## 3 Sampling Kantorovich by Difference  
From now on, our model focuses on the averages of the sampled values and not on the kernels of the operators. In fact, the proposed method, which uses multiple measurements, allows us to reconstruct the signal/image, working only on the sampled values of the signal. With the expression Sampling Kantorovich by Difference (SKD), we design the sampling process that measures the differences between mean values in repeated shifted acquisition. Using the formalism introduced in the previous section, we start working in a one-dimensional setting, introducing two sequences of the domain of the signal \(f\), i.e., \(\Pi_{A}=(t_{k_{A}})_{k_{A}\in\mathbb{Z}}\), \(\Pi_{B}=(t_{k_{B}})_{k_{B}\in\mathbb{Z}}\), such that \(t_{k_{B}+h}<t_{k_{A}+h}\), for every \(k_{A},k_{B},h\in\mathbb{Z}\) (see Figure 1). In the equi-spaced case, with  
\[t_{k_{A}+h}=\frac{t_{k_{B}+h+1}+t_{k_{B}+h}}{2}\]  
and  
\[t_{k_{B}+h}=\frac{t_{k_{A}+h}+t_{k_{A}+h-1}}{2},\]  
for a locally integrable function \(f\), the following equality stands  
\[\frac{w}{t_{k_{B}+1}-t_{k_{A}}}\int_{\frac{t_{k_{B}+1}}{w}}^{\frac{t_{k_{B}+1} }{w}}f(u)\ du=2\left(\frac{w}{\Delta k_{B}}\int_{\frac{t_{k_{B}}}{w}}^{\frac{t_ {k_{B}+1}}{w}}f(u)\ du\right)-\frac{w}{t_{k_{A}}-t_{k_{B}}}\int_{\frac{t_{k_{ B}}}{w}}^{\frac{t_{k_{A}}}{w}}f(u)\ du, \tag{1}\]  
where the first term on the left side is the mean value of \(f\) in \([\frac{t_{k_{A}}}{w},\frac{t_{k_{B}+1}}{w}]\), the first term on the right side is the mean value of \(f\) in \([\frac{t_{k_{B}}}{w},\frac{t_{k_{B}+1}}{w}]\) (multiplied by the constant 2) and the last term, the mean value of \(f\) in \([\frac{t_{k_{A}}}{w},\frac{t_{k_{A}}}{w}]\). We denote by  
\[M_{\Pi_{A}}\left(\frac{t_{k_{A}}}{w}\right):=\frac{w}{\Delta_{k_{A}}}\int_{ \frac{t_{k_{A}}}{w}}^{\frac{t_{k_{A}+1}}{w}}f(u)\ du\,,\]  
\[M_{\Pi_{B}}\left(\frac{t_{k_{B}}}{w}\right):=\frac{w}{\Delta_{k_{B}}}\int_{ \frac{t_{k_{B}+1}}{w}}^{\frac{t_{k_{B}+1}}{w}}f(u)\ du\,,\]  
where \(\Delta_{k_{A}}=\frac{t_{k_{A}+1}-t_{k_{A}}}{w}\) and \(\Delta_{k_{B}}=\frac{t_{k_{B}+1}-t_{k_{B}}}{w}\), and by  
\[M_{0}:=M_{0}\bigg{(}\bigg{[}\frac{t_{k_{B}}}{w},\frac{t_{k_{A}}}{w}\bigg{]} \bigg{)}=\frac{w}{t_{k_{A}}-t_{k_{B}}}\int_{\frac{t_{k_{B}}}{w}}^{\frac{t_{k_{ A}}}{w}}f(u)\ du.\]  
From now on, for simplicity of notation, we denote \(M_{\Pi_{A}}\left(\frac{t_{k_{A}}}{w}\right)=M_{A}\Big{(}\frac{t_{k_{A}}}{w} \Big{)}\) and \(M_{\Pi_{B}}\Big{(}\frac{t_{k_{A}}}{w}\Big{)}=M_{B}\Big{(}\frac{t_{k_{B}}}{w} \Big{)}\). The values \(M_{A}\) and \(M_{B}\) are the results of a measurement process, in which the acquired values are means of the input. To a certain extent, in the most of the practical cases, the measurement process is reduced to measure a mean of the input, in space or in time. Now, by (1), if \(M_{0}\) is known, it is possible to recursively calculate, by difference, the integral mean of \(f\) in the intervals \([\frac{t_{k_{A}}}{w},\frac{t_{k_{B}+1}}{w}],[\frac{t_{k_{B}+1}}{w},\frac{t_{k _{A}+1}}{w}]\), and so on, such to achieve a resolution of the measurements depending on the shifting distance \(\varepsilon\) between the samples of the two  
Figure 1: A generic one dimensional function whose mean values are calculated between the nodes of the given subdivisions, \((t_{k_{A}})_{k_{A}\in\mathbb{Z}}\) (in green), \((t_{k_{B}})_{k_{B}\in\mathbb{Z}}\) (in red). From the left: sampling intervals (delimited by the respective dotted vertical lines) for each subdivision; mean values (continuous red lines) of \(f\) for the subdivision \((t_{k_{B}})_{k_{B}\in\mathbb{Z}}\); mean values (continuous green lines) of \(f\) for the subdivision \((t_{k_{A}})_{k_{A}\in\mathbb{Z}}\). subdivisions grouped together. Similar procedures are commonly used in the framework of super resolution type techniques ([32]). A simple example of SKD is given by multiple measurements performed with the same sensor (built by contiguous cells of sizes \(d\)) shifted in space.