System and method for image reconstruction for helical cone beam computed tomography with factorized redundancy weighting
View Patent ↗A method and system for computed tomography were weighted data is used to reconstruct an image. The apparatus includes an x-ray source producing a cone-beam of x-rays, an x-ray detector disposed to receive x-rays from the x-ray source, a data collection unit, and a processing unit for processing the data using a weighting function using contributions from fan-beam and cone-beam weighting functions. The apparatus can produce images with reduced artifacts.
1. A computed-tomography apparatus, comprising:
an x-ray source;
an x-ray detector disposed to receive x-rays from said x-ray source;
a data collection unit;
a processing unit for processing said data using a weighting function given as
w
CB
(
β
,
γ
,
v
)
=
u
FB
(
β
)
·
u
CB
(
v
,
γ
)
∑
n
=
-
N
N
(
u
FB
(
β
n
(
β
,
γ
)
)
·
u
CB
(
v
n
)
)
where β is a projection angle of said x-rays, γ is a fan angle of said x-rays, N is a number of helical half-turns, n is an index of complementary rays and ν is a detector coordinate parallel to axis of rotation of said x-ray source, and
wherein said processing uses said weighting function where:
u
CB
(
v
,
γ
)
=
{
0
,
v
<
v
-
(
γ
)
-
δ
v
-
(
γ
)
p
(
v
-
v
-
(
γ
)
+
δ
v
-
(
γ
)
2
δ
v
-
(
γ
)
)
,
v
-
(
γ
)
-
δ
v
-
(
γ
)
≤
v
<
-
v
-
(
γ
)
+
δ
v
-
(
γ
)
1
,
v
-
(
γ
)
+
δ
v
-
(
γ
)
≤
v
≤
v
+
(
γ
)
-
δ
v
+
(
γ
)
p
(
v
+
(
γ
)
+
δ
v
+
(
γ
)
-
v
2
δ
v
+
(
γ
)
)
,
v
+
(
γ
)
-
δ
v
+
(
γ
)
<
v
≤
v
+
(
γ
)
+
δ
v
+
(
γ
)
0
,
v
>
v
+
(
γ
)
+
δ
v
+
(
γ
)
where ν + (γ) and ν − (γ) are boundaries of an n-PI window and δν + (γ) and δν − (γ) are smoothing intervals.
2. An apparatus as recited in claim 1 , wherein said processing uses said weighting function where:
u
FB
(
β
)
=
{
0
,
β
<
β
start
p
(
β
-
β
start
Δβ
)
,
β
start
≤
β
<
β
start
+
Δβ
1
,
β
start
+
Δβ
≤
β
≤
β
end
-
Δβ
p
(
β
end
-
β
Δβ
)
,
β
end
-
Δβ
<
β
≤
β
end
0
,
β
>
β
end
where β start and β end are the start and end angles of a scan, respectively, p( ) is any function that satisfies: p(0)=0, p(1)=1, and p monotonically increases from 0 to 1.
3. An apparatus as recited in claim 1 , wherein said processing unit comprises a filtering sub-unit that applies filtering along detector rows.
4. An apparatus as recited in claim 1 , wherein said processing unit comprises a filtering sub-unit that applies filtering in directions other than along detector rows.
5. An apparatus as recited in claim 1 , wherein said processing uses said weighting function where:
βν + (γ)= C (ν max −ν + (γ))
βν − (γ)= C (ν max +ν − (γ))
v + (γ, n)=H(2γ+nπ)/4πcosγ
v − (γ, n)=H(2γ−nπ)/4πcosγ
C is a proportionality constant, 0<C<1, n equals 1 or 3, and H is a helical pitch.
6. A computed tomography method, comprising:
exposing a subject to x-rays from an x-ray source;
collecting data;
processing said data comprising weighting said data using a weighting function given as
w
CB
(
β
,
γ
,
v
)
=
u
FB
(
β
)
·
u
CB
(
v
,
γ
)
∑
n
=
-
N
N
(
u
FB
(
β
n
(
β
,
γ
)
)
·
u
CB
(
v
n
)
)
where β is a projection angle of said x-rays, γ is a fan angle of said x-rays, N is a number of helical half turns, n is an index of complementary rays and ν is a detector coordinate parallel to axis of rotation of said x-ray source; and
reconstructing an image of said subject using said weighted data,
wherein said processing comprises using said weighting function where:
u
CB
(
v
,
γ
)
=
{
0
,
v
<
v
-
(
γ
)
-
δ
v
-
(
γ
)
p
(
v
-
v
-
(
γ
)
+
δ
v
-
(
γ
)
2
δ
v
-
(
γ
)
)
,
v
-
(
γ
)
-
δ
v
-
(
γ
)
≤
v
<
-
v
-
(
γ
)
+
δ
v
-
(
γ
)
1
,
v
-
(
γ
)
+
δ
v
-
(
γ
)
≤
v
≤
v
+
(
γ
)
-
δ
v
+
(
γ
)
p
(
v
+
(
γ
)
+
δ
v
+
(
γ
)
-
v
2
δ
v
+
(
γ
)
)
,
v
+
(
γ
)
-
δ
v
+
(
γ
)
<
v
≤
v
+
(
γ
)
+
δ
v
+
(
γ
)
0
,
v
>
v
+
(
γ
)
+
δ
v
+
(
γ
)
where ν + (γ) and ν − (γ) are boundaries of an n-PI window, δν + (γ) and δν − (γ) are smoothing intervals and p( )is any function that satisfies: p(0)=0, p(0.5)=0.5, p(1)=1, and p monotonically increases from 0 to 1.
7. A method as recited in claim 6 , wherein said processing comprises using said weighting function where:
u
FB
(
β
)
=
{
0
,
β
<
β
start
p
(
β
-
β
start
Δβ
)
,
β
start
≤
β
<
β
start
+
Δβ
1
,
β
start
+
Δβ
≤
β
≤
β
end
-
Δβ
p
(
β
end
-
β
Δβ
)
,
β
end
-
Δβ
<
β
≤
β
end
0
,
β
>
β
end
where p( )is any function that satisfies: p(0)=0, p(1)=1, β start and β end are the start and end angles of a scan, respectively, and p monotonically increases from 0 to 1.
8. A method as recited in claim 6 , wherein said processing comprises filtering along detector rows.
9. A method as recited in claim 6 , wherein said processing comprises filtering in directions other than along detector rows.
10. A method as recited in claim 6 , wherein said processing comprises using said weighting function where:
βν + (γ)= C (ν max −ν + (γ))
βν − (γ)= C (ν max +ν − (γ))
v + (γ, n)=H(2γ+nπ)/4πcosγ
v − (γ, n)=H(2γ−nπ)/4πcosγ
C is a proportionality constant, 0<C<1, n equals 1 or 3, and H is a helical pitch.
11. A non-transitory computer-readable medium containing instructions that may be executed by a computer to perform a method comprising:
collecting data associated with exposing a subject to x-rays from an x-ray source;
weighting said data using a weighting function given as
w
CB
(
β
,
γ
,
v
)
=
u
FB
(
β
)
·
u
CB
(
v
,
γ
)
∑
n
=
-
N
N
(
u
FB
(
β
n
(
β
,
γ
)
)
·
u
CB
(
v
n
)
)
where β is a projection angle of said x-rays, γ is a fan angle of said x-rays, N is a number of helical half turns, n is an index of complementary rays and ν is a detector coordinate parallel to axis of rotation of said x-ray source; and
reconstructing an image of said subject using said weighted data, where:
u
CB
(
v
,
γ
)
=
{
0
,
v
<
v
-
(
γ
)
-
δ
v
-
(
γ
)
p
(
v
-
v
-
(
γ
)
+
δ
v
-
(
γ
)
2
δ
v
-
(
γ
)
)
,
v
-
(
γ
)
-
δ
v
-
(
γ
)
≤
v
<
-
v
-
(
γ
)
+
δ
v
-
(
γ
)
1
,
v
-
(
γ
)
+
δ
v
-
(
γ
)
≤
v
≤
v
+
(
γ
)
-
δ
v
+
(
γ
)
p
(
v
+
(
γ
)
+
δ
v
+
(
γ
)
-
v
2
δ
v
+
(
γ
)
)
,
v
+
(
γ
)
-
δ
v
+
(
γ
)
<
v
≤
v
+
(
γ
)
+
δ
v
+
(
γ
)
0
,
v
>
v
+
(
γ
)
+
δ
v
+
(
γ
)
and where ν + (γ) and ν − (γ) are boundaries of an n-PI window, βν + (γ) and βν − (γ) are smoothing intervals and p( ) is any function that satisfies: p(0)=0, p(0.5) =0.5, p(1)=1, and p monotonically increases from 0 to 1.
12. A medium as recited in claim 11 , wherein said method comprises using said weighting function where:
u
FB
(
β
)
=
{
0
,
β
<
β
start
p
(
β
-
β
start
Δβ
)
,
β
start
≤
β
<
β
start
+
Δβ
1
,
β
start
+
Δβ
≤
β
≤
β
end
-
Δβ
p
(
β
end
-
β
Δβ
)
,
β
end
-
Δβ
<
β
≤
β
end
0
,
β
>
β
end
where p( ) is any function that satisfies: p(0)=0, p(1)=1, β start and β end are the start and end angles of a scan, respectively, and p monotonically increases from 0 to 1.
13. A medium as recited in claim 11 , wherein said method comprises filtering along detector rows.
14. A medium as recited in claim 11 , wherein said method comprises filtering in directions other than along detector rows.
15. A medium as recited in claim 11 , wherein said method comprises using said weighting function where:
βν + (γ)= C (ν max −ν + (γ))
βν − (γ)= C (ν max +ν − (γ))
v + (γ, n)=H(2γ+nπ)/4πcosγ
v − (γ, n)=H(2γ−nπ)/4πcosγ
C is a proportionality constant, 0<C<1, n equals 1 or 3, and H is a helical pitch.