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.
This book was written in tribute to our colleague Guy Demoment, who was a researcher at the CNRS from 1977 to 1988, then Professor at the University of Paris-Sud until 2008, member of the Laboratoire des Signaux et Systèmes (L2S, UMR 8506, Gif-sur-Yvette) and its director from 1997 to 2001, and the founder of a research group on inverse problems in signal and image processing at the beginning of the 1980s.
Guy Demoment's research activities began in 1970, at the interface between biological and medical engineering, automatic control and the still fledgling field of signal processing. Guy was particularly interested in cardiac function and in the cardiovascular system [DEM 77]. He derived a mathematical model of the functioning of the cardiovascular hemodynamic loop which was subsequently used to develop the control law of cardiac replacement prostheses. He also focused on aspects closer to theoretical biology such as left ventricle modeling and questions closer to physics such as the determination of vascular impedance [DEM 81].
This latter aspect naturally leads us to confront models with reality by means of measurements. The idea makes sense whereas in practice these measurements provide only indirect and degraded information on the quantities of interest. These degradations are generally considered in two forms: structure (resolution limitations, dynamics, sampling, etc.) and uncertainty (measurement noise, model approximation, etc.). The restitution of the quantity of interest then raises a real ill-posed inversion or inference problem. By creating the Groupe Problèmes Inverses (GPI - Inverse Problems Group), Guy promoted this scientific approach within the L2S then to the whole of the signal-image community within the engineering sciences. Having shared this approach with him within the GPI is a fortunate opportunity that most of the co-authors of this book have benefited from, as doctoral students or beginner colleagues.
Undoubtedly, Guy Demoment was an essential contributor to the field of inverse problems in signal and image processing, and its main instigator in the French community. He has also been passionate about related issues such as that of the effective implementation of a number of algorithms. In particular, in the context of linear deconvolution and adaptive spectral analysis, Kalman filtering and smoothing algorithms were given particular attention. Guy made several significant contributions concerning fast versions of these algorithms [DEM 85, DEM 91].
The exploitation of probabilistic models for detection-estimation has also been a subject of choice for Guy and his collaborators since the end of the 1980s, resulting in recursive [GOU 89], and then iterative [GOU 90] computational structures. It is interesting to note that the latter are very competitive precursors to the well-known greedy algorithms in parsimonious approximation, as is clearly stated in the book.
Regarding more fundamental subjects, he has been interested in the issues of information modeling and Bayesian inference, inspired by E.T. Jaynes' works. He contributed to the use of a maximum entropy principle for the synthesis of a priori models and to their application to tomography [MOH 87, MOH 88a], and then he studied the principle of maximum entropy on the mean in the context of inverse problems [LEB 99]. He then further explored these issues and, during his last period of scientific activity, became interested in variational approaches for Bayesian inference.
Guy's scientific sensitivity has also been visible in a significant way in his teaching activities. He created several courses ranging from undergraduate to PhD levels, as well as in continuing education, and always dedicated to them a lot of energy and creativity. Some particular examples include a course on Bayesian inference and the basics of probabilities, and among the most in-depth themes, Kalman algorithms and their fast versions, as well as the deconvolution of signals.
Beyond his scientific activities, researches and teachings, Guy has also been involved in a remarkable way in community life. On a national level, he has been a member of the Conseil National des Universités (www.cpcnu.fr), a particularly active member of scholar and research networks, e.g. club EEA (www.clubeea.org) and the GdR ISIS (gdr-isis.fr). Within the University of Paris-Sud, he has chaired the pedagogy commission, he has been vice president of the Department of Physics in Orsay, responsible for bachelor-level diploma, and co-creator of a masters-level diploma.
With regard to the present book, it concerns "ill-posed inverse problems". The readers can refer to the widely cited article [DEM 89] or to a previous collective book [IDI 08] on this subject, of which Guy Demoment is one of the main contributors. It is concerned with problems that cannot be resolved on the basis of the observed data only and the construction of solutions requires other information, referred to as a priori. These solutions are then specific to the information taken into account. The recognition and the explanation of this information are necessary to appreciate the range of validity and the scope of application of the constructed solutions. Over the 1980s, the scientific community has greatly acknowledged the significance of this problematic, and contributions have become very abundant not only in the signal-image community but also in that of mathematics, computer science and physics.
As a direct response to this thematic abundance concerning inverse problems, we have chosen to address a broad spectrum of data processing problems and application domains, with a particular focus on the diversity of mathematical tools.
From the point of view of application fields, an important part of the book is dedicated to different scientific fields, which present a large number of inversion problems: biological and medical imaging, and more specifically X-ray tomography (Chapters 1, 2 and 7), astronomy (Chapters 6 and 9) as well as non-destructive evaluation (Chapter 8). At least one other has been added: video sequence processing (Chapters 4 and 5). Two other applications that are more rarely met in the field of inversion: target tracking and sensor networks (Chapter 10) as well as digital communications (Chapter 11).
The diversity of chapters is also evident when the considered acquisition modalities come under scrutiny: from the more traditional ones such as tomography (Chapters 1 and 8) and MRI (Chapter 7), optical imaging in the visible (Chapters 4 and 5) or in the infrared spectrum (Chapters 5 and 9) to more recent modalities such as atomic force imaging (Chapter 2) and polarized optical imaging (Chapter 3).
Throughout the chapters, the duality between the approaches known as "energetic" and "probabilistic" emerges. The first type of approach is based on deterministic construction leading to criteria and to numerical optimization issues as typically in Chapters 1, 2, 3, 4 and 6. The second type of approach is based on a Bayesian construction, often hierarchical, which probabilizes unknown objects in addition to data. It thus leads to a joint distribution; therefore, optimal strategies are available and performance characterization right from the start becomes possible as in Chapter 5. The remaining chapters make use of a posteriori distributions to generate an estimation: they are explored by using stochastic sampling as in Chapters 6, 7 and 8 or by an approximated maximization as in Chapters 7, 8, 9 and 10. The latter also introduces a notion of learning and relies on informational principles discussed in Chapter 11, which presents more theoretical aspects related to entropy criteria.
[DEM 72] DEMOMENT G., Modèle de la boucle cardiovasculaire: évaluation de l'autorégulation mécanique et de la fonction ventriculaire gauche, PhD thesis, no.169, Orsay center, University of Paris-sud, 29 June 1972.
[DEM 77] DEMOMENT G., Contribution à l'étude du fonctionnement ventriculaire gauche par des méthodes d'identification paramétriques. obtention d'un observateur de l'état du ventricule, PhD thesis, no.1810, Orsay center, University of Paris-sud, 15 March 1977.
[DEM 82] DEMOMENT G., Introduction à la statistique, Lecture notes, École supérieure d'électrité, no. 2906, 1982.
[DEM 83] DEMOMENT G., Déconvolution des signaux, Lecture notes, École supérieure d'électrité no. 2964, 1983.
[DEM 85] DEMOMENT G., REYNAUD R., "Fast minimum-variance deconvolution", IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-33, pp. 1324-1326, 1985.
[DEM 87] DEMOMENT G., Algorithme rapides, Lecture notes, École supérieure d'électrité, no. 3152, 1987.
[DEM 89a] DEMOMENT G., "Equations de Chandrasekhar et algorithmes rapides pour le traitement du signal et des images", Traitement du Signal, vol. 6, pp. 103-115, 1989.
[DEM 89b] DEMOMENT G., "Image reconstruction and restoration: Overview of common estimation structures and problems", IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-37, no. 12, pp. 2024-2036, December 1989.
[DEM 91] DEMOMENT G., REYNAUD R., "Fast RLS algorithms and Chandrasekhar equations", HAYKIN S., (ed.), SPIE Conference on Adaptive Signal Processing, San Diego, CA, pp. 357-367, July 1991.
[DEM 05a] DEMOMENT...