17-09-2012, 05:00 PM
TheMathematical Foundations of 3D Compton Scatter Emission Imaging
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ABSTRACT
The mathematical principles of tomographic imaging using detected (unscattered) X- or gamma-rays are based on the twodimensional
Radon transform and many of its variants. In this paper, we show that two new generalizations, called conical Radon
transforms, are related to three-dimensional imaging processes based on detected Compton scattered radiation. The first class of
conical Radon transform has been introduced recently to support imaging principles of collimated detector systems. The second
class is new and is closely related to the Compton camera imaging principles and invertible under special conditions. As they are
poised to play a major role in future designs of biomedical imaging systems, we present an account of their most important properties
which may be relevant for active researchers in the field.
INTRODUCTION
During the last fifty years, progress in imaging systems using
penetrating radiation for biomedical purposes has brought
about new topics in mathematics and fueled intense research
activities with far reaching results. Themathematics of imaging
science has evolved to a full fledged discipline [1]. Transmission
computer assisted tomography (CAT-scanning) is
based on an integral transformin two dimensions discovered
in the sixties by Cormack [2–4], who did not realize that J.
Radon had already introduced and studied it in his seminal
paper [5] in 1917. Subsequently, in an effort to reconstruct
directly in three dimensions an object without having to assemble
its two-dimensional sections, one is led to consider
the so-called X-ray transform or cone beam transform. This
transform is an off-spring of the Radon transform and maps
a function in R3 to its (straight) line integrals in R3. Oneway
to reconstruct an object is to convert the line data into planar
data in R3 using the Grangeat technique [6]. Then application
of the inversion formula of the three-dimensional Radon
transform [7] yields the answer.
WORK SETTING
In this article, we consider the emission imaging problem,
that is, the problem of reconstructing in R3 a gamma-ray
radiating object from its Compton scattered radiation data.
This object is described by its activity volume density function
f . Detection of scattered radiation is performed by a
gamma camera in two instances: with or without collimator.
The recorded data consists of the coordinates of the detection
site, the surface flux density of photons at this site (pixel), and
the value of their energy (list mode). Between the radiating
object and the detector stands a scattering medium: it may
be a volume or a layer as illustrated by Figure 1. Note that the
object itself may also be a scattering medium, and for photon
energies above 25 keV, over 50% of the interactions in
biological tissues are scatterings [26]. Higher-order scattering
events (of much lower probability of occurrence) will be
the object in future studies.
Compton scattering
As Compton scattering plays a key role, we will recall some
of its properties. The Compton effect discovered in 1923 [27]
had served to confirm the particle (photon) nature of radiation,
as proposed by A Einstein. Thus, energetic radiation
under the form of X- or gamma-rays behave like particles
and scatter with electrically charged particles in matter. In
biomedical domains, X- or gamma-photons scatter electrons
in the biological media they traverse.
THE C1-CONICAL RADON TRANSFORM
In this section, we consider the possibility of imaging a threedimensional
object by collecting data on its scattered radiation
on a gamma camera equipped with a collimator and
show how the C1-conical Radon transform arises. Figure 1
shows the experimental arrangement with the location of the
radiating object, the scattering medium, and the collimated
gamma camera.