Biomedical Engineering Reference
In-Depth Information
The algorithm ART ( Algebraic Reconstruction Technique ) was proposed in 1970
[ 18 ] and is one of the simplest iterative methods of reconstruction. In this method
the estimates under a given line are compared with the measured projection and
corrected using a simple subtraction. The process follows projection after projection
iterating a certain number of times. With a suitable choice of the over-relaxation
parameters, it has been shown that this algorithm is capable of producing high
quality images [ 19 ].
In a Bayesian approach, the reconstruction ( 14 ) is seen as an optimization
problem where the values X are the ones that better explain the observable data,
meaning that the reconstruction algorithm is guided to determine the most likely
values of the image, X , given its projections, P , or to maximizing the conditional
probability, PŒXjP , which is the probability of occurring X given P [ 10 ].
According to Bayes' formula, we can rewrite the conditional probability as follows:
PŒP j X P ŒX
PŒP
PŒXjP D
:
(15)
As the denominator in ( 15 ) is constant, maximizing PŒXjP is equivalent to
maximizing only the numerator. The term PŒPjX is called likelihood and signifies
how close the data and the image are or, again, the probability of P given X .Inthe
reconstruction by maximum likelihood the probability PŒPjX is maximized.
In the case a Poisson model is considered for the emission, the conditional
probability is given by
P a ij f j p i
p i Š
e P a ij f j
Y
PŒPjX D
:
(16)
i
Applying logarithms to ( 16 ) and maximizing it:
n p i X a ij f j
ln .p i Š/ o :
X
X a ij f j
ln .P ŒP jX/ D
(17)
i
Although there are several methods to maximize the likelihood ( 16 )themost
used is expectation maximization [ 20 ]. This method (ML-EM) involves an iterative
technique whose convergence is guaranteed [ 21 ]. The method follows the general
procedure explained above (Fig. 10 ) and can be summarized by the following
equation:
f .n/
j
X
p i
f .n C 1/
j
D
P
a ij
:
(18)
P
a `j
a ik f .n/
k
i
`
k
One of the most important variants of the ML-EM algorithm is the OSEM
algorithm (Ordered Subset Expectation Maximization) [ 22 ], which converges faster
than its predecessor, however, there is no proof that OSEM converges to the same
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