PET Image Reconstruction - Correction Techniques in Emission Tomography - page 32

geometrical deficiencies or missing data. On the other hand, discretization
algorithms are typically hard to handle numerically, since they involve the
inversion of big matrices, but the flexibility makes up for that deficiency,
making them the standard algorithm for high{quality images to date.
We do not aim for an exhaustive coverage of the subject, but rather lay
the groundwork for many other articles in this topic. The existing literature
is vast, including the following references.
For a detailed classical mathematical introduction to medical image recon-
struction, we refer the reader to [10]. For an approach oriented toward current
algorithms, see [11]. For a classical engineering approach, see [8]. For a more
up-to-date engineering approach, see [2]. For a historical overview of inversion
formulas, see [9].

3.2 Analytical algorithms
In the most basic model of PET, the number of detected events on a line
of response (LOR) is proportional to the amount of radioactivity on that line.
Mathematically, that is given by the line integral over the activity distribution
function f over the LOR. The fixed proportionality coecent is derived from
the total number of counts and the measurement time. Since the object under
investigation is nite, we can assume that f has compact support .
Let us assume for the moment that measurements for all lines passing
through are available, and that everything is 2D. Our problem can then be
restated as follows:
Given measurements

Z

m L = f(x) dx

L

for all lines L though , compute f from theses measurements.
This is the classical formulation of the Radon inversion problem, solved by
Radon in 1910.
In this chapter, we leave out all mathematical delicacies, in particular with
respect to appropriate function spaces, and all proofs. More exact formulations
can be found in the provided literature.

3.2.1 Mathematical basis
Denote by S n1 the unit ball in IR n . Then for xed 2 S n1 , s 2 IR,
the set L(;x) = fx 2 IR n : x = sg is a hyperplane in IR n with normal
vector and distance jsj to the origin. In particular, if n = 2, L(;s) is a line
perpendicular to , with signed oset s from the origin.