# CTprintNet: An Accurate and Stable Deep Unfolding Approach for Few-View CT Reconstruction
## 5 Results and Discussion
### Discussion of the Learned Parameters  
In this section, we study how the choice of \(\lambda\) affects the performance of the network by examining a further learning technique and providing a brief excursus on the behavior of the three learned parameters. To this aim, we initially fixed \(\lambda\), as it happens in general in the iterative method, while the remaining parameters (\(\mu,\gamma\)) were learned from the architecture. We performed Experiment C with different values of \(\lambda\) set by analyzing those learned from the network in previous experiments, and we saved the best-performing values.  
We then utilized a different technique, described in [42; 43], for learning the regularization parameter, leaving the techniques used to learn \(\mu\) and \(\gamma\) untouched. The idea behind this technique is to provide a local TV regularization by setting a specific parameter for each pixel, thus obtaining a weighted TV (WTV), as follows:  
\[WTV(x)=\sum_{i=1}^{N}\lambda_{i}\sqrt{(D_{h}x)_{i}^{2}+(D_{v}x)_{i}^{2}}. \tag{31}\]  
This choice allows to diversify the level of the regularization within the single image, e.g., by regularizing more on constant patches and less on patches with complex textures. From the implementation point of view, the parameters \(\lambda_{i}\) (\(i=1,\dots,N\)) in the \(k\)-th layer for the \(i\)-th pixel are computed as  
\[\lambda_{i}^{(k)}=\frac{\text{Softplus}(d_{i})}{2N}\frac{||Hx_{k}-y||_{2}^{2}} {\sqrt{(D_{h}x_{k})_{i}^{2}+(D_{v}x_{k})_{i}^{2}}}, \tag{32}\]  
where \(d_{i}\) is a scalar parameter of the architecture learned during the training. In this way, the smaller the gradient magnitude is, the greater the regularization provided at pixel \(i\) is.  
We modified the architecture using this procedure for the computation of \(\lambda\) in the substructure \(\mathcal{S}^{(\lambda)}\) in Figure 3 and carried out several trainings. What is evident is that, although the architecture with this modification is able to yield slightly better performances than those obtained with the technique described in (32), it becomes worse in terms of stability.  
We plot in Figure 9 the results obtained from Experiment C with a fixed \(\lambda\) (black line), with Equation (29) (red line) and with Equation (32) (green line). They show that the proposed new learning rule weakens the network stability.  
Next, we analyzed the behavior of the parameters learned from the architecture within the different layers. We observed from all the performed experiments that these parameters show very similar trends regardless of the image and the presence or absence of noise. In Figure 10, we plot the values, obtained from Experiment B, of the three parameters as the iterations increase. Concerning the step size, it initially increases, and then it decreases and  
Figure 9: Trend of average values of PSNR (**left**) and SSIM (**right**) obtained by CTprintNet considering the COULE dataset and Experiment C with the regularization parameter manually fixed (black), learned through the technique described in (29) (red), and learned through the technique described in (32) (green). Bars and shades indicate standard deviation values over the test set images.  
stabilizes. The central panel of Figure 10 shows that the regularization parameter tends to vanish as the layers advance since the presence of noise decreases; as a result of this, the contribution of the regularization term becomes weaker. Finally, the barrier parameter vanishes along the minimization process according to the theoretical issues presented in Section 3.