Optimized weighting for Fourier rebinning of three-dimensional time-of-Flight PET data to non-time-of-flight

Abstract

Time-of-flight (TOF) PET scanners provide the potential for significantly improved signal-to-noise ratio (SNR) and lesion detectability in clinical PET. Therefore, it is likely that TOF will become the standard for clinical whole body PET in the near future. However, fully 3D TOF PET image reconstruction is a challenging task due to the huge data size. One solution to this problem is to rebin TOF data into a lower dimensional format. We have recently developed Fourier rebinning methods for mapping TOF data into non-TOF formats and achieved substantial SNR advantages over sinograms acquired without TOF information. However, such mappings for rebinning into non-TOF formats are not unique and optimization of rebinning methods has not been widely investigated. In this paper we address the question of optimal rebinning in order to make full use of TOF information and consequently to maximize image quality. We focus on FORET-3D, which rebins 3D TOF data into 3D non-TOF sinogram formats without requiring a Fourier transform in the axial direction. We optimize the weighting for FORET-3D using a uniformly minimum variance unbiased (UMVU) estimator under reasonable approximations. We show that the rebinned data with optimal weights are a sufficient statistic for the unknown image, implying that any information loss due to rebinning is as a result only of the approximations used in developing the optimal weighting. We demonstrate using simulated and real phantom TOF data that the optimal rebinning method achieves significant variance reduction and better contrast recovery compared to other rebinning weightings.

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Optimized Weighting for Fourier Rebinning of
Three-Dimensional Time-of-Flight PET Data to
Non-Time-of-Flight
Sangtae Ahn, Member, IEEE, Sanghee Cho, Member, IEEE, Quanzheng Li, Member, IEEE, and
Richard M. Leahy∗, Fellow, IEEE
Abstract—Time-of-flight (TOF) PET scanners provide the
potential for significantly improved signal-to-noise ratio (SNR)
and lesion detectability in clinical PET. Therefore, it is likely that
TOF will become the standard for clinical whole body PET in the
near future. However, fully 3D TOF PET image reconstruction
is a challenging task due to the huge data size. One solution
to this problem is to rebin TOF data into a lower dimensional
format. We have recently developed Fourier rebinning methods
for mapping TOF data into non-TOF formats and achieved
substantial SNR advantages over sinograms acquired without
TOF information. However, such mappings for rebinning into
non-TOF formats are not unique and optimization of rebinning
methods has not been widely investigated. In this paper we
address the question of optimal rebinning in order to make full
use of TOF information and consequently to maximize image
quality. We focus on FORET-3D, which rebins 3D TOF data
into 3D non-TOF sinogram formats without requiring a Fourier
transform in the axial direction. We optimize the weighting
for FORET-3D using a uniformly minimum variance unbiased
(UMVU) estimator under reasonable approximations. We show
that the rebinned data with optimal weights are a sufficient
statistic for the unknown image, implying that any information
loss due to rebinning is as a result only of the approximations
used in developing the optimal weighting. We demonstrate
using simulated and real phantom TOF data that the optimal
rebinning method achieves significant variance reduction and
better contrast recovery compared to other rebinning weightings.
I. INTRODUCTION
T
provement [1]–[5] and better lesion detectability [6]. There-
fore, TOF PET technology including scintillators, system hard-
ware and image reconstruction is attracting increasing interest
[7]–[11]. However, fully 3D TOF PET image reconstruction
is challenging due to the huge data sizes involved.
Analytical reconstruction methods based on a line-integral
model were used in [2], [12]–[15], a natural extension of the
backprojection-filtering approach for 2D non-TOF PET, and
extended to 3D TOF PET in [16]. Despite low computation
costs and linearity, which facilitates the analysis of the recon-
structed images, the analytical reconstruction methods have
IME-OF-FLIGHT (TOF) PET scanners provide the po-
tential for substantial signal-to-noise ratio (SNR) im-
S. Ahn, Q. Li and R. M. Leahy are with the Signal and Image Processing
Institute, University of Southern California, Los Angeles, CA 90089 (e-mail:
leahy@sipi.usc.edu).
S. Cho was with the Signal and Image Processing Institute, University of
Southern California, Los Angeles, CA 90089. He is now with the Department
of Radiology, Massachusetts General Hospital, Boston, MA 02114.
limitations in terms of the accuracy of the implicit physical
and statistical models assumed in their development.
Iterative methods including EM and OSEM have been used
for TOF PET image reconstruction, showing higher SNR
than analytical methods [2], [15], [17]–[22]. The iterative
reconstruction methods outperform the analytical methods at
the expense of substantially increased computation cost. In
the DIRECT (direct image reconstruction for TOF) approach,
each event is deposited into the image space using the TOF
information to produce histo-images so that the huge size of
the 3D TOF list mode data is greatly reduced [23]; angular
and co-polar groupings further reduce the data size [20].
An alternative approach to reducing the computation cost
is to rebin 3D TOF data into a lower dimensional space [15].
Single slice rebinning (SSRB-TOF) [24] combines the TOF
oblique sinograms to form a set of stacked TOF direct sino-
grams in a similar manner to SSRB for non-TOF data [25]. As
an alternative to SSRB-TOF, an approximate Fourier rebinning
method mapping 3D TOF into 2D TOF data was proposed,
where the rebinning is performed in the Fourier domain [26].
A similar approximate rebinning was also derived in the native
coordinates of the TOF sinograms rather than the Fourier
domain [27]. An exact rebinning equation was derived based
on a consistency condition expressed by a partial differential
equation in the continuous data domain, where rebinning is
performed with respect to the axial variables [28]. This result
motivated the development of an approximate discrete axial
rebinning method [28]. We have also developed an alternative
exact rebinning method, which is based on a Fourier transform
in the TOF variable, by using a generalized projection slice
theorem [29].
All of the rebinning methods described above rebin 3D
TOF to 2D TOF data and specifically retain the TOF com-
ponent in the rebinned data. We have recently developed
new rebinning methods that rebin 3D TOF to non-TOF data
in either 3D or 2D forms and shown that rebinning into
non-TOF sinograms retains significant SNR advantages over
sinograms collected without TOF information [30]. The results
include approximate rebinning methods that do not require
estimation of missing data and we have shown that they have
accuracy similar to that of Fourier rebinning for non-TOF data.
Surprisingly, the approximate rebinning methods can map all
TOF bins for a single oblique sinogram into a single non-
TOF sinogram with improved SNR relative to the non-TOF
case and minimal approximation error.
2009 IEEE Nuclear Science Symposium Conference RecordM07-4
9781-4244-3962-1/09/$25.00 ©2009 IEEE2989
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Fig. 1.
object along a LOR weighted by a TOF kernel h. Each LOR is specified by
s, φ, z and δ = tanθ.
Geometry of data formation. 3D TOF data are line integrals of a 3D
Mappings that rebin into non-TOF formats are not unique
and there exist infinitely many rebinnings, depending on which
sinogram each TOF oblique sinogram is rebinned into and also
depending on weights used in combining the rebinned data. It
is important to optimize rebinning methods in order to make
full use of TOF information and consequently to maximize the
quality of images reconstructed from the rebinned data. In this
paper we address the problem of finding an optimal method
to rebin TOF data into non-TOF formats.
We focus on FORET-3D [30], which rebins 3D TOF data
into 3D non-TOF sinogram formats without requiring a Fourier
transform in the axial direction and hence avoiding the missing
data problem [31], [32]. We formulate the rebinning problem
as estimating 3D non-TOF sinograms from noisy 3D TOF data
and show a uniformly minimum variance unbiased (UMVU)
estimator turns out to be a FORET-3D mapping with optimal
weights under reasonable approximations. Furthermore, we
show the optimal linear estimator is a sufficient statistic for
the unknown image, implying that any information loss due
to rebinning is as a result only of the approximations used in
developing the optimal weighting.
II. BACKGROUND
A. Data Model
Three-dimensional (3D) time-of-flight (TOF) data p from a
cylindrical PET scanner can be modeled using line integrals
along lines of response (LORs) weighted by a 1D TOF kernel
h [26], [30]:
?
f(scosφ − lsinφ,ssinφ + lcosφ,z + lδ)dl
where f ∈ R3denotes a 3D object; s, φ, z and δ specify
each LOR, that is, s and φ are radial and angular coordinates,
respectively, z represents the axial midpoint and δ is the
tangent of the oblique angle; and t is the TOF variable, the
time difference of the arrival times, which is converted to
distance by multiplying the speed of light (see Fig. 1 for
the geometry of data formation). The TOF kernel is assumed
to be shift invariant so that the integral in (1) is written in
the form of a convolution. The expression for p in (1) is
general enough to include 3D non-TOF data when h(·) = 1;
stacked 2D TOF sinograms when δ = 0; and stacked 2D
p(s,φ,z,δ,t) =
1 + δ2
?∞
−∞
h(t − l
?
1 + δ2)·
(1)
non-TOF sinograms when h(·) = 1 and δ = 0. The TOF
kernel models the uncertainty in TOF measurements. Here
we use a Gaussian function for the TOF kernel [15], [19],
[33], [34]. Since h is an even function, that is, h(t) = h(−t),
the following symmetry and periodicity property of p can
be shown [26]: p(s,φ,z,δ,t) = p(−s,φ + π,z,−δ,−t).
Therefore, p is completely characterized once it is defined on
the set R×[0,π)×R3. Since?∞
DC component of the data in t, yields non-TOF data.
−∞h(t)dt = 1, integrating the
TOF data p with respect to the TOF variable t, which is the
B. Generalized Projection Slice Theorem
By taking the Fourier transform of (1) with respect to s, z
and t, one can obtain the generalized projection slice theorem
for TOF data in the 3D cylindrical scanner geometry, which
is a key ingredient for deriving the mappings between 3D/2D
and TOF/non-TOF datasets [30]:
?
F(ωscosφ − χsinφ,ωssinφ + χcosφ,ωz)
where F and H are 3D and 1D Fourier transforms of f and
h, respectively; ℘ is the 3D Fourier transform of p(s,φ,z,δ,t)
with respect to s, z and t; ωs, ωz and ωt are the frequency
variables corresponding to s, z and t, and
℘(ωs,φ,ωz,δ,ωt) =1 + δ2H(ωt)·
(2)
χ = χ(δ,ωt,ωz) = ωt
?
1 + δ2− ωzδ.
(3)
The relationship in (2) also applies to 3D non-TOF data
when ωt = 0 (recall that the DC component of TOF data
in t is equivalent to non-TOF data); 2D TOF data when
δ = 0; and 2D non-TOF data when ωt = 0 and δ = 0. A
symmetry and periodicity property also holds for ℘ such that
℘(ωs,φ,ωz,δ,ωt) = ℘(−ωs,φ+π,ωz,−δ,−ωt). We use the
set R+× [0,2π) × R3, that is, ωs≥ 0, as the domain of ℘
without loss of generality where R+= {x ≥ 0|x ∈ R}.
C. Sinogram Mapping Equation
We have derived an approximate sinogram mapping
equation for mapping between a TOF oblique sinogram
˜ ℘δ,ωt,z(ωs,φ) = ˜ ℘(ωs,φ,z,δ,ωt) and a non-TOF oblique
sinogram ˜ ℘?
oblique plane, using the projection slice theorem in (2) and
the Taylor series truncation with respect to δωz/ωs, where
˜ ℘ is the 2D Fourier transform of p(s,φ,z,δ,t) with respect
to s and t [30]. Henceforth, we omit the tilde in ˜ ℘δ,ωt,z
for notational simplicity; however, one can easily distinguish
℘δ,ωt,z= ˜ ℘δ,ωt,zfrom ℘δ,ωt,ωzby the subscripts z and ωz.
The equation for mapping a TOF oblique sinogram ℘δ,ωt,z
to a non-TOF oblique sinogram ℘?
?
0,
δ,z(ω?
s,φ?) = ˜ ℘?(ω?
s,φ?,z,δ,ω?
t= 0) in the same
δ,zis given by
℘?
δ,,z(ω?
s,φ?) ≈
H(0)
H(ωt)℘δ,ωt,z(ωs,φ),ω?2
ω?2
s≥ ω2
s< ω2
t(1 + δ2)
t(1 + δ2)
(4)
where
ωs
φ
=
=
?ω?2
s− ω2
t(1 + δ2)
φ?− arctan(ωt
√1 + δ2/ωs).
(5)
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Note that if ω?
for any φ?does not have any corresponding value of ℘δ,ωt,z.
The inverse mapping from a non-TOF oblique sinogram to a
TOF oblique sinogram, is given by
℘δ,ωt,z(ωs,φ) ≈H(ωt)
where
=
φ?
=
φ + arctan(ωt
s< ω2
t(1 + δ2) then the ℘?
δ,zvalue at (ω?
s,φ?)
H(0)℘?
δ,z(ω?
s,φ?)
(6)
ω?
s
?ω2
s+ ω2
t(1 + δ2)
√1 + δ2/ωs).
(7)
D. FORET-3D
One can map 3D TOF data ℘δ,ωt,z into a 3D non-TOF
data format ℘?
We refer to this mapping as FORET-3D (FOurier REbinning
of Time-of-flight data to 3D non time-of-flight) [30]. The
procedure for the mapping is as follows. First, we map
each TOF oblique sinogram ℘δ,ωt,z into a non-TOF oblique
sinogram using (4). We denote the non-TOF oblique sinogram
by ℘?
average of ℘?
?
δ,zusing the sinogram mapping equation in (4).
δ,z;ωt. Next, we estimate ℘?
δ,z;ωtover ωt:
δ,zby taking a weighted
ˆ ℘?
δ,z(ω?
s,φ?) =
ωtαδ,z,ωt℘?
?
δ,z;ωt(ω?
s,φ?)
ωt∈Bδ,ω?sαδ,z,ωt
(8)
where α are nonnegative weights and Bδ,ω?
(ωt
for which the TOF oblique sinogram ℘δ,ωt,z is mapped into
a non-TOF oblique sinogram ℘?
non zero (see (4)).
The rebinning mapping given in (8) is a function of the
weights αδ,z,ωtand its performance depends heavily on the
weights as shown in Sec. IV. The simplest approach is to
take an unweighted average. A rebinning mapping in (8) with
αδ,z,ωt= 1 is called “unweighted FORET-3D.” However, this
naive weighting approach performs even worse than using non-
TOF sinograms acquired without TOF information as shown
in Sec. IV. The main reason is that the sinogram mapping
equation for FORET-3D given in (4) includes a scaling factor
1/H(ωt), which will amplify noise when the H(ωt) values
are small.
To reduce the noise amplification, we have previously used
αδ,z,ωt= H(ωt), which is heuristically chosen to cancel
out the scaling factor 1/H(ωt) contained in the sinogram
mappings, although we inadvertently omitted a detailed de-
scription of such weights used in [30]. We call the mapping
with αδ,z,ωt= H(ωt) “H-weighted FORET-3D.” We have
shown the H-weighted FORET-3D achieves significant SNR
improvements over non-TOF data acquisition [30]. However,
the heuristic weights are still not optimal and we optimize
these weights in Sec III.
s= {ωt : ω?2
s ≥
√1 + δ2)2}. The set Bδ,ω?
srepresents a TOF frequency bin,
δ,z;ωtwhose values at ω?
sare
III. THEORY
A. Forward Model Using Inverse Rebinning Mapping
We next develop a discrete model for rebinning. First we
represent the 3D TOF data in the Fourier domain using an in-
verse rebinning mapping from 3D non-TOF data. Let a vector
pi
sinogram ℘δ,ωt,z for i = 1,...,NzNδ and k = 1,...,Nt
where i and k denote an oblique plane index for (δ,z) and
a TOF frequency variable index for ωt, respectively, and Ns,
Nφ, Nz, Nδand Ntare the number of sample points through
the radial (s or ωs), angular (φ), axial (z or ωz), oblique angle
(δ) and TOF variable (t or ωt) direction, respectively. We
use different indices for an oblique plane (i) and for a TOF
frequency variable (k) since we rebin TOF oblique sinograms
into a single oblique sinogram for the same oblique plane, that
is, with the same i, by collapsing the TOF frequency variable
index k. Suppose k0corresponds to ωt= 0 and pi
a non-TOF oblique sinogram. Then a TOF oblique sinogram
pi
k∈ CNsNφrepresent a discrete version of a TOF oblique
k0represents
kcan be modeled as
pi
k= Ti
kpi
k0
(9)
where a NsNφ× NsNφ matrix Ti
mapping operator given in (6), which maps a non-TOF sino-
gram pi
We now investigate the properties of the inverse rebinning
matrix Ti
ning method. First we investigate the structure of the matrix
product (Ti
matrix for the geometry of the clinical PET scanner we use for
simulation studies in Section IV. The matrix is nearly diagonal.
Each column of the matrix (Ti
oblique sinogram. Fig. 2(b) shows one column arranged in
sinogram format. Nonzero values are concentrated within a
3 × 3 region, which is also true of all other columns (not
shown here). Therefore, (Ti
matrix Di
Next we examine the diagonal elements of (Ti
the sum of each column of (Ti
for a robust estimate of each diagonal element, and arrange
them in a sinogram format. The calculated diagonal elements
are shown in Fig. 2(c) and its vertical profile is shown in
Fig. 2(d). We compare the diagonal elements with H2
Hkrepresents the kth sample of H(ωt). As shown in Fig. 2(d),
the diagonal elements agree well with H2
region. Note the central part with zero diagonal elements in
Fig. 2(a) and with zero values in Fig. 2(d) corresponds to a
region where ω2
ω2
krepresents the sinogram
k0to a TOF sinogram pi
k.
ksince we exploit them when optimizing the rebin-
k)?Ti
k. Fig. 2(a) shows a central portion of such a
k)?Ti
khas the same size as an
k)?Ti
k≈ Di
kfor some diagonal
k.
k)?Ti
k. We take
k)?Ti
k, as shown in Fig. 2(a),
kwhere
kexcept in the central
s< ω2
t(1+δ2), the size of which decreases as
tor δ2decreases (see (4)). Therefore, we approximate
(Ti
k)?Ti
k≈ H2
kdiagj
?χki
j
?INsNφ
(10)
?is a
where Imis an m × m identity matrix and diagj
diagonal matrix whose jth diagonal element is χki
the jth diagonal element Ti
and χki
?χki
t(1+δ2),
j
j; χki
s< ω2
j= 0 if
kcorresponds to ω2
j= 1 otherwise. Then one can obtain, from (10),
?χki
where the NsNφ× NsNφmatrix Ri
Ti
matrix Ri
pi
and facilitate our subsequent analysis of optimal rebinning
mappings.
(Ti
k)?≈ H2
kdiagj
j
?(Ti
k)+≈ H2
kis the pseudoinverse of
kRi
k
(11)
k, that is, the sinogram mapping operator given in (4). The
kmaps a TOF sinogram pi
k0. The approximations in (10) and (11) are key results
kto a non-TOF sinogram
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(a) (b)
(c)(d)
Fig. 2.
Central portion of the matrix (Ti
mapping matrix used for inverse rebinning in FORET-3D. The Ti
TOF oblique sinogram in a central plane with the maximum ring difference
into a TOF sinogram for the same oblique plane with the maximum TOF
frequency variable. (b) A column of (Ti
(c) Calculated diagonal elements of (Ti
(d) A vertical profile through (c).
Illustration of the properties of an inverse rebinning matrix. (a)
k)?Ti
kwhere Ti
krepresents a sinogram
kmaps a non-
k)?Ti
k)?Ti
k, arranged in a sinogram format.
k, arranged in sinogram format.
B. UMVU Estimators
A noisy TOF sinogram yi
(9) as
k∈ RNsNφcan be modeled using
yi
k= pi
k+ ei
k= Ti
kpi
k0+ ei
k
(12)
where ei
uncorrelated across k and follows a Gaussian distribution:
k∈ RNsNφrepresents noise. We assume the noise is
ei
k∼ N(0,Ki
k)
(13)
where Ki
covariance matrix as
kis the noise covariance matrix. We model the noise
Ki
k≈ σ2
iINsNφ,
(14)
assuming that the noise is uncorrelated across the radial
frequency and the TOF frequency direction and is identical
across the angular direction in the Fourier domain. The validity
of this noise model is examined in [35].
Suppose we are to estimate a non-TOF sinogram pi
noisy TOF sinograms yi
minimum variance unbiased (UMVU) estimator [36], [37] can
be obtained as
??
This UMVU estimator provides a sufficient statistic for the
unknown image. In other words, the estimate contains all the
k0from
kfor k = 1,...,Nt. Then a uniformly
ˆ pi
k0=
k
(Ti
k)?(Ki
k)−1Ti
k
?−1?
k
(Ti
k)?(Ki
k)−1yi
k. (15)
information about the unknown image contained in the noisy
TOF sinograms. To show this, we represent the measured data
using a forward projection operator [35]:
yi
k= Ti
kpi
k0+ ei
k= Ti
kPif + ei
k
where f ∈ CNxNyNzrepresents the object with Nxand Ny
being the number of sample points through the x (or ωx) and
y (or ωy) direction, respectively, and a NsNφ× NxNyNz
matrix Pirepresents a forward projection operator mapping
the object into the non-TOF oblique sinogram pi
Gaussian noise model, the probability density function for data
y = {yi
pf(y)
∝
=
k0. Under the
k}i,kcan be written as
e−1
h1(f)h2(y)e−?
for some functions h1, independent of y, and h2, independent
of f. By the factorization criterion, {?
provided the matrix inverse exists in (15).
Although we show the optimal linear estimator in (15) is
a sufficient statistic, it may not be clear at this point how
to implement it efficiently and how it is related to rebinning
mappings. We next show that the optimal estimator can be
approximated by a rebinning mapping, based on the noise
models and the properties of inverse rebinning mappings.
2
?
i,k(yi
k−Ti
kPif)?(Ki
k)−1(yi
k−Ti
kPif)
if?(Pi)?(?
k(Ti
k)?(Ki
k)−1yi
k)
k(Ti
k)?(Ki
k)−1yi
k}i
is a sufficient statistic [36] and so is the UMVU estimator,
C. Optimal Rebinning Mapping
Using (10), (11) and (14) in the UMVU estimator (15)
yields
?
k
which turns out to be a discrete version of the rebin-
ning mapping equation given in (8) for FORET-3D with
H2
weighted FORET-3D.” For comparison purposes, we also
implement “unweighted FORET-3D,”
?
k
and “H-weighted FORET-3D,”
?
k
as described in Sec. II-D.
ˆ pi
k0= diagj
1/
?
H2
kχki
j
??
k
H2
kRi
kyi
k,
(16)
kweights. We call this optimal rebinning mapping “H2-
ˆ pi
k0= diagj
1/
?
χki
j
??
k
Ri
kyi
k,
(17)
ˆ pi
k0= diagj
1/
?
Hkχki
j
??
k
HkRi
kyi
k,
(18)
D. Image Reconstruction from Rebinned Data
Before applying rebinning mappings, the measured data
should be corrected for detector efficiency, attenuation and
scatter and randoms since the rebinning methods are based on
line integral based projection models. Data correction destroys
the Poisson distribution and the variance of corrected data
is no longer proportional to its mean. After rebinning, we
undo the correction by dividing the rebinned data by the
correction factors as in [38]. One can show that the rebinned
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(a)(b)
(c)(d)
Fig. 3. (a) Axial center plane of the NCAT torso phantom used for simulation
studies. The resolution and variance properties were studied at the four
points denoted by A, B, C and D. (b) Sample mean profiles and (c) sample
variance profiles for the rebinned data and the non-TOF sinogram in an axial
center plane with maximum ring difference. (d) Profiles of the covariance
of the rebinned data. The covariance was calculated between a sinogram bin
(radial index=169, angular index=169) and all other sinogram bins in the
same oblique plane. For (b)–(d), the profiles were taken at the 169th angle
corresponding to φ = π/2.
data after undoing correction turn out to have approximately
the same covariance structure as the raw non-TOF sinograms
before correction, within the SNR gain scaling factor which
depends on the rebinning weights [35]. This finding is similar
to previous observations for rebinning of non-TOF data, where
they were used to justify use of EM or OSEM reconstruction
of reweighted Fourier rebinned data [38].
Since the raw non-TOF sinogram data before correction fol-
low the Poisson distribution, the variance of the de-corrected
rebinned non-TOF data will be proportional to its mean.
Therefore, one can use PWLS [39] or Poisson likelihood based
reconstruction methods with the de-corrected rebinned data.
In Sec. IV, we apply MAP reconstruction methods based on
Poisson models to the rebinned data.
IV. RESULTS
A. 3D TOF PET Simulation
1) Simulation setup: To evaluate the performance of the
optimal rebinning method, we simulated the Siemens Biograph
PET/CT TruePoint TrueV scanner [40] as in [30]. The scanner
had 672 detectors per ring and 55 rings with a ring radius
of 421 mm and an axial field of view of 216 mm. We
generated 3D TOF data using line integral based projectors
and a Gaussian TOF kernel with timing resolution 500 ps,
and the data were sampled with a sampling period of 250
ps, leading to 15 TOF bins. The maximum ring difference
was 54 and the number of LORs per angle was 336. We had
639 oblique and direct sinogram planes with span 11. The
image size was 256 × 256 × 109 with a voxel size of 2 mm.
The NCAT torso phantom was used as a 3D object for data
generation (see Fig. 3(a)) [41]. We generated noisy 3D TOF
data with a total of 20M counts for trues and also simulated
randoms corresponding to a uniform field of 15% of the total
count. We simulated Poisson noise and randoms-precorrected
data were used for rebinning. Scatter, attenuation and detector
efficiencies were not considered.
Noisy 3D TOF data were rebinned into 3D non-TOF data by
FORET-3D with the three different weightings given in (16),
(17) and (18). To reduce approximation errors [31], for small
|ωs|, only the data for small |ωt| were used for rebinning. As
in [30], when the index for |ωs| was less than 7, we only
rebinned the data corresponding to ωt= 0. The 3D non-TOF
sinograms acquired without TOF information were obtained
by summing the 3D TOF data over the TOF bins.
A fully 3D MAP reconstruction method [42] was applied
to the rebinned data and also to the non-TOF data obtained
by summing the TOF data in the TOF bin direction. For
each reconstruction, 2 iterations of OSEM (ordered subsets
expectation maximization) with 6 subsets [43] followed by 30
iterations of PCG (preconditioned conjugate gradient) were
performed. Forward and back-projectors were implemented
based on inverse Fourier rebinning for fast image reconstruc-
tion [44].
2) Comparison of Rebinned Data: To evaluate the perfor-
mance of FORET-3D with different weights, we compared the
rebinned data and also the non-TOF data by a Monte Carlo
simulation using 50 noisy data sets. We calculated the sample
mean and variance for the rebinned data and the non-TOF
data.
Fig. 3(b) shows the profiles of the sample mean for the
rebinned data and the non-TOF sinogram in an axial center
plane with the maximum ring difference. The mean profiles for
FORET-3D agreed well with that for the non-TOF sinogram.
This implies that FORET-3D is nearly free of systematic bias.
Fig. 3(c) shows the profiles of the sample variance for
the rebinned data and the non-TOF sinogram. Unweighted
FORET-3D showed even larger variance than the non-TOF
sinogram. However, using H-weights and H2-weights sub-
stantially reduced the variance and the optimal H2-weighted
FORET-3D gave the smallest variance.
As can be seen in Fig. 3(c), the shapes of the variance
profiles for FORET-3D look similar to that for the non-
TOF sinogram. The variance of H2-weighted FORET-3D was
highly correlated with that of the non-TOF sinogram across
sinogram bins (Pearson correlation coefficient = 0.9265).
Also, the correlation coefficient between the variance of H-
weighted FORET-3D and the non-TOF sinogram was 0.8935,
and the correlation coefficient was 0.8719 for unweighted
FORET-3D. The variance of FORET-3D was approximately
a scaled version of the variance of the non-TOF sinogram.
We also calculated the covariance of the rebinned data to
test whether there is a spatial correlation in the rebinned
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