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# A nonconvex truncated regularization and box-constrained model for CT reconstruction

• * Corresponding author: Yiming Gao
This work was supported by Zhejiang Provincial Natural Science Foundation of China (No. LQ20A010007) the National Natural Science Foundation of China (NSFC) (Nos. 11871035, 11531013, 11971138), Recruitment Program of Global Young Expert, and Postdoctoral Science Foundation of China (No. 2019M651002)
• X-ray computed tomography has been a useful technology in cancer detection and radiation therapy. However, high radiation dose during CT scans may increase the underlying risk of healthy organs. Usually, sparse-view X-ray projection is an effective method to reduce radiation. In this paper, we propose a constrained nonconvex truncated regularization model for this low-dose CT reconstruction. It preserves sharp edges very well. Although this model is quite complicated to analyze, we establish two useful theoretical results for its minimizers. Motivated by them, an iterative support shrinking algorithm is introduced. To handle more nondifferentiable points of the regularization function except zero point, we use a general proximally linearization technique at them, which is helpful to implement our algorithm. For this algorithm, we prove the convergence of the objective function, and give a lower bound theory of the iterative sequence. Numerical experiments and comparisons demonstrate that our model with the proposed algorithm performs good for low-dose CT reconstruction.

Mathematics Subject Classification: Primary: 68U10, 65K10, 94A08, 90C26.

 Citation: • • Figure 1.  Three different $\varphi(t)$. (a) $\varphi(t) = t,\ 0 \leq t \leq 0.5; \ \varphi(t) = 0.5,\ t > 0.5$. (b) $\varphi(t) = t^{0.5},\ 0 \leq t \leq 0.5; \ \varphi(t) = 0.5^{0.5},\ t > 0.5$. (c) $\varphi(t) = \mathrm{log}(t+1),\ 0 \leq t \leq 1; \ \varphi(t) = \mathrm{log}(t^{0.5}+1),\ 1 < t \leq 2; \ \varphi(t) = \mathrm{log}(2^{0.5}+1),\ t >2$

Figure 2.  Test images. (a) $128\times 128$; (b) $256\times 256$

Figure 3.  The visualization of PSNR values versus parameters $a$ and $\tau$ of trunc-LN, $p$ and $\tau$ of trunc-$\ell_p$ for "Shepp-Logan'' image with 36 projections

Figure 4.  CT reconstruction comparisons for "Shepp-Logan'' image with $\sigma = 0.01\|g\|_\infty$. The first, second and third rows are the reconstructed results when $N = 18$, 36 and 72, respectively. The PSNR and SSIM values are attached in the brackets

Figure 5.  CT reconstruction comparisons for "NCAT'' image with $\sigma = 0.01\|g\|_\infty$. The first, second and third rows are the reconstructed results when $N = 18$, 36 and 72, respectively. The PSNR and SSIM values are attached in the brackets

Figure 6.  The first row shows the ground truth. The second and third rows show the zoom-in views of the "Shepp-Logan'' and "NCAT'' image reconstructions with 36 projections

Figure 7.  The residual errors of the reconstructions by three methods with 36 projections for "Shepp-Logan'' and "NCAT''

Figure 8.  The reconstruction comparisons of 60th and 80th rows of "Shepp-Logan'' image for the three methods with 36 projections

Figure 9.  The values $F(u_k)$ versus the iteration number for "Shepp-Logan'' image with 36 projections

Figure 10.  CT reconstruction comparisons for real chest CT image with projection number $N = 60$. The PSNR and SSIM values are attached in the brackets

Figure 11.  A simple example. (a) The illustration of an image and its gradient. (b) Pixel values in black and norms of gradient in red

Table 1.  More quantitative comparisons of reconstruction on "Shepp-Logan'' in terms of PSNR and SSIM

 $N_p$ noise level TV TW-$\ell_0$ trunc-LN trunc-$\ell_p$ PSNR/SSIM PSNR/SSIM PSNR/SSIM PSNR/SSIM $18$ $0.005\|g\|_\infty$ 36.28/0.9688 39.31/0.9871 43.36/0.9902 43.66/0.9924 $0.01\|g\|_\infty$ 31.88/0.9159 34.67/0.9729 39.19/0.9800 38.24/0.9794 $0.02\|g\|_\infty$ 27.52/0.8389 29.54/0.9460 32.08/0.9507 31.53/0.9561 $36$ $0.005\|g\|_\infty$ 39.49/0.9860 44.52/0.9941 46.99/0.9959 49.52/0.9991 $0.01\|g\|_\infty$ 34.45/0.9564 39.84/0.9860 42.80/0.9897 42.87/0.9910 $0.02\|g\|_\infty$ 29.71/0.8972 33.56/0.9695 38.09/0.9776 37.43/0.9769 $72$ $0.005\|g\|_\infty$ 43.27/0.9916 49.93/0.9984 50.20/0.9980 52.92/0.9989 $0.01\|g\|_\infty$ 37.80/0.9779 43.95/0.9942 45.61/0.9954 46.41/0.9954 $0.02\|g\|_\infty$ 32.54/0.9393 37.36/0.9758 41.07/0.9862 40.58/0.9868

Table 2.  More quantitative comparisons of reconstruction on "NCAT'' in terms of PSNR and SSIM

 $N_p$ noise level TV TW-$\ell_0$ trunc-LN trunc-$\ell_p$ PSNR/SSIM PSNR/SSIM PSNR/SSIM PSNR/SSIM $18$ $0.005\|g\|_\infty$ 30.89/0.9580 29.97/0.9704 33.27/0.9875 33.90/0.9887 $0.01\|g\|_\infty$ 28.15/0.9141 26.74/0.9399 28.52/0.9630 28.44/0.9658 $0.02\|g\|_\infty$ 25.39/0.8220 24.12/0.8956 24.69/0.9189 24.72/0.9216 $36$ $0.005\|g\|_\infty$ 33.49/0.9703 34.01/0.9877 38.36/0.9954 38.99/0.9964 $0.01\|g\|_\infty$ 30.20/0.9339 29.48/0.9668 31.32/0.9816 31.79/0.9834 $0.02\|g\|_\infty$ 27.08/0.8448 25.88/0.9210 27.02/0.9520 27.19/0.9548 $72$ $0.005\|g\|_\infty$ 35.78/0.9721 37.81/0.9943 42.29/0.9980 42.76/0.9985 $0.01\|g\|_\infty$ 31.70/0.9340 31.86/0.9795 34.25/0.9902 34.40/0.9908 $0.02\|g\|_\infty$ 27.89/0.8276 27.44/0.9410 28.94/0.9694 29.18/0.9714
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