The focus of this book is on "ill-posed inverse problems". These problems cannot be solved only on the basis of observed data. The building of solutions involves the recognition of other pieces of a priori information. These solutions are then specific to the pieces of information taken into account. Clarifying and taking these pieces of information into account is necessary for grasping the domain of validity and the field of application for the solutions built.  For too long, the interest in these problems has remained very limited in the signal-image community. However, the community has since recognized that these matters are more interesting and they have become the subject of much greater enthusiasm. From the application field’s point of view, a significant part of the book is devoted to conventional subjects in the field of inversion: biological and medical imaging, astronomy, non-destructive evaluation, processing of video sequences, target tracking, sensor networks and digital communications. The variety of chapters is also clear, when we examine the acquisition modalities at stake: conventional modalities, such as tomography and NMR, visible or infrared optical imaging, or more recent modalities such as atomic force imaging and polarized light imaging.

Table of Contents:
INTRODUCTION xi Jean-Francois GIOVANNELLI and Jerome IDIER CHAPTER 1. 3D RECONSTRUCTION IN X-RAY TOMOGRAPHY: APPROACH EXAMPLE FOR CLINICAL DATA PROCESSING 1 Yves GOUSSARD 1.1. Introduction 1 1.2. Problem statement 2 1.3. Method 7 1.4. Results 15 1.5. Conclusion 26 1.6. Acknowledgments 27 1.7. Bibliography 28 CHAPTER 2. ANALYSIS OF FORCE-VOLUME IMAGES IN ATOMIC FORCE MICROSCOPY USING SPARSE APPROXIMATION 31 Charles SOUSSEN, David BRIE, Gregory FRANCIUS, Jerome IDIER 2.1. Introduction 31 2.2. Atomic force microscopy 32 2.3. Data processing in AFM spectroscopy 40 2.4. Sparse approximation algorithms 43 2.5. Real data processing 49 2.6. Conclusion 52 2.7. Bibliography 53 CHAPTER 3. POLARIMETRIC IMAGE RESTORATION BY NON-LOCAL MEANS 57 Sylvain FAISAN, Francois ROUSSEAU, Christian HEINRICH, Jihad ZALLAT 3.1. Introduction 57 3.2. Light polarization and the Stokes–Mueller formalism 58 3.3. Estimation of the Stokes vectors 61 3.4. Results 72 3.5. Conclusion 77 3.6. Bibliography 78 CHAPTER 4. VIDEO PROCESSING AND REGULARIZED INVERSION METHODS 81 Guy LE BESNERAIS, Frederic CHAMPAGNAT 4.1. Introduction 81 4.2. Three applications 82 4.3. Dense image registration 88 4.4. A few achievements based on direct formulation 92 4.5. Conclusion 104 4.6. Bibliography 106 CHAPTER 5. BAYESIAN APPROACH IN PERFORMANCE MODELING: APPLICATION TO SUPERRESOLUTION 109 Frederic CHAMPAGNAT, Guy LE BESNERAIS, Caroline KULCSAR 5.1. Introduction 109 5.2. Performance modeling and Bayesian paradigm 111 5.3. Superresolution techniques behavior 113 5.4. Application examples 126 5.5. Real data processing 130 5.6. Conclusion 136 5.7. Bibliography 137 CHAPTER 6. LINE SPECTRA ESTIMATION FOR IRREGULARLY SAMPLED SIGNALS IN ASTROPHYSICS 141 Sebastien BOURGUIGNON, Herve CARFANTAN 6.1. Introduction 141 6.2. Periodogram, irregular sampling, maximum likelihood 144 6.3. Line spectra models: spectral sparsity 146 6.4. Prewhitening, CLEAN and greedy approaches 151 6.5. Global approach and convex penalization 155 6.6. Probabilistic approach for sparsity 159 6.7. Conclusion 164 6.8. Bibliography 165 CHAPTER 7. JOINT DETECTION-ESTIMATION IN FUNCTIONAL MRI 169 Philippe CIUCIU, Florence FORBES, Thomas VINCENT, Lotfi CHAARI 7.1. Introduction to functional neuroimaging 169 7.2. Joint detection-estimation of brain activity 171 7.3. Bayesian approach 178 7.4. Scheme for stochastic MCMC inference 183 7.5. Alternative variational inference scheme 184 7.6. Comparison of both types of solutions 190 7.7. Conclusion 194 7.8. Bibliography 195 CHAPTER 8. MCMC AND VARIATIONAL APPROACHES FOR BAYESIAN INVERSION IN DIFFRACTION IMAGING 201 Hacheme AYASSO, Bernard DUCHENE, Ali MOHAMMAD-DJAFARI 8.1. Introduction 201 8.2. Measurement configuration 204 8.3. The forward model 206 8.4. Bayesian inversion approach 211 8.5. Results 220 8.6. Conclusions 220 8.7. Bibliography 222 CHAPTER 9. VARIATIONAL BAYESIAN APPROACH AND BI-MODEL FOR THE RECONSTRUCTION-SEPARATION OF ASTROPHYSICS COMPONENTS 225 Thomas RODET, Aurelia FRAYSSE, Hacheme AYASSO 9.1. Introduction 225 9.2. Variational Bayesian methodology 228 9.3. Exponentiated gradient for variational Bayesian 229 9.4. Application: reconstruction-separation of astrophysical components 232 9.5. Implementation of the variational Bayesian approach 236 9.6. Results 240 9.7. Conclusion 246 9.8. Bibliography 246 CHAPTER 10. KERNEL VARIATIONAL APPROACH FOR TARGET TRACKING IN A WIRELESS SENSOR NETWORK 251 Hichem SNOUSSI, Paul HONEINE, Cedric RICHARD 10.1. Introduction 251 10.2. State of the art: limitations of existing methods 252 10.3. Model-less target tracking 254 10.4. Simulation results 261 10.5. Conclusion 264 10.6. Bibliography 264 CHAPTER 11. ENTROPIES AND ENTROPIC CRITERIA 267 Jean-Francois BERCHER 11.1. Introduction 267 11.2. Some entropies in information theory 268 11.3. Source coding with escort distributions and Renyi bounds 273 11.4. A simple transition model 277 11.5. Minimization of the Renyi divergence and associated entropies 281 11.6. Bibliography 289 LIST OF AUTHORS 293 INDEX 297