# A Graduated Non-Convexity Technique for Dealing Large Point Spread Functions
## 3 CATIED Technique  
This section presents the CATIED (_Convex Approximation Technique for Interacting Line Elements Deblurring_) algorithm presented in [5]. Such an algorithm is a GNC (_Graduated Non-Convexity_) technique (cf. [4; 5; 8; 41; 42; 43]) that allows minimization of the energy function \(E\) given in (1). It is simple to verify that such a function is non-convex. In order to use a gradient descent technique, it is necessary to determine an appropriate initial point near the globular optimum. For this purpose, a GNC technique constructs a family of approximations of the energy function \(\{E^{(p)}\}_{p}\) such that the first approximation is convex and the last corresponds to the non-convex function. Then, the following algorithm finds an approximation of the global minimum using the family \(\{E^{(p)}\}_{p}\). initialize \(\mathbf{x}\);  
while \(E^{(p)}\
eq E\) do  
find the minimum of the function \(E^{(p)}\) starting from the initial point \(\mathbf{x}\);  
\(\mathbf{x}=\arg\min E^{(p)}\);  
update the parameter \(p\). It is immediate to verify that the first term of the energy function \(E\) in (1), called the _data consistency term_, is convex. Then, finding a first convex approximation of the function \(E\) reduces to determining a convex approximation of the second additive term in (1), called the _smoothness term_. In [5], the authors first determine a \(C^{1}(\mathbb{R}^{2})\) family of approximation of the function \(\psi\) in (2). Namely, \(\psi^{(0)}\equiv\psi\), and for \(p\in(0,1]\),  
\[\psi^{(p)}(t_{1},t_{2})=\left\{ \begin{array}{ll}\overline{s}^{(p)}(t_{1},0),& \text{if }|\ t_{2}\mid\leq s,\\ a^{(p)}(t_{1})(|\ t_{2}\mid-s)^{2}+\overline{s}^{(p)}(t_{1},0),& \text{if }s<|\ t_{2}\mid\leq\frac{u(p)+s}{2},\\ -a^{(p)}(t_{1})(|\ t_{2}\mid-u(p))^{2}+\overline{s}^{(p)}(t_{1},\widetilde{e}),&\text{if }\frac{u(p)+s}{2}<|\ t_{2}\mid<u(p),\\ \overline{s}^{(p)}(t_{1},\widetilde{e}),&\text{otherwise},\end{array} \right. \tag{3}\]  
where \(u(p)=s+pz\), with an arbitrary \(z>0\). The function \(\overline{s}^{(p)}(t,k)\) is  
\[\overline{s}^{(p)}(t,k)=\left\{ \begin{array}{ll}\widehat{\lambda}^{2}t^{2}& \text{if }|t|<q_{p}(k),\\ \\ \widehat{\alpha}+k-\frac{\tau^{(p)}}{2}(|t|-r_{p}(k))^{2}& \text{if }q_{p}(k)\leq|t|\leq r_{p}(k),\\ \\ \widehat{\alpha}+k&\text{if }|t|>r_{p}(k),\end{array} \right.\]  
where  
\[q_{p}(k)=\frac{\sqrt{\widehat{\alpha}+k}}{\widehat{\lambda}^{2}}\bigg{(}\frac {2}{\tau^{(p)}}+\frac{1}{\widehat{\lambda}^{2}}\bigg{)}^{-1/2},\]  
with \(\tau^{(p)}=\tau^{*}/p\), where \(\tau^{*}>0\) is an arbitrary constant, and  
\[r_{p}(k)=\frac{\widehat{\alpha}+k}{\widehat{\lambda}^{2}\,q_{p}(k)}.\]  
The function \(a^{(p)}(t)\) is  
\[a^{(p)}(t)=2\,\frac{\overline{s}^{(p)}(t,\widetilde{e})-\overline{s}^{(p)}( t,0)}{[u(p)-s]^{2}}.\]  
Thus, the first convex approximation of \(\psi\) of class \(C^{1}(\mathbb{R}^{2})\) in the CATILED technique is  
\[\psi^{(2)}(t_{1},t_{2})==\left\{ \begin{array}{ll}\widehat{\lambda}^{2}t_{1 }^{2},&\text{if }|\ t_{1}\mid<q_{1}(0),\\ \\ 2\widehat{\lambda}^{2}q_{1}(0)\,|t_{1}|-\widehat{\lambda}^{2}q_{1}^{2}(0),& \text{if }|\ t_{1}\mid\geq q_{1}(0).\end{array} \right. \tag{4}\]  
For \(p\in[1,2]\),  
\[\psi^{(p)}=(p-1)\,\psi^{(2)}+(2-p)\psi^{(1)},\]  
where \(\psi^{(1)}\) and \(\psi^{(2)}\) are given in (3) and (4) respectively. In the CATILED algorithm, the parameter \(p\) varies from \(2\) to \(0\) with a fixed step \(h\). Note that all variables used in this section are necessary to make all approximations \(\psi^{(p)}\) of the function \(\psi\), for \(p\in(0,2]\), of class \(C^{1}(\mathbb{R}^{2})\) (see [5] for details). In order to obtain a graphical view, Figure 3a-c show the graphs of the functions \(\psi^{(2)}\), \(\psi^{(1)}\), and \(\psi^{(0)}\equiv\psi\) when \(\widehat{\lambda}=1\), \(\widehat{\alpha}=80\) and \(\widehat{e}=80\). The different approximations are minimized by the NL-SOR (_Non-Linear Successive Over Relation_) algorithm (cf. [5,8]).