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:

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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
A CIP record for this book is available from the British Library
ISBN: 978-1-84821-099-8

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
ℜn | set of real vectors of dimension n
ℜ⁺ | set of non-negative real numbers
| set of natural numbers, set of integers
Ω₂ | unit disk
Sn−1 | n-dimensional unit sphere
Ω | subset of the unit sphere S²; equatorial band
θ ⊥ ={x ∈ ℜn,<x, θ>= 0} | orthogonal subspace to θ for θ ∈ Sn−1
Zn= Sn−¹× ℜ | unit cylinder in ℜn+1
i | root of (− 1)
x* | conjugate complex
ν | frequency
Pe, h | sampling interval
νc | cut-off frequency
ln | natural logarithm
log | logarithm to base 10
e | base of the natural logarithm

Coordinates

S | point on X-ray source
M | point in image space
A | point on detector
r = (x, y, z) | Cartesian coordinates in image space
ψ,φ | azimuthal angles of projections in fan-beam, cone-beam, and parallel-beam geometry
θ | colatitude angle of oblique projections
p, q | Cartesian coordinates on detector
D (S,A) | line through points S and A
D (φ, p) | line defined by normalized polar coordinates (φ,p)
D(n, s), D(a, S) | line defined by the unit vector n and the point to which the vector s points from the origin; line defined by the unit vector a and the point S

Specific signals

χA(x) | indicator function of the set A: χA(x) = 1 if x ∈ A, = 0 otherwise
δ(x) | Dirac distribution
sign( ) | sign function
images | sine cardinal function
Jn | Bessel function of order n

Transformations

f | function describing the image to be reconstructed
Ff | Fourier transform of the function f
FpRf, FρRf | 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
Hc, HΩ | 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 pf, X f f, Xc 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
images | partial derivative
images | 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, hHD, hHDW | ramp filter, non-apodized ramp filter, ramp filter apodized with the operator W
hp, hf | ramp filter in parallel-beam and fan-beam geometry

Sets of functions

L¹(ℜn) | space of integrable functions
L²(ℜn) | space of square-integrable functions
L¹p([0, a]n), (a>0) | space of periodic functions integrable over [0, a]n.
images | space of infinitely differentiable functions whose support is contained in Ω
S(ℜn) or S | Schwartz space
S′(ℜn) or S′ | space of tempered distributions
ε′(ℜn) 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
mx | 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, ФM | potential function of Markov fields
μ | a priori distribution in the MEM approach
Fμ | 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
xΛy | vector product of two vectors
<x,y> | scalar product of two vectors
images | norm of a vector
|x| | absolute value
Sup (f) | support of a function

Matrix operations

R | matrix
rij = [R]ij | element of a matrix
ri. ,r.j | i-th row and j-th column of a matrix
f | column vector representing an image or measurements
fk | element of a column vector of samples (pixel, dexel, …)
dim(f) | dimension of a vector
f(n) | n-th iteration during the calculation of a vector
B⁻¹ | inverse of a matrix B
det B | determinant of a matrix B
B⁺ | pseudo-inverse of a matrix B
xT | transpose of a vector x
I | identity matrix

Optimization

J | minimization criterion
grad, ∇ | gradient operator
∇xJ | gradient of J with respect to x
∇² | Laplace operator
div, <∇,.> | divergence operator

Electromagnetic and optical notation

ω | pulsation
k (x,y,z) | wave vector
(kx, ky, kz) | spatial frequencies of the wave vector in the reciprocal space or k-space
r(x,y,z), R(x,y,z) | spatial vectors pointing to coordinate (x, y, z)
ei(<k,r>-ωt) | plane wave
E, E | electrical field vector, electrical field scalar
V | electrical potential
Q = (Qx, Qy, Qz) | equivalent current dipole
H | gain matrix
Γ(r₁−r₂, t₁−t₂) | cross-correlation function of fields E(r₁), E(r₂)
Gr (r₁−r₂,t₁−t₂) | Green function
Wi(r, t) | Wigner distribution
B, B | magnetic field vector, magnetic field scalar
μ = μr+i μi | permeability
μ₀ | vacuum permeability
ε = εr+iεi | dielectric constant
ε₀ | vacuum dielectric constant
j, j | current density vector, current density scalar
[σ] | conductivity tensor
M | vector of magnetic or electric data
c | speed of light
ħ | Planck’s constant
λ | wavelength
NA | numerical aperture of an optical system
n | refractive index
μa | absorption coefficient
μs | diffusion coefficient
μ′s | reduced diffusion coefficient
U (r,t) | photon density wave scalar
D | diffusion coefficient of photon density waves
Mz | longitudinal magnetization
M⊥ | transverse magnetization
ρ(x, y, z) | nuclear magnetization density at location (x, y, z)
Φ | phase of NMR signal
T₁ | longitudinal relaxation time
T₂ | transverse relaxation time
T₂* | effective transverse relaxation time
Dl | free diffusion coefficient of water
Θb | flip angle
images | mean free path
Da | apparent diffusion coefficient
TE | echo time
TR | repetition time
TI | inversion time
G = (Gx,Gy,Gz) | gradient field with x, y, and z components
(x, y, z) | laboratory frame of reference
(x′, y′, z′) | rotating frame of reference
γ | gyromagnetic ratio
B₀ | static magnetic field
ω₀ | Larmor pulsation associated with the field B₀
B₁ | rotating magnetic field
ω₁ | Larmor pulsation associated with the field B₁
dν | diameter of vessels
χm | magnetic susceptibility
M | magnetization
kB | Boltzmann constant
Θa | absolute temperature

Ionizing radiation notation

Ep | photon energy
Np₀, Np | number of emitted and transmitted photons, or pairs of photons in positron tomography
φ p0,φp | flux of emitted and transmitted photons
μ, μT | linear attenuation coefficient
μE | linear attenuation coefficient at energy E
μC | linear attenuation coefficient for Compton effect
μPE | linear attenuation coefficient for photoelectric effect
μR | linear attenuation coefficient for Rayleigh effect
μCP | linear attenuation coefficient for pair production
μAE | linear energy absorption coefficient
α | diffusion angle
m₀ | rest mass of the electron
ρν | density
ρZ | electron density
Z | atomic number
Zeff | effective atomic number
gg | geometric magnification
dX | diameter or size of the focal spot of the X-ray source
Tr | half-life of a radioactive element
Ar | activity of a radioactive element
Da | absorbed dose
De | equivalent dose
K-edge | edges in absorption spectrum due to atomic transitions to the first free energy levels
εc | 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
BaF₂ | 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
Gd₂O₂S: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 Tomography ¹

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.

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