# A Graduated Non-Convexity Technique for Dealing Large Point Spread Functions
## 4 E-CATILED Technique
### Choice of the Blur Matrix Approximation  
In order to test the goodness of the proposed approximations, we have proceeded as follows: fixing the dimension \(n\) and the range of values which the entries of a considered Toeplitz matrices can assume, we created 10,000 different instances of Toeplitz symmetric matrices \(T_{n}\), whose entries were randomly and uniformly chosen in the prefixed range. Then, we computed \(G_{n}(T_{n})\), \(V_{n}(T_{n})\), \(C_{n}(T_{n})\), and \(H_{n}(T_{n})\), given in (14)-(16). Then, we computed the mean of the Frobenius norm of the difference between the matrices \(T_{n}\) and the approximating matrices. The considered range in Table 1 is \([0,1]\). Note that the approximations given via \(\beta\)-matrices are always the best since the class \(\mathcal{V}_{n}\) contains the other three classes considered. We focus on figuring out which class of matrices gives results most similar to those obtained via \(\beta\)-matrices. In this case, \(G_{n}(T_{n})\) is the second-best approximation in the mean. In Table 2, the considered interval is \([-1,1]\), and the obtained results are analogous to the previous ones. In Table 3, we have generated the first row of the Toeplitz symmetric matrix as follows. We set the value of the first entry equal to \(1\). To determine the value of the \(i\)-th entry, we multiplied the value of the \(i-1\)-th entry by a random constant chosen uniformly in \([0.9,1]\). Such a choice allows us to simulate better the Toeplitz matrices present in the blur operators that, in many cases, have a Gaussian shape. The behavior of the errors is similar to that of the previous cases. Moreover, it is possible to see in Tables 1-3 that, for large numbers, the \(C_{n}(T_{n})\) and \(H_{n}(T_{n})\) approximations give similar results, and that the \(G_{n}(T_{n})\) and \(V_{n}(T_{n})\) approximations are similar too. Furthermore, as seen in Table 4, for large numbers, the \(G_{n}(T_{n})\) approximations are always better than the \(H_{n}(T_{n})\) approximations. Since the multiplication of \(V_{n}(T_{n})\) by a vector needs more fast discrete transforms than the multiplication of \(G_{n}(T_{n})\) by a vector, \begin{table}
\begin{tabular}{c c c c c} \hline  & \(\|T_{n}-C_{n}(T_{n})\|_{F}\) & \(\|T_{n}-H_{n}(T_{n})\|_{F}\) & \(\|T_{n}-G_{n}(T_{n})\|_{F}\) & \(\|T_{n}-V_{n}(T_{n})\|_{F}\) \\ \hline \(n=20\) & 3.1389 & 3.1156 & 3.0770 & 3.0532 \\ \hline \(n=25\) & 4.1076 & 4.0885 & 3.9591 & 3.9392 \\ \hline \(n=30\) & 4.8062 & 4.7903 & 4.7369 & 4.7207 \\ \hline \(n=35\) & 5.7528 & 5.7390 & 5.5989 & 5.5847 \\ \hline \(n=40\) & 6.4536 & 6.4416 & 6.3811 & 6.3689 \\ \hline \(n=45\) & 7.4243 & 7.4135 & 7.2649 & 7.2538 \\ \hline \(n=50\) & 8.1211 & 8.1114 & 8.0471 & 8.0373 \\ \hline \(n=100\) & 16.46786 & 16.46293 & 16.38939 & 16.38444 \\ \hline \(n=1000\) & 166.48101 & 166.48051 & 166.39821 & 166.39771 \\ \hline \end{tabular}
\end{table} 
Table 1: Mean error obtained by the various approximations with respect to 10,000 instances of randomly generated Toeplitz matrices \(T_{n}\) with entries in \([0,1]\). \begin{table}
\begin{tabular}{c c c c c} \hline  & \(\|T_{n}-C_{n}(T_{n})\|_{F}\) & \(\|T_{n}-H_{n}(T_{n})\|_{F}\) & \(\|T_{n}-G_{n}(T_{n})\|_{F}\) & \(\|T_{n}-V_{n}(T_{n})\|_{F}\) \\ \hline \(n=20\) & 6.2564 & 6.2098 & 6.1313 & 6.0838 \\ \hline \(n=25\) & 8.2016 & 8.1633 & 7.8982 & 7.8584 \\ \hline \(n=30\) & 9.6160 & 9.5842 & 9.4776 & 9.4453 \\ \hline \(n=35\) & 11.517 & 11.489 & 11.210 & 11.182 \\ \hline \(n=40\) & 12.915 & 12.891 & 12.771 & 12.747 \\ \hline \(n=45\) & 14.835 & 14.813 & 14.521 & 14.499 \\ \hline \(n=50\) & 16.292 & 16.272 & 16.141 & 16.121 \\ \hline \(n=100\) & 32.92819 & 32.91833 & 32.76966 & 32.75976 \\ \hline \(n=1000\) & 332.72496 & 332.72396 & 332.56154 & 332.56054 \\ \hline \end{tabular}
\end{table} 
Table 2: Mean error obtained by the various approximations concerning 10,000 instances of randomly generated Toeplitz matrices \(T_{n}\) with entries in \([-1,1]\).