# CTprintNet: An Accurate and Stable Deep Unfolding Approach for Few-View CT Reconstruction
## 3 Proximal Interior Point Method for CT Reconstruction  
Problem (8) involves the constrained minimization of a smooth and convex objective function, which can be performed by means of a large variety of optimization methods. In our paper, we consider the proximal interior point approach [32], in which the original constrained minimization problem (8) is replaced by a sequence of unconstrained problems  
\[\operatorname*{argmin}_{x\in\mathbb{R}^{N}}\|Hx-y\|_{2}^{2}+\lambda TV_{\delta}( x)+\mu\mathcal{B}(x), \tag{9}\]  
where \(\mathcal{B}:\mathbb{R}^{N}\to\mathbb{R}\cup\{+\infty\}\) is the logarithmic barrier function, defined as  
\[\mathcal{B}(x)= \begin{cases}-\sum_{j=1}^{N}(\ln(x_{j})+\ln(1-x_{j}))&x\in(0,1) ^{N}\\ +\infty&\text{otherwise},\end{cases}  \tag{10}\]  
and \(\mu\in(0,+\infty)\) is the barrier parameter, which goes to zero along the minimization. Following [21], we address the minimization of (9) by means of the forward-backward proximal interior point (FBPIP) method, whose iterations are defined in Algorithm 1, where  
\[\operatorname*{prox}_{\mu\mathcal{B}}(x)=\operatorname*{argmin}_{u\in\mathbb{ R}^{n}}\frac{1}{2}\|x-u\|^{2}+\mu\mathcal{B}(u) \tag{11}\]  
is the proximity operator of \(\mu\mathcal{B}\) at \(x\)[33]. The following proposition directly derives from ([21], Proposition 2); we report its statement and proof adapted to problem (9) and (10) to show the closed-form expression of the proximal point defining \(x_{k+1}\) in Algorithm 1, together with its derivatives with respect to \(x\), \(\mu\), and \(\gamma\). ```
Let \(x_{0}\in\text{int}\mathcal{C}\), \(\gamma>0\) and \((\gamma_{k})_{k\in\mathbb{N}}\) be a sequence such that \((\forall\,k\in\mathbb{N})\,\underline{\gamma}\leq\gamma_{k}\); for\(k=0,1,\dots\)do \(x_{k+1}=\operatorname*{prox}_{\gamma_{k}\mu_{k}\mathcal{B}}\bigl{(}x_{k}-\gamma_ {k}\bigl{(}H^{\top}(Hx_{k}-y)+\lambda\
abla TV_{\delta}(x_{k})\bigr{)}\bigr{)}\) endfor
```  
**Algorithm 1** Proximal interior point algorithm. **Proposition 1**. : _Let \(\gamma,\mu>0\) and let \(\mathcal{B}_{i}(u)\) be the barrier function associated with the set \([0,1]\), defined as_  
\[\mathcal{B}_{i}(u)= \begin{cases}-\ln(1-u_{i})-\ln(u_{i})&\text{if }0<u_{i}<1 \\ +\infty&\text{otherwise}\end{cases} . \tag{12}\]  
_Then, for every \(x\in\mathbb{R}^{N}\), the proximity operator associated with \(\gamma\mu\mathcal{B}_{i}\) is given by_  
\[\varphi_{i}(x,\mu,\gamma):=\operatorname*{prox}_{\gamma\mu\mathcal{B}_{i}}(x) =x+(\kappa(x,\mu,\gamma)-u_{i})e_{i}, \tag{13}\]  
_where \(\kappa(x,\mu,\gamma)\) is the unique solution in \((0,1)\) of the cubic equation_  
\[z^{3}-(1+u_{i})z^{2}+(u_{i}-2\gamma\mu)z+\gamma\mu=0 \tag{14}\]  
_and \(e_{i}\) is the i-th vector of the canonical basis of \(\mathbb{R}^{N}\). The Jacobian matrix of \(\varphi\) with respect to \(x\) and the gradients of \(\varphi\) with respect to \(\mu\) and \(\gamma\) are given by_  
\[J_{(x)}\varphi_{i}(x,\mu,\gamma)=\mathbb{I}_{n}+\biggl{(}\frac{\kappa(x,\mu, \gamma)(\kappa(x,\mu,\gamma)-1)}{\eta(x,\mu,\gamma)}-1\biggr{)}e_{i}e_{i}^{T} \tag{15}\]  
\[\
abla_{(\mu)}\varphi_{i}(x,\mu,\gamma)=\frac{-\gamma(1-2\kappa(x,\mu,\gamma ))}{\eta(x,\mu,\gamma)}e_{i} \tag{16}\]  
\[\
abla_{(\gamma)}\varphi_{i}(x,\mu,\gamma)=\frac{-\mu(1-2\kappa(x,\mu,\gamma ))}{\eta(x,\mu,\gamma)}e_{i} \tag{17}\]_where_  
\[\eta(x,\mu,\gamma)=\kappa(x,\mu,\gamma)(\kappa(x,\mu,\gamma)-1)-(1-2\kappa(x,\mu, \gamma))(\kappa(x,\mu,\gamma)-x_{i})-2\gamma\mu. \tag{18}\]  
**Proof. ** The formal derivation of the proximity operator (13) can be found in [34]. Let \(x\in\mathbb{R}^{N}\), \(\gamma,\mu\in(0,+\infty)\), and \(F\) be defined as  
\[F(x,\mu,\gamma,z) =z(z-1)(z-x_{i})+\gamma\mu(1-2z) \tag{19}\] \[=z^{3}-(x_{i}+1)z^{2}+(x_{i}-2\gamma\mu)z+\gamma\mu \tag{20}\]  
for all \(z\in(0,1)\). By definition of \(\kappa(x,\mu,\gamma)\), \(F(x,\mu,\gamma,\kappa(x,\mu,\gamma))=0\).