Bayesian Approach to Inverse Problems

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  • Edition: 1st
  • Format: eBook
  • Copyright: 2010-01-27
  • Publisher: Wiley-ISTE

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Many scientific, medical or engineering problems raise the issue of recovering some physical quantities from indirect measurements; for instance, detecting or quantifying flaws or cracks within a material from acoustic or electromagnetic measurements at its surface is an essential problem of non-destructive evaluation. The concept of inverse problems precisely originates from the idea of inverting the laws of physics to recover a quantity of interest from measurable data.
Unfortunately, most inverse problems are ill-posed, which means that precise and stable solutions are not easy to devise. Regularization is the key concept to solve inverse problems.
The goal of this book is to deal with inverse problems and regularized solutions using the Bayesian statistical tools, with a particular view to signal and image estimation.
The first three chapters bring the theoretical notions that make it possible to cast inverse problems within a mathematical framework. The next three chapters address the fundamental inverse problem of deconvolution in a comprehensive manner. Chapters 7 and 8 deal with advanced statistical questions linked to image estimation. In the last five chapters, the main tools introduced in the previous chapters are put into a practical context in important applicative areas, such as astronomy or medical imaging.

Author Biography

Jérôme Idier was born in France in 1966. He received the diploma degree in electrical engineering from the Ecole Superieure d'Electricité, Gif-sur-Yvette, France, in 1988, the Ph.D. degree in physics from the Universite de Paris-Sud, Orsay, France, in 1991, and the HDR (Habilitation a diriger des recherches) from the same university in 2001. Since 1991, he is a full time researcher at CNRS (Centre National de la Recherche Scientifique). He has been with the Laboratoire des Signaux et Systemes from 1991 to 2002, and with IRCCyN (Institut de Recherches en Cybernetique de Nantes (IRCCyN) since september 2002.
His major scientific interest is in statistical approaches to inverse problems for signal and image processing. More specifically, he studies probabilistic modeling, inference and optimization issues yielded by data processing problems such as denoising, deconvolution, spectral analysis, reconstruction from projections. The investigated applications are mainly non destructive testing, astronomical imaging and biomedical signal processing, and also radar imaging and geophysics. Dr Idier has been involved in joint research programs with several specialized research centers: EDF (Electricite de France), CEA (Commissariat a l'Energie Atomique), CNES (Centre National d'Etudes Spatiales), ONERA (Office National d'Etudes et de Recherches Aerospatiales), Loreal, Thales, Schlumberger.

Table of Contents

Part 1: Fundamental problems and tools.

Chapter 1. Inverse problems, ill-posed problems (Guy Demoment, Jérôme Idier).

Chapter 2. Main approaches to the regularization of ill-posed problems (Guy Demoment, Jérôme Idier).

Chapter 3. Inversion within the probabilistic framework (Guy Demoment, Yves Goussard).

Part 2: Deconvolution.

Chapter 4. Inverse filtering and other linear methods (Guy Le Besnerais, Jean-François Giovannelli, Guy Demoment).

Chapter 5. Deconvolution of spike trains (Frédéric Champagnat, Yves Goussard, Stéphane Gautier, Jérôme Idier).

Chapter 6. Deconvolution of images (Jérôme Idier, Laure Blanc-Féraud).

Part 3: Advanced problems and tools.

Chapter 7. Gibbs-Markov image models (Jérôme Idier).

Chapter 8. Unsupervised problems (Xavier Descombes, Yves Goussard).

Part 4: Some applications.
Chapter 9. Deconvolution applied to ultrasonic non-destructive evaluation (Stéphane Gautier, Frédéric Champagnat, Jérôme Idier).

Chapter 10. Inverse problems in optical imaging through atmospheric turbulence (Laurent Mugnier, Guy Le Besnerais).

Chapter 11. Spectral characterization in ultrasonic Doppler velocimetry (Jean-François Giovannelli, Alain Herment).

Chapter 12. Tomographic reconstruction from few projections (Ali Mohammad-Djafari, Jean-Marc Dinten).

Chapter 13. Diffraction tomography (Hervé Carfantan, Ali Mohammad-Djafari).

Chapter 14. Imaging from low-intensity data (Ken Sauer, Jean-Baptiste Thibault).

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