Biomedical Engineering Reference
In-Depth Information
An iterative reconstruction algorithm consists of three components: (1) a
data model which describes the data and acquisition artifacts, (2) an objective
function that quantifies the agreement between the image estimate and the mea-
sured data, and (3) an optimization algorithm that determines the next image
estimate based on the current estimate. The measured data can be modeled by
p
=
C
λ
(2.11)
where
p
={
p
j
,
j
=
1
,
2
,...,
M
}
is a vector containing values of the measured
projection data (i.e. sinogram);
λ
={
λ
i
,
i
=
1
,
2
,...,
N
}
is a vector containing
all the voxel values of the image to be reconstructed; and
C
={
C
ij
}
is a transfor-
mation (or system) matrix which defines a mapping (forward-projection) from
f
to
p
. The elements of the matrix
C
ij
is the probability that a positron annihilation
event that occurred at voxel
i
is detected at projection ray
j
. Other physical pro-
cesses such as nonuniform attenuation and scattered and random effects can be
incorporated into the data model in the form of additive noise that corrupted the
acquired projection data. Detailed discussion of more complex data models is
considered beyond the scope of this chapter. The objective function can include
any
a priori
constraints such as nonnegativity and smoothness. Depending on
the assumed number of counts, the objective function can include the Poisson
likelihood or the Gaussian likelihood for maximization. The iterative algorithm
seeks successive estimates of the image that best match the measured data and
it should converge to a solution that maximizes the objective function, with the
use of certain termination criteria.
Iterative reconstruction methods based on the maximum-likelihood (ML)
have been studied extensively, and the expectation maximization (EM) algo-
rithm [38, 39] is the most popular. The ML-EM algorithm seeks to maximize the
Poisson likelihood. In practical implementation, the logarithm of the likelihood
function is maximized instead for computational reasons:
N
M
N
L
(
p
|
λ
)
=
ln
C
ij
λ
i
C
ij
λ
i
(2.12)
−
j
=
1
i
=
1
i
=
1
The EM algorithm updates the image values by
M
i
j
=
1
C
ij
λ
p
j
i
=
1
C
i
j
λ
k
+
1
i
λ
=
C
ij
(2.13)
k
i
j
=
1
