# Inpainting in Discrete Sobolev Spaces: Structural Information for Uncertainty Reduction
## 6 Numerical Results
We perform the same, achieving that our new version \(E_{T}=E_{C_{T}}+E_{S_{T}}\) not only has a decreasing trend but reaches the minimum value faster (see Figure 20, that is qualitatively representative of all the performed tests). Figure 16: From top to down, left to right: the candidate points for each different pattern (from 3.1 to 3.9). The squares with the same color in the 9 chessboards constitute the sets \(\mathcal{S}_{3,1}\), \(\mathcal{S}_{3,2}\),..., \(\mathcal{S}_{3,9}\). Figure 17: Inpainting of the missing part \(\Omega\) in a chessboard, using \(2L+1=3\), \(l=1\). Figure 18: Inpainting of \(\Omega\) using [13], for different values of \(\text{supp}\,\chi\) and different initial conditions. On the left: initial condition with random white noise; on the right: initial condition equal to zero. Both images achieved using \(2L+1=3\). This is an example of missing causality condition. **Remark 1**. : _The value of \(E_{T}\) is meaningful when it is calculated at the end of each scan and if the causality condition is respected. Otherwise, in the inpainting algorithm, a potentially dangerous (for stability) positive feedback is introduced. During the first scan, the causality property addresses the algorithm to the direction of that minimum characterized by the lowest uncertainty. Subsequent scans refine the calculation._  
The introduction of \(E_{S_{T}}\) allows to carry more structural information in the similarity computation. Other examples of reconstructions for quasi-periodic textures are provided in Figures 21 and 22. Moreover, we compared the proposed method with results achieved by Matlab(r)_inpaintExemplar_ function ([35; 47]) and with ones coming from an available implementation of [35] (see Figures 23-26). In Figure 24, the reconstruction of the Kanizsa triangle is shown: the connectivity principle is respected. Figure 15.0 Inpainting of texture D1 taken from [PERSON] database. From left to right: image to be inpainted (\(\Omega\) in black, size \(16\times 16\)); inpainting by mean of \(E_{T}=E_{C_{T}}\) with \(2L+1=23\) (second column) and \(E_{T}=E_{C_{T}}+E_{S_{T}}\) (\(2L+1=3,l=1\)) (third column). The case using the first order finite difference reconstructs visually better. Moreover the minimum value of \(L\) to inpaint correctly is 23 when \(E_{T}=E_{C_{T}}\), against \(L=3\) when introducing \(E_{S_{T}}\) (reduction in the number of needed points of around 98%). Figure 15.1 From left to right: initial \(\Omega\) of size \(12\times 14\) pixels (**left**); inpainting result using our method (\(2L+1=3,l=1\)) (**center**); inpainting result using method [13] with \(2L+1=11\) (**right**). The better performance deriving from the introduction of \(E_{S_{T}}\) and the causality of the new proposed method are visually evident. The image in the center is achieved at the first iteration and does not change meaningfully in the subsequent ones, highlighting a fast convergence too. Our method needs less than 8 points to provide suitable results, while for [13], \(11^{2}-1=120\) points are not enough. Figure 15.2 Figure shows \(E_{T}\) trends, calculated at the end of each scanning iteration of the whole \(\Omega\): in red \(E_{T}=E_{C_{T}}\), in blue \(E_{T}=E_{C_{T}}+E_{S_{T}}\). Finite difference contributions determine a faster convergence to the steady state.