## 3 Results and Discussion
### Comparison of Deblurring Results.
The SSIM [39] is a full-reference metric for measuring the similarity between two images, \(\mathbf{x}\) and \(\mathbf{y}\), and is defined as follows:  
\[SSIM(\mathbf{x},\mathbf{y})=\frac{(2\overline{\mathbf{x}\overline{y}}+C_{1}) \Big{(}2\sigma_{\mathbf{x}\mathbf{y}}+C_{2}\Big{)}}{\Big{(}\overline{\mathbf{ x}^{2}+\overline{y}^{2}+C_{1}\Big{)}(\sigma_{x}^{2}+\sigma_{y}^{2}+C_{2} \Big{)}}} \tag{25}\]  
Figure 4: Comparison results for blur kernel estimation. (**a-h**): 8 blur kernels listed in Table 1. First row: ground truth blur kernel. Second row: method in [35]. Third row: method in [36]. Fourth row: method in [37]. Fifth row: method in [38]. Last row: proposed method. where **x** and **y** are the mean values, \(\sigma_{x}\) and \(\sigma_{y}\) are the standard deviations, and \(\sigma_{xy}\) is the cross-covariance. Furthermore, \(C_{1}=(k_{1}L)^{2}\) and \(C_{2}=(k_{2}L)^{2}\) are factors used to ensure stability in case of division with a small denominator, with \(k_{1}=0.01\) and \(k_{2}=0.03\) in general. An example of \"Img02\" blurred by kernel \"(b)\" and the deblurring results are shown in Figure 5. The white ball and the latch, as well as the wood grains and cracks, are blurred and indistinct, as shown in Figure 5a. The deblurring method described in [35] significantly reduces the blur in Figure 5b, but the result is still unsatisfying. The methods described in [36; 37] introduce the ringing effect, both around the ball and wood cracks, as they are obvious in Figure 5c,d. This phenomenon is reduced by the method described in [38] as shown in Figure 5e. However, wood grains in the image are not clear enough. In contrast, as it can be seen in Figure 5f, the proposed method restores both the texture and the background. Figure 6 shows an image, namely, \"Img07,\" blurred by kernel \"(a).\" As can be seen in Figure 6b, the deblurring method described in [35] reduced some of the blurriness; however, the visual effect is still lacking. Particularly, the shutters in the background are still not clear enough, and the estimated blur kernel shown in the top left is a little rough when compared with others. Moreover, as it can be seen in Figure 6c,d, the method described in [36; 37] restored the sharpness, but introduced strong artifacts, especially around the petals and leaves. These limitations were overcome by the method described in [38] (see Figure 6e), as well as by the proposed method (see Figure 6f), which restored most of the details, and the shape of its estimated blur kernel is the closest to the ground truth data. Figure 3:0 Comparison results for the “RedDoor.png” image: (**a**) blurry image; (**b**) method in [35]; (**c**) method in [36]; (**d**) method in [37]; (**e**) method in [38]; (**f**) proposed method. Figure 3:1 Comparison results for the “Flowers.png” image: (**a**) blurry image; (**b**) method in [35]; (**c**) method in [36]; (**d**) method in [37]; (**e**) method in [38]; (**f**) proposed method. Figure 7 shows another example blurred by kernel \"(f)\"; the image here is \"Img21.\" The lighthouse and cabins are blurred and unclear in Figure 7a. As can be seen in Figure 7b, the deblurring method described in [35] reduced some of the blurriness; however, the visual effect is still lacking. In addition, as can be seen in Figure 7c, the method described in [36] restored the sharpness but introduced strong artifacts along the roofs and the white tower. In Figure 7d, because the method described in [37] estimated a blur kernel with numerous outliers, its deblurring result is not as good as the others. The limitation was partially overcome by the method described in [38]. There were still, however, some outliers in the estimated blur kernel shown in Figure 7e; the general outlines of the kernel were estimated correctly. In contrast, the proposed method not only estimated the most appropriate blur kernel, but also obtained a clear result with high quality. Table 2 lists the original FRIQA data of a comparative analysis performed on the examples.