Bayesian Approach to Inverse Problems

Jérôme Idier

• Publisher: Wiley-ISTE
• Year: 2013
• Print ISBN:
  9781848210325
• Edition: 1
• eBook ISBN:
  9781118623695
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

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.

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.

A very rough distinction can be made between these two communities by saying that the former deals with the problem in an infinite dimension, with the questions of existence, uniqueness and stability, which become very complicated for nonlinear direct problems, and solves it numerically in finite dimensions, while the latter starts with a problem for which the discretization has already been performed and is not called into question, and takes advantage of the finite nature of the problem to introduce prior information built up from probabilistic models.

In this chapter, we propose to use a basic example to point out the difficulties that arise when we try to solve these inverse problems.

1.2. Basic example

We will now illustrate the basic concepts introduced in this chapter by an artificial example that mixes the essential characteristics of several types of inverse problems.

We are looking for a spectrum, the square of the modulus of a function

but, because of the experimental constraints, we only have access to the dual domain of the variable v, through the function x(t) of which

is the Fourier transform (FT). What is more, imperfections in the apparatus mean that the function x(t) is only observable as weighted by a “window” h(t), which gives the observable function y(t):

(1.1)

To make our ideas clear, let us think of a visible optical interferometry device like that by Michelson. To have access to the emission spectrum of the light source, we measure an energy flux as a function of the phase difference between two optical paths. The interferogram obtained is, ignoring the additional constant, the Fourier transform of the function we are looking for but the limitati...

Table of contents

• Cover
• Title Page
• Copyright
• Introduction
• PART I: Fundamental Problems and Tools
• PART II: Deconvolution
• PART III: Advanced Problems and Tools
• PART IV: Some Applications
• List of Authors
• Index