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

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