Graphics Reference
In-Depth Information
In order to correct for AC (or RC) events, the recent generation of commercial PET
systems use prompt (or real-time) window coincidence to subtract the random co-
incidences from delay windows (Spinks et al.,
; Fessler,
). he following no-
tation is used:
n
p
: the number of coincident photon pairs collected in the prompt windows
for detector tube d
n
d
(
d
)
: the number of coincident photon pairs collected in the delay windows for
detector tube d.
(
d
)
Furthermore, n
p
and n
d
areassumedtobestatistically independentandPois-
son-distributed with different means:
(
d
)
(
d
)
p
d
n
d
Poisson
λ
d
λ
d
,
(
.
)
(
)
(
(
)+
(
))
n
d
λ
d
(
d
)
Poisson
(
(
d
))
(
.
)
” means that the random variable “is distributed as,” λ
d
where “
d
is the mean in-
(
)
tensity of n
d
d
,and
(
)
λ
(
d
)=
b
p
(
b, d
)
λ
(
b
)
(
.
)
One can define the precorrecting value in detector tube d, n
(
d
)
by subtracting
n
d
from n
p
(
d
)
(
d
)
:
n
n
p
n
d
(
d
)=
(
d
)−
(
d
)
(
.
)
However, n
can take a negative value. Also, the mean and variance of n
(
d
)
(
d
)
are different when λ
d
. herefore, n
(
d
)
(
d
)
is not Poisson-distributed. When
λ
λ
d
, n
with probability
. Otherwise, by approximating
the moment-generating function in the neighborhood of
, n
(
d
)=
(
d
)=
(
d
)=
d
is approximately
(
)
distributed as a normal distribution with mean λ
and variance λ
λ
d
.
herefore, the maximum likelihood estimate of this approximate model turns out to
be the weighted least square estimate (WLSE).
WLSEmethodscanbeachievedbyapplyingfiniteseriesexpansionreconstruction
methods(Censor,
)like the algebraic reconstruction technique (ART)(Herman,
; Herman et al.,
). he idea of ART is to solve the system of equations suc-
cessively, but not simultaneously. Because it is an iterative method based on row op-
erations, it ise
cient in terms of computation time andstorage requirements. Range
limit constraints, such as nonnegativeness, can be easily implemented during every
iteration as well.
Due to the ill-posedness of this inverse problem, the reconstruction procedure
needstoberegularized.OnewaytoregularizetheWLSEistocombinethe(weighted)
least square term and a penalty term into a functional object to be optimized. In
a Bayesian framework, the penalty term is related to the prior distribution. Various
methods of Bayesian estimation with Markov chain Monte Carlo (MCMC) meth-
ods have been proposed in the literature. For example, it was proposed that the local
d
d
d
(
)
(
)+
(
)
