Performance of sparse-view CT reconstruction with multi-directional gradient operators

Abstract

To further reduce the noise and artifacts in the reconstructed image of sparse-view CT, we have modified the traditional total variation (TV) methods, which only calculate the gradient variations in x and y directions, and have proposed 8- and 26-directional (the multi-directional) gradient operators for TV calculation to improve the quality of reconstructed images. Different from traditional TV methods, the proposed 8- and 26-directional gradient operators additionally consider the diagonal directions in TV calculation. The proposed method preserves more information from original tomographic data in the step of gradient transform to obtain better reconstruction image qualities. Our algorithms were tested using two-dimensional Shepp-Logan phantom and three-dimensional clinical CT images. Results were evaluated using the root-mean-square error (RMSE), peak signal-to-noise ratio (PSNR), and universal quality index (UQI). All the experiment results show that the sparse-view CT images reconstructed using the proposed 8- and 26-directional gradient operators are superior to those reconstructed by traditional TV methods. Qualitative and quantitative analyses indicate that the more number of directions that the gradient operator has, the better images can be reconstructed. The 8- and 26-directional gradient operators we proposed have better capability to reduce noise and artifacts than traditional TV methods, and they are applicable to be applied to and combined with existing CT reconstruction algorithms derived from CS theory to produce better image quality in sparse-view reconstruction.

Fig 1. Procedure of compressed-sensing-based reconstruction algorithm using the 8-directional gradient operator.

FBP: filtered back projection. SIRT: simultaneous iterative reconstruction technique.

Fig 2. Relative positions of voxels in a 3D image.

m and n represent the voxel row and column of the image. i is the level in the z direction. The voxel f_i,m,n has 26 neighboring voxels around it.

Fig 3. Reconstruction results after six iterations obtained for a sampling interval of 5°.

(a) Original Shepp-Logan phantom. (b)–(f) Results reconstructed using FBP, ART, 2-, 4-, and 8-directional gradient operators, respectively. Areas marked by dotted ellipses are the differences of the results between 4-, and 8-directional gradient operators. As it can be seen in (f), when sampling interval is 5°, 8-directional gradient operator gave the fewest artifacts.

Fig 4. As in Fig 3, this figure shows the reconstruction results after six iterations obtained for a sampling interval of 10°.

(a) Original Shepp-Logan phantom. (b)–(f) Results reconstructed using FBP, ART, 2-, 4-, and 8-directional gradient operators, respectively. Areas marked by dotted ellipses are the differences of the results between 4-, and 8-directional gradient operators. In this scanning circumstance, artifacts and noise in the reconstructed images are more vivid than Fig 3. However, the image obtained from 8-directional gradient operator still has the best image quality.

Fig 5. Reconstruction results approaching convergence obtained for a sampling interval of 5°.

Columns from left to right show the reconstructed image, Log(RMSE), PSNR and UQI after two thousand iterations. Rows from top to bottom: images reconstructed using ART, 2-, 4-, and 8-directional gradient operators. When the algorithms use more number of directions in gradient operators, then all three figures of merit are better.

Fig 6. Reconstruction results approaching convergence obtained for a sampling interval of 10°.

Columns from left to right show the reconstructed image, Log(RMSE), PSNR and UQI after two thousand iterations. Rows from top to bottom: images reconstructed using ART, 2-, 4-, and 8-directional gradient operators. As the same in Fig 5, even if the sampling interval is 10°, the gradient operators with more number of directions have better image quality.

Fig 7. Reconstruction results of Shepp-Logan phantom by using EPTV combining with the multi-directional gradient operators when the sampling interval is 5°.

(a)–(c) Results reconstructed using EPTV combining with 2-, 4-, and 8-directional gradient operators, respectively. Areas marked by dotted ellipses are the differences of the results between 4-, and 8-directional gradient operators. Even if combined with EPTV, the images reconstructed from more number of directions in gradient operators still have less artifacts.

Fig 8. Reconstruction results of an abdomen image after six iterations obtained for a sampling interval of 5°.

First row: ground truth; subsequent rows from top to bottom: images reconstructed using FDK, ART, and the 3-, 6-, and 26-directional gradient operators. Images from left to right show the sagittal, transaxial, and coronal sections of the abdomen. Areas marked by dotted rectangles are enlarged and displayed in Fig 9. Subsequent rows from top to bottom, the lower images in the figure, the smoother they are, and are closer to the original images.

Fig 9. The zoom-in views of the images displayed in previous one figure.

First row: ground truth; subsequent rows from top to bottom: images reconstructed using FDK, ART, and the 3-, 6-, and 26-directional gradient operators. Subsequent rows from top to bottom, the more number of directions in gradient operators, the less streak artifacts the reconstructed images have.

Fig 10. Reconstruction results of an abdomen image after six iterations obtained for a sampling interval of 10°.

First row: ground truth; subsequent rows from top to bottom: images reconstructed using FDK, ART, and the 3-, 6-, and 26-directional gradient operators. Images from left to right show the sagittal, transaxial, and coronal sections of the abdomen. Areas marked by dotted rectangles are enlarged and displayed in Fig 11. Artifacts in the reconstructed images are more obvious than Fig 8. However, as seen in Fig 8, subsequent rows from top to bottom, the lower images in the figure, the smoother they are.

Fig 11. The zoom-in views of the images displayed in Fig 10.

As the same in Figs 8–10, the images reconstructed from the 26-directional gradient operators have the least artifacts and noise.

Fig 12. Reconstruction results of an abdomen image by using EPTV combined with the multi-directional gradient operators when the sampling interval is 5°.

First row: ground truth; subsequent rows from top to bottom: images reconstructed using EPTV combined with the 3-, 6-, and 26-directional gradient operators, respectively. Images from left to right show the sagittal, transaxial, and coronal sections of the abdomen. Areas marked by dotted rectangles are enlarged and displayed in Fig 13. Subsequent rows from top to bottom, the lower images in the figure, the smoother they are, and are closer to the original images.

Fig 13. The zoom-in views of the images displayed in Fig 12.

First row: ground truth; subsequent rows from top to bottom: images reconstructed using EPTV combined with the 3-, 6-, and 26-directional gradient operators. Even if combined with EPTV, the more number of directions in gradient operators, the less streak artifacts the reconstructed images have.