## 4 Application of the SKD Model to a Super-Resolution-Type Problem  
In this section, the proposed mathematical model is applied to a super resolution type problem in images, and we will always consider multiple acquisitions shifting the sensor by \(\varepsilon=d/2\). The super resolution problem is an ill-posed problem whose goal is to obtain Higher Resolution (HR) images starting from Lower Frequency (LR) acquisitions. In case of super resolution from multiple images, the missing information is retrieved from repeated acquisitions of the same static scene. The data are then merged together on the same grid and some interpolation or quasi-interpolation methods, with their own kernels, are used to perform the final reconstruction [33] (see Figure 3). Once the method and the kernel have been chosen, the reconstruction depends exclusively on the measured values. In the following, our interest is focused on the measured values with different measurement methods: pointwise or average. We assume that the cells of which the measuring sensor is made, have square shape, as it happens in practice. From now on, we refer to the classic and generalized sampling models as the WKSG model. The main difference between the SKD and the WKSG models is that the first one uses the \(w\int_{\frac{k}{2}}^{\frac{k+1}{2}}f(u)du\) mean values while the second one employs the point values \(f(\frac{k}{u})\) (in this application \(t_{k}=k,k\in\mathbb{Z}\)). Of course, the application of the results achieved in the one-dimensional case previously investigated, can be extended to the two-dimensional case, with the same procedure of shifting along the \(x\) and \(y\) axes. All the results below presented have been achieved writing codes in Python, vers. 3.9.13. One set of six HR images of artworks (resolution \(H\times W\)) has been used. To avoid the non-linearity of the Analog to Digital (A/D) converters, the measurement data are simulated, such that the HR images are halved in size, performing a mean over each \(2\times 2\) square of not overlapping pixels. The mean process then, artificially generates four different images, each one of size \(\frac{H}{2}\times\frac{W}{2}\), (see Figure 4). We denote the values of the pixels in the new smaller images with \(M_{R}(i,j),M_{B}(i,j),M_{G}(i,j),M_{Y}(i,j)\), where the letters \(R,B,G,Y\) are connected with the color of the grid used in the meaning (\(M\)) process (see Figure 4 again). The halving process, to which the WKSG model is applied, is made through the following equations, which link the original image (HR) and the halved image (LR). Here, they follow,  
Figure 4: Different positions of the \(2\times 2\) grid to calculate the mean vales on the original HR image (grid in black). From left to right, top to down: in red, the not shifted measurement giving \(M_{R}\) values; in blue, the measurement shifted by one cell to the right (\(M_{B}\)); in green, the measurement shifted down by one cell (\(M_{G}\)); in yellow, the measurement diagonally shifted by one cell. Figure 3: Merging of shifted measurements on the same grid. Red, blue, green, and yellow points represent the set of values acquired in four different sampling acquisitions. \[if\;\;i\;\text{odd}\;\;\text{and}\;\;j\;\text{odd},\;\;M_{\text{R}}\left(\frac{i+1}{ 2},\frac{j+1}{2}\right)=\frac{I(i,j)+I(i,j+1)+I(i+1,j)+I(i+1,j+1)}{4}\;,\]  
\[if\;\;i\;\text{odd}\;\;\text{and}\;\;j\;\text{even},\;\;M_{\text{B}}\left(\frac {i+1}{2},\frac{j}{2}\right)=\frac{I(i,j)+I(i,j+1)+I(i+1,j)+I(i+1,j+1)}{4}\;,\]  
\[if\;\;i\;\text{even}\;\;\text{and}\;\;j\;\text{odd},\;\;M_{\text{G}}\left(\frac {i}{2},\frac{j+1}{2}\right)=\frac{I(i,j)+I(i,j+1)+I(i+1,j)+I(i+1,j+1)}{4}\;,\]  
\[if\;\;i\;\text{even}\;\;\text{and}\;\;j\;\text{even},\;\;M_{\text{Y}}\left( \frac{i}{2},\frac{j}{2}\right)=\frac{I(i,j)+I(i,j+1)+I(i+1,j)+I(i+1,j+1)}{4}\;,\]  
where \(I\) is the \(W\times H\) pristine version of a generic low-frequency image in the set.