Inverse Problems in Vision and 3D Tomography E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 753

Veröffentlichungsjahr: 2013

Beschreibung

The concept of an inverse problem is a familiar one to most scientists and engineers, particularly in the field of signal and image processing, imaging systems (medical, geophysical, industrial non-destructive testing, etc.), and computer vision. In imaging systems, the aim is not just to estimate unobserved images but also their geometric characteristics from observed quantities that are linked to these unobserved quantities by a known physical or mathematical relationship. In this manner techniques such as image enhancement or addition of hidden detail can be delivered. This book focuses on imaging and vision problems that can be clearly described in terms of an inverse problem where an estimate for the image and its geometrical attributes (contours and regions) is sought. The book uses a consistent methodology to examine inverse problems such as: noise removal; restoration by deconvolution; 2D or 3D reconstruction in X-ray, tomography or microwave imaging; reconstruction of the surface of a 3D object using X-ray tomography or making use of its shading; reconstruction of the surface of a 3D landscape based on several satellite photos; super-resolution; motion estimation in a sequence of images; separation of several images mixed using instruments with different sensitivities or transfer functions; and much more.

Leseprobe

Table of Contents

Preface

Chapter 1. Introduction to Inverse Problems in Imaging and Vision

1.1. Inverse problems

1.2. Specific vision problems

1.3. Models for time-dependent quantities

1.4. Inverse problems with multiple inputs and multiple outputs (MIMO)

1.5. Non-linear inverse problems

1.6. 3D reconstructions

1.7. Inverse problems with multimodal observations

1.8. Classification of inversion methods: analytical or algebraic

1.9. Standard deterministic methods

1.10. Probabilistic methods

1.11. Problems specific to vision

1.12. Introduction to the various chapters of the book

1.13. Bibliography

Chapter 2. Noise Removal and Contour Detection

2.1. Introduction

2.2. Statistical segmentation of noisy images

2.3. Multi-band multi-scale Markovian regularization

2.4. Bibliography

Chapter 3. Blind Image Deconvolution

3.1. Introduction

3.2. The blind deconvolution problem

3.3. Joint estimation of the PSF and the object

3.4. Marginalized estimation of the impulse response

3.5. Various other approaches

3.6. Multi-image methods and phase diversity

3.7. Conclusion

3.8. Bibliography

Chapter 4. Triplet Markov Chains and Image Segmentation

4.1. Introduction

4.2. Pairwise Markov chains (PMCs)

4.3. Copulas in PMCs

4.4. Parameter estimation

4.5. Triplet Markov chains (TMCs)

4.6. TMCs and non-stationarity

4.7. Hidden Semi-Markov chains (HSMCs) and TMCs

4.8. Auxiliary multivariate chains

4.9. Conclusions and outlook

4.10. Bibliography

Chapter 5. Detection and Recognition of a Collection of Objects in a Scene

5.1. Introduction

5.2. Stochastic approaches

5.3. Variational approaches

5.4.Bibliography

Chapter 6. Apparent Motion Estimation and Visual Tracking

6.1. Introduction: from motion estimation to visual tracking

6.2. Instantaneous estimation of apparent motion

6.3. Visual tracking

6.4. Conclusions

6.5. Bibliography

Chapter 7. Super-resolution

7.1. Introduction

7.2. Modeling the direct problem

7.3. Classical SR methods

7.4. SR inversion methods

7.5. Methods based on a Bayesian approach

7.6. Simulation results

7.7. Conclusion

7.8.Bibliography

Chapter 8. Surface Reconstruction from Tomography Data

8.1. Introduction

8.2. Reconstruction of localized objects

8.3. Use of deformable contours for 3D reconstruction

8.4. Appropriate surface models and algorithmic considerations

8.5. Reconstruction of a polyhedric active contour

8.6. Conclusion

8.7. Bibliography

Chapter 9. Gauss-Markov-Potts Prior for Bayesian Inversion in Microwave Imaging

9.1. Introduction

9.2. Experimental configuration and modeling of the direct problem

9.3. Inversion in the linear case

9.4. Inversion in the non-linear case

9.5. Conclusion

9.6. Bibliography

Chapter 10. Shape from Shading

10.1. Introduction

10.2. Modeling of shape from shading

10.3. Resolution of shape from shading

10.4. Conclusion

10.5. Bibliography

Chapter 11. Image Separation

11.1. General introduction

11.2. Blind image separation

11.3. Bayesian formulation

11.4. Stochastic algorithms

11.5. Simulation results

11.6. Conclusion

11.7. Appendix 1: a posteriori distributions

11.8. Bibliography

Chapter 12. Stereo Reconstruction in Satellite and Aerial Imaging

12.1. Introduction

12.2. Principles of satellite stereovision

12.3. Matching

12.4. Regularization

12.5. Numerical considerations

12.6. Conclusion

12.7. Bibliography

Chapter 13. Fusion and Multi-modality

13.1. Fusion of optical multi-detector images without loss of information

13.2. Fusion of multi-spectral images using hidden Markov trees

13.3. Segmentation of multimodal cerebral MRI using an a priori probabilistic map

13.4. Bibliography

List of Authors

Index

First published 2009 in France by Hermes Science/Lavoisier in 2 volumes entitled: Problèmes inverses en imagerie et en vision 1 et 2 © LAVOISIER 2009

First published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

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:

ISTE Ltd 27-37 St George’s Road London SW19 4EU UK

www.iste.co.uk

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

www.wiley.com

© ISTE Ltd 2010

The rights of Ali Mohammad-Djafari 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

Problèmes inverses en imagerie et en vision. English

Inverse problems in vision and 3D tomography / edited by Ali Mohammad-Djafari.

     p. cm.

Includes bibliographical references and index.

ISBN 978-1-84821-172-8

1. Three-dimensional imaging--Mathematical models. 2. Image processing--Mathematics. 3. Tomography--Mathematics. 4. Inverse problems (Differential equations) I. Mohammad-Djafari, Ali.

TA1637.P7813 2009

621.36'7--dc22

2009038814

British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-84821-172-8

Preface

The notion of inverse problems has now become a familiar concept to most scientists and engineers, particularly in many imaging systems and in computer vision. In brief, solving an inverse problem requires the estimate of a quantity (mono- or multivariate) which is not directly observable, starting from the observation of another quantity (which is also a mono- or multi-variate function) which is linked to the first one through the intermediary of a mathematical model known as the forward model. The main difficulty is that, in general, the forward problem is well posed whereas the corresponding inverse problem is ill posed. Among the most well-known examples of inverse problems, we can mention deconvolution, image restoration and tomographic image reconstruction. Many other image processing problems, however, can also be described as an inverse problem. This is the case, for example, for segmentation – the reconstruction of a 3D scene from its shadows or from a series of aerial or satellite photographs – or alternatively the creation of a high-resolution image from a sequence of low resolution images.

In what are known as imaging problems, the unknown quantity is often a bivariate function (an image); the deconvolution and reconstruction of tomographic images are two popular examples. The solution to both these problems has occupied a large number of researchers since the beginning of the twentieth century, using the theory of regularization in its most general form as its basis. Today, many books exist on the concept of regularization.

Probabilistic approaches have also shown their relevance and effectiveness by providing tools for the generalization of deterministic regularization, and particularly by providing measurements of the uncertainty of the solution. A number of books have also recently been published on this subject, including the book entitled Bayesian Approach to Inverse Problems in the ISTE Digital Signal and Image Processing series. In this book we will discuss the basis of, and the difficulties of, inverse problems treated with the help of standard inversion methods, both deterministic and probabilistic.

However, the formulation of problems posed in other fields in terms of an inverse problem, particularly in the field of computer vision, and new demand for the improvement of inversion methods in imaging prompted us to start work on this book.

In fact, today, in most imaging techniques the demand is not only to be able to construct an image, but also to be able to directly access the geometric information contained within it. This is why the main aim of this book is to focus on those problems of imaging and vision for which the problem can be written clearly in the form of an inverse problem involving the determination of an estimate for a mono- or multivariate function and its geometric attributes (contours or labels of the regions of an image), starting from the observation of another quantity which may also be a monoor multi-variate function linked to the first one through a forward model.

In this book, then, we will treat inverse problems that are encountered in many imaging systems and computer vision problems. These problems include:

– noise removal;

– restoration by deconvolution;

– 2D or 3D reconstruction in X-ray or tomography or microwave imaging;

– the reconstruction of the surface of a 3D object using X-ray tomography or making use of its shading;

– the reconstruction of the surface of a 3D landscape based on several satellite photos.

It also includes the construction of a high resolution image based on several low resolution images (super-resolution), the estimate ofmotion in a sequence of images or alternatively the separation of several images mixed using instruments with different sensitivities or transfer functions. All these subjects will be presented in the various chapters of this book, using a consistent methodology.

I hope you will enjoy reading this book, and I would be delighted to hear your views on its content, as I am sure the authors of the various chapters would be.

Ali MOHAMMAD-DJAFARI

Orsay, November 2009

Chapter 1

Introduction to Inverse Problems in Imaging and Vision1

The concept of an inverse problem is now a familiar concept to most scientists and engineers, particularly in the field of signal and image processing. In a nutshell, it involves the estimation of an unknown quantity, a mono-or multi-variate function f (r), starting from another observable quantity g(s) which is linked to it through a mathematical relationship known as the forward model. The main difficulty is that often such problems are ill posed [HAD 01]. The basic tools are therefore the theory of regularization [TIK 63, TIK 76] and its probabilistic counterpart of Bayesian estimation [HAN 83, TAR 82]. An earlier book on the subject in this same series, entitled Bayesian Approach to Inverse Problems [IDI 08], presents the basis of inversion methods, whether they be deterministic or probabilistic. However, the formulation of problems encountered in other communities in terms of an inverse problem, particularly in computer vision, as well as recent advances concerning inversion methods in imaging systems, prompted us to produce this book.

These days, in most imagery techniques, the aim is not only to construct images, but also to directly access the geometric characteristics of those images. This is why the main objective of this book is to focus on imagery and vision problems for which the problem can clearly be written in terms of an inverse problem. In the inverse problem an estimate for a function f (r) and its geometrical attributes is sought, in other words its contours q(r) or labels for its regions z(r) are to be determined from the observation g(s), which is linked to f (r) through what is known as the forward model.

The links between f(r) and g(s), on one hand, and between f(r) and its geometrical attributes q(r) and z(r) on the other hand, will be specified later. The main object of this introductory chapter is to present examples of inverse problems with different forward models and the bases of inversion methods.

1.1. Inverse problems

The unknown function f(r) and the observable function g(s) will not necessarily be defined in the same space. In fact, r and s can represent a position in space (x in 1D, (x, y) in 2D or (x, y, z) in 3D) or even a coordinate (x, y, z, t) in space-time or (x, y, z, ) in space-wavelength (4D), etc. The two spaces may have the same dimensions, as is the case in image restoration, or different dimensions, as is the case for tomographic reconstruction.

The link between f(r) and g(s) is described, in the most general case, by an operator known as the forward operator which, when applied to the function f(r), gives:

(1.1)

This equation is also known as the observation equation. In most cases, this relationship is not linear. However, a linear approximation can often be found which makes it possible to solve the problem more easily. In the case of a linear operator we have:

(1.2)

where h(r,s) represents the response of the measurement system.

At this point, we should note that we are very often working in finite dimensions, and consequently we must discretize this equation. It is then easy to show that, in the general case, the discretized form of this equation can be written:

(1.3)

(1.4)

where i(s) and j (r) are the appropriate test functions and basis functions in the spaces of functions g(s) and f(r) respectively. This is equivalent to assuming:

(1.5)

With these assumptions, the elements of the matrix H are given by:

(1.6)

The main object of this introductory chapter is to prepare the ground for the notations which will be used in this book throughout the examples of inverse problems examined in detail. This is why, in the rest of this chapter, the various problems will only be briefly presented.

1.1.1. 1D signal case

The case of a standard instrument, with an input f(t) and an output g(t), can be written:

(1.7)

(1.8)

The corresponding inverse problem is known as deconvolution [DEM 85, HUN 72, HUN 73, RIC 72]. h(t) in this case is called the impulse response of the system.

Figure 1.1. Signal deconvolution problem

In what follows, we will consider a certain number of classical inverse image processing problems.

1.1.2. Convolution model for image restoration

(1.9)

or alternatively:

(1.10)

This is the case, for example, in image restoration:

(1.11)

(1.12)

Figure 1.2. image restoration problem

1.1.3. General linear model

When observed quantities g(s) and unknown quantities f(r) are defined in spaces of the same dimensions, but not of the same nature, the linear model takes the form:

(1.13)

Here, the observed quantity g(s) is defined over a space and the unknown quantity f(r) is defined over another space of a different nature, but and have the same dimensions. This is, for example, the case for Fourier synthesis:

(1.14)

or alternatively:

(1.15)

where the measured quantity g() is the Fourier transform of the unknown quantity f(r). This is an equation which can for example be found in magnetic resonance imaging [BRA 86, MOH 87, MOH 03].

Figure 1.3.Fourier synthesis problem in microwave imaging

There are also situations where the observed quantity g(s) and the unknown quantity f(r) are defined in spaces of different dimensions and nature. This is, for example, the case in X-ray tomography where the relationship between g and f is a Radon transform in 2D and an X-transform in 3D (see Figure 1.4).

Figure 1.4.X-ray tomography: a) 3D case and b) 2D case