Tomography E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 680

Veröffentlichungsjahr: 2013

Beschreibung

The principle of tomography is to explore the structure and composition of objects non-destructively along spatial and temporal dimensions, using penetrating radiation, such as X- and gamma-rays, or waves, such as electromagnetic and acoustic waves. Based on computer-assisted image reconstruction, tomography provides maps of parameters that characterize the emission of the employed radiation or waves, or their interaction with the examined objects, for one or several cross-sections. Thus, it gives access to the inner structure of inert objects and living organisms in their full complexity. In this book, multidisciplinary specialists explain the foundations and principles of tomographic imaging and describe a broad range of applications. The content is organized in five parts, which are dedicated to image reconstruction, microtomography, industrial tomography, morphological medical tomography and functional medical tomography.

Leseprobe

Table of Contents

Preface

Notation

Chapter 1. Introduction to Tomography

1.1. Introduction

1.2. Observing contrasts

1.3. Localization in space and time

1.4. Image reconstruction

1.5. Application domains

1.6. Bibliography

Part 1. Image Reconstruction

Chapter 2. Analytical Methods

2.1. Introduction

2.2. 2D Radon transform in parallel-beam geometry

2.3. 2D Radon transform in fan-beam geometry

2.4. 3D X-ray transform in parallel-beam geometry

2.5. 3D Radon transform

2.6. 3D positron emission tomography

2.7. X-ray tomography in cone-beam geometry

2.8. Dynamic tomography

2.9. Bibliography

Chapter 3. Sampling Conditions in Tomography

3.1. Sampling of functions in ℜn

3.2. Sampling of the 2D Radon transform

3.3. Sampling in 3D tomography

3.4. Bibliography

Chapter 4. Discrete Methods

4.1. Introduction

4.2. Discrete models

4.3. Algebraic methods

4.4. Statistical methods

4.5. Example of tomographic reconstruction

4.6. Discussion and conclusion

4.7. Bibliography

Part 2. Microtomography

Chapter 5. Tomographic Microscopy

5.1. Introduction

5.2. Projection tomography in electron microscopy

5.3. Tomography by optical sectioning

5.4. 3D data processing, reconstruction and analysis

5.5. Bibliography

Chapter 6. Optical Tomography

6.1. Introduction

6.2. Interaction of light with matter

6.3. Propagation of photons in diffuse media

6.4. Optical tomography methods

6.5. Optical tomography in highly diffuse media

6.6. Bibliography

Chapter 7. Synchrotron Tomography

7.1. Introduction

7.2. Synchrotron radiation

7.3. Quantitative tomography

7.4. Microtomography using synchrotron radiation

7.5. Extensions

7.6. Conclusion

7.7. Bibliography

Part 3. Industrial Tomography

Chapter 8. X-ray Tomography in Industrial Non-destructive Testing

8.1. Introduction

8.2. Physics of the measurement

8.3. Sources of radiation

8.4. Detection

8.5. Reconstruction algorithms and artifacts

8.6. Applications

8.7. Conclusion

8.8. Bibliography

Chapter 9. Industrial Applications of Emission Tomography for Flow Visualization

9.1. Industrial applications of emission tomography

9.2. Examples of applications

9.3. Physical model of data acquisition

9.4. Definition and characterization of a system

9.5. Conclusion

9.6. Bibliography

Part 4. Morphological Medical Tomography

Chapter 10. Computed Tomography

10.1. Introduction

10.2. Physics of helical tomography

10.3. Applications of volume CT

10.4. Conclusion

10.5. Bibliography

Chapter 11. Interventional X-ray Volume Tomography

11.1. Introduction

11.2. Example of 3D angiography

11.3. Clinical examples

11.4. Conclusion

11.5. Bibliography

Chapter 12. Magnetic Resonance Imaging

12.1. Introduction

12.2. Nuclear paramagnetism and its measurement

12.3. Spatial encoding of the signal and image reconstruction

12.4. Contrast factors and examples of applications

12.5. Tomography or volumetry?

12.6. Bibliography

Part 5. Functional Medical Tomography

Chapter 13. Single Photon Emission Computed Tomography

13.1. Introduction

13.2. Radiopharmaceuticals

13.3. Detector

13.4. Image reconstruction

13.5. Example of myocardial SPECT

13.6. Conclusion

13.7. Bibliography

Chapter 14. Positron Emission Tomography

14.1. Introduction

14.2. Data acquisition

14.3. Data processing

14.4. Research and clinical applications of PET

14.5. Conclusion

14.6. Bibliography

Chapter 15. Functional Magnetic Resonance Imaging

15.1. Introduction

15.2. Functional MRI of cerebrovascular responses

15.3. fMRI of BOLD contrasts

15.4. Different protocols

15.5. Bibliography

Chapter 16. Tomography of Electrical Cerebral Activity in Magneto- and Electro-encephalography

16.1. Introduction

16.2. Principles of MEG and EEG

First published in France in 2002 by Hermes Science/Lavoisier entitled: La tomographie and La tomographie médicale © LAVOISIER, 2002

First published in Great Britain and the United States in 2009 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, 2009

The rights of Pierre Grangeat 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

Tomographie. English. Tomography / edited by Pierre Grangeat. p. cm. Includes bibliographical references and index. Translation of: La tomographie and La tomographie médicale, originally published by Hermes Science/Lavoisier, 2002. ISBN 978-1-84821-099-8 1. Tomography. I. Grangeat, Pierre. II. Tomographie médicale. English. III. Title. [DNLM: 1. Tomography. WN 206 T661 2009a] RC78.7.T6T65513 2009 616.07′57--dc22

2009017099

British Library Cataloguing-in-Publication Data

Preface

Since visible light is reflected by most of the objects around us, our perception of the environment is mainly determined by the properties of their surfaces. To lift this restriction and explore their interior, we have to develop dedicated instruments that rely on penetrating radiation, such as X- and γ-rays, and certain electromagnetic and acoustic waves.

Tomography constitutes the culmination of this endeavor. By combining a set of measurements and performing a reconstruction, it provides a map of a characteristic parameter of the employed radiation for one or more cross sections. It thus enables us to see the interiors of objects on a screen, whereas this was previously only possible either by imagination, based on the measurements, or by direct observation, based on a physical sectioning of the objects. In the case of medical imaging, the latter involved surgical intervention. Tomography is a remarkable invention, which allows us to discover the interiors of the world and the body, their organization in space and time, without destroying them. It is the favored tool for analyzing and characterizing matter, be it dead or alive, static or dynamic, of microscopic or of astronomical scale. By giving access to its structure and the form of its components, it enables us to understand the complexity of the studied object.

Computer aided tomography is a digital image acquisition technique. It produces an encoding, i.e. a digital representation on a computer, of a region of interest inside a patient, a structure, or an object. It thus provides a virtual representation of reality. The digital representation also facilitates subsequent exploitation, exchange, and storage of the associated information. By suitable processing, it then becomes possible to detect the presence of defects, to identify internal structures and to study their form and position, to quantify density variations, to model the components, the body, or the organs, and to guide interventional devices. Moreover, the user may benefit from the assistance of digital image processing, analysis, and visualization software.

A tomographic imaging system comprises several components and technologies. It requires the participation of the final users, such as physicians, physicists, and biologists, for its specification, of researchers and engineers for its development, and of industrial manufacturing and marketing experts for its production and commercialization. These participants are usually educated in medical and engineering schools, as well as at universities.

Throughout this book, we wish to provide help equally to students interested in the scientific and technological background of tomography and to the above mentioned group of people directly involved in the conception and application of tomographic systems. First of all, we focus on explaining the different fundamentals and principles of the formation of tomographic images and on illustrating their aim. Since it is the subject of the series IC2 and of the corresponding English books published by ISTE Ltd. and John Wiley & Sons Inc., we emphasize signal processing and only touch upon the components of the acquisition systems, such as the radiation sources, the detectors, the processing platforms, and the mechanics. Signal processing in tomography forms the intersection between physics for modeling the acquisition systems, mathematics for solving the measurement equations, and computer science for efficiently implementing and executing the image reconstruction. The analysis, visualization, and transmission of these images are addressed in other French books in the series IC2 and the corresponding English books.

This book is compiled from two French books. La tomographie1 corresponds to the first three parts of this book, and La tomographie médicale2 corresponds to the last two parts. For the translation of these two books, several chapters have been updated to reflect advances in the respective domains since their publication.

This book is the result of collective work. Therefore, I would like to dearly thank all authors for their contribution. Tomographic imaging is the “heart” of our work and our research. Each one of us committed him- or her-self to introducing the reader to tomography, in such a way that the origin of the images and the information in these images are comprehensible. By gathering engineers, physicists, mathematicians, and physicians, I formed a multidisciplinary editing team, which allows the reader to benefit from explanations by experts in the respective fields.

The translation of this book was carried out by Holger Eggers. I would like to express my gratitude for his work, which demonstrates not only perfect knowledge of the covered technical aspects but also a good command of both French and English.

In this book, we have compiled in five distinct parts the mathematical foundations associated with image reconstruction, the applications linked to microscopic and industrial imaging, and the applications of medical tomography, separated into morphological and functional imaging.

The book begins with an introduction to tomography, a summary of the domain. This chapter describes the large variety of tomographic systems across the range of accessible contrasts, the choice of acquisition strategies to localize information in space and time, the different approaches to define reconstruction algorithms, and the variety of application domains.

Since the series IC2 and the corresponding English books address signal processing, we have compiled in the first part of the book the mathematical foundations, which serve the development of reconstruction algorithms. Analytical approaches, data sampling, and discrete approaches are discussed successively.

Attempting to cover the applications of tomography exhaustively in a limited number of pages is unrealistic. Therefore, we have selected a set of contributions that illustrate the domain with examples, primarily from French-speaking experts who are actively involved in research in the respective fields. For all chapters devoted to the applications of tomographic systems, we have been committed to describing the physical, physiological, and technological principles that underlie data acquisition and contrast creation. These acquisition strategies lead to the direct problem, which describes the relation between the image to be reconstructed and the performed measurements. The reconstruction algorithms, which attempt to solve the inverse problem, are only mentioned in these chapters, and references are given to the first part of the book for the mathematical derivation. However, the specific problems of each modality, such as pre- and post-processing of data for the correction of parasitic physical effects, are covered in more detail. Finally, several chapters contain sections in which one or several typical applications of the covered imaging modality are described.

The exploration of matter naturally leads to the investigation of smaller and smaller structures, the enhancement of spatial resolution, and the reduction of the scale of the images. Thus, we leaves the dimensions of the human body and looks at samples, cells, proteins, or genes. This is the domain of microtomography. Certain instruments applied in this domain are simply a miniaturization of tomographic systems employed in medical imaging, such as micro CT, MRI, SPECT, and PET scanners. In this book, however, we are more interested in the instruments that are unique to this domain. The first chapter of the second part of this book is devoted to microscopic tomography and describes in particular the confocal scanning microscope in more detail. The second chapter deals with optical imaging in diffuse media. The last chapter covers tomography with synchrotrons, which are very intense and spatially coherent X-ray sources.

The third part of this book addresses industrial applications of tomography. These must respond to the increasing demands on quality control in manufacturing, on security, and on design. Tomography thus assists design, control and maintenance engineers. In analogy to medical imaging, we have selected a chapter on X-ray tomography for the imaging of containers, which may be associated with morphological medical imaging. This chapter notably describes several examples to illustrate different uses. The second chapter covers emission tomography applied to the visualization of industrial flow, i.e. the imaging of contents, like functional medical imaging.

Medical imaging constitutes the domain in which tomographic systems are developed the furthest. Two parts of this book are devoted to the modalities that are applicable to humans, either for clinical purposes or for cognitive studies. Tomography assists physicians in diagnosis, planning and intervention.

The fourth part of this book deals with morphological medical imaging and covers successively computed tomography, X-ray volume tomography, and magnetic resonance imaging. Since ultrasound imaging is mostly applied to observation of the surface of internal organs, this modality is not addressed in this book. Another book in the series IC2 is devoted to depth imaging, to which ultrasound imaging naturally belongs.

The fifth part of this book covers functional medical imaging in its different forms, namely single photon emission computed tomography, positron emission tomography, functional cerebral tomography by magnetic resonance imaging, and tomography of electrical activity by magneto- and electro-encephalography.

By enabling us to see the invisible, to look inside matter, tomographic systems have a magical, mysterious aspect. They are routinely used tools, which open up the possibility for physicians, researchers, and engineers to answer fundamental questions on the organisms or objects that they examine. Throughout this book, we invite the reader to understand the magic of these tools and thus to discover the exciting world of tomography.

Pierre GRANGEAT

1 GRANGEAT P. (Ed.), La Tomographie: Fondements Mathématiques, Imagerie Microscopique et Imagerie Industrielle, Hermes, 2002.

2 GRANGEAT P. (Ed.), La Tomographie Médicale: Imagerie Morphologique et Imagerie Fonctionnelle, Hermes, 2002.

Notation

ℜ

set of real numbers

ℜ

n

set of real vectors of dimension

n

ℜ

+

set of non-negative real numbers

set of natural numbers, set of integers

Ω

2

unit disk

S

n−1

n

-dimensional unit sphere

Ω

subset of the unit sphere S

2

; equatorial band

θ

⊥ ={x ∈ ℜ

n

,<x,

θ

>= 0}

orthogonal subspace to

θ

for

θ ∈

S

n−1

Z

n

S

n−

1

× ℜ

unit cylinder in ℜ

n

+1

i

root of (− 1)

x

*

conjugate complex

ν

frequency

P

e

, h

sampling interval

ν

c

cut-off frequency

ln

natural logarithm

log

logarithm to base 10

e

base of the natural logarithm

Coordinates

Specific signals

Transformations

f

function describing the image to be reconstructed

Ff

Fourier transform of the function

f

F

p

Rf, F

ρ

Rf

Fourier transforms of

Rf

with respect to variables

p

and

ρ

F

−1

m

inverse Fourier transform of the function

m

Hf

Hilbert transform of the function

f

Xf

X-ray, or line integral, transform of the function

f

Yf

weighted X-ray transform of the function

f

H

c

, H

Ω

2D filter of the 3D X-ray transform, Colsher filter

FRf

2D Fourier transform of

Rf

with respect to the variables

ψ

and

p

FXf

2D Fourier transform of

Xf

with respect to

s

D

line in space

P

plane in space

PX

detection plane in cone-beam geometry

X

p

f

,

X

f

f

,

X

c

f

X-ray transform in parallel-beam, fan-beam, and conebeam geometry

X μf

attenuated line integral transform of the function

f

associated with the attenuation map

μ

Rf

Radon transform of the function

f

partial derivative

first and second derivative of the Radon transform with respect to the variable

ρ

Bm

backprojection operator applied to measurements

m

w

apodization window

HDm

,

HDWm

measurements

m

, filtered by the ramp filter, either non-apodized or apodized with the operator

W

h

,

h

HD

,

h

HDW

ramp filter, non-apodized ramp filter, ramp filter apodized with the operator

W

h

p

,

h

f

ramp filter in parallel-beam and fan-beam geometry

Sets of functions

L

1

(ℜ

n

)

space of integrable functions

L

2

(ℜ

n

)

space of square-integrable functions

L

1

p

([0,

a

]

n

), (

a

>0)

space of periodic functions integrable over [0,

a

]

n.

space of infinitely differentiable functions whose support is contained in Ω

S(ℜ

n

) or S

Schwartz space

S′(ℜ

n

) or S′

space of tempered distributions

ε

′(ℜ

n

) or ε′

space of distributions with compact support

Characteristics of a variable, a vector, or a random signal

X

random variable

E()

expectation

σ

standard deviation of a random variable

var(

x

)

variance

ρ

correlation coefficient

m

x

mean of vector

x

Σ

covariance matrix

e

noise in measurements

m

Probability, uncertainty

pdf

probability density function

Pr()

probability for discrete variables

p

(

f

),

p

(

f

)

probability distribution for a continuous variable or vector

p

(

m / f

)

conditional probability distribution

Statistics

φ, Ф, Ф

M

potential function of Markov fields

μ

a priori

distribution in the MEM approach

F

μ

criterion associated with the

μ

distribution in the MEM approach

Operations, operators

f*h

convolution of the function

f

with the kernel

h

g*

s

h(θ,s)

convolution of the function

g

with the kernel

h

with respect to the variable

s

x

Λ

y

vector product of two vectors

<

x,y

>

scalar product of two vectors

norm of a vector

|

x

|

absolute value

Sup (f)

support of a function

Matrix operations

Optimization

J

minimization criterion

grad, ∇

gradient operator

∇

x

J

gradient of

J

with respect to

x

∇

2

Laplace operator

div, <∇,.>

divergence operator

Electromagnetic and optical notation

Ionizing radiation notation

E

p

photon energy

N

p

0

,

N

p

number of emitted and transmitted photons, or pairs of photons in positron tomography

φ

p

0,φ

p

flux of emitted and transmitted photons

μ, μ

T

linear attenuation coefficient

μ

E

linear attenuation coefficient at energy E

μ

C

linear attenuation coefficient for Compton effect

μ

PE

linear attenuation coefficient for photoelectric effect

μ

R

linear attenuation coefficient for Rayleigh effect

μ

CP

linear attenuation coefficient for pair production

μ

AE

linear energy absorption coefficient

α

diffusion angle

m

0

rest mass of the electron

ρ

ν

density

ρ

Z

electron density

Z

atomic number

Z

eff

effective atomic number

g

g

geometric magnification

d

X

diameter or size of the focal spot of the X-ray source

T

r

half-life of a radioactive element

A

r

activity of a radioactive element

D

a

absorbed dose

D

e

equivalent dose

K-edge

edges in absorption spectrum due to atomic transitions to the first free energy levels

ε

c

critical energy of the electromagnetic radiation spectrum of a synchrotron

K

deflection parameter for electromagnetic structures on a synchrotron

Acronyms and abbreviations

3DRP

3D reprojection algorithm

3D-RA

3D rotational angiography

ADC

analog-to-digital converter

APD

avalanche photo diode

ART

algebraic reconstruction technique

AVM

arteriovenous malformation

BaF

2

barium fluoride

BGO

germinate of bismuth

BOLD

blood oxygenation level dependent

CAD

computer-aided design

CCD

charge-coupled device

CDET

coincidence detection emission tomography

CdTe

cadmium telluride

CG

conjugate gradient

CLSM

confocal laser scanning microscopy

CT

computed tomography

CV

coefficient of variation

CZT

cadmium zinc telluride

dexel

detector element

DNA

deoxyribonucleic acid

DQE

detection quantum efficiency

ECG

electrocardiogram

EEG

electro-encephalography

EM

expectation maximization

EPI

echo-planar imaging

ESRF

European synchrotron radiation facility

FBP

filtered backprojection

FDG

fluorodeoxyglucose

FISH

fluorescence

in situ

hybridization

FLASH

fluorescein arsenical helix binder

fMRI

functional magnetic resonance imaging

FORE

Fourier rebinning

FRELON

fast readout low noise

FWHM

full width at half maximum

FWTM

full width at tenth maximum

Gd

2

O

2

S:Tb

terbium-doped gadolinium oxysulfide

GE

gradient echo

GeHP

high purity germanium

GFP

green fluorescent protein

ICM

iterated conditional mode

IR

infrared

L-DOPA

precursor of dopamine

LOR

line of response

LORETA

low resolution electrical tomography

LSO

orthosilicate of lutetium

LUT

look-up table

MAP

maximum

a posteriori

MART

multiplicative algebraic reconstruction technique

MCMC

Markov chain Monte Carlo

MEG

magneto-encephalography

MEM

maximum entropy on the mean

MIP

maximum intensity projection

ML

maximum likelihood

MPR

multiplanar reformat

MRI

magnetic resonance imaging

MTF

modulation transfer function

NA

numeric aperture

NaI

sodium iodide

NDT

non-destructive testing

NEC

noise equivalent count rate

NEQ

noise equivalent quantum

NMR

nuclear magnetic resonance

OCT

optical coherence tomography

OS

ordered subsets

OSL

one-step-late

OTF

optical transfer function

PET

positron emission tomography

pixel

picture element

PM

photomultiplier

PMMA

polymethyl methacrylate

POCS

projection onto convex sets

PSF

point-spread function

RF

radiofrequency

RGB

red green blue

RMS

root mean square

RTD

residence time distribution

SAR

synthetic aperture radar

SE

spin echo

SI

international system of units

SIRT

simultaneous iterative reconstruction technique

SNR

signal-to-noise ratio

SPECT

single photon emission computed tomography

SPM

statistical parametric mapping

SQUID

superconducting quantum interference device

SSP

section sensitivity profile

SSRB

single-slice rebinning

STEM

scanning transmission electron microscopy

TDM

tomodensitometry

TEM

transmission electron microscopy

TFT

thin film transistor

TOF

time of flight

UV

ultraviolet

voxel

volume element

Chapter 1

Introduction to Tomography1

1.1. Introduction

Tomographic imaging systems are designed to analyze the structure and composition of objects by examining them with waves or radiation and by calculating virtual cross-sections through them. They cover all imaging techniques that permit the mapping of one or more physical parameters across one or more planes. In this book, we are mainly interested in calculated, or computer-aided, tomography, in which the final image of the spatial distribution of a parameter is calculated from measurements of the radiation that is emitted, transmitted, or reflected by the object. In combination with the electronic measurement system, the processing of the collected information thus plays a crucial role in the production of the final image. Tomography complements the range of imaging instruments dedicated to observation, such as radar, sonar, lidar, echograph, and seismograph. Currently, these instruments are mostly used to detect or localize an object, for instance an airplane by its echo on a radar screen, or to measure heights and thicknesses, for instance of the earth’s surface or of a geological layer. They mainly rely on depth imaging techniques, which are described in another book in the French version of this series [GAL 02]. By contrast, tomographic systems calculate the value of the respective physical parameter at all vertices of the grid that serves the spatial encoding of the image. An important part of imaging systems such as cameras, camcorders, or microscopes is the sensor that directly delivers the observed image. In tomography, the sensor performs indirect measurements of the image by detecting the radiation with which the object is examined. These measurements are described by the radiation transport equations, which lead to what mathematicians call the , i.e. the measurement or signal equation. To obtain the final image, appropriate algorithms are applied to solve this equation and to reconstruct the virtual cross-sections. The reconstruction thus solves the [OFT 99]. Tomography therefore yields the desired image only indirectly by calculation.