Fourier re-binning of time-of-flight positron emission tomography data
Fast reconstruction methods are provided for 3D time-of-flight (TOF) positron emission tomography (PET), based on 2D data re-binning. Starting from pre-corrected 3D TOF data, a re-binning algorithm estimates for each transaxial slice the 2D TOF sinogram. The re-binned sinograms can then be reconstructed using any algorithm for 2D TOF reconstruction. A TOF-FORE (Fourier re-binning of TOF data) algorithm is provided as an approximate re-binning algorithm obtained by extending the Fourier re-binning method for non-TOF data. In addition, two partial differential equations are identified that must be satisfied by consistent 3D TOF data, and are used to derive exact re-binning algorithms and to characterize the degree of the approximation in TOF-FORE. Numerical simulations demonstrate that TOF-FORE is more accurate than two different TOF extensions of the single-slice re-binning method, and suggest that TOF-FORE will be a valuable tool for practical TOF PET in the range of axial apertures and time resolutions typical of current scanners.
1. A method for reconstructing a nuclear medical image from TOF-PET imaging data, comprising the steps of:
obtaining three dimensional TOF-PET data from a PET scanner;
storing said TOF-PET data in a plurality of TOF time bins, each bin corresponding to a time difference value of photons arriving at opposite detectors of said PET scanner;
estimating, for each time bin, a two dimensional TOF sinogram of each transaxial slice z within an axial field of view of said scanner, by averaging independent estimates of redundant three dimensional TOF-PET data; and
reconstructing an image from said estimated two dimensional TOF sinograms using a two dimensional TOF reconstruction algorithm.
2. The method of claim 1 , wherein said three dimensional TOF-PET data is represented by the following equation:
p
t
m
(
s
,
ϕ
,
z
,
δ
)
=
1
+
δ
2
∫
-
∞
∞
ⅆ
lf
(
s
cos
ϕ
-
l
sin
ϕ
,
s
sin
ϕ
+
l
cos
ϕ
,
z
+
l
δ
)
h
(
t
,
l
1
+
δ
2
)
.
3. The method of claim 2 , wherein the step of estimating comprises the step of modifying said three dimensional TOF-PET data to
p
t
(
s
,
ϕ
,
z
,
δ
)
=
∫
-
∞
∞
ⅆ
lf
(
s
cos
ϕ
-
l
sin
ϕ
,
s
sin
ϕ
+
l
cos
ϕ
,
z
+
l
δ
)
h
(
t
,
l
)
.
4. The method of claim 3 , wherein said modified data is estimated as
p reb,t ( s,φ,z+t δ)= p t ( s,φ,z ,δ).
5. The method of claim 4 , wherein said two dimensional TOF sinogram is obtained by solving for the equation
p
reb
,
t
(
s
,
ϕ
,
z
)
≃
1
2
δ
max
(
z
)
∫
-
δ
max
(
z
)
δ
max
(
z
)
ⅆ
δ
p
t
(
s
,
ϕ
,
z
,
δ
)
.
6. The method of claim 5 , wherein said two dimensional TOF sinogram is obtained by solving for the equation
P
t
(
ω
,
k
,
z
,
δ
)
≃
P
t
(
ω
,
k
,
z
-
k
δ
ω
,
0
)
.
7. The method of claim 5 , wherein said two dimensional TOF sinogram is obtained by solving for the equation
P
t
(
ω
,
k
,
z
0
,
0
)
=
P
t
(
ω
,
k
,
z
1
,
δ
1
)
-
∫
0
δ
1
ⅆ
δ
ⅆ
P
t
(
ω
,
k
,
z
0
+
k
δ
/
ω
,
δ
)
ⅆ
δ
=
P
t
(
ω
,
k
,
z
0
+
k
δ
1
/
ω
,
δ
1
)
+
1
ω
∫
0
δ
1
ⅆ
δ
(
δ
∂
3
P
t
∂
ω
∂
z
2
+
t
σ
2
∂
2
P
t
∂
ω
∂
z
-
1
σ
2
∂
2
P
t
∂
ω
∂
δ
)
(
ω
,
k
,
z
0
+
k
δ
ω
,
δ
)
.
8. The method of claim 5 , wherein said two dimensional TOF sinogram is obtained by solving for the equation
p
t
(
s
,
ϕ
,
z
0
,
0
)
=
p
t
(
s
,
ϕ
,
z
0
-
t
δ
1
,
δ
1
)
-
σ
2
∫
0
δ
1
ⅆ
δ
∂
2
p
t
(
s
,
ϕ
,
z
0
-
t
δ
,
δ
)
∂
z
∂
t
.