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Chapter 1
Introduction to tomography
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 computed 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 the radar,
the sonar, the lidar, the echograph, and the 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 of 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 the mathematicians call the direct problem, i.e. the measurement
Chapter written by Pierre GRANGEAT.
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Tomography
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 inverse problem [OFT 99]. Tomography evidently yields the desired
image only indirectly by calculation.
1.2. Observing contrasts
A broad range of physical phenomena may be exploited to examine objects.
Electromagnetic waves, acoustic waves, or photonic radiation are used to carry the
information to the sensor. The choice of the exploited physical phenomenon and of
the associated imaging instrument depends on the desired contrast, which must assure a good discrimination between the different structures present in the objects.
This differentiation is characterized by its specificity, i.e. by its ability to discriminate between inconsequential normal structures and abnormal structures, and by its
sensitivity, i.e. its capacity for measuring the weakest possible intensity level of the
relevant abnormal structures. In medical imaging, for example, tumors are characterized by a metabolic hyperactivity, which leads notably to an increased glucose consumption. A radioactive marker such as fluorodeoxyglucose (FDG) permits detecting
this increase in the metabolism, but it is not absolutely specific, since other phenomena, like an inflammation, also entail a local increase in glucose consumption. Therefore, the physician must interpret the physical measure in the context of the results of
other clinical examinations.
It is preferable to use coherent radiation whenever possible, which is the case for
ultrasound, microwaves, laser radiation, and optical waves. Coherent radiation permits measuring not only its attenuation but also its dephasing. The latter serves associating a depth with the measured information, because the propagation time difference results in a dephasing. Each material is characterized by its attenuation coefficient and its refractive index, which describe the speed of propagation of waves in it.
In diffraction tomography, one essentially aims at reconstructing the surfaces of the
interfaces between materials with different indices. In materials with complex structure, however, the multiple interferences rapidly render the phase information unusable. Moreover, sources of coherent radiation are often more expensive, like
lasers. With X-rays, only phase contrast phenomena that are linked to the spatial
coherence of photons are currently observable, using microfocus sources or synchrotrons. This concept of spatial coherence reflects the fact that an interference phenomenon may only be observed behind two slits if they are separated by less than the
coherence length. In such a situation, a photon interferes solely with itself. Truly coherent X-ray sources, such as the X-FEL (X-ray Free Electron Laser), are only
emerging.
Introduction to tomography
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In case of non-coherent radiation, like γ-rays, and in most cases of X-ray imaging
and optical imaging with conventional light sources, each material is mainly characterized by its attenuation and diffusion coefficients. The elastic or Rayleigh diffusion is
distinguished from the Compton diffusion, which results from inelastic photon-electron
collisions. In γ- and X-ray imaging, one tries to keep the attenuated direct radiation
only, since it propagates, in contrast to the diffused radiation, along a straight line and
thus permits obtaining a very high spatial resolution. The attenuation coefficients reflect
in first approximation the density of the traversed material. Diffusion tomography is, for
instance, considered in infrared imaging to study the blood and its degree of oxygenation. Since diffusion spreads light in all directions, the spatial resolution of such systems is limited.
After the interaction between radiation and matter, we now consider the principle of
generating the radiation. We distinguish between active and passive systems. In the
former, the source of radiation is controlled by an external generator that is activated at
the moment of the measurement. The object is explored by direct interrogation if the
generated incident radiation and the emerging measured radiation are of the same type.
This is the case in X-ray, optical, microwave, and ultrasound imaging, for example. The
obtained contrast consequently corresponds to propagation parameters of the radiation.
For these active systems with direct interrogation, we distinguish measurement systems
based on reflection or backscattering, where the emerging radiation is measured on the
side of the object the generator is placed, and measurement systems based on transmission, where the radiation is measured on the opposite side. The choice depends on
the type of radiation employed and on constraints linked to the overall dimensions of
the object. Magnetic resonance imaging (MRI) is a system with direct interrogation,
since the incident radiation and the emerging radiation are both radiofrequency waves.
However, the temporal variations of the excitation and receive signals are very different. The object may also be explored by indirect interrogation, which means that the
generated incident radiation and the emerging measured radiation are of different type.
This is the case in fluorescence imaging, where the incident wave produces fluorescent
light with a different wavelength (see Chap. 6). The observed contrast corresponds in
this case to the concentration of the emitter.
In contrast to active systems, passive systems rely on internal sources of radiation, which are inside the analyzed matter or patient. This is the case in magnetoand electro-encephalography (MEG and EEG), where the current dipoles linked to
synaptic activity are the sources. This is also the case in nuclear emission or photoluminescence tomography, where the tracer concentration in tissues or materials is
measured. The observed contrast again corresponds to the concentration of the
emitter. Factors such as attenuation and diffusion of the emitted radiation consequently lead to errors in the measurements.
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On these different systems, the contrasts may be natural or artificial. Artificial
contrasts correspond to the injection of contrast agents, such as agents based on iodine in X-ray imaging and gadolinium in MRI, and of specific tracers dedicated to
passive systems, such as tracers based on radioisotopes and fluorescence. Within the
new field of nanomedicine, new activatable markers are investigated, such as luminescent or fluorescent markers, which become active when the molecule is in a given
chemical environment, exploiting for instance the quenching effect, or when the
molecule is activated or dissociated by an external signal.
Finally, studying the kinetics of contrasts provides complementary information,
such as uptake and redistribution time constants of radioactive compounds. Moreover, multispectral studies permit characterizing flow velocity by the Doppler effect,
and dual-energy imaging allows decomposing matter into two reference materials
like water and bone, and thus enhancing the contrast.
In Tab. 1.1, principal tomographic imaging modalities are listed, including in
particular those addressed in this book. In the second column, entitled “contrast”, the
physical parameter visualized by each modality is stated. This parameter serves in
certain cases, as in functional imaging, the calculation of physiological or biological
parameters via a model, which describes the interaction of the tracer with the organism. In the third column, we propose a classification of these systems according to
their accuracy. We differentiate between very accurate systems, which are marked
by + and may reach a reproducibility of the basic measurement of 0.1 %, thus permitting quantitative imaging, accurate systems, which are marked by = and attain a
reproducibility in the order of 1 %, and less accurate systems, which are marked by –
and provide a reproducibility in the order of 10%, thus allowing qualitative imaging
only. Finally, in the fourth and fifth column, typical orders of magnitude are given
for the spatial and temporal resolution delivered by these systems. In view of the
considerable diversity of existing systems, they primarily allow a distinction between
systems with a high spatial resolution and systems with a high temporal resolution.
They also illustrate the discussion in Sec. 1.3. The spatial and temporal resolutions
are often conflicting parameters that depend on how the data acquisition and the
image reconstruction are configured. For example, increasing the time of exposure
improves the signal-to-noise ratio and thus the accuracy at the expense of the temporal resolution. Likewise, spatial smoothing alleviates the effect of noise and ameliorates the statistical reproducibility at the expense of the spatial resolution. Another
important parameter is the sensitivity of the imaging system. For instance, nuclear
imaging systems are very sensitive and are able to detect traces of radioisotopes, but
need very long acquisition times to reach a good signal-to-noise ratio. Therefore, the
accuracy is often limited. The existence of disturbing effects often corrupts the measurement in tomographic imaging and makes it difficult to attain an absolute quantification. One is then satisfied with a relative quantification, according to the accuracy
of the measurement.
Introduction to tomography
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Tomographic
imaging
modality
Contrast
Accuracy
Spatial
resolution
Temporal
resolution
CT
attenuation coefficient,
density of matter
+
axial
 0.4 mm
0.1 s – 2.0 s
per set of slices
associated with
180° rotation
transaxial
 0.6 - 5 mm
X-ray tomography in radiology
attenuation coefficient,
density of matter
=
 0.25 mm
3-6s
by volumetric
acquisition
(180° rotation)
MRI
amplitude of transverse
magnetization
=
 0.5 mm
0.1 – 10.0 s
SPECT
tracer concentration,
associated physiological or biological parameters
-
10 - 20 mm
600 - 3 600 s
per acquisition
PET
tracer concentration,
associated physiological or biological parameters
=
3 - 10 mm
300 - 600 s
per acquisition
fMRI
amplitude of transverse
magnetization, concentration of oxy-deoxyhemoglobine
=
 0.5 mm
0.1 s
MEG
current density
-
some mm
10-3 s
EEG
current density
-
10 - 30 mm
10-3 s
TEM
attenuation coefficient
-
10-6 mm
some 100 s
(function of
image size)
CLSM
concentration of
fluorophore
-
2 · 10-4 mm
some 100 s
(function of
image size)
Coherent optical
tomography
refractive index
-
10-2 mm
10-1 s
Non-coherent
optical tomography
attenuation and diffusion coefficients
-
10 mm
some 100 s
Synchrotron
tomography
attenuation coefficient,
density of matter
+
10-3 - 10-2 mm
800 - 3 600 s
per acquisition
(180° rotation)
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Tomographic
imaging
modality
Contrast
Accuracy
Spatial
resolution
Temporal
resolution
X-ray tomography in NDT
attenuation coefficient,
density of matter
+
10-2 - 102 mm
(dependent on
system)
1 - 1 000 s
per acquisition
(dependent on
system)
Emission
tomography in
industrial flow
visualization
concentration of tracer
-
some 10 % of
the diameter of
the object
(some 10 mm)
10-1 s
Impedance
tomography
electric impedance
(resistance, capacitance, inductance)
-
3 - 10 % of the
diameter of the
object
(1 - 100 mm)
10-3 - 10-2 s
Microwave
tomography
refractive index
-
 - /10
with wavelength
10-2 - 10 s

(1 - 10 mm)
Industrial
ultrasound
tomography
ultrasonic refractive
index, reflective index,
speed of propagation,
attenuation coefficient
-

with wavelength
10-2 - 10-1 s

(0.4 – 10.0 mm)
+ : very accurate systems, reproducibility of the basic measurement in the order of 0.1%, quantitative imaging.
= : accurate systems, reproducibility of the basic measurement in the order of 1%.
- : less accurate systems, reproducibility of the basic measurement in the order of 10%, qualitative
imaging.
Table 1.1. Comparison of principal tomographic imaging modalities
In most cases, the interaction between radiation and matter is accompanied by
energy deposit, which may be associated with diverse phenomena, like a local increase of the thermal agitation, a change in state, an ionization of atoms, and a
breaking of chemical bonds. Improving image quality in terms of contrast- or signalto-noise ratio unavoidably leads to an increase in the applied dose. In medical imaging, a compromise has to be found to assure that image quality is compatible with
the demands of the physicians and the dose tolerated by the patient. After stopping
the irradiation, the storage of energy may be emptied by returning to the equilibrium,
by thermal dissipation, or by biological mechanisms that try to repair or replace the
defective elements.
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1.3. Localization in space and time
Tomographic systems provide images, i.e. a set of samples of a quantity on a
spatial grid. When this grid is two-dimensional (2D) and associated with the plane of
a cross section, one speaks of 2D imaging, where each sample represents a pixel.
When the grid is three-dimensional (3D) and associated with a volume of interest,
one speaks of 3D imaging, where each element of the volume is called a voxel.
When the measurement system provides a single image at a given moment in time,
the imaging system is static. One may draw an analogy with photography, in which
an image is instantaneously captured. When the measurement system acquires an
image sequence over a period of time, the imaging system is dynamic. One may draw
an analogy with cinematography, in which image sequences of animated scenes are
recorded. In each of these cases, the basic information associated with each sample
of the image is the local value in space and time of a physical parameter characteristic of the employed radiation. The spatial grid serves as support of the mapping
of the physical parameter. The spatial sampling step is defined by the spatial resolution of the imaging system and its ability to separate two elementary objects. The
acquisition time, which is equivalent to the time of exposure in photography, determines the temporal resolution. Each tomographic imaging modality possesses its
own characteristic spatial and temporal resolution.
When exploring an object non-destructively, it is impossible to access the desired,
local information directly. Therefore, one has to use penetrating radiation to measure
the characteristic state of the matter. The major difficulty is that the emerging radiation
will provide a global, integral measure of contributions from all traversed regions. The
emerging radiation thus delivers a projection of the desired image, in which the contribution of each point of the explored object is weighted by an integration kernel, whose
support geometrically characterizes the traversed regions. To illustrate these projective
measurements, let us consider X-ray imaging as an example, where all observed structures superimpose on a planar image. On an X-ray image of the thorax seen from the
front, the anterior and posterior parts superimpose, the diaphragm is combined with the
vertebrae. On these projection images, only the objects with high contrast are observable. In addition, only their profile perpendicular to the direction of observation may be
identified. Their 3D form may not be recognized.
To obtain a tomographic image, a set of projections is acquired under varying
measurement conditions. These variations change the integration kernel, in particular
the position of its support, to completely describe a projective transformation that is
characteristic of the employed physical means of exploration. In confocal microscopy, for example, a Fredholm integral transform characterizes the impulse response of the microscope. In MRI, it is a Fourier transform, and in X-ray tomography, it is a Radon transform. In the last case, describing the Radon transform
implies acquiring projections, i.e. radiographs, with repeated measurements under
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angles that are regularly distributed around the patient or the explored object. Once
the transformation of the image is described, the image may be reconstructed from
the measurements by applying the inverse transformation, which corresponds to
solving the signal equation. By these reconstruction operations, global projection
measurements are transformed into local values of the examined quantity. This localization has the effect of increasing the contrast. It thus becomes possible to discern
organs, or defects, with low contrast. This is typically the case in X-ray imaging,
where one may basically observe the bones on the radiographs, while one may identify the soft organs on the tomographs. This increase in contrast permits being more
sensitive in the search for anomalies characterized by either hyper- or hypo-attenuation, or by hyper- or hypo-uptake of a tracer. Thanks to the localization, it also becomes possible to separate organs from their environment and thus to gain characteristic information on them, such as the form of the contour, the volume, the mass,
and the number, and to position them with respect to other structures in the image.
This localization of information is of primary interest and justifies the success of
tomographic techniques. A presentation of the principles of computerized tomographic imaging can be found in [KAK 01, HSI 03, KAL 06] for X-rays and [BEN
98, BAI 05] for positron emission tomography (PET).
The inherent cost of tomographic procedures is linked to the necessity of completely describing the space of projection measurements to reconstruct the image.
Tomographic systems are thus “scanners”, i.e. systems that measure, or scan, each
point in the transformed domain. The number of basic measurements is of the same
order of magnitude as the number of vertices of the grid on which the image is calculated, typically between 642 and 10242 in 2D and between 643 and 10243 in 3D. The
technique used for data acquisition thus greatly influences the temporal resolution
and the cost of the tomographic system. The higher the number of detectors and generators placed around the object gets, the better the temporal resolution will be, but
the higher the price will rise. A major evolution in the detection of nuclear radiation
is the replacement of one-dimensional (1D) by 2D, either multi-line or surface, detectors. In this way, the measurements may be performed in parallel. In PET, one
observes a linear increase in the number of detectors available on high-end systems
with the year of the product launch. However, scanning technology must also be
taken into account for changing the conditions of the acquisition.
The most rapid techniques are those of electronic scanning. This is the case in
electromagnetic measurements and in MRI [HAA 99]. In the latter, only a single detector, a radiofrequency antenna, is used and the information is gathered by modifying the ambient magnetic fields, in such a way that the whole measurement domain is
covered. The obtained information is intrinsically 3D. The temporal resolution
achieved with these tomographic systems is very good, permitting dynamic imaging
at video rates, i.e. at about 10 images/s. When electronic scanning is impossible,
mechanical scanning is employed instead. This is the case for rotating gantries in X-
Introduction to tomography
9
ray tomography, which rotate the source-detector combination around the patient. To
improve the temporal resolution, the rotation time must be decreased - the most
powerful systems currently complete a rotation in the order of 0.3 s - and the rotation
must be combined with an axial translation of the bed to efficiently cover the whole
volume of interest. Today, it is possible to acquire the whole thoracic area in less
than 10 s, i.e. in a time that patients can hold their breath.
The principle of acquiring data by scanning, adopted by tomographic systems,
poses problems of motion compensation when objects that evolve over time are observed. This is the case in medical imaging when the patient moves or when organs
which are animated by continuous physiological motion like the heart or the lungs
are studied. This is also the case in non-destructive testing on assembly or luggage
inspection lines, or in process monitoring, where chemical or physical reaction are
followed. The problem must be considered as a reconstruction problem in space and
time, and appropriate reconstruction techniques for compensating the motion or the
temporal variation of the measured quantities must be introduced.
The evolution in tomographic techniques manifests itself mainly in the localization
in space and time. Growing acquisition rates permit increasing the dimensions of the
explored regions. Thus, the transition from 2D to 3D imaging has been accomplished.
In medical imaging, certain applications like X-ray and MR angiography and oncological imaging by PET require a reconstruction of the whole body. Accelerating the
acquisition also makes possible dynamic imaging, for instance to study the kinetics of
organs and metabolism or to guide interventions based on images. An interactive tomographic imaging is thus approached. Finally, the general trend towards miniaturization
leads to growing interest in microtomography, where an improved spatial resolution of
the systems is the primary goal. In this way, one may test micro-systems in non-destructive testing (NDT), study animal models like mice in biotechnology, or analyze
biopsies in medicine. Nanotomography of molecules such as proteins, molecule assemblies, or atomic layers within integrated microelectronic devices is also investigated
using either electron microscopy or synchrotron X-ray sources.
1.4. Image reconstruction
Tomographic systems combine the examination of matter by radiation with the calculation of characteristic parameters, which are either linked to the generation of this
radiation or the interaction between this radiation and matter. Generally, each measurement of the emerging radiation provides an analysis of the examined matter, like a
sensor. Tomographic systems thus acquire a set of partial measurements by scanning.
The image reconstruction combines these measurements to assign a local value, which
is characteristic of the employed radiation, to each vertex of the grid that supports the
spatial encoding of the image. For each measurement, the emerging radiation is the sum
of the contributions of the quantity to be reconstructed along the traversed path, each
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Tomography
elementary contribution being either an elementary source of radiation or an elementary
interaction between radiation and matter. It is thus an integral measurement which
defines the signal equation. The set of measurements constitutes the direct problem,
linking the unknown quantities to the measurements. The image reconstruction amounts
to solving this system of equations, i.e. the inverse problem [OFT 99].
The direct problem is deduced from the fundamental equations of the underlying
physics, such as the Boltzmann equation for the propagation of photons in matter, the
Maxwell equation for the propagation of electromagnetic waves [DEH 95], and the
elastodynamic equation for the propagation of acoustic waves [DEH 95]. The measurements provide integral equations, often entailing numerically instable differentiations in
the image reconstruction. In this case, the reconstruction corresponds to an inverse
problem that is ill-posed in the sense of Hadamard [HAD 32].
The acquisition is classified as complete when it provides all measurements that are
necessary to calculate the object, which is in particular the case when the measurements
cover the whole support of the transformation modeling the system. The inverse problem is considered weakly ill-posed, if the principal causes of errors are instabilities
associated with noise or the bias introduced by an approximate direct model. This is the
case in X-ray tomography, for instance. If additionally different solutions may satisfy
the signal equation, even in absence of noise, the inverse problem is considered strongly ill-posed. This is the case in magneto-encephalography and in X-ray tomography
using either a small angular range or a small number of positions of the X-ray source,
for example.
The acquisition is classified as incomplete when some of the necessary measurements are missing, in particular when technological constraints reduce the number of
measurements that can be made. This situation may be linked to a subsampling of
measurements or an absence of data caused by missing or truncated acquisitions. In this
case, the inverse problem is also strongly ill-posed, and the reconstruction algorithm
must choose between several possible solutions.
To resolve these ill-posed inverse problems in the sense of Hadamard, the reconstruction must be regularized by imposing constraints on the solution. The principal
techniques of regularization reduce the number or the range of the unknown parameters – for instance by using a coarser spatial grid to encode the image, by parameterizing the unknown image, or by imposing models on the dynamics –, introduce
regularity constraints on the image to be reconstructed, or eliminate from the inverse
operator the smallest spectral or singular values of the direct operator. All regularization techniques require that a compromise is made between the fidelity to the
given data and the fidelity to the constraints on the solution. This compromise reflects
the uncertainty relations between the localization and the quantification in images, applied to tomographic systems.
Introduction to tomography
11
Several approaches may be pursued to define algorithms for image reconstruction,
for which we refer the reader to [HER 79, NAT 86, NAT 01]. Like for numerical
methods in signal processing, deterministic and statistical methods, and continuous and
discrete methods are distinguished. For a review on recent advances in image reconstruction, we refer the reader to [CEN 08].
The methods most widely used are continuous approaches, also called analytical
methods. The direct problem is described by an operator over a set of functions. In the
case of systems of linear measurements, the unknown image is linked to the measurements by a Fredholm transform of the first kind. In numerous cases, this equation simplifies either to a convolution equation, a Fourier transform, or a Radon transform. In
these cases, the inverse problem admits an explicit solution, either in form of an inversion formula, or in form of a concatenation of explicit transformations. A direct method
of calculating the unknown image is obtained in this way. These methods are thus more
rapid than the iterative methods employed for discrete approaches. In addition, these
methods are of general use, because the applied regularization techniques, which perform a smoothing, do not rely on specific a priori knowledge about the solution. The
discretization is only introduced for the numerical implementation.
However, when the direct operator becomes more complex to better describe the
interaction between radiation and matter, or when the incorporation of a priori information is necessary to choose a solution in case of incomplete data, an explicit formula
for calculating the solution does not exist anymore. In these specific cases, discrete approaches are preferable. They describe the measurements and the unknown image as
two finite, discrete sets of sampled values, represented by vectors. When the measurement system is linear or can be linearized by a simple transformation of the data, the
relation between the image and the measurements may be expressed by a matrix-vector
product. The problem of image reconstruction then leads to the problem of solving a
large system of linear equations. In the case of slightly underdetermined, complete
acquisitions, regularization is introduced by imposing smoothness constraints. The
solution is formulated as the argument of a composite criterion that combines the
fidelity to the given data and the regularity of the function. In the case of highly underdetermined, incomplete acquisitions, regularization is introduced by choosing an optimal solution among a set of possible solutions, which are, for example, associated with
a discrete subspace that depends on a finite number of parameters. The solution is thus
defined as the argument of a linear system under constraints. In both cases, the solution
is calculated with iterative algorithms.
When the statistic fluctuation of the measurements becomes important, as in
emission tomography, it is preferable to explicitly describe the images and measurements statistically, following the principle of statistical methods. The solution is expressed as a statistical estimation problem. In the case of weakly ill-posed problems,
Bayesian estimators corresponding to the minimization of a composite criterion have
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Tomography
been proposed. They choose the solution with the highest probability for the processed measurement, or on average for all measurements, taking the measurements and
the a priori information about the object into account. In the case of strongly illposed problems, constrained estimators, for example linked to Kullback distances,
have been envisioned. Also in these cases, one resorts to iterative reconstruction algorithms.
When the ill-posed character becomes too severe, for example in case of very
limited numbers of measurement angles, constraints at low levels of the image samples are no longer sufficient. One must then employ a geometric description of the
object to be reconstructed, in form of a set of elementary objects, specified by a
graphical model or a set of primitives. The image reconstruction amounts to matching the calculated and the measured projections. In this work, these approaches,
which do not directly concern the problem of mapping a characteristic physical parameter, but rather problems in computer vision and pattern recognition, are not considered any further. We refer the reader to [GAR 06].
The image reconstruction algorithms require considerable processing performance. The elementary operations are the backprojection, or the backpropagation, in
which a basic measurement is propagated through a volume by following the path
traversed in the acquisition in reverse direction. For this reason, the use of more and
more sophisticated algorithms, and the reconstruction of larger and larger images,
with higher spatial and temporal resolution, are only made possible by advances in
the processing performance and the memory size of the employed computers. Often,
the use of dedicated processors and of parallelization and vectorization techniques is
necessary. In view of these demands, dynamic tomographic imaging in real-time
constitutes a major technological challenge today.
1.5. Application domains
By using radiation to explore the matter and calculations to compute the map of the
physical parameter characteristic of the employed radiation, tomographic imaging
systems provide virtual cross sections through objects without resorting to destructive
means. These virtual cross sections may easily be interpreted and enable the identification and localization of internal structures. Computed tomography techniques directly
provide a digital image, thus offering the possibility of applying the richness of image
processing algorithms. The examination of these images notably permits a characterization of each image element for differentiation, a detection of defects for control and
testing, a quantification for measurement, and a modeling for description and understanding. The imaging becomes 3D when the tomographic system provides a set of
parallel cross sections that cover the region of interest. Thus, cross sections in arbitrary
planes may be calculated, and surfaces, shapes, and the architecture of internal struc-
Introduction to tomography
13
tures may be represented. In applications associated with patients or materials with
solid structures in which fluids circulate, a distinction is made between imaging systems
that primarily map the solid architecture, i.e. the structures – one then speaks of morphological imaging – and imaging systems that mainly follow the exchange of fluids,
i.e. the contents – one then speaks of functional imaging.
In optical imaging, the development of the first, simple microscopes, which were
equipped with a single lens, dates back to the 17th century. From a physical point of
view, the use of X-rays for the examination of matter started with their discovery by
Röntgen in 1895. In the course of experiments conducted to ascertain the origin of this
radiation, Becquerel found the radiation that uranium spontaneously emits in 1896
[BIM 99]. After the discovery of polonium and radium in 1898, Pierre and Marie Curie
characterized the phenomenon that produces this radiation and called it “radioactivity”.
For these discoveries, Henri Becquerel, Marie Curie and Pierre Curie received the
Nobel price for physics in 1903. The use of radioactive tracers was initiated in 1913 by
Hevesy, Nobel price for chemistry in 1943. The discovery of artificial radioactivity by
Irène and Frédéric Joliot-Curie in 1934 opened the way for the use of numerous tracers
to study the functioning of the living, from cells to entire organisms. Important research
has then been carried out on the production and the detection of these radiations. The
first microscope using electrons to study the structure of an object was developed by
Ernst Ruska in 1929. From a mathematical point of view, the Fourier transform
constitutes an important tool in the study of the analytical methods used in tomography.
Several scholars like Maxwell, Helmholtz, and Born have studied the equation of wave
propagation. The name of Fredholm remains linked to the theory of integral equations.
The problem of reconstructing an image from line integrals goes back to the original
publication by Radon in 1917 [RAD 17], reissued in 1986 [RAD 86], and the studies of
John published in 1934 [JOH 34].
During the second half of the 20th century, a rapid expansion of tomographic techniques took place. In X-ray imaging, the first tomographic systems were mechanic
devices which combined movements of the source and of the radiological film to eliminate information off the plane of the cross section by a blurring effect. This is called
longitudinal tomography, since the plane of the cross section is parallel to the patient
axis. The first digitally reconstructed images were obtained by radio astronomers at the
end of the 1950s [BRA 79]. The first studies of Cormack at the end of the 1950s and
the beginning of the 1960s showed that it is possible to precisely reconstruct the image
of a transverse cross section of a patient from its radiographic projections [COR 63,
COR 64]. In the 1960s, several experiments with medical applications were carried out
[COR 63, COR 64, KUH 63, OLD 61]. The problem of reconstructing images from
projections in electron microscopy was also tackled in this time [DER 68, DER 71]. In
1971, Hounsfield built the first prototype of an X-ray tomograph dedicated to cerebral
imaging [HOU 73]. Hounsfield and Cormack received the Nobel price for physiology
and medicine in 1979 [COR 80, HOU 80]. The beginnings of MRI go back to the first
14
Tomography
publication by Lauterbur in 1973 [LAU 73]. In nuclear medicine, the pioneering work
in single photon emission computed tomography (SPECT) date back to Kuhl and Edwards in 1963 [KUH 63], followed by Anger in 1967 [ANG 67], Muehllehner in 1971
[MUE 71], and Keyes et al. [BOW 73, KEY 73]. The first PET scanner was built at the
beginning of the 1960s by Rankowitz et al. [RAN 62], followed in 1975 by TerPogossian and Phelps [PHE 75, TER 75]. For more references covering the beginning
of tomography, we refer the interested reader to the works of Herman [HER 79], Robb
[ROB 85], and Webb [WEB 90], and to the following papers summarizing the developments achieved during the past 50 years in X-ray computed tomography [KAL 06],
MRI [MAL 06], SPECT [JAS 06], PET [MUE 06], and image reconstruction [DEF
06].
In France, we keep of this time the pioneering work of the group of Robert Di
Paola at the Institute Gustave Roussy (IGR) on gamma tomography and of the
groups of the Laboratory of Electronics, Technology, and Instrumentation (LETI) at
the Atomic Energy Commission (CEA). In 1976, under the direction of Edmond
Tournier and Robert Allemand, the first French prototype of an X-ray tomograph
was installed at the Grenoble hospital, in collaboration with the company CGR [PLA
98]. In 1981, the LETI installed at the Service Hospitalier Frédéric Joliot (SHFJ) of
the CEA its first prototype of a time-of-flight PET scanner (TTV01), followed by
three systems of the next generation (TTV03). The SHFJ participated in the specification of these systems and carried out their experimental evaluation. The SHFJ was
the first center in France having a cyclotron for medical use and a radiochemistry
laboratory for synthesizing the tracers used in PET. In 1990, two prototypes of 3D
X-ray scanners called morphometers were developed by the LETI, one of which was
installed at the hospital of Rennes and the other at the neurocardiological hospital of
Lyon, in collaboration with the company General Electric-MSE and the groups of
these hospitals. In parallel, several prototypes of X-ray scanners were built for applications in non-destructive testing.
The 1980s saw the development of 3D imaging and digital detection technology, as
well as tremendous advances in the processing performance of computers. One pioneer
project was the DSR project (Dynamic Spatial Reconstructor at the Mayo Clinic in
Rochester [ROB 85], which aimed at imaging the beating heart and the breathing lung
using several pairs of X-ray sources and detectors. The evolution continued in the
1990s, focusing notably on increasing the sensitivity and the acquisition rates of the
systems, the latter in particular linked with the development of 2D multi-line or surface
radiation detectors. Several references describe the advances achieved in this period
[BEN 98, GOL 95, KAL 00, LIA 00, ROB 85, SHA 89].
Medical imaging is an important application domain for tomographic techniques.
Currently, X-, -, and -ray tomography, and MRI together make up a third of the
medical imaging market. Tomographic systems are heavy and expensive devices that
Introduction to tomography
15
often require specific infrastructure, like protection against ionizing radiation or
magnetic fields or facilities for preparing radiotracers. Nevertheless, tomography
plays a crucial role in diagnostic imaging today, and it has a major impact on public
health.
Medical imaging and the life sciences thus constitute the primary application domains of tomographic systems. In the life sciences, the scanning transmission electron
microscope permits studying fine structures, like those of proteins or viruses, and, at a
larger scale, the confocal laser scanning microscope (CLSM) allows visualizing the
elements of cells. In neuroscience, PET, fMRI (functional MRI), MEG, and EEG have
contributed to the identification and the localization of cortical areas and thus to the
understanding of the functioning of the brain. By permitting a tracking of physiological
parameters, PET contributes to the study of primary and secondary effects of drugs.
Finally, the technique of gene reporters opens up the possibility of genetic studies in
vivo. For these studies on drugs or in biotechnology, animal models are necessary,
which explains the current development of platforms for tomographic imaging on
animals. For small animal studies, optical tomography associated either with luminescence or laser-induced fluorescence is also progressing. In particular, it allows linking
in vitro studies on cell cultures and in vivo studies on small animal using the same
markers. Generally speaking, through MRI, nuclear and optical imaging, molecular
imaging is an active field of research nowadays.
In medicine, X-ray tomography and MRI are anatomic, morphological imaging
modalities, while SPECT, PET, fMRI, and magneto- and electroencephalography
(MEG and EEG) are employed in functional studies. X-ray tomography is well suited
for examining dense structures, like the bones or the vasculature after injection of a
contrast agent. MRI is better suited for visualizing contrast based on the density of
hydrogen atoms, as between the white and gray matter in the brain [HAA 99]. SPECT
is used a lot to study the blood perfusion of the myocardium, and PET is about to establish itself for staging of cancer in oncology. For medical application, optical tomography [MÜL 93] remains for the moment restricted to a few dedicated applications,
such as the imaging of the cornea or the skin by optical coherence tomography. Other
applications like the exploration of the breast or prostate using laser-induced fluorescence tomography are still under investigation. These aspects are elaborated in the following chapters that are dedicated to the individual modalities.
Tomographic system are used in all stages of patient care, including the diagnosis,
the therapy or surgical planning, the image-based guidance of interventions for minimal invasive surgery, and the therapy follow-up. Digital assistants may help with the
image analysis. Moreover, robots use these images for guidance. These digital images
are easily transmitted over imaging networks and stored to establish a reference database, to be shared among operators or students. In the future, veritable virtual patients
16
Tomography
and digital anatomic atlases will serve the training of certain medical operations, or
even surgical procedures. Such digital models also facilitate the design of prostheses.
The second application domain of tomographic imaging systems are astronomy
and the material sciences. They often involve large instruments, such as telescopes,
synchrotrons, and plasma chambers, and nuclear simulation devices, such as the installations Mégajoule and Airix in France. Geophysics is another important domain in
which, on a large scale, oceans or geological layers are examined and, on a smaller
scale, rocks and their properties, such as their porosity. Astrophysics, associated with
telescopes operating at different ranges of wavelengths, is also concerned, which
exploits the natural rotation of the stars for tomographic acquisitions [BRA 79, FRA
05].
The last application domain is the industry and services. Several industries are
interested, like the nuclear, aeronautic, automobile, agriculture, food, pharmaceutical,
petroleum, and chemical industry [BAR 00, BEC 97, MAY 94]. Here, a broad range of
materials in different phases is concerned. Tomography is first of all applied in design
and development departments, for the development of fabrication processes, the characterization and modeling of the behavior of materials, reverse engineering, and the
adjustment of graphical models in computer aided design (CAD). Second, tomography
is also applied in process control [BEC 97], for instance in mixers, thermal exchangers,
multi-phase transport, fluidified bed columns, motors, and reactors. It concerns in
particular the identification of functioning regimes, the determination of resting times
and of the kinetics of products, and the validation of thermo-hydraulic models. However, in non-destructive testing, once the types of defects or malfunction are identified
on a prototype, one often tries to reduce the number of measurements and thus the cost,
and to simplify the systems. Therefore, the use of tomographic systems remains limited
in this application domain. Lastly, tomography is applied to security and quality
control. This includes for instance production control, such as on-line testing after assembly to check dimensions or to study defects, or the inspection of nuclear waste
drums. Security control is also becoming a major market. This concerns in particular
luggage inspection with identification of materials to detect drugs or explosives.
In conclusion, from the mid of the 20th century up to now, tomography has become
a major technology with a large range of applications.
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Introduction to tomography
17
[BAR 00] BARUCHEL J, BUFFIÈRE JY, MAIRE E, MERLE P, PEIX G (Eds.), X-Ray Tomography
in Material Science, Hermès, 2000.
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CORFIELD JR, MALLARD JR, “A radioisotope scanner for rectilinear arc, transverse section,
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[FRA 05] FRAZIN RA, KAMALABADI F, “Rotational tomography for 3D reconstruction of the
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[GAL 02] GALLICE J (Ed.), Images de Profondeur, Hermès, 2002.
[GAR 06] GARDNER RJ, Geometric Tomography, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2006.
18
Tomography
[GOL 95] GOLDMAN LW, FOWLKES JB (Eds.), Medical CT and Ultrasound: Current Technology and Applications, Advanced Medical Publishing, 1995.
[HAA 99] HAACKE E, BROWN RW, THOMPSON MR, VENKATESAN R, Magnetic Resonance
Imaging. Physical Principles and Sequence Design, John Wiley and Sons, 1999.
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Linéaires Hyperboliques, Hermann, 1932.
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[HSI 03] HSIEH J, Computed Tomography: Principles, Design, Artifacts, and Recent Advances, SPIE Press, 2003.
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Introduction to tomography
19
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vol. 51, n° 13, pp. R117-R137, 2006.
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20
Tomography
accuracy, 4
acoustic waves, 2
acquisition time, 7
active system, 3
analytical methods, 11
anatomic imaging, 15
Anger, 14
astronomy, 16
attenuation coefficient, 2, 3
backprojection, 12
backpropagation, 12
backscattering, 3
Boltzmann equation, 10
CEA, 14
coherent radiation, 2
complete, 10
computed tomography, 12
computer vision, 12
concentration, 3
continuous method, 11
contrast, 3, 4, 8
contrast agent, 4
Cormack, 13
cost, 8
density, 3
design, 16
deterministic method, 11
diagnosis, 15
diffusion coefficient, 3
direct interrogation, 3
direct problem, 1, 10
discrete method, 11
dual-energy, 4
dynamic, 7
dynamic imaging, 9, 12
Dynamic Spatial Reconstrutor, 14
Edwards, 14
elastodynamic equation, 10
electromagnetic waves, 2
electronic scanning, 8
estimation, 11
fluorescence, 3
Fourier transform, 7, 11
Fredholm transform, 7, 11
functional imaging, 13
gamma ray, 3
Hounsfield, 13
ill-posed, 10
image-based guidance, 15
incomplete, 10
indirect interrogation, 3
indirect measurements, 1
industry, 16
integral measurement, 10
inverse problem, 2, 10
iterative algorithm, 11, 12
kinetics, 4, 9
Kuhl, 14
Lauterbur, 14
LETI, 14
life science, 15
localization, 8
magnetic resonance imaging, 3
material science, 16
matrix-vector product, 11
Maxwell equation, 10
mechanical scanning, 8
medical imaging, 14, 15
microtomography, 9
minimal invasive surgery, 15
molecular imaging, 15
morphological imaging, 13, 15
morphometer, 14
motion compensation, 9
non-coherent radiation, 3
passive system, 3
pattern recognition, 12
Phelps, 14
photonic radiation, 2
pixel, 7
process control, 16
projection, 7
Introduction to tomography
public health, 15
quality control, 16
Radon, 13
Radon transform, 7, 11
Rankowitz, 14
reconstruction, 8, 9, 12
reflection, 3
refractive index, 2
regularization, 10, 11
security, 16
sensitivity, 2
service, 16
SHFJ, 14
spatial grid, 7
spatial resolution, 4, 7
specificity, 2
static, 7
21
statistical method, 11
strongly ill-posed, 10
surgical planning, 15
temporal resolution, 4, 8
Ter-Pogossian, 14
therapy, 15
therapy follow-up, 15
three-dimensional imaging, 7, 9, 14
tomographic imaging systems, 1
tomography, 1, 15
transmission, 3
two-dimensional imaging, 7, 9
uncertainty relation, 10
voxel, 7
weakly ill-posed, 10
whole body, 9
X-ray, 3, 7