The integral equations of Compton Scatter Tomography
M. K. Nguyen
Equipes Traitement de l’Information et Systèmes (ETIS)
UMR CNRS 8051/ ENSEA / University of Cergy-Pontoise
F-95014 Cergy-Pontoise, France
T. T. Truong
Laboratoire de Physique Théorique et Modélisation (LPTM)
UMR CNRS 8089 / University of Cergy-Pontoise
F-95302 Cergy-Pontoise
Abstract — Two new Compton Scatter Tomography modalities, which are aimed at imaging hidden structures in bulk
matter for industrial non-destructive control (or testing) and for medical diagnostics are shown to be based on the solutions
of a special class of Chebyshev integral transforms. Besides their remarkable analytic properties, they can be inverted by
existing methods which lend themselves nicely to numerical treatment and provide convergent, stable and fast computation
algorithms. The existence of explicit inversion formulas implies that viable new imaging techniques can be developed, which
may take over the current ones in a near future.
Keywords-Integral transforms; Radon transforms; Compton scatter imaging; Non-destructive testing; Medical imaging.
1. Introduction
Integral equations arise naturally in a wide range of
fields as the proper way to represent a relevant phenomena
to be exploited for applications. Examples are numerous in
wave propagation, transport theory, optics,
electromagnetism, acoustics, etc. Quite often, the
conversion of differential processes into integral processes
turns out to be beneficial in the sense that a solution can be
worked out by integral operator techniques. This is in
particular the case of aerodynamics, in which, for the first
time an integral equation of the first kind appears with a
Chebyshev integral kernel. Ta Li [1], who discovered this
class of integral equations, has also realized that the
equation he had discovered belongs to a much larger family
with a hypergeometric function kernel, including numerous
special cases involving Legendre, Jacobi, Gegenbauer
polynomial kernels. What should be pointed out is the fact
that their inversion, essential for applications, relies on a
very peculiar property of the Gauss hypergeometric
function, which was most elegantly derived by R.
Buschman in [2].
In this paper, we show how the process of measuring
energy flux densities of scattered gamma rays by bulk
matter leads to the reconstruction its structure and
composition without having to dismember it or take it apart.
The image is called tomographic when such operation is
done in a two-dimensional slice, perpendicular to some
main axis of the object. A three-dimensional image can be
then obtained by juxtaposition of a large number of
tomographic images. To this end, in the next section, we
recall the basic principle of Compton Scatter Tomography
and introduce two recently suggested operating modalities.
Section 3 handles the resulting integral transforms by
describing the derivation of their inversion. The paper ends
with a short conclusion and some perspectives for
2. Compton Scatter Tomography
One of the most efficient way of probing the inner
structure of an objects the use of penetrating radiation (X-
or gamma-rays). Originally one exploits the phenomena of
radiation attenuation inside matter from emission to
detection along a straight line path. This has given rise to
the widely known X-ray scanner, which is nowadays
commonly used in hospitals and in industrial protocols.
However as radiation propagates inside matter, scattering
with distributed electric charges - or Compton effect - is the
main cause for its attenuation. So a smart idea would be to
collect information carried by scattered radiation to try to
image the internal structure of matter. Of course there are
many ways to put this idea to work. One of them is called
Compton Scatter Tomography (or CST).
In CST the object is illuminated by a monochromatic
source of penetrating radiation S, placed at some spatial
position. An energy sensitive detector D will measure the
radiation flux density for a given scattered energy at all
accessible sites outside the object. If the totality of such
measurements allows to reconstruct the electric charge
density inside the object, a corresponding image of this
object is therefore obtained. This is the principle of CST,
when this is done in a plane containing the line SD. As the
object electric charge density is a smooth real-valued
function f(x,y) with compact support in R2, the set of
measurements is required to have two independent variables.
One natural variable is the radiation scattered energy, which
is directly related to the scattering angle Ȧ by the well-
known Compton relation. So when D registers scattered
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radiation at fixed energy and position in space, this means
that it collects the scattered radiation flux density from all
the scattering sites lying on a circular arc subtending an
angle ʌ-Ȧ provided that selected collimators are set up so
as to force the radiation rays to be inside a fixed plane.
Hence the measurement at Dis essentially an integral of
f(x,y) along this circular arc starting from Sand ending at D.
The second natural variable to be taken is simply the
angular position of SD with respect to a fixed reference
direction. It may be labeled by an angle ij, i.e. The data
registered at Dis thus a function JȦij Omitting all
unnecessary quantities, the mathematical problem at hand
may be formulated as
So for all allowable values of Ȧij, how can one
reconstruct the electric charge density f(x,y) of an object
from the data JȦij? This is precisely the problem solved
by J. Radon in 1917 for straight lines in a plane [3]. But
here the same problem is posed for circular arcs C going
from S to D. (1) has the structure of an integral equation of
the first kind with a delta-function kernel į& concentrated
on C. Such a problem is usually ill-posed and occurs
recurrently in imaging theory.
p be a circle of radius p, centered at the coordinate
system origin O. We suggest two ways of varying the
relative positions of S and of D 
p, such that the
inversion problem of this type of Radon transform can be
solved exactly.
In modality 1, the circular arcs C1 p, and the
segment SD       p [4]. Thus the
radiation source and the detector move rigidly around the
apparatus center O. This imposes some restrictions on the
positioning of the object inside the scanning zone, unless
the value of the parameter pis changed.
Fig.1 CST modality 1.
In modality 2, the circular arcs C2 p,
 p at right angles [5]. The source Sand
the detector D are no longer at a constant distance from
each other. They are sepap    0 which
is linked to the scattering angle Ȧ by Ȧ Ȗ0+ʌ This
gives a larger flexibility for positioning objects inside the
scanning zone as compared to the previous modality.
Fig. 2 CST modality 2
3. Chebyshev Integral Transforms
Since there is no preferred physical orientation in the
plane, it is advantageous to work in polar coordinates and
decompose the unknown function and its transform into
angular Fourier series
Then, for both modalities, (1) reduces to an ordinary
integral equation of the first kind with a Chebyshev kernel
for the Fourier components. Let Tl(x) be the Chebyshev
polynomial of the first kind or order l. Respectively we have
for modality 1, with IJ FRWȦ, and
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for modality 2, with IJ FRVHFȦ.
To get their inverses we bring these equations to the
form of the Chebyshev transform of A. M. Cormack [6]
through the following change of variables
and functions
for modality 1,
for modality 2.
Then (3,4) are rewritten as
which has the form of the Chebyshev transform in [6]. Here
the inversion of (8) can be achieved by a clever application
of (9). The resulting explicit inversion formula reads
However (10), as such is still improper for setting up a
computational algorithm. It needs to be regularized as
indicated in [7]. This is so because the transforms GlIJhave
to verify the co-called consistency conditions, which consist
of the vanishing of a finite number of moments of the data.
Then the proper form of (10) is made up of two parts
where Ul-1(x) is the Chebyshev polynomial of the second
kind and of order (l-1) 
Using the Fourier series
we reconstruct the functions
Then (11) may be used to derive a close form for the
inversion formula first for the intermediate functions given
in (12)
A final form of the inversion formula can be given in terms
of the original functions (IUș the object electric charge
density and JIJij the measured radiation flux density on
the detector) by “extracting” IUșand JIJij from (13)
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–integral should be understood as a Cauchy
principal value. The structure of this formula is analogous
to that of the classical Radon transform. Yet it belongs to
two types of Radon transforms on finite circular arcs which
have appeared for the first time in the literature of integral
geometry in the sense of I. M. Gelfand. This is in contrast to
other types of Radon transforms which are defined on
closed (or unbounded open) curves in the plane.
In practice a computing algorithm can be set up from
(11) following the approach of Chapman and Carey [9]. It
-integration range into a finite
number of integrals and then making the proper change of
variables so that each term is represented by an exactly
calculated indefinite integral. The derivative of GlIJ is
simply calculated by linear interpolation. Of course the
discretization steps must be optimized to save computing
time. This way of reconstructing the unknown function
turns out to be efficient, consistent with the data and give
satisfactory results [10].
4. Conclusion and Perspectives
The idea of CST goes back to the early 50’s. However
the proposed scanning processes operate either point by
point or line by line. Much later, to improve sensitivity,
wide angle radiation collimators have been introduced. This
has led naturally to the concept of integral measurements
along arcs of circles. But no true inversion method was then
at hand.
In the mid 90’s, a first true CST modality was
introduced by S. J. Norton [8]. It has a fixed source of
radiation and a moving detector running on a straight line
containing the source site. Such a CST scanner is, in
particular appropriate for large objects which can only be
studied from one side (such as a wall or a long metal beam).
Here one may also formulate the problem in terms of
circular harmonic components of the unknown electron
density and end up with an integral equation with a
Chebyshev kernel as shown independently by A. M.
Cormack in [6]. The inversion procedure is thus simpler
because one actually ends up with (14), in which q+or q- is
replaced by q.
Consequently the two CST scanning modalities
proposed here appear to be complementary to Norton’s CST
modality. All three use Compton scattered radiation as
imaging agent to probe the hidden parts of objects of
interest in non-destructive control or in medical diagnostics.
All three are based on the inversion of a Chebyshev integral
transform with different degrees of complexity but share
many common features which usher them into a near future
as tomorrow imaging scanners for medical as well as
industrial applications.
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