# US 8,787,646 B2 · Method and apparatus for computed tomography image reconstruction

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- Number: US 8,787,646 B2
- Document ID: US8787646B2
- Status: Granted
- Kind: B2
- Application number: 13/254,001
- Filed: 2010-02-18
- Published: 2012-01-19
- Granted: 2014-07-22
- Earliest claimed: 2009-03-26
- Assignees: Koninklijke Philips N.V.
- Applicants: Dirk Schaefer; Michael Grass
- Inventors: Dirk Schaefer; Michael Grass
- Web page: https://patentgrep.com/patent/US8787646B2

## CPC

- G06T 12/10: Image preprocessing, e.g. calibration, positioning of sources or scatter correction
- A61B 6/032: {Transmission computed tomography [CT]}
- A61B 6/027: {characterised by the use of a particular data acquisition trajectory, e.g. helical or spiral}
- G06T: IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
- A61B: DIAGNOSIS; SURGERY; IDENTIFICATION (usefulness limited to only animals A61D)

## Abstract

A method and apparatus are provided to reconstruct projection data obtained from CT imaging devices with offset detector geometries. According to one aspect of the present invention, a method is provided to reconstruct projection data obtained from CT imaging devices with offset detector geometries that includes the following steps: (i) matching projection data measured at opposing sides of the acquisition trajectory and splicing them together to generate a full, non-truncated projection data set; (ii) differentiation of the projection data; (iii) filtering the differentiated projection data with a filter, such as for example a Hilbert filter; (iv) applying redundancy weighting to the filtered projection data; and (v) back-projecting the redundancy weighted projection data to generate image data.

## Claims

### Claim 1

An imaging method, comprising the steps of:
acquiring projection data of an object with an imaging apparatus comprising a detector which is offset from a center of rotation;
matching at least one first projection from the projection data with at least one second projection from the projection data, wherein the at least one second projection is measured at an approximately opposing side of an acquisition trajectory from the at least one first projection;
splicing estimated projection data derived from the projection data of the at least one second projection with projection data of the first projection to generate a full, non-truncated projection data set;
differentiating the non-truncated projection data set with respect to the acquisition trajectory to generate differentiated projection data;
applying a filter to the differentiated projection data to generate differentiated and filtered projection data;
applying redundancy weighting to the differentiated and filtered projection data to generate differentiated, filtered and weighted projection data; and
back-projecting the differentiated, filtered and weighted projection data to generate volumetric data indicative of the object;
wherein the at least one second projection is selected in a region μmin≦μ≦μ−0 with μmin=−μmax according to {tilde over (X)}1(α,ν,β)={tilde over (X)}(−α,ν,β+2α+π) for α(μmin)≦α≦α(μ−0); and
wherein {tilde over (X)}(α,ν,β)=X(atan(μ/D),ν,λ/R), D is a distance from a source to the detector, R is a distance from the source to a rotation axis, a circular orbit is parameterized by a path length λεΛ=[0, 2πR), a detector ν-axis is parallel to the rotational axis, and a u-axis is parallel to a trajectory tangent vector.

### Claim 2 (depends on 1)

The method of claim 1, wherein the non-truncated projection data set corresponds to a virtual projection measured by a single virtual detector capable of spanning the imaged object.

### Claim 3 (depends on 1)

The method of claim 1, wherein the redundancy weighting is applied to pixels of the differentiated and filtered projection data that were redundantly measured by the first projection and the at least one second projection at the approximately opposing sides of the acquisition trajectory.

### Claim 4 (depends on 1)

The method of claim 1, wherein the redundancy weighting comprises applying a weight of zero to the estimated projection data spliced from the at least one second projection.

### Claim 5 (depends on 1)

The method of claim 1, wherein the redundancy weighting comprises applying a redundancy weight ω(μ) according to:
ω
⁡
(
u
)
=
{
0
,
u
min
≤
u
<
u
-
0
sin
2
⁡
(
π
2
⁢
u
-
u
-
0
Δ
⁢
⁢
u
)
,
u
-
0
≤
u
≤
u
+
0
1
,
u
+
0
<
u
≤
u
max
.

### Claim 6 (depends on 1)

The method of claim 1, wherein the filter is a Hilbert filter or a modified Hilbert filter.

### Claim 7 (depends on 1)

The method of claim 1, wherein the imaging method comprises a reconstruction algorithm according to:
f
new
⁡
(
x
)
=
1
2
⁢
π
⁢
∫
Λ
⁢
ω
⁡
(
u
)
R
-
(
x
-
I
)
·
d
^
⁢
∫
-
∞
∞
⁢
D
SE
_
⁡
(
x
)
⁢
X
d
⁡
(
u
′
,
v
,
λ
)
⁢
h
H
⁡
(
u
-
u
′
)
⁢
⁢
ⅆ
u
′
⁡
(
1
/
R
)
⁢
⁢
ⅆ
λ
,
⁢
⁢
with
⁢
h
H
⁡
(
ρ
)
=
-
∫
-
∞
∞
⁢
i
⁢
⁢
sgn
⁡
(
P
)
⁢
ⅇ
j
⁢
⁢
2
⁢
πρ
⁢
⁢
P
⁢
ⅆ
P
,
and
⁢
X
d
⁡
(
u
,
v
,
λ
)
=
(
∂
χ
2
∂
λ
+
∂
χ
2
∂
u
⁢
∂
u
∂
λ
+
∂
χ
2
∂
v
⁢
∂
v
∂
λ
)
⁢
(
u
,
v
,
λ
)
;
wherein I is an iso-center of the imaging apparatus, the detector center is offset from the center of rotation in a transaxial plane by a distance d, and SE is a length from the source to a detector element.

### Claim 8 (depends on 1)

The method of claim 1, wherein the imaging apparatus is a cone-beam computed tomography imaging device.

### Claim 9

An apparatus for acquiring projection data at a plurality of angular positions relative to an object disposed in an examination region, the apparatus comprising:
a radiation source;
a radiation sensitive detector which detects radiation emitted by the source that has traversed the examination region, wherein the detector is offset from a center of rotation; and
a reconstructor;
wherein the apparatus is adapted to perform a scanning procedure of the object to obtain projection data;
wherein at least one first projection from the projection data is matched with at least one second projection from the projection data, wherein the at least one second projection is measured at an approximately opposing side of an acquisition trajectory from the at least one first projection;
wherein projection data from the at least one second projection is used to derive estimated projection data that is spliced with projection data of the first projection to generate a full, non-truncated projection data set;
wherein the full, non-truncated projection data set is then differentiated with respect to the acquisition trajectory to generate differentiated projection data;
wherein the differentiated projection data is then filtered to generate differentiated and filtered projection data;
wherein redundancy weighting is applied to the differentiated and filtered projection data to generate differentiated, filtered and weighted projection data;
wherein the differentiated, filtered and weighted projection data is then back-projected by the reconstructor to generate volumetric data indicative of the object; and
wherein the at least one second projection is selected in a region μmin≦μ≦μ−0 with μmin=−μmax according to {tilde over (X)}1(α,ν,β)={acute over (X)}(−α,ν,β+2α+π) for α(μmin)≦α≦α(μ−0); and
wherein {acute over (X)}(α,ν,β)=X(atan(μ/D),ν,λ/R), D is a distance from a source to the detector, R is a distance from the source to a rotation axis, a circular orbit is parameterized by a path length λεΛ=[0, 2πR), a detector v-axis is parallel to the rotational axis, and a μ-axis is parallel to a trajectory tangent vector.

### Claim 10 (depends on 9)

The apparatus of claim 9, wherein the full, non-truncated projection data set corresponds to a virtual projection measured by a single virtual detector capable of spanning the imaged object.

### Claim 11 (depends on 9)

The apparatus of claim 9, wherein the redundancy weighting is applied to pixels of the differentiated and filtered projection data that were redundantly measured by the first projection and the at least one second projection at the approximately opposing sides of the acquisition trajectory.

### Claim 12 (depends on 9)

The apparatus of claim 9, wherein the redundancy weighting comprises applying a weight of zero to the estimated projection data spliced from the at least one second projection.

### Claim 13 (depends on 9)

The apparatus of claim 9, wherein the redundancy weighting comprises applying a redundancy weight ω(μ) according to:
ω
⁡
(
u
)
=
{
0
,
u
min
≤
u
<
u
-
0
sin
2
⁡
(
π
2
⁢
u
-
u
-
0
Δ
⁢
⁢
u
)
,
u
-
0
≤
u
≤
u
+
0
1
,
u
+
0
<
u
≤
u
max
.

### Claim 14 (depends on 10)

The apparatus of claim 10, wherein the filter is a Hilbert filter or a modified Hilbert filter.

### Claim 15 (depends on 9)

The apparatus of claim 9, wherein the radiation sensitive detector is a flat detector.

### Claim 16 (depends on 9)

The apparatus of claim 9, wherein the apparatus is a cone-beam computed tomography imaging device.

### Claim 17 (depends on 9)

The apparatus of claim 9, further comprising a mechanical drive for moving the radiation sensitive detector with respect to the radiation source.

### Claim 18 (depends on 9)

The apparatus of claim 9, further comprising an image processor, a user interface and a user input, and wherein the image processor processes the volumetric data for display on the user interface.

### Claim 19 (depends on 9)

The apparatus of claim 9, wherein the reconstructor applies a reconstruction algorithm according to:
f
new
⁡
(
x
)
=
1
2
⁢
π
⁢
∫
Λ
⁢
ω
⁡
(
u
)
R
-
(
x
-
I
)
·
d
^
⁢
∫
-
∞
∞
⁢
D
SE
_
⁡
(
x
)
⁢
X
d
⁡
(
u
′
,
v
,
λ
)
⁢
h
H
⁡
(
u
-
u
′
)
⁢
⁢
ⅆ
u
′
⁡
(
1
/
R
)
⁢
⁢
ⅆ
λ
,
⁢
⁢
with
⁢
h
H
⁡
(
ρ
)
=
-
∫
-
∞
∞
⁢
i
⁢
⁢
sgn
⁡
(
P
)
⁢
ⅇ
j
⁢
⁢
2
⁢
πρ
⁢
⁢
P
⁢
⁢
ⅆ
P
,
⁢
and
⁢
X
d
⁡
(
u
,
v
,
λ
)
=
(
∂
χ
2
∂
λ
+
∂
χ
2
∂
u
⁢
∂
u
∂
λ
+
∂
χ
2
∂
v
⁢
∂
v
∂
λ
)
⁢
(
u
,
v
,
λ
)
;
wherein I is an iso-center of the imaging apparatus, the detector center is offset from the center of rotation in a transaxial plane by a distance d, and SE is a length from the source to a detector element.

### Claim 20

An imaging method, comprising the steps of:
acquiring projection data of an object with an imaging apparatus comprising a detector which is offset from a center of rotation;
matching at least one first projection from the projection data with at least one second projection from the projection data, wherein the at least one second projection is measured at an approximately opposing side of an acquisition trajectory from the at least one first projection;
splicing estimated projection data derived from the projection data of the at least one second projection with projection data of the first projection to generate a full, non-truncated projection data set;
differentiating the non-truncated projection data set with respect to the acquisition trajectory to generate differentiated projection data;
applying a filter to the differentiated projection data to generate differentiated and filtered projection data;
applying redundancy weighting to the differentiated and filtered projection data to generate differentiated, filtered and weighted projection data; and
back-projecting the differentiated, filtered and weighted projection data to generate volumetric data indicative of the object;
wherein the redundancy weighting comprises applying a redundancy weight ω(μ) according to:
ω
⁡
(
u
)
=
{
0
,
1
,
⁢
sin
2
⁡
(
π
2
⁢
u
-
u
-
0
Δ
⁢
⁢
u
)
,
u
min
≤
u
<
u
-
0
u
-
0
≤
u
≤
u
+
0
;
⁢
and
u
+
0
<
u
≤
u
max
wherein a μ-axis is parallel to a trajectory tangent vector.

### Claim 21

An imaging method, comprising the steps of:
acquiring projection data of an object with an imaging apparatus comprising a detector which is offset from a center of rotation;
matching at least one first projection from the projection data with at least one second projection from the projection data, wherein the at least one second projection is measured at an approximately opposing side of an acquisition trajectory from the at least one first projection;
splicing estimated projection data derived from the projection data of the at least one second projection with projection data of the first projection to generate a full, non-truncated projection data set;
differentiating the non-truncated projection data set with respect to the acquisition trajectory to generate differentiated projection data;
applying a filter to the differentiated projection data to generate differentiated and filtered projection data;
applying redundancy weighting to the differentiated and filtered projection data to generate differentiated, filtered and weighted projection data; and
back-projecting the differentiated, filtered and weighted projection data to generate volumetric data indicative of the object;
wherein the imaging method comprises a reconstruction algorithm according to:
f
new
⁡
(
x
)
=
1
2
⁢
⁢
π
⁢
∫
Λ
⁢
ω
⁡
(
u
)
R
-
(
x
-
I
)
·
d
^
⁢
∫
-
∞
∞
⁢
D
SE
_
⁡
(
x
)
X
d
⁡
(
u
′
,
υ
,
λ
)
⁢
h
H
⁡
(
u
-
u
′
)
⁢
ⅆ
u
′
⁡
(
1
/
R
)
⁢
d
⁢
⁢
λ
,
with
h
H
⁡
(
ρ
)
=
-
∫
-
∞
∞
⁢
i
⁢
sgn
⁡
(
P
)
⁢
ⅇ
j
⁢
2
⁢
⁢
πρ
⁢
⁢
P
⁢
ⅆ
P
,
and
X
d
⁡
(
u
,
υ
,
λ
)
=
(
∂
χ
2
∂
λ
+
∂
χ
2
∂
u
⁢
∂
u
∂
λ
+
∂
χ
2
∂
υ
⁢
∂
υ
∂
λ
)
⁢
(
u
,
υ
,
λ
)
;
wherein D is a distance from a source to the detector, R is a distance from the source to a rotation axis, a circular orbit is parameterized by a path length λεΛ=[0, 2πR), a detector ν-axis is parallel to the rotational axis, a μ-axis is parallel to a trajectory tangent vector, I is an iso-center of the imaging apparatus, the detector center is offset from the center of rotation in a transaxial plane by a distance d, and SE is a length from the source to a detector element.

### Claim 22

An apparatus for acquiring projection data at a plurality of angular positions relative to an object disposed in an examination region, the apparatus comprising:
a radiation source;
a radiation sensitive detector which detects radiation emitted by the source that has traversed the examination region, wherein the detector is offset from a center of rotation; and
a reconstructor;
wherein the apparatus is adapted to perform a scanning procedure of the object to obtain projection data;
wherein at least one first projection from the projection data is matched with at least one second projection from the projection data, wherein the at least one second projection is measured at an approximately opposing side of an acquisition trajectory from the at least one first projection;
wherein projection data from the at least one second projection is used to derive estimated projection data that is spliced with projection data of the first projection to generate a full, non-truncated projection data set;
wherein the full, non-truncated projection data set is then differentiated with respect to the acquisition trajectory to generate differentiated projection data;
wherein the differentiated projection data is then filtered to generate differentiated and filtered projection data;
wherein redundancy weighting is applied to the differentiated and filtered projection data to generate differentiated, filtered and weighted projection data;
wherein the differentiated, filtered and weighted projection data is then back-projected by the reconstructor to generate volumetric data indicative of the object; and
wherein the redundancy weighting comprises applying a redundancy weight ω(μ) according to:
ω
⁡
(
u
)
=
{
0
,
1
,
⁢
sin
2
⁡
(
π
2
⁢
u
-
u
-
0
Δ
⁢
⁢
u
)
,
u
min
≤
u
<
u
-
0
u
-
0
≤
u
≤
u
+
0
;
⁢
and
u
+
0
<
u
≤
u
max
wherein a μ-axis is parallel to a trajectory tangent vector.

### Claim 23

An apparatus for acquiring projection data at a plurality of angular positions relative to an object disposed in an examination region, the apparatus comprising:
a radiation source;
a radiation sensitive detector which detects radiation emitted by the source that has traversed the examination region, wherein the detector is offset from a center of rotation; and
a reconstructor;
wherein the apparatus is adapted to perform a scanning procedure of the object to obtain projection data;
wherein at least one first projection from the projection data is matched with at least one second projection from the projection data, wherein the at least one second projection is measured at an approximately opposing side of an acquisition trajectory from the at least one first projection;
wherein projection data from the at least one second projection is used to derive estimated projection data that is spliced with projection data of the first projection to generate a full, non-truncated projection data set;
wherein the full, non-truncated projection data set is then differentiated with respect to the acquisition trajectory to generate differentiated projection data;
wherein the differentiated projection data is then filtered to generate differentiated and filtered projection data;
wherein redundancy weighting is applied to the differentiated and filtered projection data to generate differentiated, filtered and weighted projection data;
wherein the differentiated, filtered and weighted projection data is then back-projected by the reconstructor to generate volumetric data indicative of the object; and
wherein the reconstructor applies a reconstruction algorithm according to:
f
new
⁡
(
x
)
=
1
2
⁢
π
⁢
∫
Λ
⁢
ω
⁡
(
u
)
R
-
(
x
-
I
)
·
d
^
⁢
⁢
∫
-
∞
∞
⁢
D
SE
_
⁡
(
x
)
⁢
⁢
X
d
⁡
(
u
′
,
υ
,
λ
)
⁢
h
H
⁡
(
u
-
u
′
)
⁢
ⅆ
u
′
⁡
(
1
/
R
)
⁢
ⅆ
λ
,
⁢
with
h
H
⁡
(
ρ
)
=
-
∫
-
∞
∞
⁢
i
⁢
⁢
sgn
(
P
)
⁢
ⅇ
j2πρ
⁢
⁢
P
⁢
ⅆ
P
,
and
X
d
⁡
(
u
,
υ
,
λ
)
=
(
∂
χ
2
∂
λ
+
∂
χ
2
∂
u
⁢
∂
u
∂
λ
+
∂
χ
2
∂
υ
⁢
∂
υ
∂
λ
)
⁢
(
u
,
υ
,
λ
)
;
and
wherein D is a distance from the source to the detector, R is a distance from the source to a rotation axis, a circular orbit is parameterized by a path length λεΛ=[0, 2πR), a detector ν-axis is parallel to the rotational axis, a μ-axis is parallel to a trajectory tangent vector, I is an iso-center of the imaging apparatus, the detector center is offset from the center of rotation in a transaxial plane by a distance d, and SE is a length from the source to a detector element.