Reconstructing a tomographic image with reduced artifacts
A method of reconstructing a tomographic image of a region of investigation with reduced artifacts, said method comprises the steps of (a) reconstructing a first partial image and a second partial image of the region of investigation from first and second projection profiles each of which including projection data collected at first and second different groups of parallel projection lines, resp., wherein the first and second projection profiles are provided such that streak aliasing artifacts in the first and second partial images have different spatial phases, and (b) generating the tomographic image of the region of investigation by superimposing the first and second partial images. Preferably, the first and second projection profiles are constructed such that streak aliasing artifacts in the first and second partial images have opposite spatial phases relative to each other. Furthermore, an imaging method and an imaging device for imaging a region of investigation in an object are described.
1. Method of reconstructing a tomographic image of a region of investigation with reduced artifacts, the method comprising:
(a) reconstructing a first partial image and a second partial image of the region of investigation from first and second projection profiles each of which including projection data collected at first and second different groups of parallel projection lines, respectively, wherein the first and second projection profiles are provided such that streak aliasing artifacts in the first and second partial images have different spatial phases, and
(b) generating the tomographic image of the region of investigation by superimposing the first and second partial images, wherein:
the projection lines represent a scanning geometry such that parallel projection lines are distributed corresponding to zeros of Tschebycheff polynomials of the second kind of order [N−1],
the projection lines in each of the first projection profiles are distributed corresponding to zeros of Tschebycheff polynomials of the second kind of order [(N/2)−1], and
the projection lines in each of the second projection profiles are distributed corresponding to zeros of Tschebycheff polynomials of the first kind of order [N/2].
2. Reconstructing method according to claim 1 , wherein:
the first and second projection profiles are constructed such that streak aliasing artifacts in the first and second partial images have opposite spatial phases relative to each other.
3. Reconstructing method according to claim 1 , wherein:
the reconstructing steps include determining image functions of the first and second partial images as sums of polynomials multiplied with the projection data of each of the first and second groups of parallel projection lines, respectively, wherein the polynomials are sums of orthogonal ridge polynomials.
4. Reconstructing method according to claim 1 , wherein:
the step of generating the image of the region of investigation comprises adding the first and second partial images.
5. Reconstructing method according to claim 1 , wherein:
the first and second projection data are collected with an X-ray computer tomography (CT) device or a PET imaging device.
6. Reconstructing method according to claim 1 , wherein:
the projection lines represent a scanning geometry such that intersection points of the projection lines with a circle including the region of investigation are spaced by equal angles Δ relative to the circle centre, and
both the intersection points of the projection lines of each of the first groups of parallel projections lines and the intersection points of the projection lines of each of the second groups of parallel projection lines being spaced by Δ, while an offset of the projection angles of the first group of projection profiles is shifted at Δ/2 relative to the angular offset of the second group of projection profiles.
7. Reconstructing method according to claim 1 , wherein:
the projection lines represent a scanning geometry such that intersection points of the projection lines with a circle including the region of investigation are spaced by equal angles Δ relative to the circle centre,
the intersection points of the projection lines of each of the first groups of parallel projection lines being spaced by 2Δ,
the intersection points of the projection lines of each of the second groups of parallel projection lines being spaced by 2Δ, and
the intersection points of the projection lines of the first projection profiles differ from the intersection points of the projection lines of the second projection profiles by odd multiples of the angle Δ.
8. Reconstructing method according to claim 1 , wherein:
the first and second projection profiles are provided such that the projection data d of the first and second projection profiles {d ν,j 1 } j=0, . . . , N-1 ν=0, . . . , N-1 and {d ν,j 2 } j=0, . . . , N-1 ν=0, . . . , N-1 fulfill
d
v
,
j
1
=
R
(
φ
v
1
,
cos
ψ
j
1
)
,
φ
v
1
=
2
π
v
N
or
φ
v
1
=
π
v
N
d
v
,
j
2
=
R
(
φ
v
2
,
cos
ψ
j
2
)
,
φ
v
2
=
2
π
(
v
+
α
)
N
or
φ
v
2
=
π
(
v
+
α
)
N
ψ
j
1
=
(
β
+
j
)
π
N
,
ψ
j
2
=
(
β
+
1
2
+
j
)
π
N
,
0
≤
α
≤
1
0
≤
β
≤
1
wherein R is the Radon transformation.
9. Reconstructing method according to claim 1 , further comprising at least one of the steps of:
representing an approximation of the tomographic image as a visualized image to be obtained,
storing image data representing the tomographic image, and
subjecting the tomographic image to an image processing step.
10. Method of reconstructing a tomographic image of a region of investigation with reduced artifacts, the method comprising:
(a) reconstructing a first partial image and a second partial image of the region of investigation from first and second projection profiles each of which including projection data collected at first and second different groups of parallel projection lines, respectively, wherein the first and second projection profiles are provided such that streak aliasing artifacts in the first and second partial images have different spatial phases, and
(b) generating the tomographic image of the region of investigation by superimposing the first and second partial images, wherein:
the projection lines represent a scanning geometry such that parallel projection lines are distributed corresponding to zeros of Tschebycheff polynomials of the first kind of order N,
the projection lines in each of the first projection profiles are distributed corresponding to
t
2
j
=
cos
(
j
+
1
/
4
)
π
N
/
2
,
j=0, 2, . . . N/2−1, and
the projection lines in each of the second projection profiles are distributed corresponding to
t
2
j
+
1
=
cos
(
j
+
3
/
4
)
π
N
/
2
,
j=0, . . . N/2−1.
11. Reconstructing method according to claim 10 , wherein:
the first and second projection profiles are constructed such that streak aliasing artifacts in the first and second partial images have opposite spatial phases relative to each other.
12. Reconstructing method according to claim 10 , wherein:
the reconstructing steps include determining image functions of the first and second partial images as sums of polynomials multiplied with the projection data of each of the first and second groups of parallel projection lines, respectively, wherein the polynomials are sums of orthogonal ridge polynomials.
13. Reconstructing method according to claim 10 , wherein:
the step of generating the image of the region of investigation comprises adding the first and second partial images.
14. Reconstructing method according to claim 10 , wherein:
the projection lines represent a scanning geometry such that intersection points of the projection lines with a circle including the region of investigation are spaced by equal angles Δ relative to the circle centre,
the intersection points of the projection lines of each of the first groups of parallel projection lines being spaced by 2Δ,
the intersection points of the projection lines of each of the second groups of parallel projection lines being spaced by 2Δ, and
the intersection points of the projection lines of the first projection profiles differ from the intersection points of the projection lines of the second projection profiles by odd multiples of the angle Δ.
15. Reconstructing method according to claim 10 , wherein:
the first and second projection profiles are provided such that the projection data d of the first and second projection profiles {d ν,j 1 } j=0, . . . , N-1 ν=0, . . . , N-1 and {d ν,j 2 } j=0, . . . , N-1 ν=0, . . . , N-1 fulfill
d
v
,
j
1
=
R
(
φ
v
1
,
cos
ψ
j
1
)
,
φ
v
1
=
2
π
v
N
or
φ
v
1
=
π
v
N
d
v
,
j
2
=
R
(
φ
v
2
,
cos
ψ
j
2
)
,
φ
v
2
=
2
π
(
v
+
α
)
N
or
φ
v
2
=
π
(
v
+
α
)
N
ψ
j
1
=
(
β
+
j
)
π
N
,
ψ
j
2
=
(
β
+
1
2
+
j
)
π
N
,
0
≤
α
≤
1
0
≤
β
≤
1
wherein R is the Radon transformation.
16. Reconstructing method according to claim 10 , further comprising at least one of the steps of:
representing an approximation of the tomographic image as a visualized image to be obtained,
storing image data representing the tomographic image, and
subjecting the tomographic image to an image processing step.
17. Imaging method for imaging a region of investigation in an object, comprising:
directing a plurality of energy input beams at predetermined projection directions through the region of investigation,
determining projection data measured with the plurality of energy input beams, and
subjecting the projection data to a reconstructing method according to claim 1 .
18. Imaging device for imaging a region of investigation in an object, the imaging device comprising:
a measuring device being adapted for directing a plurality of energy input beams at predetermined projection directions through the region of investigation, and for determining projection data measured with the plurality of energy input beams, and
a reconstruction circuit, connected with the measuring device, for reconstructing an image function, wherein the reconstruction circuit is configured to reconstruct a tomographic image of the region of investigation with reduced artifacts by:
(a) reconstructing a first partial image and a second partial image of the region of investigation from first and second projection profiles each of which including projection data collected at first and second different groups of parallel projection lines, respectively, wherein the first and second projection profiles are provided such that streak aliasing artifacts in the first and second partial images have different spatial phases, and
(b) generating the tomographic image of the region of investigation by superimposing the first and second partial images, wherein:
the projection lines represent a scanning geometry such that parallel projection lines are distributed corresponding to zeros of Tschebycheff polynomials of the second kind of order [N−1],
the projection lines in each of the first projection profiles are distributed corresponding to zeros of Tschebycheff polynomials of the second kind of order [(N/2)−1], and
the projection lines in each of the second projection profiles are distributed corresponding to zeros of Tschebycheff polynomials of the first kind of order [N/2].
19. Computer program residing on at least one non-transitory computer-readable medium, comprising computer-executable instructions for performing a method for reconstructing a tomographic image of a region of investigation with reduced artifacts, the method comprising:
(a) reconstructing a first partial image and a second partial image of the region of investigation from first and second projection profiles each of which including projection data collected at first and second different groups of parallel projection lines, respectively, wherein the first and second projection profiles are provided such that streak aliasing artifacts in the first and second partial images have different spatial phases, and
(b) generating the tomographic image of the region of investigation by superimposing the first and second partial images, wherein:
the projection lines represent a scanning geometry such that parallel projection lines are distributed corresponding to zeros of Tschebycheff polynomials of the second kind of order [N−1],
the projection lines in each of the first projection profiles are distributed corresponding to zeros of Tschebycheff polynomials of the second kind of order [(N/2)−1], and
the projection lines in each of the second projection profiles are distributed corresponding to zeros of Tschebycheff polynomials of the first kind of order [N/2].
20. Apparatus comprising at least one non-transitory computer-readable storage medium, comprising computer-executable instructions for performing a method for reconstructing a tomographic image of a region of investigation with reduced artifacts, the method comprising:
(a) reconstructing a first partial image and a second partial image of the region of investigation from first and second projection profiles each of which including projection data collected at first and second different groups of parallel projection lines, respectively, wherein the first and second projection profiles are provided such that streak aliasing artifacts in the first and second partial images have different spatial phases, and
(b) generating the tomographic image of the region of investigation by superimposing the first and second partial images, wherein:
the projection lines represent a scanning geometry such that parallel projection lines are distributed corresponding to zeros of Tschebycheff polynomials of the second kind of order [N−1],
the projection lines in each of the first projection profiles are distributed corresponding to zeros of Tschebycheff polynomials of the second kind of order [(N/2)−1], and
the projection lines in each of the second projection profiles are distributed corresponding to zeros of Tschebycheff polynomials of the first kind of order [N/2].
21. Imaging method for imaging a region of investigation in an object, comprising:
directing a plurality of energy input beams at predetermined projection directions through the region of investigation,
determining projection data measured with the plurality of energy input beams, and
subjecting the projection data to a reconstructing method according to claim 10 .
22. Imaging device for imaging a region of investigation in an object, the imaging device comprising:
a measuring device being adapted for directing a plurality of energy input beams at predetermined projection directions through the region of investigation, and for determining projection data measured with the plurality of energy input beams, and
a reconstruction circuit, connected with the measuring device, for reconstructing an image function, wherein the reconstruction circuit is configured to reconstruct a tomographic image of the region of investigation with reduced artifacts by:
(a) reconstructing a first partial image and a second partial image of the region of investigation from first and second projection profiles each of which including projection data collected at first and second different groups of parallel projection lines, respectively, wherein the first and second projection profiles are provided such that streak aliasing artifacts in the first and second partial images have different spatial phases, and
(b) generating the tomographic image of the region of investigation by superimposing the first and second partial images, wherein:
the projection lines represent a scanning geometry such that parallel projection lines are distributed corresponding to zeros of Tschebycheff polynomials of the first kind of order N,
the projection lines in each of the first projection profiles are distributed corresponding to
t
2
j
=
cos
(
j
+
1
/
4
)
π
N
/
2
,
j=0, 2, . . . N/2−1, and
the projection lines in each of the second projection profiles are distributed corresponding to
t
2
j
+
1
=
cos
(
j
+
3
/
4
)
π
N
/
2
,
j=0, . . . N/2−1; and wherein the reconstruction circuit is connected with the measuring device.
23. At least one non-transitory computer-readable medium, comprising computer-executable instructions for performing a method for reconstructing a tomographic image of a region of investigation with reduced artifacts, the method comprising:
(a) reconstructing a first partial image and a second partial image of the region of investigation from first and second projection profiles each of which including projection data collected at first and second different groups of parallel projection lines, respectively, wherein the first and second projection profiles are provided such that streak aliasing artifacts in the first and second partial images have different spatial phases, and
(b) generating the tomographic image of the region of investigation by superimposing the first and second partial images, wherein:
the projection lines represent a scanning geometry such that parallel projection lines are distributed corresponding to zeros of Tschebycheff polynomials of the first kind of order N,
the projection lines in each of the first projection profiles are distributed corresponding to
t
2
j
=
cos
(
j
+
1
/
4
)
π
N
/
2
,
j=0, 2, . . . N/2−1, and
the projection lines in each of the second projection profiles are distributed corresponding to
t
2
j
+
1
=
cos
(
j
+
3
/
4
)
π
N
/
2
,
j=0, . . . N/2−1.