# Super-Resolution Reconstruction of Depth Image Based on Kriging Interpolation
## 3 Fractional-Order Differential Edge Recognition  
Image enhancement processing is an important branch of digital image processing. Under real-world environmental conditions, the visual effectiveness of image capture is poor, which necessitates image enhancement techniques. Such techniques include highlighting certain characteristics of the target object in the image and extracting the characteristic parameters of the target object from the digital image. The goal is to improve the recognition, tracking, and understanding of the target [36]. The main objective of image enhancement processing is to highlight the parts of interest in an image and attenuate or remove unwanted information so that the useful information is enhanced to obtain a more useful image, or the image can be made more suitable for analysis and processing by humans or machines. The main features of an image (i.e., the main information) include edges, texture details, and other features. The enhancement of image features plays an important role in improving the visual effectiveness of an image and also lays a good foundation for subsequent image processing (e.g., image recognition, image segmentation, or super resolution). Figure 7: Kriging interpolation results for depth images (PSNR = 20.629 dB). (a), depth image; (b), depth image after Kriging interpolation; (c), point cloud map orthomosaic of the depth image. Fractional-order differentiation significantly enhances the high- and medium-frequency components of the signal while nonlinearly strengthening the medium-frequency components of the signal to a certain extent as well as nonlinearly preserving the low-frequency and DC components of the signal. Therefore, fractional-order differentiation is beneficial in terms of both significantly enhancing the image edge and contour information and strengthening the texture details in the smooth regions of the image. Due to this property of fractional-order differentiation, the fractional calculus theory in image processing operations has received increasing attention [13; 14; 15; 16; 17]. It is generally believed that the main benefit of applying methods based on fractional-order calculus to digital images is to add a degree of freedom, i.e., the fractional-order v (0 < [v] < 1). By choosing the appropriate order and constructing the corresponding mask operator, better image enhancement (v > 0) and denoising (v < 0) can be achieved. [PERSON] et al. derived fractional-order calculus and verified the application of the fractional-order integral in image processing [37]. They also proposed a fractional-order differential enhancement and fractional-order integral denoising model based on the Grunwald-Letnikov (G-L) definition of 2D images. Introducing the G-L definition of the fractional-order calculus, the \(-v\) order integral of the G-L expression [17] is given by the following equation:  
\[{}^{\rm G}_{\rm a}D^{-\rm v}_{\rm x}=\lim_{\rm h\to 0}\rm{h}^{\rm v}\sum \
olimits_{\rm m=0}^{\rm\frac{\rm t-a}{\rm h}}\binom{-\rm v}{\rm j}\rm f(x-jh),v\in R \tag{34}\]  
The fractional-order integral can be approximately expanded by defining the one-dimensional (1D) signal f(t) of the duration interval as [a,t]. According to the unit h = 1 equipartition, we have \(\rm m=\left[\frac{t-a}{\rm h}\right]=[t-a]\), implying that there is a 1D signal of v (v \(\geq\) 0). The equivalent expression of the fractional-order differentiation of order \(v\) is  
\[\frac{\rm d^{\rm v}t}{\rm d\rm v^{\rm v}}\approx f(t)+(-1)^{1}(v)f(t-1)+(-1)^{ 2}\Big{(}\frac{\rm v(v+1)}{2}\Big{)}f(t-2)+\ldots+(-1)^{1}\frac{\rm\Gamma(v+1) }{\rm\Gamma(j+1)\Gamma(v-j+1)}f(t-j) \tag{35}\]  
Define the 2D image signal \(\rm I(x,y)\), and assume that the fractional-order derivatives of \(\rm I(x,y)\) are separable for the x-axis and y-axis directions under certain conditions. Furthermore, we use the separability of the Fourier transform to extend the fractional-order calculus from one dimension to two dimensions.