Table of Contents

Introduction

PART I. FUNDAMENTAL PROBLEMS AND TOOLS

Chapter 1. Inverse Problems, Ill-posed Problems

1.1. Introduction

1.2. Basic example

1.3. Ill-posed problem

1.4. Generalized inversion

1.5. Discretization and conditioning

1.6. Conclusion

1.7. Bibliography

Chapter 2. Main Approaches to the Regularization of Ill-posed Problems.

2.1. Regularization

2.2. Criterion descent methods

2.3. Choice of regularization coefficient

2.4. Bibliography

Chapter 3. Inversion within the Probabilistic Framework

3.1. Inversion and inference

3.2. Statistical inference

3.3. Bayesian approach to inversion

3.4. Links with deterministic methods

3.5. Choice of hyperparameters

3.6. A priori model

3.7. Choice of criteria

3.8. The linear, Gaussian case

3.9. Bibliography

PART II. DECONVOLUTION

Chapter 4. Inverse Filtering and Other Linear Methods

4.1. Introduction

4.2. Continuous-time deconvolution

4.3. Discretization of the problem

4.4. Batch deconvolution

4.5. Recursive deconvolution

4.6. Conclusion

4.7. Bibliography

Chapter 5. Deconvolution of Spike Trains

5.1. Introduction

5.2. Penalization of reflectivities, L2LP/L2Hy deconvolutions

5.3. Bernoulli-Gaussian deconvolution

5.4. Examples of processing and discussion

5.5. Extensions

5.6. Conclusion

5.7. Bibliography

Chapter 6. Deconvolution of Images

6.1. Introduction

6.2. Regularization in the Tikhonov sense

6.3. Detection-estimation

6.4. Non-quadratic approach

6.5. Half-quadratic augmented criteria

6.6. Application in image deconvolution

6.7. Conclusion

6.8. Bibliography

PART III. ADVANCED PROBLEMS AND TOOLS

Chapter 7. Gibbs-Markov Image Models

7.1. Introduction

7.2. Bayesian statistical framework

7.3. Gibbs-Markov fields

7.4. Statistical tools, stochastic sampling

7.5. Conclusion

7.6. Bibliography

Chapter 8. Unsupervised Problems

8.1. Introduction and statement of problem

8.2. Directly observed field

8.3. Indirectly observed field

8.4. Conclusion

8.5. Bibliography

PART IV. SOME APPLICATIONS

Chapter 9. Deconvolution Applied to Ultrasonic Non-destructive Evaluation

9.1. Introduction

9.2. Example of evaluation and difficulties of interpretation

9.3. Definition of direct convolution model

9.4. Blind deconvolution

9.5. Processing real data

9.6. Conclusion

9.7. Bibliography

Chapter 10. Inversion in Optical Imaging through Atmospheric Turbulence

10.1. Optical imaging through turbulence

10.2. Inversion approach and regularization criteria used

10.3. Measurement of aberrations

10.4. Myopic restoration in imaging

10.5. Image reconstruction in optical interferometry (OI)

10.6. Bibliography

Chapter 11. Spectral Characterization in Ultrasonic Doppler Velocimetry

11.1. Velocity measurement in medical imaging

11.2. Adaptive spectral analysis

11.3. Tracking spectral moments

11.4. Conclusion

11.5. Bibliography

Chapter 12. Tomographic Reconstruction from Few Projections

12.1. Introduction

12.2. Projection generation model

12.3. 2D analytical methods

12.4. 3D analytical methods

12.5. Limitations of analytical methods

12.6. Discrete approach to reconstruction

12.7. Choice of criterion and reconstruction methods

12.8. Reconstruction algorithms

12.9. Specific models for binary objects

12.10. Illustrations

12.11. Conclusions

12.12. Bibliography

Chapter 13. Diffraction Tomography

13.1. Introduction

13.2. Modeling the problem

13.3. Discretization of the direct problem

13.4. Construction of criteria for solving the inverse problem

13.5. Solving the inverse problem

13.6. Conclusion

13.7. Bibliography

Chapter 14. Imaging from Low-intensity Data

14.1. Introduction

14.2. Statistical properties of common low-intensity image data

14.3. Quantum-limited measurements in inverse problems

14.4. Implementation and calculation of Bayesian estimates

14.5. Conclusion

14.6. Bibliography

List of Authors

Index

First published in France in 2001 by Hermes Science/Lavoisier entitled “Approche bayésienne pour les problèmes inverses”

First published in Great Britain and the United States in 2008 by ISTE Ltd and John Wiley & Sons, Inc. Translation made by Mrs Becker

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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© LAVOISIER, 2001

The rights of Jérôme Idier to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Bayesian approach to inverse problems / edited by Jérôme Idier.

  p. cm.

  Includes bibliographical references and index.

  ISBN: 978-1-84821-032-5

 1. Inverse problems (Differential equations) 2. Bayesian statistical decision theory I. Idier, Jérôme.

  QA371.B365 2008

  515'.357--dc22

2007047723

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN: 978-1-84821-032-5

Introduction ¹

When a physical quantity is not directly accessible for measurement, it is common to proceed by observing other quantities that are connected with it by physical laws. The notion of an inverse problem corresponds to the idea of inverting these physical laws to gain indirect access to the quantity we are interested in.

For example, in electromagnetism, calculating the electric field induced by a known distribution of electric charges is a direct problem, i.e., a problem posed in “the natural direction” of physics as we are used to practising and controlling it. Deducing the distribution of the electric charges from measurements of the field is, on the other hand, an inverse problem.

Similarly, in signal processing, modeling a transmission channel that introduces distortion, interference and parasitic signals corresponds to solving a direct problem. Reconstructing the shape of a signal input to the channel from measurements made at the output is an inverse problem.

The situation where the quantity of interest is directly accessible for measurement is obviously more favorable. Nevertheless, direct measurement does not signify perfect measurement: the instrumental response of a piece of measuring apparatus and the various error sources connected with the observation process (systematic error, fluctuations connected with the physical sensors or electronic components, quantization, etc.) are degradations that can also be encompassed in the question of inversion.

When all is said and done, the concept of inverse problems underlies the processing of experimental data in its broadest sense. In the experience of the authors of this book, making it explicit that a data processing chain actually carries out an inversion is also often a very worthwhile exercise. It brings to light ad hoc hypotheses and arbitrary

Figure 1. A simple example of the direct problem/inverse problem pair in electromagnetism: the direct problem consists of finding the field at the circular boundary of the domain from the distribution of the sources; the inverse problem consists of deducing the distribution of the sources from measurements of the field at the boundary

constraints, and provides a rational, modular framework for designing data processing methods and analyzing their efficiency and their faults.

When approached with no special precautions, the inversion problems we meet with in practice, unlike direct problems, have a nasty tendency to be “naturally unstable”: if there are errors, however tiny, on the data, the behavior of “naive” inversion methods is not robust.

Let us take the example of inverse filtering (or deconvolution), a classic in signal processing. Figure 2 proposes two digital experiments:

– The first consists of inverting a discrete convolution relationship, y = h * x. The input signal x is triangular, of length M = 101 (Figure 2a), and the impulse response (IR) h is a discretized, truncated Gaussian of length L = 31 (Figure 2b). The output y, of length N = M + L — 1 = 131, is calculated using the Matlab language in the form y=conv(h,x) (Figure 2c). Matlab also offers a deconvolution method (by polynomial division) that, here, faithfully gives x again from y, in the form deconv(y,h) (Figure 2d).

– Now let us suppose that output y was measured imperfectly, e.g. with no measurement error but uniformly quantized to 10 bits, i.e., 1,024 levels: z is the quantized output (Figure 2e), calculated in Matlab in the form z=round(y*2^10)/2^10. The difference between y and z is imperceptible; and yet, deconv(z,h) (Figure 2f) is very different from x. Note in passing that the on-line help provided by Matlab (version 7.5) for deconv contains no warning of the unstable nature of this operation.

Figure 2. An example of a “naturally unstable” inverse problem: deconvolution. A tiny difference between z and conv(x,h) is enough to make deconv(z,h) very different from x. Like “conv”, “deconv” is an instruction in Matlab, version 7.5; it uses a non-regularized deconvolution method which is thus not robust. In comparison, (g) corresponds to a robust solution, obtained by Tikhonov regularization. Finally, (h) is an “exact” solution in the same way as x, in the sense that it reproduces z exactly by quantized convolution

In the early 20th century, Hadamard characterized these unstable problems mathematically, qualifying them as ill posed, in the sense that they did not lend themselves to being satisfactorily solved mathematically (and physically) [HAD 01]. Chapter 1 of this book develops the notion of ill posed problems and makes the non-robust behavior of “naive” inversion methods mathematically explicit.

In the 1960s, the Russian mathematician Tikhonov laid down the theoretical basis of modern inversion methods by introducing the concept of regularized solutions [TIK 63]. These solutions result from two ingredients being brought together. They are faithful to the data but this ingredient is not discriminating enough if the problem is ill posed. Among other solutions that are faithful to the data, they are the most regular, in a pragmatic sense that depends on the context. Tikhonov formalizes this trade-off between fidelity to the data and regularity by defining regularized solutions as those that minimize a composite criterion. He shows that the problem thus reformulated is well-posed. The principle of regularization in Tikhonov’s sense is one of the main subjects of Chapter 2.

Figure 2g illustrates the use of a method, regularized in Tikhonov’s sense, that reproduces input x very acceptably from imperfect data z. The calculation of this type of solution is dealt with in Chapters 3 and 4. The only imperfection visible on Figure 2g concerns the point of the triangle, which is slightly blunt. This is a logical effect of the regularity imposed uniformly on a signal which is, in fact, locally irregular at its midpoint (its derivative is discontinuous). The case of signals or images that are globally regular but have localized irregularities is a very important one in practice. It is treated in the context of deconvolution in Chapter 5 for irregularities that are “bright spots” and in Chapter 6 for irregularities that are borders between homogeneous areas. The latter situation is precisely the one we have in Figure 2.

The problem of Figure 2 could also be solved by positioning straight lines through the minimization of a least squares criterion. A parametric approach of this kind, which regularizes the problem by dimension control (Chapter 2, section 2.1.1), can be considered as the oldest of the inversion methods as it was invented around the same time as the least squares method in the late 18th century. One of the first times the principle was put into practice was when Gauss estimated the coefficient of ellipticity of the Earth from arc length measurements, having modeled the Earth’s profile in the form of an ellipse [STI 81]. This was indeed a case of an inverse problem being solved by a parametric approach, even though the concept actually appeared much later.

Finally, we could think that the inversion would naturally become stable if the direct problem could be modeled with no errors. If this were true, it is a more detailed description of the direct problem that would lead to stabilization of the inversion. Let us take the example of Figure 2, for which the exact mathematical relation linking x and z includes quantization: the inversion of this exact relation remains unstable. In fact, x is only one of an infinite number of solutions, some of which stay remarkably far from x; Figure 2h is an example.

Figure 2 is instructive but simplistic. In a more realistic situation, attaining a description of a direct problem – including the measuring system – in a mathematically perfect form is, in any case, more than we can hope for. In the inversion field, it is widely accepted that a credible inverse method must possess a minimum of robustness with respect to imperfect modeling of the direct problem. Adding pseudo-random noise to simulated data is a way of testing this robustness. Testing inversion only with “exact” simulated data is sometimes called the inverse crime.

Regularization in the Tikhonov sense, dimensionality control, adoption of such or such a parametric model, etc., there is no universal way of stabilizing an ill-posed inversion problem. The regularity of the solution must be defined case by case, in a form that may therefore appear subjective. Due to this, the concept of regularization is sometimes criticized or misunderstood. In fact, regularization is part of an application-oriented approach: it is not a question of inverting abstract problems that can be characterized by an input-output equation, but of solving real problems, where there is always advantage to be drawn from a few general characteristics of the quantity we are interested in, which may have been neglected in the initial formulation. The practical success of Tikhonov’s regularized approach and its subsequent evolutions has demonstrated that the approach is well founded. It is now accepted that an ill-posed inverse problem cannot be satisfactorily solved without some prior information, and this prior information is often qualitative or partial. For example, in an image restoration problem, it is desirable to take into consideration the fact that an image is generally composed of homogeneous regions, but this characteristic is qualitative and does not directly correspond to a mathematical model.

The encoding of uncertain or partial information can be envisaged within a Bayesian probabilistic framework. Work published a considerable time ago [FRA 70] showed that Tikhonov’s contribution could be interpreted in this framework. For direct problems formulated deterministically, the handling of probabilistic rules for the inversion gives rise to comprehension difficulties. It has to be understood that these probabilistic rules are inference rules: they enable states of knowledge to be quantified and their evolution, through measurements, is itself uncertain because of errors. So we are not judging the fundamentally deterministic or random nature of the observed phenomena or even of the measurement errors. In this respect, Jaynes’ work, brought together in [JAY 03], provides a reference for understanding the Bayesian approach in the data processing field. Chapter 3 of this book takes its inspiration from this work.

Modern methods for solving inverse problems have been arousing increasing interest since their beginnings in the 1960s. The question of inversion concerns a variety of sectors of activity, such as Earth and space sciences, meteorology, the aerospace industry, medical imaging, the nuclear electricity industry and civil engineering. Added up over all these sectors, its scientific and economic impact is enormous.

As far as the structure of inverse problems is concerned, very different fields may have very similar needs. However, the compartmentalization of scientific disciplines makes it difficult for ideas and methods to circulate. In this respect, it is the role of the signal processing community to respond to needs common to other disciplines in terms of data processing methods and algorithms. It was with this in mind that this book was written. Its 14 chapters are grouped together in four parts.

Part I is devoted to introducing the problems and the basic inversion tools and is the most abstract. It comprises three chapters:

– Chapter 1 introduces the problem of inversion as a whole, in a structured mathematical framework. It gives the characteristics of inverse problems posed in a continuous or discrete framework, and of ill-posed problems. It introduces the ideas of the pseudo-solutions and generalized inverse;

– Chapter 2 introduces the essential characteristics of regularization theory, and describes Tikhonov’s approach and its subsequent evolutions together with other approaches to regularization. Reminders are then given of methods for minimizing criteria, followed by techniques for estimating the regularization parameter, so that an automatic choice can be made for the trade-off between fidelity to data and regularity;

– Chapter 3 deals with solving inverse problems in the framework of statistical inference. It becomes apparent that the conventional estimation technique, known as maximum likelihood, corresponds to a non-regularized solution, whereas a Tikhonov regularized approach finds a natural interpretation in the framework of Bayesian estimation. A number of questions are then re-examined in this context: the automatic choice of parameters, and the building of models and criteria. The end of the chapter is devoted to the Gaussian linear framework, which constitutes a fundamental special case.

Part II is made up of Chapters 4, 5 and 6, and is entirely given over to deconvolution, as a case that is very widespread in practice and also as a very instructive case where many of the tools introduced in these chapters can be adapted to the inversion of problems that are structured differently:

– Chapter 4 deals with deconvolution methods yielding solutions that are linear functions of the data. It first studies the general properties of the solutions, then the various algorithm structures that enable them to be calculated. Traditional signal processing tools, such as Wiener and Kalman filters, figure among these structures;

– Chapter 5 looks at the more specific problem of deconvolution when the signal of interest is a series of pulses. This situation is very common in numerous domains such as non-destructive evaluation and medical imaging. Taking the pulse character of the input signal into account leads us to two classes of nonlinear solutions according to the data. One follows a detection-estimation approach and the other uses the minimization of convex criteria and robust estimation;

– Chapter 6 is devoted to deconvolution when the unknown signal is “regular almost everywhere” and, in particular, takes this characteristic into consideration for images rather than monovariate signals. As in the previous chapter, we find two classes of solutions, according to whether the problem is approached in terms of detection of edges or as a problem of image restoration in robust form.

Part III groups together two chapters introducing “advanced tools” specific to the Bayesian framework presented in Chapter 3:

– Chapter 7 is concerned with imaging from a probabilistic point of view. The composite criteria of Chapter 6 are reinterpreted in this framework, which leads us to the Gibbs-Markov models. Various sub-classes are introduced as models for images. We next look at the statistical aspects connected with calculating the estimators and evaluating their performance. Finally, the principle of iterative methods for random sampling of the Gibbs-Markov models is presented;

– Chapter 8 is entirely devoted to the problem of non-supervised inversion, i.e., inversion without a regularization parameter fixed by the user. This question, already mentioned in Chapters 2 and 3, is of considerable theoretical and practical interest but brings together several types of methodological and algorithmic difficulties. Chapter 8 studies the case of a linear Markov penalization function with respect to the parameters to be estimated and proposes, in particular, deterministic or stochastic techniques for implementing maximum likelihood, exact or approximate estimators.

Part IV, in six chapters, presents some inverse problems in their applications. This is by no means a complete review of all the domains involving inversion; several important fields such as heat, mechanics and geophysics are not covered¹. For the applications that are mentioned, we do not give an overall synthesis of the inversion problems encountered but rather some typical examples chosen by the authors as concrete illustrations of how regularized solutions are implemented in a particular domain. This last part also contains some important methodology extensions - myopic deconvolution (Chapters 9 and 10), Fourier synthesis (Chapter 10), spectral estimation and handling of hidden Markov chains (Chapter 11), and tomography problem solving (Chapters 12, 13 and 14):

– Chapter 9 concerns industrial non-destructive evaluation using ultrasound. It compares the implementation of the spike train deconvolution methods presented in Chapter 5. The problem of an impulse response that is poorly known or that introduces deformation is specifically studied. The result of this is some extended versions of the algorithms looked at in Chapter 5;

– Chapter 10 is about the inversion problems encountered in optical imaging in astronomy and, more specifically, for ground-based telescopes. In this case, atmospheric turbulence considerably reduces the resolution of the images. Various configurations intended to limit this degradation are considered. On the one hand, it is possible to approach the image restoration problem through myopic deconvolution; on the other, the effects of turbulence can be partially compensated by a technique known as adaptive optics, where the deconvolution of the images thus acquired remains a helpful step. The end of Chapter 10 is devoted to optical interferometry, which leads to a Fourier synthesis problem complicated by aberration due to atmospheric turbulence;

– Chapter 11 is devoted to Doppler ultrasound velocimetry, an imaging technique that is widespread in medicine. Two data inversion problems are seen to arise: time-frequency analysis and frequency tracking. These are spectral characterization problems that are particularly difficult for two reasons: firstly, the number of observed data points is very small and, secondly, the Shannon sampling conditions are not always respected. Chapter 11 covers these two problems in the regularization framework in order to compensate, at least partially, for the lack of information in the data;

– Chapter 12 considers the problem of reconstruction in X-ray tomography using a small number of projections. The Radon transform is introduced, and after a brief reminder of the various conventional approaches for its inversion, the algebraic and probabilistic methods are developed more specifically. In fact these are the only methods that can be used effectively in cases where the projections are limited in number and contain noise;

– Chapter 13 looks at how the Bayesian approach can be used to solve the problem of diffraction tomography. For this type of problem, the measurements collected are the waves scattered by an object and depend nonlinearly on the physical parameters we are trying to image. This chapter deals with diffraction tomography without the usual linear approximations, whose domains of validity do not cover all the situations encountered in practice. The wave propagation equations provide a integral direct model in the form of two coupled equations. These are discretized by the method of moments. The Bayesian approach then allows the inversion to be approached by minimizing a penalized criterion. However, the nonlinearity of the direct model leads to non-convexity of this criterion. In particularly difficult situations where local minima exist, the use of global optimization techniques is recommended;

– Chapter 14 studies, in the framework of medical imaging techniques such as positron emission tomography, the situations in which the corpuscular character of the counting measurements needs to be taken into account. Poisson’s law serves as the reference statistical distribution here to define the likelihood of the observations. There are also composite cases, in which Gaussian noise is added to the Poisson variables. It is sometimes possible to approximate the Poisson likelihood by a Gaussian law. When this is not the case, the properties of the log-likelihood criterion are studied and algorithms are put forward for various situations: emission or transmission tomography and composite cases.

Bibliography

[BEC 85] BECK J., BLACKWELL B., ST. CLAIR C., Inverse Heat Conduction, Ill-posed Problems, Wiley Interscience, New York, NY, 1985.

[BUI 94] BUI H. D., Inverse Problems in the Mechanics of Materials: An Introduction, CRC Press, Boca Raton, FL, 1994.

[DES 90] DESAUBIES Y. , TARANTOLA A., ZINN-JUSTIN J., Eds., Oceanographic and Geophysical Tomography, Amsterdam, The Netherlands, Elsevier Science Publishers (North-Holland), 1990.

[FRA 70] FRANKLIN J. N., “Well-posed stochastic extensions of ill-posed linear problems”, J. Math. Anal. Appl., vol. 31, p. 682-716, 1970.

[HAD 01] HADAMARD J., “Sur les problèmes aux dérivées partielles et leur signification physique”, Princeton University Bull., vol. 13, 1901.

[JAY 03] JAYNES E. T., Probability Theory – The Logic of Science, Cambridge University Press, Apr. 2003.

[STI 81] STIGLER S. M., “Gauss and the invention of least squares”, Annals Statist., vol. 9, num. 3, p. 465-474, 1981.

[TIK 63] TIKHONOV A., “Regularization of incorrectly posed problems”, Soviet. Math. Dokl., vol. 4, p. 1624-1627, 1963.

¹ Introduction written by Jérôme IDIER.

1. See [BEC 85, BUI 94, DES 90] as respective entry points to these fields.

PART I

Fundamental Problems and Tools

Chapter 1

Inverse Problems, Ill-posed Problems ¹

1.1. Introduction

In many fields of applied physics, such as optics, radar, heat, spectroscopy, geophysics, acoustics, radioastronomy, non-destructive evaluation, biomedical engineering, instrumentation and imaging in general, we are faced with the problem of determining the spatial distribution of a scalar or vector quantity - we often talk about an object - from direct measurements - called an image - or indirect measurements - called projections in the case of tomography, for example - of this object. Solving such imaging problems can habitually be broken down into three stages [HER 87, KAK 88]:

a direct problem where, knowing the object and the observation mechanism, we establish a mathematical description of the data observed. This model needs to be accurate enough to provide a correct description of the physical observation phenomenon and yet simple enough to lend itself to subsequent digital processing;

an instrumentation problem in which the most informative data possible must be acquired so that the imaging problem can be solved in the best conditions;

an inverse problem where the object has to be estimated from the preceding model and data.

Obtaining a good estimate of the object obviously requires these three sub-problems to be studied in a coordinated way. However, the characteristic that these image reconstruction or restoration problems have in common is that they are often ill-posed or ill-conditioned. Higher level problems that are found in computer vision, such as image segmentation, optical flow processing and shape reconstruction from shading, are also inverse problems and suffer from the same difficulties [AND 77, BER 88, MAR 87]. In the same way, a problem such as spectral analysis, which has similarities with the Fourier synthesis used in radio-astronomy, for example, and which is not usually treated as an inverse problem, can gain from being approached this way, as we will see later.

Schematically, there are two brohad communities that are interested in these inverse problems from a methodological point of view:

the mathematical physics community, with the seminal works of Phillips, Twomey and Tikhonov in the 1960s [PHI 62, TIK 63, TWO 62]. Sabatier was one of the pioneers in France [SAB 78]. A representative journal is Inverse Problems;

the statistical data processing community, which can be linked to the work of Franklin in the late 1960s [FRA 70], although the ideas involved - the basis of Wiener filtering - had been bubbling beneath the surface in many works for several years [FOS 61]. The Geman brothers gave a major boost to image processing about twenty years ago [GEM 84] A representative journal is IEEE Transactions on Image Processing.

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