# Super-Resolution Reconstruction of Depth Image Based on Kriging Interpolation
## 2 Depth Image super-resolution Algorithm
### Interpolation Algorithm Based on Kriging Interpolation  
A depth image is itself a distance image and is similar to a geological/topographic image after conversion to a point cloud. Considering the concept of inverse distance used in adaptive weighting, an algorithm suitable for geology/topography is introduced. The algorithm is based on inverse distance interpolation and Kriging interpolation, both of which take into account the spatial relationship between the point to be interpolated and the valid points around them [20; 22]. Kriging interpolation is a method for the unbiased optimal estimation of regionalised variables in a finite region (used to estimate values that are spatially correlated, such as air quality, where values at closely spaced locations are similar). 'Unbiased' means that the expectation of the difference between the estimated and actual values is equal to zero, and 'optimal' means that the variance between the estimated and actual values is minimised. Based on this feature, Kriging interpolation is much more effective than other interpolation methods. In simple terms, Kriging interpolation does the following. Given some coordinates and the real values at those coordinates, which we call sampling points, these points are fed into the Kriging algorithm to estimate the values at other unknown locations [22]. Kriging interpolation analyses the regional variation of variables to achieve an optimal linear unbiased estimation [20; 22]. The method consists of two steps: (1) the variational function model is established using the values of the spatial attributes of the regional variables, which are used to express the structural and stochastic characteristics of those attributes; (2) on the basis of the analysis of the spatial attributes, the domain range is determined and the Kriging equation is solved to estimate the values of the attributes with interpolation points. \[\hat{Z}_{o}=\sum_{i=1}^{n}\lambda_{i}z_{i} \tag{1}\]  
where \(z_{i}\) is the estimated value at the point \((x,y)\) and \(\lambda_{i}\) is the weight coefficient, which is a weighted sum of the data at all known points in the space and is used to estimate the value of unknown points. However, the weight coefficients are not the inverse of the distance but are the optimal set of coefficients that can satisfy the minimum difference between the valuation \(z_{o}\) of the point \((x,y)\) and the true value of \(z_{o}\), i.e.,  
\[\min_{\lambda_{i}}\mathrm{Var}(\hat{z}_{o}-z_{o}) \tag{2}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Interpolation** & \begin{tabular}{c} **Nearest** \\ **Neighbor** \\ **Interpolation** \\ \end{tabular} & \begin{tabular}{c} **Bilinear** \\ **Interpolation** \\ \end{tabular} & \begin{tabular}{c} **Resampling** \\ **Interpolation** \\ \end{tabular} & \
\begin{tabular}{c} **Cubic** \\ **Interpolation** \\ \end{tabular} &
\begin{tabular}{c} **Lanczos** \\ **Interpolation** \\ \end{tabular} \\ \hline PSNR (dB) & 29.795 & 34.154 & 33.570 & 34.515 & 33.654 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of classical interpolation algorithms. Additionally, the unbiased condition must be satisfied:  
\[\mathrm{E}(\mathrm{z}_{\mathrm{o}}-\mathrm{z}_{\mathrm{o}})=0 \tag{3}\]  
The main difference between different Kriging interpolation methods is the underlying assumptions. In this paper, only the assumptions and applications of ordinary Kriging interpolation are presented. Ordinary Kriging interpolation assumes that the spatial properties \(\mathrm{z}\) are homogeneous. All points in space \((\mathrm{x},\mathrm{y})\) have the same expectation \(\mathrm{c}\) with variance \(\sigma^{2}\).