Fast iterative image reconstruction method for 3D computed tomography
This invention relates to a high resolution and low dosage tomographic imaging in three dimensions, and more particularly, to a fully analytical fast iterative statistical algorithm for image reconstruction from projections obtained in a spiral cone-beam x-ray scanner is described. The presented method allows to improve the resolution of the reconstructed images and/or to decrease the x-ray intensity while maintaining the quality of the obtained CT images, because the signals obtained are adapted to the specific statistics for this imaging technique. The location of pixels in a reconstructed image and the location of detectors in a detector array in this new approach are described. The topology of pixels and detectors presented here avoids an inconsistency in the distribution of the coefficients assigned to the pixels in the image, which appears in the formulation of the analytical iterative statistical reconstruction problem.
1. A method of performing of an iterative reconstruction process, forming part of a method for reconstructing an image of an examined object using a spiral computed tomographic scanner, is carried out by a procedure comprising of:
performing of a 2D FFT of a reconstructed image μ i,j t ; a resulting matrix M k,l t is a frequency representation of the reconstructed image μ i,j t ;
performing multiplication of elements of the resulting matrix M k,l t which is obtained in a previous step of the iterative reconstruction process by corresponding elements of a matrix H k,l ; a resulting matrix A k,l t is a frequency representation of a referential image a k,l t for an image {tilde over (μ)} i,j t obtained after a back-projection operation;
performing of a 2D IFFT of the matrix A k,l t obtained in a previous step of the iterative reconstruction process; a resulting matrix represents the referential image a k,l t for the image {tilde over (μ)} i,j t obtained after the back-projection operation;
transforming, by a nonlinear function, of differences between corresponding pixels of a referential image a k,l t obtained as a result of a previous step of the iterative reconstruction process and pixels of the image {tilde over (μ)} i,j t obtained after the back-projection operation; a resulting matrix is a matrix b k,l t ;
performing of the 2D FFT of the matrix b k,l t obtained in a previous step of the iterative reconstruction process; a resulting matrix B k,l t is a frequency representation of the matrix b k,l t ;
performing multiplication of elements of the matrix B k,l t obtained in a previous step of the iterative reconstruction process by corresponding elements of a matrix H k,l ; a resulting matrix C k,l t is a frequency representation of a matrix of corrections for the reconstructed image μ i,j t ;
performing of the 2D IFFT of the matrix C k,l t obtained in a previous step of the iterative reconstruction process; a resulting matrix c k,l t represents a matrix of corrections for the reconstructed image μ i,j t ;
performing a correction of the reconstructed image μ i,j t ;
using of a criterion of an iterative process for stopping.
2. The method in accordance with claim 1 further comprising
orienting a plane;
placing an image in the plane by a coordinate system (x,y);
placing a topology of pixels in the reconstructed image μ i,j t according to the following description:
x
=
…
,
-
5
Δ
s
2
,
-
3
Δ
s
2
,
-
Δ
s
2
,
Δ
s
2
,
3
Δ
s
2
,
5
Δ
s
2
,
…
,
for an x direction, and
y
=
…
,
-
5
Δ
s
2
,
-
3
Δ
s
2
,
-
Δ
s
2
,
Δ
s
2
,
3
Δ
s
2
,
5
Δ
s
2
,
…
for a y direction, where Δ s is a raster of pixels in the reconstructed image μ i,j t , for both x and y directions.
3. The method in accordance with claim 1 wherein a transformation by a nonlinear function is performed using the following relation:
b
i
,
j
t
=
g
i
,
j
·
tanh
(
a
i
,
j
t
-
μ
~
i
,
j
λ
)
where i and j are indexes indicating coordinates pixels in the reconstructed image μ i,j t in x and y directions, respectively; a i,j t is a pixel from a referential image for the image {tilde over (μ)} i,j obtained after the back-projection operation; b i,j t is a pixel from an image obtained after transformation of the pixel a i,j t ; g i,j is a coefficient established for a given pixel indicated by indices i and j; {tilde over (μ)} i,j is a pixel indicated by indices i and j in the image obtained after the back-projection operation based on non-filtered projections; λ is a constant slope coefficient.
4. A method of performing of an iterative reconstruction process, forming part of a method of tomographic imaging, is carried out by steps comprising of:
performing of a 2D FFT of a reconstructed image μ i,j t ; a resulting matrix M k,l t is a frequency representation of the reconstructed image μ i,j t ;
performing multiplication of elements of the resulting matrix M k,l t which is obtained in a previous step of the iterative reconstruction process by corresponding elements of a matrix H k,l ; a resulting matrix A k,l t is a frequency representation of a referential image a k,l t for an image {tilde over (μ)} i,j t obtained after a back-projection operation;
performing of a 2D IFFT of the matrix A k,l t obtained in a previous step of the iterative reconstruction process; a resulting matrix represents the referential image a k,l t for the image {tilde over (μ)} i,j t obtained after the back-projection operation;
transforming, by a nonlinear or linear function, of differences between corresponding pixels of a referential image a k,l t obtained as a result of a previous step of the iterative reconstruction process and pixels of the image {tilde over (μ)} i,j t obtained after the back-projection operation; a resulting matrix is a matrix b k,l t ;
performing of the 2D FFT of the matrix b k,l t obtained in a previous step of the iterative reconstruction process; a resulting matrix B k,l t is a frequency representation of the matrix b k,l t ;
performing multiplication of elements of the matrix B k,l t obtained in a previous step of the iterative reconstruction process by corresponding elements of a matrix H k,l ; a resulting matrix C k,l t is a frequency representation of a matrix of corrections for the reconstructed image μ i,j t ;
performing of the 2D IFFT of the matrix C k,l t obtained in a previous step of the iterative reconstruction process; a resulting matrix c k,l t presents a matrix of corrections for the reconstructed image μ i,j t ;
performing a correction of the reconstructed image μ i,j t ;
using of a criterion of an iterative process for stopping.
5. The method in accordance with claim 4 further comprising:
orienting a plane;
placing an image in the plane by a coordinate system (x,y);
placing a topology of pixels in the reconstructed image μ i,j t according to the following description:
x
=
…
,
-
5
Δ
s
2
,
-
3
Δ
s
2
,
-
Δ
s
2
,
Δ
s
2
,
3
Δ
s
2
,
5
Δ
s
2
,
…
,
for an x direction, and
y
=
…
,
-
5
Δ
s
2
,
-
3
Δ
s
2
,
-
Δ
s
2
,
Δ
s
2
,
3
Δ
s
2
,
5
Δ
s
2
,
…
for a y direction, where Δ s is a raster of pixels in the reconstructed image μ i,j t , for both x and y directions.
6. The method in accordance with claim 4 wherein a transformation by a nonlinear function is performed using the following relation:
b
i
,
j
t
=
g
i
,
j
·
tanh
(
a
i
,
j
t
-
μ
~
i
,
j
λ
)
where i and j are indexes indicating coordinates pixels in the reconstructed image μ i,j t in x and y directions, respectively; a i,j t is a pixel from a referential image for the image {tilde over (μ)} i,j obtained after the back-projection operation; b i,j t is a pixel from an image obtained after transformation of the pixel a i,j t ; g i,j is a coefficient established for a given pixel indicated by indices i and j; {tilde over (μ)} i,j is a pixel indicated by indices i and j in the image obtained after the back-projection operation based on non-filtered projections; λ is a constant slope coefficient.