doi: 10.1038/srep41218.
Alignment Solution for CT Image Reconstruction using Fixed Point and Virtual Rotation Axis
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- PMID:28120881
- PMCID: PMC5264594
- DOI: 10.1038/srep41218
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Alignment Solution for CT Image Reconstruction using Fixed Point and Virtual Rotation Axis
Kyungtaek Jun et al. Sci Rep..
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Abstract
Since X-ray tomography is now widely adopted in many different areas, it becomes more crucial to find a robust routine of handling tomographic data to get better quality of reconstructions. Though there are several existing techniques, it seems helpful to have a more automated method to remove the possible errors that hinder clearer image reconstruction. Here, we proposed an alternative method and new algorithm using the sinogram and the fixed point. An advanced physical concept of Center of Attenuation (CA) was also introduced to figure out how this fixed point is applied to the reconstruction of image having errors we categorized in this article. Our technique showed a promising performance in restoring images having translation and vertical tilt errors.
Figures
Notice that we marked the stage with the red dot at the bottom to indicateθ is zero degree. (a) The sinogram when the specimen (right panel) is translated in parallel with the beam from the center of the stage atθ = 0°. (b) The sinogram and its reconstruction with which we figured
of each column in sinogram a and align them on the functionT50,30°.
was marked in black on each column of the sinogram. Because the
was off the center of the stage, the black line showed sinusoidal function. The reconstruction is moved to the upward side of the original stage. (c) The sinogram and its reconstruction with which we figured
of each column in sinogram a and align them on the functionT0,φ.
was marked in blue on each column of the sinogram. Because
was moved to the center of stage this time, it appeared as a straight line across the center. The center of reconstruction of specimen is moved to the center of the original stage. This modification shows the same result just like moving the specimen in real space.
of each column in sinogram a and align them on the functionT50,30°. was marked in black on each column of the sinogram. Because the was off the center of the stage, the black line showed sinusoidal function. The reconstruction is moved to the upward side of the original stage. (c) The sinogram and its reconstruction with which we figured of each column in sinogram a and align them on the functionT0,φ. was marked in blue on each column of the sinogram. Because was moved to the center of stage this time, it appeared as a straight line across the center. The center of reconstruction of specimen is moved to the center of the original stage. This modification shows the same result just like moving the specimen in real space.
(a) An image sample (right panel) and its sinogram (left panel) (The object is on the lower right side of the stage). The
s of all projections are marked in black in the sinogram. They follow the circular trajectory in real space, therefore show the sinusoidal graph in the sinogram. (b) A sinogram (left panel) with artificial translation errors added to the sinogram (a) including vertical and horizontal movement at each projection angle and its reconstruction (right panel). The
s, the black marks, are all scattered. (c) A sinogram that we aligned
of the sinogram (b) onT0,φ and its reconstruction. The
s are arranged linearly on the center of the sinogram.
is on the center of the stage and the image is ideally restored. (d) The cases when the x-ray density of projection is changed during the beam time. The sinogram and its reconstruction when the x-ray density of projection ina is decreased by half afterθ = 90°. The pattern of ideal sinogram ofa was maintained even though there was 50% decrease in x-ray density from the 90 degrees ofθ while preserving the linear relationship. (e) The sinogram and its reconstruction when the CA was applied to (d). The reconstructed using the sinogram with the
and the Tr,φ function applied showed no difference in terms of image itself when compared with the reconstruction of (d), and the image of specimen was laid in the center.
s of all projections are marked in black in the sinogram. They follow the circular trajectory in real space, therefore show the sinusoidal graph in the sinogram. (b) A sinogram (left panel) with artificial translation errors added to the sinogram (a) including vertical and horizontal movement at each projection angle and its reconstruction (right panel). Thes, the black marks, are all scattered. (c) A sinogram that we aligned of the sinogram (b) onT0,φ and its reconstruction. Thes are arranged linearly on the center of the sinogram. is on the center of the stage and the image is ideally restored. (d) The cases when the x-ray density of projection is changed during the beam time. The sinogram and its reconstruction when the x-ray density of projection ina is decreased by half afterθ = 90°. The pattern of ideal sinogram ofa was maintained even though there was 50% decrease in x-ray density from the 90 degrees ofθ while preserving the linear relationship. (e) The sinogram and its reconstruction when the CA was applied to (d). The reconstructed using the sinogram with the and the Tr,φ function applied showed no difference in terms of image itself when compared with the reconstruction of (d), and the image of specimen was laid in the center.
The third and fourth pictures in (a,b andc) show the trajectories of
. When the
is on the RA (orange dot in the picture), it is expressed as a dot. In other cases, it is expressed as a circular trajectory (blue circle). (a) The sinogram and the
trajectory when an object is tilted. Even though the object is tilted, the
is on the parallel line with the stage in the projection. This is quite a typical occasion where no error is found. (b) The sinogram and the
trajectory when the RA is vertically tilted. The object rotates around the RA and the line that
makes are perpendicular to the RA on the projection. (c) The sinogram, its reconstruction and the
trajectory when the RA has a parallel tilt. The pattern of the sinogram is linear around the center. However, its reconstruction is flawed (shown better in the magnified figure), since the layers are all mixed up at each angle θ. In this case, the collection of
s in the projection makes an elliptical shape, not a line. The RA is also perpendicular to the major axis of
trajectory in this case.
. When the is on the RA (orange dot in the picture), it is expressed as a dot. In other cases, it is expressed as a circular trajectory (blue circle). (a) The sinogram and the trajectory when an object is tilted. Even though the object is tilted, the is on the parallel line with the stage in the projection. This is quite a typical occasion where no error is found. (b) The sinogram and the trajectory when the RA is vertically tilted. The object rotates around the RA and the line that makes are perpendicular to the RA on the projection. (c) The sinogram, its reconstruction and the trajectory when the RA has a parallel tilt. The pattern of the sinogram is linear around the center. However, its reconstruction is flawed (shown better in the magnified figure), since the layers are all mixed up at each angle θ. In this case, the collection ofs in the projection makes an elliptical shape, not a line. The RA is also perpendicular to the major axis of trajectory in this case.
(a) The white parts in the projection image represent the high density areas. The marked rectangle represents a common layer including high density areas. (b) This reconstruction was made by ideal sinogram pattern, but the density of the grain depends on the position of the reconstructed object. (c) Two magnified images in the reconstruction of (b).
The third pictures in (a,b, andc) are the magnified images of the yellow frames of the corresponding reconstruction s in the middle. (a) From the analysis of this sinogram, the specificfn(t) value was found at the discontinued spot; it was expected that a vertical translation error arose between the 360th and the 361th projections. For the middle panel, we used the RA of part B to minimize the error from two distinct RAs. (b) We corrected a vertical translation error (see Supplementary Section 2.2) to the part A of the sinogram (a), which brought us a sinogram with the continuity and a better reconstruction. The area with high x-ray impermeability is moving along inside the sinogram showing a sinusoidal pattern (black arrows in left panel). In its reconstruction, this area is placed in the upper right part (white arrow) in the middle panel. (c) The center of the x-ray impermeable material is set as the fixed point and it is applied to the functionT0,φ representing the virtual rotational axis of the sinogram; the FP is now shifted on the line across the center in the sinogram (black arrows). It is placed on the center of the image in the reconstruction image (white arrow).
(a) Projection images at θ = 45° (left) and atθ = 90° (middle). Right panel shows the sinogram with errors (θe = 60°) at the blue colored axial level of the CCD. These projection images (atθ = 45° andθ = 90°) contain two cleared fixed points (red and orange circles indicate the first and second fixed points, respectively). (b) Adjusted projection images via the second fixed point. The region between upper and lower red colored lines is a common layer set. A sinogram of each common layer can be converted to the ideal sinogram pattern. The right panel shows the sinogram at the projected common layer containing the second fixed point. The black arrows indicate the trajectory of the second fixed point in the sinogram.
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