
\eeqn 
The initial convexity of the criterion is insured for 
$a_{0}>\frac{c_{0}}{2}$. Many details, discussions and more 
extensions, specially for 2D case, are given in 
\cite{Nikolova94b,Nikolova95b,Nikolova95d}.  

\subsection{Link with compound Markov models}
Compound Markov modeling in image processing became popular after the 
works of Geman \& Geman \cite{Geman84} and Besag 
\cite{Besag74,Besag86}. To see the link with these model briefly, let 
consider a 1-D case. When $x(t)$ is assumed piecewise 
continuous or piecewise Gaussian, it can be modeled 
with a set of coupled variables $(\xb,\lb)$, where $\lb$ is a 
vector of binary-valued variables and $\xb$ is Gaussian. 
Now, consider the MAP estimate of $(\xb,\lb)$:
\beq \label{map1}
(\wh{\xb},\wh{\lb})=\argmins{(\xb,\lb)}{J(\xb,\lb)=-\log p(\xb,\lb|\yb)}
\eeq
where
\beq
 -\log p(\xb,\lb|\yb)=-\log p(\yb|\xb) -\log p(\xb|\lb) -\log p(\lb) + 
 cte,
\eeq
with the following prior laws:
\beqnn
 -\log p(\yb|\xb)&=&\|\yb-\Ab \xb\|^{2} \\ 
 -\log p(\xb|\lb)&=&\sum_{j} \phi(t_{j}) (1-l_{j}), \quad 
 t_{j}=x_{j}-x_{j-1},\quad  \phi(t)=(\lambda t)^{2} \\ 
 -\log p(\lb)  &=&\alpha \sum_{j} l_{j} 
\eeqnn
we obtain:
\beq \label{map2}
(\wh{\xb},\wh{\lb}) = \argmins{(\xb,\lb)}{J(\xb,\lb)}
\eeq
with
\beq
J(\xb,\lb)=\|\yb-\Ab\xb\|^{2}+
\sum_{j}\left[ \lambda^{2}(x_{j}-x_{j-1})^{2}(1-l_{j})+\alpha l_{j}\right].
\eeq
Note that the line variables $l_{j}$ are assumed non-interacting 
(mutually independent). With this hypothesis, it is easy to show that 
the solution $(\wh{\xb},\wh{\lb})$ obtained by (\ref{map2}) is 
equivalent to the solution obtained by
\beq  
\wh{\xb}=\argmins{\xb}{J(\xb)}
\quad\hbox{with}\quad 
J(\xb)=\|\yb-\Ab\xb\|^{2}+\sum_{j} \phi(x_{j}-x_{j-1})^{2}
\eeq
with $\phi(t)$ defined by (\ref{quadt}) and 
\beq  
\wh{l}_{j}=\left\{\barr{lll} 
1 & \hbox{if} & |\wh{x}_{j}-\wh{x}_{j-1}|>T \\ 
0 & \hbox{if} & |\wh{x}_{j}-\wh{x}_{j-1}|<T
\earr\right.
\eeq

\section{A non linear inverse problem}
To show how the GNC principle can be used for nonlinear inverse 
problems we consider the case of inverse scattering and more 
specifically the diffraction tomography. To be short in presentation 
of the application, we give here an abstract presentation of the 
problem. For more details on derivation of 
the inverse scattering and diffraction tomography application see 
\cite{Carfantan95a,Carfantan95b,Carfantan96a}. To summarize, considering  
the geometry of Fig.~\ref{config}, we have the following relations:
\beqn
y(\rb_i)&=&\intd_{D} G_m(\rb_i,\rb') \phi(\rb') x(\rb') \d{\rb'}, 
\quad \rb_i\in S, 
\\ 
\phi(\rb)&=&\phi_0(\rb)+\intd_{D} G_o(\rb,\rb') \phi(\rb') x(\rb') \d{\rb'}, 
\quad \rb\in D.
\eeqn
The discretized version of these two equations can be written with the 
following compact notations:
\beqn 
 \yb   &=& \Gb_{m} \Xb \phib + \bb,       \label{diff1} \\ 
 \phib &=& \phib_{0} + \Gb_{o} \Xb \phib, \label{diff2}
\eeqn
where $\phib_{0}$ is the incident field, 
$\Gb_{m}, \Gb_{o}$ are matrices related to the Green functions, $\Xb$ is a 
diagonal matrix with the components of the vector $\xb$ (a $n$ length vector) 
as its diagonal elements and $\yb, \phib$ are respectively $m$ and $n$ length 
vectors representing the measured data (scattered field) and the total 
field on the object. Note that $n$ may be greater than $m$.

\begin{figure}[htb] \label{config}
   \begin{center}
      \input{config.tex} 
   \end{center}
   \caption{Diffraction tomography geometry configuration}
\end{figure}

\bigskip 
These two equations can be combined to obtain a symbolic explicit relation
between the data $\yb$ and the unknowns $\xb$: 
\beq
\yb = \Gb_{m} \Xb \left( \Ib - \Gb_{o} \Xb \right)^{-1}\phib_{0} + \bb
    = \Ab(\xb)+ \bb,
\eeq
where the considered matrix is assumed to be invertible. 
Now, the inverse problem we are faced is to find $\xb$ given $\yb$. 
Note that, $\phib_{0}$, $\Gb_{o}$ and $\Gb_{m}$ are known and the 
relation between $\xb$ and $\yb$ is non linear. In fact, given 
$\phib$, $\yb$ is linear in $\xb$ (\ref{diff1}), but $\phib$ depends on 
$\xb$ through the second equation (\ref{diff2}).  

Here also, using the Bayesian approach, the MAP estimate is defined 
as the minimizer of 
\beq
J(\xb)=\|\yb-\Ab(\xb)\|^{2}+\Omega(\xb)
\eeq
and, even when $\Omega(\xb)$ is chosen to be convex, $J(\xb)$ may 
not due to the fact that $\|\yb-\Ab(\xb)\|^{2}$ is no more quadratic 
in $\xb$. Carfantan $et~al.$ in~\cite{Carfantan95a,Carfantan95b,Carfantan96a} 
proposed to use the GNC idea in this case. 

To introduce the GNC technique, they considered the following 
relaxation sequence:
\beq
\Ab_{c_{k}}(\xb) = \Gb_{m} \Xb (\Ib - c_{k} \Gb_{o} \Xb)^{-1}\phib_{0},
\eeq
with $c_0 = 0$, and $\lim_{k \rightarrow \infty} c_k=1$. 

Note that the first term ($c_0=0$) corresponds to a linearized 
model for the problem named the Born approximation which consists 
in neglecting partially the diffraction effects. 
This results to the following convex criterion:
\[
J_{0}(\xb)=\|\yb-\Gb_{m} \Xb \phib_{0}\|^2 + \Omega(\xb).
\]
Note also that for $c_{k}=1$ the criterion $J_{1}(\xb)=J(\xb)$. 
The main practical problem is then the choice of sequences 
$\{c_{k}=0,\cdots,1\}$ which is done by experiment. 

\section{Some simulation results and applications}
\subsection{1-D noise filtering}
Fig.~\ref{filt} shows an example of results obtained in noise filtering. 
In this figure we see the original signal, noisy data, restoration 
using a Gaussian model ($\phi(t)$ quadratic) and restoration 
obtained by GNC when $\phi(t)$ is chosen to be truncated 
quadratic.  

\begin{figure}[htb] \label{filt}
 \btab{cc}
  \includegraphics[width=6cm,height=5cm]{filta} & 
  \includegraphics[width=6cm,height=5cm]{filtb}
  \\ a & b 
 \etab
 \caption[Noise filtering]{Noise filtering: \\ 
   a) Original and data, 
   b) Original, Gaussian restoration and GNC restoration}
\end{figure}

\bigskip
\subsection{1-D signal deconvolution}
Fig.~\ref{dec1d} shows an example of results in signal deconvolution. 
In this figure we see the original signal, noisy data, restoration 
using a Gaussian model and restoration obtained by GNC.  

\begin{figure}[htb] \label{dec1d}
 \btab{cc}
  \includegraphics[width=6cm,height=5cm]{dec1da} & 
  \includegraphics[width=6cm,height=5cm]{dec1db}
  \\ a & b 
 \etab
 \caption[Deconvolution]{Deconvolution: \\ 
   a) Original and data, 
   b) Original, Gaussian restoration and GNC restoration}
\end{figure}

\bigskip
\subsection{Image restoration}
Fig.~\ref{dec2d} shows an example of results in Image restoration. 
In this figure 
a) is the original image, 
b) is the blurred and noisy data, 
c) is the restoration using a Gaussian model and 
d) is the restoration obtained by GNC with truncated quadratic regularization.  

\begin{figure}[hb] \label{dec2d}
 \btab{cc}
  \includegraphics[width=6cm,height=6cm]{dec2da} &  
  \includegraphics[width=6cm,height=5cm]{dec2db} 
  \\ a & b \\  
  \includegraphics[width=6cm,height=5cm]{dec2dc} &  
  \includegraphics[width=6cm,height=5cm]{dec2dd}
  \\  c & d 
 \etab
 \caption[Image restoration]{Image restoration: \\ 
   a) original, b) data, 
   c) Gaussian restoration, d) GNC restoration}
\end{figure}

\newpage
\subsection{Image reconstruction in X-ray tomography}
Fig.~\ref{tomox} shows results obtained in X-ray tomography 
image reconstruction. In this figure 
a) shows the original image, 
b) shows the projections (data),  
c) shows the backprojection reconstruction, 
d) shows a reconstruction using a Gaussian prior, 
e) shows a reconstruction using a Gamma prior, and 
f) shows a reconstruction using GNC with truncated quadratic regularization.  

\begin{figure}[htb] \label{tomox}
\begin{tabular}{ccc}
\includegraphics[width=3.8cm,height=3.8cm]{orig2} & 
\includegraphics[width=3.8cm,height=3.8cm]{back2} & 
\includegraphics[width=3.8cm,height=3.8cm]{melogx} \\  
      a & c & e \\
\includegraphics[width=3.8cm,height=3.8cm]{proj} & 
\includegraphics[width=3.8cm,height=3.8cm]{regq1} & 
\includegraphics[width=3.8cm,height=3.8cm]{gnc1} \\  
      b & d & f
\end{tabular}
\caption[Image reconstruction in X-ray tomography]{
   Image reconstruction in X-ray tomography: \\ 
   a) original, 
   b) projections (data), 
   c) Backprojection, 
   d) Gaussian reconstruction, 
   d) Gamma prior reconstruction, and 
   e) GNC reconstruction
   }
\end{figure}

\bigskip
\subsection{Inverse scattering and diffraction tomography}
Fig.~\ref{tomodiff} shows an example of results in non linear diffraction 
tomography image reconstruction. In this figure 
a) is the original image, 
b) is the measured scattered field data,  
d) is a reconstruction using the linear Born approximation, and  
e) is a reconstruction using GNC.  

\begin{figure}[hbt] \label{tomodiff}
\begin{tabular}{ccc}
\includegraphics[width=3.8cm,height=3.8cm]{dt1}
 & 
\includegraphics[width=3.8cm,height=3.8cm]{dt3}
 & 
\includegraphics[width=3.8cm,height=3.8cm]{dt2} \\ 
a & b & c 
\end{tabular}
 \caption[Diffraction tomography image reconstruction]{
   Diffraction tomography image reconstruction: \\�
   a) original, 
   b) linear Born approximation reconstruction, 
   c) GNC based reconstruction}
\end{figure}

\section{Conclusions}
The Bayesian maximum {\em a posteriori} (MAP) estimates requires   
the minimization of a compound criterion which, in general, has two parts: 
a data fitting part and a prior part. 
In many situations in inverse problems the criterion to be minimized is 
multimodal. The cost of the Simulated Annealing (SA) based techniques is 
in general huge for these problems. 

We reported here recently proposed new techniques, based on Graduated 
Non Convexity (GNC), 
to overcome this difficulty and showed two specific implementations of 
this technique for: \\ 
-- The linear inverse problems such as: noise filtering, 
deconvolution, image restoration and tomographic image reconstruction, 
where the solution is modeled as a piecewise 
continuous function and where, the non convexity of the criterion is due 
to the special choice of the prior; and \\ 
-- A nonlinear inverse inverse scattering and diffraction 
tomography where the non convexity of the criterion is due to 
the likelihood part. 

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

\edoc

two.bbl
\begin{thebibliography}{10}

\bibitem{Geman84}
S.~Geman and D.~Geman, ``Stochastic relaxation, \uppercase{G}ibbs
  distributions, and the \uppercase{B}ayesian restoration of images,'' {\em
  \ieeePAMI}, vol.~PAMI-6, pp.~721--741, Nov. 1984.

\bibitem{Younes88}
L.~Youn\`es, ``Estimation and annealing for \uppercase{G}ibbsian fields,'' {\em
  Annales de l'institut Henri Poincar\'e}, vol.~24, pp.~269--294, Feb. 1988.

\bibitem{Jeng90}
F.~Jeng and J.~Woods, ``Simulated annealing in compound \uppercase{G}aussian
  random fields,'' {\em \ieeeIT}, vol.~IT-36, pp.~94--107, Jan. 1990.

\bibitem{Blake87}
A.~Blake and A.~Zisserman, {\em Visual reconstruction}.
\newblock Cambridge: The MIT Press, 1987.

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\end{thebibliography}
