# A Graduated Non-Convexity Technique for Dealing Large Point Spread Functions
## 5 Experimental Results  
In this section, we show, by some experimental results, how the E-CATILED algorithm achieves similar quantitative and qualitative results to CATILED, given in [5], in reduced computational time. We test the algorithms by implementing them in C language and running them in a Linux Ubuntu environment on a computer with an i5-9400F processor at 2.90 GHz. We consider both synthetic and actual data. To obtain the synthetic data, we apply to a test image a blur operator with Gaussian shape PSF (_Point Spread Function_) of standard deviation \(\widehat{\sigma}\), and sometimes, we add an uncorrelated Gaussian noise of zero mean and variance \(\widehat{\sigma}^{2}\). We use the fast transforms proposed in [38] to multiply between gamma-matrices and vectors. These transforms are explicitly designed to deal with gamma-matrices and have a small number of multiplicative operations. In the examples below, we empirically choose the involved free parameters \(\widehat{\lambda}\), \(\widehat{\alpha}\), and \(\widehat{\varepsilon}\). On the other hand, in the literature, there are available algorithms for estimating the values for the free parameters (cf. [40]). In our first experiment, we use the ideal synthetic test image in Figure 3(a). We blur this image with a Gaussian shape PSF of standard deviation \(\widehat{\sigma}=1.5\), and Figure 3(b) presents the blurred image. Figure 3(c) shows the reconstruction obtained with a standard non-edge-preserving Tikhonov regularization technique (cf. [51]), where the regularization parameter \(\widehat{\lambda}\) is fixed at 1. Whereas, Figure 3(d) shows the reconstruction obtained again by Tikhonov regularization but with \(\widehat{\lambda}=0.05\). Figure 3(e),f presents the results obtained with the CATILED and E-CATILED algorithms where \(\widehat{\lambda}=1\), \(\widehat{\alpha}=5\), and \(\widehat{\varepsilon}=5\). It is possible to see both quantitatively, by the MSE (_Mean Squared Error_) from the ideal image, and qualitatively that the images obtained with CATILED and E-CATILED are equivalent \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(range=[-1,1]\) & \(range=[0,1]\) \\  & & **Decreasing Case** \\ \hline \(n=20\) & 8727 & 10,000 \\ \hline \(n=25\) & 9794 & 10,000 \\ \hline \(n=30\) & 9765 & 10,000 \\ \hline \(n=35\) & 9973 & 10,000 \\ \hline \(n=40\) & 9943 & 10,000 \\ \hline \(n=45\) & 9993 & 10,000 \\ \hline \(n=50\) & 9990 & 10,000 \\ \hline \(n=100\) & 10,000 & 10,000 \\ \hline \(n=1000\) & 10,000 & 10,000 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Number of times in which the \(G_{n}(T_{n})\) approximation gives better results than the \(H_{n}(T_{n})\) approximation concerning 10,000 instances of randomly generated Toeplitz matrices \(T_{n}\) with decreasing entries in \([0,1]\). and better than those obtained with a Tikhonov regularization. In fact, by an implicit use of line elements, it is possible to obtain a more accurate reconstruction of the edges of the objects present in the ideal image. However, the computational time for determining the solution in the case of Tikhonov regularization is about one-sixth of the time of the CATILED technique. Moreover, it is possible to obtain more accurate results by minimizing the energy function in (1) via a stochastic algorithm such as simulated annealing, but with significantly longer computation times (cf. [52]). In our next experiments, we consider a Gaussian shape PSF of standard deviation \(\widetilde{\sigma}=3.25\), and we apply the corresponding blurring operator to the two test images in Figure 5 to obtain the starting data. In Figure 6, we present the reconstructions obtained by CATILED and E-CATILED of the image in Figure 5a. Instead, the restorations obtained by the two algorithms of the image in Figure 5b are shown in Figure 7. In this case, we set \(\widehat{\lambda}=0.05\), \(\widehat{\alpha}=1\), and \(\widehat{\varepsilon}=1\). Figure 6d shows the reconstruction of the image in Figure 5a by a Tikhonov regularization with \(\widehat{\lambda}=0.05\).