Back-projection of a projection image data set with depth-dependent filtering
View Patent ↗For filtered back-projection of a projection image data set, the projection image data set is cosine-weighted. The cosine-weighted projection image data set within the image plane of the projection image data set is subjected to a two-dimensional Radon transformation. The Radon transform of the cosine-weighted projection image data set differentiated with respect to the distance from an image origin of an image coordinate system. The derivative of the Radon transform is redundancy-weighted. The redundancy-weighted derivative is subjected to a two-dimensional Radon back-transformation. The Radon back-transform is differentiated and back-projected with respect to an image column coordinate. A differentiation step width entering into the differentiation is varied depending on depth.
1. A method for filtered back-projection of a projection image data set, consisting of:
providing a projection image data set, acquired by panning an x-ray source through a plurality of projection angles with respect to an image plane, to a processor and, in said processor, cosine-weighting said projection image data set to obtain a cosine-weighted projection image data set g 1 ;
in said processor, subjecting the cosine-weighted projection image data set g 1 within said image plane to a two-dimensional Radon transformation according to a Radon transform g 2 ;
in said processor, differentiating the Radon transform g 2 of the cosine-weighted projection image data set g 1 with respect to a distance from an image origin of an image coordinate system, to obtain a differentiated Radon transform g 3 ;
in said processor, redundancy-weighting the differentiated Radon transform g 3 , to obtain a redundancy-weighted derivative ĝ 4 ,
in said processor, subjecting the redundancy-weighted derivative ĝ 4 to a two-dimensional Radon back-transformation, thereby obtaining a Radon back-transform ĝ 5 ; and
in said processor, differentiating and back-projecting the Radon back-transform ĝ 5 with respect to an image column coordinate u with differentiation step with delta u*, and varying said differentiation step with delta u* depending on depth.
2. A method as claimed in claim 1 , comprising:
redundancy-averaging the derivative (g 3 ) of the Radon transform (g 2 ) according to
g
^
4
=
g
^
4
(
λ
,
μ
,
s
)
=
-
1
4
π
2
r
d
M
(
λ
,
μ
,
s
)
·
g
3
(
λ
,
μ
,
s
)
wherein λ is the projection angle, μ is the polar angle without inclination coordinate within the image plane, s is the distance from the origin of the image coordinate system, r is the distance of a projection center to the center point of a rotation circle along which the projection center is panned during variation of the projection angle λ, d is the distance of the projection center from the image plane, M is a redundancy weighting function, g 3 is the derivative of the Radon transform (g 2 ), and ĝ 4 is the resulting redundancy-weighted derivative.
3. A method according to claim 1 , comprising:
implementing the back-projection of the Radon back-transform (ĝ 5 ) and the derivative with respect to the image line coordinate (u) according to
f
*
=
f
*
(
x
_
)
=
rd
2
ɛ
∫
λ
out
λ
in
[
g
^
5
(
λ
,
u
*
+
Δ
u
*
,
v
*
)
-
g
^
5
(
λ
,
u
*
-
Δ
u
*
,
v
*
)
]
·
ⅆ
λ
wherein x is the location of a spatial volume (back-projection location) to be reconstructed, f* is the reconstructed attenuation, r is the distance of a projection center from the center point of a rotation circle along which the projection center is panned during variation of the projection angle, d is the distance of the projection center from the image plane, λ is the projection angle, λ in and λ out are the limits of a projection angle range, u* and v* are the image column coordinates or, respectively, image line coordinates of the image point in which the back-projection beam traversing the projection origin and the back-projection location ( x ) intersects the image plane, Δu* is the derivation step width, ĝ 5 is the Radon back-transform, and ε is a constant number.
4. A method according to claim 3 , comprising:
setting the derivation step width (Δu*) according to
Δ u*=ε·[R− x · e w (λ)] −2
wherein e w is the unit vector of the line connecting the center point of the rotation circle with the projection center.
5. An apparatus for filtered back-projection of a projection image data set consisting of:
a processor having an input that receives a projection image data set, acquired by panning an x-ray source through a plurality of projection angles with respect to an image plane, to a processor, and a reconstruction module configured to cosine-weight said projection image data set to obtain a cosine-weighted projection image data set g 1 ;
processor reconstruction module being configured to subject the cosine-weighted projection image data set g 1 within said image plane to a two-dimensional Radon transformation according to a Radon transform g 2 ;
said reconstruction module being configured to differentiate the Radon transform g 2 of the cosine-weighted projection image data set g 1 with respect to a distance from an image origin of an image coordinate system, to obtain a differentiated Radon transform g 3 ;
said reconstruction module being configured to redundancy-weight the differentiated Radon transform g 3 , to obtain a redundancy-weighted derivative ĝ 4 ;
said reconstruction module being configured to subject the redundancy-weighted derivative ĝ 4 to a two-dimensional Radon back-transformation, thereby obtaining a Radon back-transform ĝ 5 ; and
said reconstruction module being configured to differentiate and back-project the Radon back-transform ĝ 5 with respect to an image column coordinate u with differentiation step with delta u*, and varying said differentiation step with delta u* depending on depth.
6. A non-transitory, computer-readable data storage medium encoded with programming instructions, said data storage medium being loaded into a computerized control system of a tomographic imaging system and said programming instructions causing said computerized control system to execute a procedure consisting of:
receive a projection image data set, acquired by panning an x-ray source through a plurality of projection angles with respect to an image plane, to a processor and cosine-weight said projection image data set to obtain a cosine-weighted projection image data set g 1 ;
subject the cosine-weighted projection image data set g 1 within said image plane to a two-dimensional Radon transformation according to a Radon transform g 2 ;
differentiate the Radon transform g 2 of the cosine-weighted projection image data set g 1 with respect to a distance from an image origin of an image coordinate system, to obtain a differentiated Radon transform g 3 ;
redundancy-weight the differentiated Radon transform g 3 , to obtain a redundancy-weighted derivative ĝ 4 ;
subject the redundancy-weighted derivative ĝ 4 to a two-dimensional Radon back-transformation, thereby obtaining a Radon back-transform ĝ 5 ; and
differentiate and back-project the Radon back-transform ĝ 5 with respect to an image column coordinate u with differentiation step with delta u*, and varying said differentiation step with delta u* depending on depth.