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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 IEEE 2989

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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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